TASK LIST...YEAR 7 MATHEMATICS o f W e s t ern A u s r a l i a I n c . T MA WA h e M a t h e m a t i...
Transcript of TASK LIST...YEAR 7 MATHEMATICS o f W e s t ern A u s r a l i a I n c . T MA WA h e M a t h e m a t i...
YEAR 7 MATHEMATICS
o f Wes te rn Austra l ia In
c.
MA WAThe
Mathematical Association
Task 11:Task 12:Task 13:Task 14:Task 15:Task 16:Task 18:Task 19:Task 24:Task 25:Task 26:
Workshop SettingsOrderingRules for CalculatingMaking Calculations EasierRearranging NumbersEstablishing LawsRounding DecimalsSquare NumbersRatiosEquationsScoring Golf
Task 27: Task 28: Task 30: Task 32: Task 37: Task 38: Task 40: Task 101: Task 102: Task 105:
Fraction GraphicsPercentagesDiscountsMoving PointsFraction OperationsFraction ActionGraphing RelationshipsConsecutive NumbersLarge ElevensIdeal Fractions
TASK LISTTASK LIST
YEAR 7 MATHEMATICSNumber & Algebra TasksSet 2
PRODUCED BY A DEPARTMENT OF EDUCATION - MAWA PARTNERSHIP PROJECT
WRITTEN FOR THE YEAR 7 AUSTRALIAN CURRICULUM
PRODUCED BY A DEPARTMENT OF EDUCATION - MAWA PARTNERSHIP PROJECT
WRITTEN FOR THE YEAR 7 AUSTRALIAN CURRICULUM
© Department of Education, Western Australia (2015)© Department of Education, Western Australia (2015) © Department of Education, Western Australia (2015)© Department of Education, Western Australia (2015)
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MATHS7TL002 | Mathematics | Number and Algebra Activity | Workshop Settings 2 © Department of Education WA 2015
TASK 11: WORKSHOP SETTINGS
Overview
In this task, students are guided in their examination of a relationship between two variables.
The process by which relationships are examined is central to this investigation. Students
should be encouraged to look at relationships by using diagrams, tables, graphs, worded
descriptions and symbolic representation.
Students will need
No special requirements
Relevant content descriptions from the Western Australian Curriculum
Create algebraic expressions and evaluate them by substituting a given value for each
variable (ACMNA176)
Investigate, interpret and analyse graphs from authentic data (ACMNA180)
Students can demonstrate
understanding when they
o use a graph of plotted points to represent the connection between the numbers of
tables and the numbers of chairs
o determine the algebraic expressions linking the numbers of tables and the
numbers of chairs
o see the relationship between squares and square roots
o recognise different ways of determining the answer
reasoning when they
o explain why it does not make sense to join the points on the graph
problem solving when they
o independently provide several representations of the relationship between the
numbers of tables and the numbers of chairs in the final activity.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Workshop Settings 3 © Department of Education WA 2015
WORKSHOP SETTINGS Solutions and Notes for Teachers
1. Chris sets up the seminar room for workshops and Leah has asked that all the tables and
chairs be arranged as shown in the diagram.
A chair is shown as and a table as
(a) Give a reason to explain the fact that there are no chairs at one end of the two tables.
Maybe to leave room to see the presenter.
(b) If there were 4 tables used and the arrangement above was repeated, how many
chairs in total would be needed?
12 chairs
(c) As the number of tables increases by 2, how does the number of chairs increase?
Describe how the answer to this question can be quickly determined.
As the number of tables increases by 2, the number of chairs increases by 6.
It is easily seen in the diagram that there are 6 chairs for every 2 tables.
(d) If there were 20 tables set out using this arrangement, how many people could be
accommodated at the workshop?
There are 10 sets of 2 tables so 10 x 6 chairs = 60 chairs.
(e) Complete the table showing numbers of tables and numbers of chairs.
Number of tables 2 4 6 8 10 12 14 16 18 20
Number of chairs 6 12 18 24 30 36 42 48 54 60
MATHS7TL002 | Mathematics | Number and Algebra Activity | Workshop Settings 4 © Department of Education WA 2015
(f) Plot the points representing number of chairs and tables as in the table above.
(g) Describe the pattern seen in the location of the points.
The points lie in a straight line
(h) Does it make sense to join up the points you have plotted? Explain.
It does NOT make sense to join the points because that would imply the values
between also exist. In this case there is no situation for 3 tables.
(i) Describe (in words) the link between the number of chairs and the number of tables.
Number of chairs = 3 x number of tables
(j) Using the symbols h to represent the number of chairs and b to represent the number
of tables, write the rule linking h and b.
h = b x 3 OR h = 3b OR h = 3 x b
(k) For the rule you have developed in (j) describe the types of numbers that h and b can
represent.
For b, the numbers it represents are even, positive, integers; and for a workshop these
numbers should not be very large (less than 100).
For h, the numbers it represents are even, positive, integers, multiples of 2, 3, 6 and
composite; and should not be more than 300 for a workshop.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Workshop Settings 5 © Department of Education WA 2015
(l) Show how you can use your rule to determine the number of chairs needed for 100
tables.
h = b x 3 = 100 x 3 = 300
(m) How many tables would you need for 72 people?
h = b x 3 so b x 3 = 72 and b = 24
2. In another room suitable for workshops the tables have a different shape and are arranged
as shown in the diagram below.
For this arrangement, describe the link between the number of chairs and the number of
tables in the following ways:
(a) As a table of values:
Number of tables 2 4 6 8 10 12 14 16 18 20
Number of chairs 8 16 24 32 40 48 56 64 72 80
(b) As a series of plotted points:
a series of plotted points
MATHS7TL002 | Mathematics | Number and Algebra Activity | Workshop Settings 6 © Department of Education WA 2015
(c) As a rule described in words:
For every two tables there are 8 people.
There 4 people per table.
(d) As a rule described in mathematical symbols:
If h represents the number of chairs and b represents the number of tables,
h = b x 4 OR h = 4b OR h = 4 x b
MATHS7TL002 | Mathematics | Number and Algebra Activity | Workshop Settings 7 © Department of Education WA 2015
STUDENT COPY WORKSHOP SETTINGS
Chris sets up the seminar room for workshops and Leah has asked that all the tables and
chairs be arranged as shown in the diagram.
A chair is shown as and a table as
(a) Give a reason to explain the fact that there are no chairs at one end of the two tables.
(b) If there were 4 tables used and the arrangement above was repeated, how many
chairs in total would be needed?
(c) As the number of tables increases by 2, how does the number of chairs increase?
Describe how the answer to this question can be quickly determined.
(d) If there were 20 tables set out using this arrangement, how many people could be
accommodated at the workshop?
MATHS7TL002 | Mathematics | Number and Algebra Activity | Workshop Settings 8 © Department of Education WA 2015
(e) Complete the table showing numbers of tables and numbers of chairs.
Number of tables 2 4 6 8 10 12 14 16 18 20
Number of chairs 6
(f) Plot the points representing number of chairs and tables as in the table above.
(g) Describe the pattern seen in the location of the points.
(h) Does it make sense to join up the points you have plotted? Explain.
(i) Describe (in words) the link between the number of chairs and the number of tables.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Workshop Settings 9 © Department of Education WA 2015
(j) Using the symbols h to represent the number of chairs and b to represent the number
of tables, write the rule linking h and b.
(k) For the rule you have developed in (j) describe the types of numbers that h and b can
represent.
(l) Show how you can use your rule to determine the number of chairs needed for 100
tables.
(m) How many tables would you need for 72 people?
2. In another room suitable for workshops the tables have a different shape and are arranged
as shown in the diagram below.
For this arrangement, describe the link between the number of chairs and the number of
tables in the following ways:
(a) A table of values:
Number of tables 2 4 6 8 10 12 14 16 18 20
Number of chairs
MATHS7TL002 | Mathematics | Number and Algebra Activity | Workshop Settings 10 © Department of Education WA 2015
(b) As a series of plotted points:
(c) As a rule described in words:
(d) As a rule described in mathematical symbols:
MATHS7TL002 | Mathematics | Number and Algebra Activity | Ordering 1 © Department of Education WA 2015
MATHS7TL002 | Mathematics | Number and Algebra Activity | Ordering 2 © Department of Education WA 2015
TASK 12: ORDERING
Overview
For this activity students need to use their knowledge and understanding of the relative sizes
of positive and negative numbers including fractions and decimals.
Students will need
no material needed
Relevant content descriptions from the Western Australian Curriculum
Compare fractions using equivalence. Locate and represent positive and negative
fractions and mixed numbers on a number line (ACMNA152)
Connect fractions, decimals and percentages and carry out simple conversions
(ACMNA157)
Students can demonstrate
fluency when they
o move flexibly between decimal and fractional representations of numbers as they
attempt to order numbers
understanding when they
o recognise equivalence of decimals and fractions
o can create comparative statements of numbers as in Activity 1 (b)-(d)
reasoning when they
o explain errors in the students‟ thinking in Activity 2
problem solving when they
o can place a set of numbers in numerical order as in Activity 1 (h) and (j)
MATHS7TL002 | Mathematics | Number and Algebra Activity | Ordering 3 © Department of Education WA 2015
ORDERING Solutions and Notes for Teachers
Activity 1
1. Kate, Liz and Jon were given a set of twelve numbers to write in numerical order. The
table below shows the original numbers, Kate‟s list, Liz‟s list and Jon‟s list.
Original numbers Kate‟s list Liz‟s list Jon‟s list
1 0.75 0 -0.7 0
2 0 -0.7 -1
3
-1 -1.8
4 -0.7 -1.8 -
5 0.079 -
0 -0.7
6
0.079
-1
7 0.5678 0.5678
-1.8
8
0.75
-
9 -
1.7 0.079 0.079
10 -1.8
0.5678 0.5678
11 -1
0.75 0.75
12 1.7
1.7 1.7
(a) What is meant by „in numerical order‟? Give an example.
Order of size from smallest to greatest; e.g., 5, 7, 9, 10
MATHS7TL002 | Mathematics | Number and Algebra Activity | Ordering 4 © Department of Education WA 2015
(b) According to Kate‟s list,
>
Is this statement TRUE or FALSE? TRUE
Create another five “greater than” (>) statements from Kate‟s list and identify for each
one if they are TRUE or FALSE.
-
> 0 FALSE
> 0.5678 TRUE
> 0 TRUE
1.7 > 0.75 TRUE
-1 > 0 FALSE
(c) From Liz‟s list create five “less than” (<) statements that Liz would make and identify for
each one if they are TRUE or FALSE.
-0.7 < -1 FALSE
-1 < -1.8 FALSE
FALSE
<
TRUE
TRUE
(d) Jon has indicated that 1.7 > 0.5678, and this is TRUE.
Locate five more TRUE statements according to Jon‟s list.
1.7 > 0.75
>
1.7 > 0
>
Locate five FALSE statements according to Jon‟s list.
-1 > 0
-1.8 > -0.7
0.079 >
-1 >
>
MATHS7TL002 | Mathematics | Number and Algebra Activity | Ordering 5 © Department of Education WA 2015
(e) Kate and Jon both think that 0 is the smallest number in the list. Are they correct?
Explain.
No, zero is larger than any negative numbers.
(f) For each of the three persons, identify, describe and show by examples from the list, at
least ONE error in their thinking about the order of numbers.
Kate:
Kate thinks 0 is smaller than all negative numbers because she has put 0 as the
smallest number.
All Kate‟s negative numbers are out of order because she thinks -1.8 > -1 and has
ignored the difference that the negative sign makes.
Kate‟s decimals and fractions are correct but they have been separated into two
sections which is not correct because
is less than 0.75. She has put all the fractions
as being greater than all the decimals.
Liz:
Liz makes the same mistake as Kate with regard to the negative numbers. All Liz‟s
negative numbers are out of order because she thinks -1 > -0.7 and has ignored the
difference that the negative sign makes.
Liz‟s decimals and fractions are correct but they have been separated into two sections
which is not correct because
is greater than 0.079. She has put all the decimals as
being greater than all the fractions.
Jon
Jon makes a mistake when he puts all the fractions first, then all the negative numbers
then all the decimals. According to his list
is less than -1, which is incorrect.
Jon thinks that 0 is less than all negative numbers, which is incorrect.
Jon‟s negative numbers are in the reverse order because -1 is not less than -1.8
(g) Write the list of numbers in the correct order.
-1.8 -
-1 -0.7 0 0.079
0.5678
0.75 1.7
(h) Display the numbers in their approximate positions on the number line.
(i) Write a list of rules that provide instructions for placing numbers on a number line.
Discuss your list with another student and refine if necessary.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Ordering 6 © Department of Education WA 2015
Zero is left of any positive numbers
To the right of zero the numbers are in the counting order with the decimals and
fractions placed according to their size.
All the negative numbers are to the left of zero.
To locate negative numbers, ignore the negative sign so the numbers start at 0 and
go left as they “increase”.
Positive fractions less than 1 go on the right between 0 and 1 while all negative
fractions are left of 0.
Decimals are placed in order of size; positive ones go right of 0 and the greater they
are the further right they will be. If they are negative, they are left of 0 and the smaller
they are, the further left they will be. Example -5 is further left than -4.
(j) Use your rules to order these ten numbers.
0.8 0 -0.81
-1.8 1.08 -
0.0008
-1.8 -0.81 -
0 0.0008
0.8
1.08
(k) Create a similar list of 12 numbers (include decimals, fractions and negative numbers)
and give them to another student to place in ascending order.
Various answers as appropriate.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Ordering 7 © Department of Education WA 2015
Activity 2
Identify the errors made by each of these students when they placed numbers into ascending
order.
Situation 1
Numbers to order 0.99 0.9 0.9099 0.09999 0.909
Student‟s answer 0.9 0.99 0.909 0.9000 0.09999
Error in thinking The more digits there are after the decimal point, the greater the number.
Correct order 0.09999 0.9000 0.9 0.909 0.99
Situation 2
Numbers to order
Student‟s answer
Error in thinking The greater the denominator, the greater the fraction.
Correct order
Situation 3
Numbers to order -4 -7 -21 -5 -8 -9
Student‟s answer -4 -5 -7 -8 -9 -21
Error in thinking Negative numbers increase in the same order as if the negative sign is not there.
Correct order -21 -9 -8 - 7 -5 -4
Situation 4
Numbers to order 56.103 56.303 56.031 56.014 56.150 56.320
Student‟s answer 56.103 56.014 56.150 56.031 56.320 56.303
Error in thinking Zeroes after the decimal point have no value
Correct order 56.014 56.031 56.103 56.150 56.303 56.320
MATHS7TL002 | Mathematics | Number and Algebra Activity | Ordering 8 © Department of Education WA 2015
STUDENT COPY ORDERING
Activity 1
1. Kate, Liz and Jon were given a set of twelve numbers to write in numerical order. The
table below shows the original numbers, Kate‟s list, Liz‟s list and Jon‟s list.
Original numbers Kate‟s list Liz‟s list Jon‟s list
1 0.75 0 -0.7 0
2 0 -0.7 -1
3
-1 -1.8
4 -0.7 -1.8 -
5 0.079 -
0 -0.7
6
0.079
-1
7 0.5678 0.5678
-1.8
8
0.75
-
9 -
1.7 0.079 0.079
10 -1.8
0.5678 0.5678
11 -1
0.75 0.75
12 1.7
1.7 1.7
(a) What is meant by „in numerical order‟? Give an example.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Ordering 9 © Department of Education WA 2015
(b) According to Kate‟s list,
>
Is this statement TRUE or FALSE?
Create another five “greater than” (>) statements from Kate‟s list and identify for each
one if they are TRUE or FALSE.
(c) From Liz‟s list create five “less than” (<) statements and identify for each one if they are
TRUE or FALSE.
(d) Jon has indicated that 1.7 > 0.5678 and this is TRUE.
Locate five more TRUE statements according to Jon‟s list.
Locate five FALSE statements according to Jon‟s list.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Ordering 10 © Department of Education WA 2015
(e) Kate and Jon both think that 0 is the smallest number in the list. Are they correct?
Explain.
(f) For each of the three persons, identify, describe and show by examples from the list, at
least ONE error in their thinking about the order of numbers.
Kate:
Liz:
Jon:
MATHS7TL002 | Mathematics | Number and Algebra Activity | Ordering 11 © Department of Education WA 2015
(g) Write the list of numbers in the correct order.
(h) Display the numbers in their approximate positions on the number line.
(i) Write a list of rules that provide instructions for placing numbers on a number line.
Discuss your list with another student and refine if necessary.
(j) Use your rules to order these ten numbers.
0.8 0 -0.81
-1.8 1.08 -
0.0008
(k) Create a similar list of 12 numbers (include decimals, fractions and negative numbers)
and give them to another student to place in ascending order.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Ordering 12 © Department of Education WA 2015
Activity 2
Identify the errors made by each of these students when they placed numbers into ascending
order.
Situation 1
Numbers to order 0.99 0.9 0.9099 0.09999 0.909
Student‟s answer 0.9 0.99 0.909 0.9000 0.09999
Error in thinking
Correct order
Situation 2
Numbers to order
Student‟s answer
Error in thinking
Correct order
Situation 3
Numbers to order -4 -7 -21 -5 -8 -9
Student‟s answer -4 -5 -7 -8 -9 -21
Error in thinking
Correct order
Situation 4
Numbers to order 56.103 56.303 56.031 56.014 56.150 56.320
Student‟s answer 56.103 56.014 56.150 56.031 56.320 56.303
Error in thinking
Correct order
MATHS7TL002 | Mathematics | Number and Algebra Activity | Rules for Calculating 2 © Department of Education WA 2015
TASK 13: RULES FOR CALCULATING
Overview
In this task students will have the opportunity to investigate and review / establish the
commutative law of arithmetic and to determine when the rule applies. Students should
attempt as many calculations as possible without using calculators. This task builds on Year
6 curriculum content and it is not necessary to have covered any other Year 7 content.
Students will need
Calculators to check answers for a few activities.
Relevant content descriptions from the Western Australian Curriculum
Apply the associative, commutative and distributive laws to aid mental and written
computation (ACMNA151)
Students can demonstrate
understanding when they
o describe and summarise the commutative rule
reasoning when they
o determine that the commutative rules also apply to decimals and fractions
MATHS7TL002 | Mathematics | Number and Algebra Activity | Rules for Calculating 3 © Department of Education WA 2015
RULES FOR CALCULATING Solutions and Notes for Teachers
In this task, there will be a series of activities in which the known rules for calculating are
reviewed and other rules are developed. Calculators should only be used to check answers
or where indicated by the following symbol.
Activity 1
(a) Complete the following table by entering the results of the calculations given.
1 21 + 8 = 39 8 + 21 = 39
2 42 + 7 = 49 7 + 42 = 49
3 8 + 66 = 74 66 + 8 = 74
4 15 + 16 = 31 16 + 15 = 31
5 22 + 32 = 54 32 + 22 = 54
6 18 + 10 = 28 10 + 18 = 28
7 100 + 200 = 300 200 + 100 = 300
8 180 + 1000 = 1180 1000 + 180 = 1180
(b) In every row, there are two sets of numbers to add. What do you notice about these
sets of numbers?
The numbers are the same but the order is different.
(c) What do you notice about the answers in each row?
In each row the answers are the same.
(d) Write a sentence to describe the rule that is highlighted in these examples.
When two whole numbers are added, the order in which the numbers are written or
added does not influence the answer.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Rules for Calculating 4 © Department of Education WA 2015
(e) Does your rule apply to decimals? Write down three examples to support your decision.
Yes. Examples:
0.6 + 0.4 = 1 and 0.4 + 0.6 = 1
1.3 + 1.2 = 2.5 and 1.2 + 1.3 = 2.5
10.1 + 11.5 = 21.6 and 11.5 + 10.1 = 21.6
(f) Does your rule apply to fractions? Write down three examples to support your decision.
Yes. Examples:
+ = 1 and + = 1
+ = and + =
+ = and + =
(g) Does the rule work when you add -
(i) a fraction to a whole number; e.g., 3 + ? Yes
(ii) a decimal to a whole number; e.g., 0.2 + 3? Yes
(iii) a decimal to a fraction; e.g., 0.5 + ? Yes
(iv) three numbers; e.g., 14 + 13 +1 2? Yes
(h) Write a number sentence to give another example of each type given in (g)
Examples:
(i) 1 + = + 1 =
(ii) 9.3 + 10 = 10 + 9.3 = 19.3
(iii) 0.5 + = + 0.5 = 0.75 or
(iv) 1 + 2 + 3 = 3 + 2 + 1
(i) Write a conclusion for this activity.
When two or more whole, fractional or decimal numbers are added the answer is the same
regardless of the order in which the numbers are added (or written).
MATHS7TL002 | Mathematics | Number and Algebra Activity | Rules for Calculating 5 © Department of Education WA 2015
Activity 2
(a) Complete the following table be entering the results of the calculations given.
1 3 x 4 = 12 4 x 3 = 12
2 5 x 10 = 50 10 x 5 = 50
3 20 x 10 = 200 10 x 20 = 200
4 6 x 11 = 66 11 x 6 = 66
5 12 x 20 = 240 20 x 12 = 240
6 8 x 7 = 56 7 x 8 = 56
7 9 x 7 = 63 7 x 9 = 63
8 150 x 10 = 1500 10 x 150 = 1500
(b) Write a sentence to describe the rule that has applied to the multiplication of these
whole numbers.
Multiplying one number by a second number gives the same answer as if you multiply
the second number by the first number.
(c) Does your rule apply to decimals? Write down five examples to support your decision.
Examples: 0.1 x 0.2 = 0.2 x 0.1 = 0.02
1.2 x 0.4 = 0.4 x 1.2 = 0.48
2.3 x 1.2 = 1.2 x 2.3 = 2.76
0.9 x 1.5 = 1.5 x 0.9 = 0.45
5.6 x 2.4 = 2.4 x 5.6 = 13.44
(d) Does your rule apply to fractions? Write down five examples to support your decision.
Examples: x = x =
x = x =
x = x =
x = x =
x = x =
MATHS7TL002 | Mathematics | Number and Algebra Activity | Rules for Calculating 6 © Department of Education WA 2015
Activity 3
(a) Complete the following table by entering the results of the calculations given.
1 20 – 8 = 12 8 – 20 = -12
2 32 – 20 = 8 20 – 32 = -8
3 4 – 50 = -46 50 – 4 = 46
4 7 – 73 = -66 73 – 7 = 66
5 81 – 90 = -9 90 – 81 = 9
6 45 – 100 = -55 100 – 45 = 55
7 68 – 12 = 56 12 – 68 = -56
8 36 – 13 = 23 13 – 36 = -23
(b) The rule for subtraction of two numbers is different from the rules for addition and
multiplication. Explain how the rule is different.
The order in which the numbers are written influences the answer.
Subtracting the second number from the first number gives a different answer from
subtracting the first number from the second number.
(c) Looking at the pattern in your answers, describe a way to find the answer when a
larger whole number is subtracted from a smaller whole number.
Take the smaller number from the larger number and use a negative sign with your
result; e.g., for 5 – 56, take 5 from 56 (51) and use a negative sign with the 51.
So 5 – 56 = -51
(d) Use your method from part (c) to determine the following answers mentally and THEN
use your calculator if necessary to check your answer.
(i) 90 – 110 = -20
(ii) 63 – 100 = -37
(iii) 45 – 90 = -45
(iv) 8 – 70 = -62
MATHS7TL002 | Mathematics | Number and Algebra Activity | Rules for Calculating 7 © Department of Education WA 2015
Activity 4
1. Complete the following table by entering the results of the calculations given.Express your answers as both fractions and decimals.
1 20 ÷ 8 = = 2.5 8 ÷ 20 = = 0.4
2 25 ÷ 4 = 6.25 4 ÷ 25 = 0.16
3 64 ÷ 4 = 16 4 ÷ 64 = 0.0625
4 50 ÷ 100 = = 0.5 100 ÷ 50 = 2
5 15 ÷ 60 = = 0.25 60 ÷ 15 = 4
6 6 ÷ 30 = = 0.2 30 ÷ 6 = 5
7 7 ÷ 70 = = 0.1 70 ÷ 7 = 10
8 8 ÷ 10 = 0.8 10 ÷ 8 = 1.25
2. Does order matter when dividing?
Yes
Activity 5
When a mathematical operation is applied to two numbers, does the order in which the two
numbers are written make a difference? Write a paragraph which provides an answer to this
question, explains when it is true and when it is not true, and which provides evidence for
your conclusion.
When a mathematical operation is applied to two numbers, the order in which the two
numbers are written makes a difference during subtraction and division but not for addition
and multiplication. Adding 5 to 3 is the same as adding 3 to 5 but subtracting 3 from 5 is not
the same as subtracting 5 from 3. Multiplying 7 by 6 gives 42 and this is the same as
multiplying 6 by 7. However, in division the order matters because 100 divided by 10 is 10,
but 10 divided by 100 is 0.1. This rule applies for all numbers - whole numbers, fractions and
decimals.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Rules for Calculating 8 © Department of Education WA 2015
STUDENT COPY RULES FOR CALCULATING
In this task, there will be a series of activities in which the known rules for calculating are
reviewed and other rules are developed. Calculators should only be used to check answers
or where indicated by the following symbol.
Activity 1
(a) Complete the following table by entering the results of the calculations given.
1 21 + 8 = 8 + 21 =
2 42 + 7 = 7 + 42 =
3 8 + 66 = 66 + 8 =
4 15 + 16 = 16 + 15 =
5 22 + 32 = 32 + 22 =
6 18 + 10 = 10 + 18 =
7 100 + 200 = 200 + 100 =
8 180 + 1000 = 1000 + 180 =
(b) In every row, there are two sets of numbers to add. What do you notice about these
sets of numbers?
(c) What do you notice about the answers in each row?
(d) Write a sentence to describe the rule that is highlighted in these examples.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Rules for Calculating 9 © Department of Education WA 2015
(e) Does your rule apply to decimals? Write down three examples to support your decision.
(f) Does your rule apply to fractions? Write down three examples to support your decision.
(g) Does the rule work when you add -
(i) a fraction to a whole number; e.g., 3 + ?
(ii) a decimal to a whole number; e.g., 0.2 + 3
(iii) a decimal to a fraction; e.g., 0.5 + ?
(iv) three numbers; e.g.,14 + 13 +1 2
(h) Write a number sentence to give another example of each type given in (g)
(i)
(ii)
(iii)
(iv)
(i) Write a conclusion for this activity.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Rules for Calculating 10 © Department of Education WA 2015
Activity 2
(a) Complete the following table be entering the results of the calculations given.
1 3 x 4 = 4 x 3 =
2 5 x 10 = 10 x 5 =
3 20 x 10 = 10 x 20 =
4 6 x 11 = 11 x 6 =
5 12 x 20 = 20 x 12 =
6 8 x 7 = 7 x 8 =
7 9 x 7 = 7 x 9 =
8 150 x 10 = 10 x 150 =
(b) Write a sentence to describe the rule that has applied to the multiplication of these
whole numbers.
(c) Does your rule apply to decimals? Write down five examples to support your decision.
(d) Does your rule apply to fractions? Write down five examples to support your decision.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Rules for Calculating 11 © Department of Education WA 2015
Activity 3
(a) Complete the following table by entering the results of the calculations given.
Use calculators to check your answers if necessary.
1 20 – 8 = 8 – 20 =
2 32 – 20 = 20 – 32 =
3 4 – 50 = 50 – 4 =
4 7 – 73 = 73 – 7 =
5 81 – 90 = 90 – 81 =
6 45 – 100 = 100 – 45 =
7 68 – 12 = 12 – 68 =
8 36 – 13 = 13 – 36 =
(b) The rule for subtraction of two numbers is different from the rules for addition and
multiplication. Explain how the rule is different.
(c) Looking at the pattern in your answers, describe a way to find the answer when a
greater whole number is subtracted from a smaller whole number.
(d) Use your method from part (c) to determine the following answers mentally and THEN
use your calculator if necessary to check your answer.
(i) 90 – 110 =
(ii) 63 – 100 =
(iii) 45 – 90 =
(iv) 8 – 70 =
MATHS7TL002 | Mathematics | Number and Algebra Activity | Rules for Calculating 12 © Department of Education WA 2015
Activity 4
Complete the following table by entering the results of the calculations given.
Express your answers as both fractions and decimals.
1 20 ÷ 8 = 8 ÷ 20 =
2 25 ÷ 4 = 4 ÷ 25 =
3 64 ÷ 4 = 4 ÷ 64 =
4 50 ÷ 100 = 100 ÷ 50 =
5 15 ÷ 60 = 60 ÷ 15 =
6 6 ÷ 30 = 30 ÷ 6 =
7 7 ÷ 70 = 70 ÷ 7 =
8 8 ÷ 10 = 10 ÷ 8 =
2. Does order matter when dividing?
Activity 5
When a mathematical operation is applied to two numbers, does the order in which the two
numbers are written make a difference? Write a paragraph which provides an answer to this
question, explains when it is true and when it is not true, and provides evidence for your
conclusion.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Making Calculations Easier 2 © Department of Education WA 2015
TASK 14: MAKING CALCULATIONS EASIER
Overview
This task involves using the associative rule for addition and multiplication. Knowing the rule
and its name is not expected. Rather, it is important that students can use the process and
can appreciate its usefulness in making calculations much easier.
Associative rule:
Adding 3 + 4 + 5 gives the same answer whether you add the 3 and the 4 first, or the 4 and
the 5 first. Thus, (3 + 4) + 5 = 3 + (4 + 5).
Multiplying 2 x 4 x 10 gives the same answer whether you multiply the 2 and the 4, and then
multiply your answer by 10, OR multiply the 4 and the 10 and then multiply this answer by 2.
Thus, (2 x 4) x 10 = 2 x (4 x 10).
Students should be familiar with commutativity (even if not the word) and
appreciate that 3 + 4 = 4 + 3, as well as 5 x 2 = 2 x 5
Students will need
Students should attempt to do all calculations without a calculator
Relevant content descriptions from the Western Australian Curriculum
Apply the associative, commutative and distributive laws to aid mental and written
computation (ACMNA151)
Multiply and divide fractions and decimals using efficient written strategies and digital
technologies (ACMNA154)
Students can demonstrate
fluency when they
o represent fractions and decimals in various ways
reasoning when they
o apply the associative and commutative laws to numbers
problem solving when they
o complete Activities 6 and 7
MATHS7TL002 | Mathematics | Number and Algebra Activity | Making Calculations Easier 3 © Department of Education WA 2015
MAKING CALCULATIONS EASIER Solutions and Notes for Teachers
The purpose of this task is to make calculations easier by rearranging numbers in ways that
can be shown to work. You may recall that when numbers are multiplied, changing the order
of the numbers does not change the answer.
Activity 1
Consider
6 x 2 = 12
60 x 2 must be 10 times the answer above because 60 = 10 times 6, and 60 = 6 x 10 x 2
600 x 2 must be 100 times 12 because 600 is a hundred times larger than 6.
1. Without using a calculator, use the patterns and the above information to complete the
table below.
6 x 2 = 12 60 x 2 = 120 600 x 2 = 1200 6000 x 2 = 12 000
7 x 2 = 14 70 x 2 = 140 700 x 2 = 1400 7000 x 2 = 14 000
8 x 2 = 16 80 x 2 = 160 800 x 2 = 1600 8000 x 2 = 16 000
9 x 2 = 18 90 x 2 = 180 900 x 2 = 1800 9000 x 2 = 18 000
10 x 2 = 20 100 x 2 = 200 1000 x 2 = 2000 10 000 x 2 = 20 000
15 x 2 = 30 150 x 2 = 300 1500 x 2 = 3000 15 000 x 2 = 30 000
18 x 2 = 36 180 x 2 = 360 1800 x 2 = 3600 18 000 x 2 = 36 000
15 000 x 2 can also be written as -
15 x 10 x 10 x 10 x 2
15 x 10 x 100 x 2
15 x 1000 x 2
2. Write 65 000 x 2 using the same pattern.
65 x 10 x 10 x 10 x 2
65 x 10 x 100 x 2
65 x 1000 x 2
MATHS7TL002 | Mathematics | Number and Algebra Activity | Making Calculations Easier 4 © Department of Education WA 2015
Activity 2
1. This approach to multiplication can be used with numbers other than 2.
Use the method outlined above to complete the table below.
It may help to refer back to the first entry in each row.
2. Consider number sentences with 6000 x 3:
6000 x 3 = 6 x 100 x 10 x 3
6 x 100 x 10 x 3 = 6 x 10 x 100 x 3 (for this exercise considered the same)
6000 x 3 = 2 x 3000 x 3 (is another example)
3. Complete four more different number sentences for 6000 x 3.
6000 x 3 = 6 x 1000 x 3
6000 x 3 = 3 x 2000 x 3
6000 x 3 = 3 x 2 x 100 x 10 x 3
6000 x 3 = 3 x 200 x 30
4. Compare your list with that of another student. Identify similarities and differences.
Summarise your observations.
6 3 = 18 60 3 = 180 30 6 = 180 600 3 = 1800 6000 3 = 18 000
7 4 = 28 70 4 = 280 40 7 = 280 700 4 = 2800 7000 4 = 28 000
9 5 = 45 90 5 = 450 50 9 = 450 900 5 = 4500 9000 5 = 45 000
3 8 = 24 30 8 = 240 80 3 = 240 300 8 = 2400 3000 8 = 24 000
6 6 = 36 60 6 = 360 6 60 = 360 600 6 = 3600 6000 6 = 36 000
4 9 = 36 40 9 =360 90 4 =360 400 9 =3600 4000 9 = 36 000
7 8 = 56 70 8 = 560 80 7 = 560 700 8 = 5600 7000 8 = 56 000
MATHS7TL002 | Mathematics | Number and Algebra Activity | Making Calculations Easier 5 © Department of Education WA 2015
Activity 3
1. Complete each of these tables using mental arithmetic.
(a)
0.5 x 2 = 1 0.5 x 20 = 10 0.5 x 200 = 100 0.5 x 2000 = 10 000
1.5 x 4 = 6 1.5 x 40 = 60 1.5 x 400 = 600 1.5 x 4000 = 6000
0.8 x 5 = 4 0.8 x 50 = 40 0.8 x 500 = 400 0.8 x 5000 = 4000
2.5 x 8 = 20 2.5 x 80 = 200 2.5 x 800 = 2000 2.5 x 8000 =20 000
1.4 x 5 = 7 1.4 x 50 = 70 1.4 x 500 = 700 1.4 x 5000 = 7000
2.8 x 5 = 14 2.8 x 50 = 140 2.8 x 500 = 1400 2.8 x 5000 = 14 000
14.5 x 2 = 29 14.5 x 20 = 290 14.5 x 200 = 2900 14.5 x 2000 = 29 000
(b) Hint: Think of the first number in each row in words.
For Row 1, what is 8 quarters?
For Row 8, what number is formed if you have 9 thirds?
x 8 = 2 x 80 = 20 x 800 = 200 x 8000 = 2000
x 8 = 4 x 80 = 40 x 800 = 400 x 8000 = 4000
x 6 = 2 x 60 = 20 x 600 = 200 x 6000 = 2000
x 4 = 6 x 40 = 60 x 400 = 600 x 4000 = 6000
x 8 = 10 x 80 = 100 x 800 = 1000 x 8000 = 10 000
6 x = 15 60 x = 150 600 x = 1500 6000 x = 15 000
7 x = 1 70 x = 10 700 x = 100 7000 x = 1000
9 x = 3 90 x = 30 900 x = 300 9000 x = 3000
2. Describe a step-by-step process (in words) to determine the answer to x 80 000.
Multiply 2 by 8 (gives 16)
Multiply a quarter by 8 (gives 2)
Add 16 plus 2 (gives 18)
Multiply 18 by 10 000 gives 180 000
MATHS7TL002 | Mathematics | Number and Algebra Activity | Making Calculations Easier 6 © Department of Education WA 2015
Activity 4
The process can be used in reverse to “break down” a calculation and make it easier.
Example: 25 x 2000 = 25 x 2 x 1000 = 50 x 1000 = 50 000
1. “Break down” these calculations as shown in the example given and calculate the
answers.
x 2000 = x 2 x 1000 = 17 x 1000 = 17 000
2.1 x 6000 = 2.1 x 6 x 1000 = 12.6 x 1000 = 12.6 x 10 x 100 = 126 x 100 = 12 600
0.9 x 300 = 0.9 x 3 x 100 = 2.7 x 100 = 270
x 12 000 = x 12 x 1000 = 3 x 1000 = 3000
25 x 4000 = 25 x 4 x 1000 = 100 x 1000 = 100 000
33 x 900 = 33 x 9 x 100 = 297 x 100 = 29 700
44 x 20 000 = 44 x 2 x 10 000 = 88 x 10 000 = 880 000
1.3 x 3000 = 1.3 x 3 x 1000 = 3.9 x 1000 = 3 900
x 60 000 = x 6 x 10 000 = 15 x 10 000 = 150 000
2. Now create some to give to another student to “break down” and answer.
Various answers
MATHS7TL002 | Mathematics | Number and Algebra Activity | Making Calculations Easier 7 © Department of Education WA 2015
Activity 5
The process can also be used when numbers are added. You may recall that when numbers
are added, changing the order of the numbers does not change the answer.
Example: 67 + 73 = 60 + 7 + 70 + 3 = 60 + 70 + 7 + 3 = 130 + 10 = 140
Break down these additions so that the answer is easily calculated.
681 + 128 = 600 + 60 + 1 + 100 + 20 + 8 = 700 + 80 + 9 = 789
+ + = + + + + + = 9 + 1 = 10
3.2 + 1.4 + 16.3 = 3 + 1 + 16 + 0.2 + 0.4 + 0.3 = 20 + 0.9 = 20.9
1010 + 7081 = 1000 + 7000 + 10 + 80 + 1 = 8091
196 + 214 = 100 + 200 + 90 + 10 + 10 = 300 + 100 + 10 = 410
+ + = + + + + + = 14 + = 14
892 + 618 = 800 + 600 + 90 + 10 + 2 + 8 = 1400 + 100 + 10 = 1510
41.7 + 55.6 + 21.4 + 80.3 = 40 + 50 + 20 + 80 + 1 + 5 + 1 + 0.7 + 0.6 + 0.4 + 0.3
= 190 + 7 + 2 = 199
+ 11 + + = + 11 + + + + + + = 28 + 2 = 30
Activity 6
In the mathematical puzzle called DocDoc the user is given a focus number.
The user has to identify the possible whole numbers which add to the focus number.
The numbers used can only be the numbers 1 to 9.
Examples: If the focus number is 7 and addition is the activity, then some possibilities are -
1 + 6, 2 + 5, 3 + 4, 1 + 2 + 4, 1 + 3 + 3, 1 + 1 + 5, 1 + 2 + 2 + 2, 1 + 1 + 2 + 3
Rearranging is not seen as a different possibility. 1 + 3 + 3 = 3 + 3 + 1 [same]
This gives 3 ways of adding 2 numbers, 3 ways of adding 3 numbers and 1 way of adding 4
numbers for a sum of 7.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Making Calculations Easier 8 © Department of Education WA 2015
Using the same rules, determine the numbers of ways that the following focus numbers can
be made with 2, 3, 4 or 5 numbers and record your answers in the table provided (only the
digits 1 to 9 can be used). Then write all the equations for each set.
Number of ways of adding
Focus number 2 numbers 3 numbers 4 numbers 5 numbers
5 2 2 1 1
8 4 5 5 2
9 4 6
16 2
17 1 13
24 0 3
27 0 1
5 = 3 + 2 = 1 + 4 = 1 + 1 + 3 = 2 + 2 + 1 = 1 + 1 + 2 + 1 = 1 + 1 + 1 + 1 + 1
8 = 1 + 7 = 6 + 2 = 5 + 3 = 4 + 4 = 1 + 2 + 5 = 1 + 3 + 4 = 1 + 1 + 6 = 2 + 2 + 4 = 3 + 3 + 2
8 = 2 + 2 + 2 + 2 = 1 + 3 + 1 + 3 = 1 + 1 + 1 + 5 = 1 + 1 + 2 + 4 = 1 + 2 + 2 + 3
8 = 1 + 1 + 1 + 1 + 4 = 1 + 1 + 1 + 2 + 3 =
9 = 1 + 8 = 2 + 7 = 3 + 6 = 4 + 5 = 2 + 3 + 4 = 1 + 2 + 6 = 1 + 3 + 5 = 1 + 1 + 7 = 1 + 4 + 4
9 = 2 + 2 + 5
16 = 8 + 8 = 7 + 9
17 = 8 + 9 = 1 + 8 + 8 = 1 + 7 + 9 = 2 + 8 + 7 = 2 + 6 + 9 = 3 + 9 + 5 = 3 + 8 + 6 = 3 + 7 + 7
17 = 4 + 6 + 7 = 4 + 5 + 8 = 4 + 4 + 9 = 5 + 9 + 3 = 5 + 6 + 6 = 5 + 7 + 5
24 = 7 + 8 + 9 = 8 + 8 + 8 = 9 + 9 + 6
27 = 9 + 9 + 9
MATHS7TL002 | Mathematics | Number and Algebra Activity | Making Calculations Easier 9 © Department of Education WA 2015
Activity 7
In the same mathematical puzzle called DocDoc the user can be given a focus number with
a different operation.
The user has to identify the possible whole numbers which multiply to produce the
focus number.
If the focus number is greater than 100, then the number 1 cannot be used. The
numbers used can then only be the numbers 2 to 9.
Examples: If the focus number is 14 then possibilities are 2 x 7 and 1 x 2 x 7
This gives 1 way with 2 numbers and 1 way with 3 numbers.
Using the same rules, determine the numbers of ways that the following focus numbers can
be made with 2, 3, or 4 numbers and record your answers in the table provided (only the
digits 1 to 9 can be used). Then write all the equations for each set.
Number of ways
Focus number 2 numbers 3 numbers 4 numbers
6 2 2 2
12 2 3 3
24 2 3 5
81 1 2 3
100 0 1 2
216 0 3 5
224 0 1 2
6 = 2 x 3 = 1 x 6 = 1 x 2 x 3 = 1 x 1 x 6 = 1 x 2 x 3 x 1 = 1 x 1 x 1 x 6
12 = 3 x 4 = 2 x 6 = 1 x 2 x 6 = 1 x 3 x 4 = 2 x 2 x 3
12 = 1 x 2 x 2 x 3 = 1 x 1 x 2 x 6 = 1 x 1 x 3 x 4
24 = 6 x 4 = 8 x 3 = 2 x 3 x 4 = 1 x 4 x 6 = 1 x 3 x 8
24 = 1 x 1 x 4 x 6 = 1 x 1 x 3 x 8 = 1 x 2 x 3 x 4 = 2 x 2 x 3 x 2 = 1 x 2 x 2 x 6
81 = 9 x 9 = 1 x 9 x 9 = 3 x 3 x 9 = 1 x 9 x 9 x 1 = 3 x 3 x 3 x 3 = 1 x 3 x 3 x 9
100 = 4 x 5 x 5 = 4 x 5 x 5 x 1 = 2 x 2 x 5 x 5
216 = 9 x 6 x 4 = 9 x 8 x 3 = 6 x 6 x 6
216 = 2 x 3 x 6 x 6 = 3 x 3 x 4 x 6 = 2 x 2 x 6 x 9 = 2 x 3 x 4 x 9 = 3 x 3 x 3 x 8
224 = 7 x 4 x 8 = 7 x 4 x 2 x 4 = 7 x 2 x 2 x 8
MATHS7TL002 | Mathematics | Number and Algebra Activity | Making Calculations Easier 10 © Department of Education WA 2015
STUDENT COPY MAKING CALCULATIONS EASIER
The purpose of this task is to make calculations easier by rearranging numbers in ways that
can be shown to work. You may recall that when numbers are multiplied, changing the order
of the numbers does not change the answer.
Activity 1
Consider
6 x 2 = 12
60 x 2 must be 10 times the answer above because 60 = 10 times 6, and 60 = 6 x 10 x 2
600 x 2 must be 100 times 12 because 600 is a hundred times greater than 6.
1. Without using a calculator, use the patterns and the above information to complete the
table below.
6 x 2 = 12 60 x 2 = 600 x 2 = 6000 x 2 =
7 x 2 = 70 x 2 = 700 x 2 = 7000 x 2 =
8 x 2 = 80 x 2 = 800 x 2 = 8000 x 2 =
9 x 2 = 90 x 2 = 900 x 2 = 9000 x 2 =
10 x 2 = 100 x 2 = 1000 x 2 = 10 000 x 2 =
15 x 2 = 150 x 2 = 1500 x 2 = 15 000 x 2 =
18 x 2 = 180 x 2 = 1800 x 2 = 18 000 x 2 =
15 000 x 2 can also be written as -
15 x 10 x 10 x 10 x 2
15 x 10 x 100 x 2
15 x 1000 x 2
2. Write 65 000 x 2 using the same pattern.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Making Calculations Easier 11 © Department of Education WA 2015
Activity 2
1. This approach to multiplication can be used with numbers other than 2.
Use the method outlined above to complete the table below.
It may help to refer back to the first entry in each row.
2. Consider number sentences with 6000 x 3:
6000 x 3 = 6 x 100 x 10 x 3
6 x 100 x 10 x 3 = 6 x 10 x 100 x 3 (for this exercise considered the same)
6000 x 3 = 2 x 3000 x 3 (is another example)
3. Complete four more different number sentences for 6000 x 3.
4. Compare your list with that of another student. Identify similarities and differences.
Summarise your observations.
6 3 = 60 3 = 30 6 = 600 3 = 6000 3 =
7 4 = 70 4 = 40 7 = 700 4 = 7000 4 =
9 5 = 90 5 = 50 9 = 900 5 = 9000 5 =
3 8 = 30 8 = 80 3 = 300 8 = 3000 8 =
6 6 = 60 6 = 6 60 = 600 6 = 6000 6 =
4 9 = 40 9 = 90 4 = 400 9 = 4000 9 =
7 8 = 70 8 = 80 7 = 700 8 = 7000 8 =
MATHS7TL002 | Mathematics | Number and Algebra Activity | Making Calculations Easier 12 © Department of Education WA 2015
Activity 3
1. Complete each of these tables using mental arithmetic.
(a)
0.5 x 2 = 1 0.5 x 20 = 0.5 x 200 = 0.5 x 2000 =
1.5 x 4 = 1.5 x 40 = 1.5 x 400 = 1.5 x 4000 =
0.8 x 5 = 0.8 x 50 = 0.8 x 500 = 0.8 x 5000 =
2.5 x 8 = 2.5 x 80 = 2.5 x 800 = 2.5 x 8000 =
1.4 x 5 = 1.4 x 50 = 1.4 x 500 = 1.4 x 5000 =
2.8 x 5 = 2.8 x 50 = 2.8 x 500 = 2.8 x 5000 =
14.5 x 2 = 14.5 x 20 = 14.5 x 200 = 14.5 x 2000 =
(b) Hint: Think of the first number in each row in words.
For Row 1, what is 8 quarters?
For Row 8, what number is formed if you have 9 thirds?
x 8 =
x 80 =
x 800 =
x 8000 =
x 8 =
x 80 =
x 800 =
x 8000 =
x 6 =
x 60 =
x 600 =
x 6000 =
x 4 =
x 40 =
x 400 =
x 4000 =
x 8 =
x 80 =
x 800 =
x 8000 =
6 x
= 60 x
= 600 x
= 6000 x
=
7 x
= 70 x
= 700 x
= 7000 x
=
9 x
= 90 x
= 900 x
= 9000 x
=
2. Describe a step-by-step process (in words) to determine the answer to
x 80 000.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Making Calculations Easier 13 © Department of Education WA 2015
Activity 4
The process can be used in reverse to “break down” a calculation and make it easier.
Example: 25 x 2000 = 25 x 2 x 1000 = 50 x 1000 = 50 000
1. “Break down” these calculations as shown in the above example and calculate the
answers.
x 2000
2.1 x 6000
0.9 x 300
x 12 000
25 x 4000
33 x 900
44 x 20 000
1.3 x 3000
x 60 000
2. Now create some to give to another student to “break down” and answer.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Making Calculations Easier 14 © Department of Education WA 2015
Activity 5
The process can also be used when numbers are added. You may recall that when numbers
are added, changing the order of the numbers does not change the answer.
Example: 67 + 73 = 60 + 7 + 70 + 3 = 60 + 70 + 7 + 3 = 130 + 10 = 140
Break down these additions so that the answer is easily calculated.
681 + 128
+ +
3.2 + 1.4 + 16.3
1010 + 7081
196 + 214
+ +
892 + 618
41.7 + 55.6 + 21.4 + 80.3
+ 11 + +
Activity 6
In the mathematical puzzle called DocDoc the user is given a focus number.
The user has to identify the possible whole numbers which add up to the focus number.
The numbers used can only be the numbers 1 to 9.
Examples: If the focus number is 7 and addition is the activity, then the possibilities are:
1 + 6, 2 + 5, 3 + 4, 1 + 2 + 4, 1 + 3 + 3, 1 + 1 + 5, 1 + 2 + 2 + 2, 1 + 1 + 2 + 3
Rearranging is not seen as a different possibility. 1 + 3 + 3 = 3 + 3 + 1 [same]
This gives 3 ways of adding 2 numbers, 3 ways of adding 3 numbers and 1 way of adding 4
numbers for a sum of 7.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Making Calculations Easier 15 © Department of Education WA 2015
Using the same rules, determine the numbers of ways that the following focus numbers can
be made with 2, 3, 4 or 5 numbers and record your answers in the table provided (only the
digits 1 to 9 can be used). Then write all the equations for each set.
Number of ways of adding
Focus number 2 numbers 3 numbers 4 numbers 5 numbers
5
8
9
16
17
24
27
MATHS7TL002 | Mathematics | Number and Algebra Activity | Making Calculations Easier 16 © Department of Education WA 2015
Activity 7
In the same mathematical puzzle called DocDoc the user can be given a focus number with
a different operation.
The user has to identify the possible whole numbers which multiply to produce the
focus number.
If the focus number is greater than 100, then the number 1 cannot be used. The
numbers used can then only be the numbers 2 to 9.
Examples: If the focus number is 14 then the only possibility is 2 x 7.
This gives 1 way with 2 numbers and no ways with 3 numbers.
Using the same rules, determine the numbers of ways that the following focus numbers can
be made with 2, 3 or 4 numbers and record your answers in the table provided. [Note: Only
the digits 2 to 9 can be used if the focus number is greater than 100]
Number of ways
Focus number 2 numbers 3 numbers 4 numbers
6
12
24
81
100
216
224
MATHS7TL002 | Mathematics | Number and Algebra Activity | Rearranging Numbers 2 © Department of Education WA 2015
TASK 15: REARRANGING NUMBERS
Overview
The aim of this task is to introduce students to the recognition and use of the distributive law,
but the law will not be named nor generalised in these activities. Students should be already
familiar with the commutative and associative laws. These ideas are developed and reviewed
in Tasks 13 and 14.
Students will need
calculators
Relevant content descriptions from the Western Australian Curriculum
Apply the associative, commutative and distributive laws to aid mental and written
computation (ACMNA151)
Multiply and divide fractions and decimals using efficient written strategies and digital
technologies (ACMNA154)
Students can demonstrate
understanding when they
o determine the equivalence of number expressions
o recognise different ways of expressing numbers
problem solving when they
o interpret and find evidence to support statements as in Activity 6
MATHS7TL002 | Mathematics | Number and Algebra Activity | Rearranging Numbers 3 © Department of Education WA 2015
REARRANGING NUMBERS Solutions and Notes for Teachers
This task supports the investigation to learn how numbers can be rearranged to make
calculations easier.
Remember that for addition and multiplication operations the order does not matter if there is
only one type of operation in the number sentence. If addition and multiplication occur in the
same expression (with no brackets) then multiplication is done first.
Examples: 6 + 7 = 7 + 6 5 x 2 = 2 x 5
3 + 8 + 10 = 11 + 10 or 3 + 8 + 10 = 3 + 18 5 x 2 x 4 = 10 x 4 or 5 x 2 x 4 = 5 x 8
Activity 1
1. Consider 61 x 3. Which of the following expressions are equivalent to 61 x 3? Use your
calculator if necessary. Write TRUE or FALSE to show your thinking.
(60 + 1) x 3 TRUE 3 x (60 + 1) TRUE
60 x 3 + 1 x 3 TRUE 60 x 3 x 60 x 3 FALSE
60 x 3 – 1 x 3 FALSE 60 x 3 ÷ 3 FALSE
3 x 60 + 3 x 1 TRUE 3 x 60 x 1 FALSE
2. Consider 51 x 4. Which of the following expressions are equivalent to 51 x 4? Use your
calculator if necessary. Write TRUE or FALSE to show your thinking.
(50 + 1 ) x 4 TRUE 4 x (50 + 1) TRUE
50 x 4 + 1 x 4 TRUE 50 x 4 x 50 x 4 FALSE
50 x 4 – 1 x 4 FALSE 50 x 4 ÷ 4 FALSE
4 x 50 + 4 x 1 TRUE 4 x 50 x 1 FALSE
3. Consider 101 x 7. Which of the following expressions are equivalent to 101 x 7? Use your
calculator if necessary. Write TRUE or FALSE to show your thinking.
(100 + 1 ) x 7 TRUE 7 x (100 + 1) TRUE
100 x 7 + 1 x 7 TRUE 100 x 7 x 100 x 7 FALSE
100 x 7 – 1 x 7 FALSE 100 x 7 ÷ 7 FALSE
7 x 100 + 7 x 1 TRUE 7 x 100 x 1 FALSE
MATHS7TL002 | Mathematics | Number and Algebra Activity | Rearranging Numbers 4 © Department of Education WA 2015
Activity 2
1. Consider 99 x 8. Which of the following expressions are equivalent to 99 x 8? Use your
calculator if necessary. Write TRUE or FALSE to show your thinking.
(100 - 1 ) x 8 TRUE 8 x (100 - 1) TRUE
100 x 8 + 1 x 8 FALSE 100 x 8 x 100 x 8 FALSE
100 x 8 – 1 x 8 TRUE 100 x 8 ÷ 8 FALSE
8 x 100 - 8 x 1 TRUE 8 x 100 x 1 FALSE
2. Consider 48 x 2. Which of the following expressions are equivalent to 48 x 2? Use your
calculator if necessary. Write TRUE or FALSE to show your thinking.
(50 - 2 ) x 2 TRUE 2 x (50 - 2) TRUE
50 x 2 + 1 x 2 FALSE 50 x 2 x 50 x 2 FALSE
50 x 2 – 1 x 2 TRUE 50 x 2 ÷ 2 FALSE
2 x 50 - 2 x 1 TRUE 2 x 50 x 1 FALSE
3. Consider 69 x 9. Which of the following expressions are equivalent to 69 x 9? Use your
calculator if necessary. Write TRUE or FALSE to show your thinking.
(70 - 1 ) x 9 TRUE 9 x (70 - 1) TRUE
70 x 9 + 1 x 9 FALSE 70 x 9 x 70 x 9 FALSE
70 x 9 – 1 x 9 TRUE 70 x 9 ÷ 9 FALSE
9 x 70 - 9 x 1 TRUE 9 x 70 x 1 FALSE
4. Consider 81 x 7. Write 4 TRUE statements and 4 FALSE statements similar to the above.
TRUE FALSE
81 x 7 = (80 + 1) x 7 81 x 7 = 80 x 1 x 7
81 x 7 = 7 x (80 + 1) 81 x 7 = 80 x 7 – 7 x 1
81 x 7 = 80 x 7 + 1 x 7 81 x 7 = 80 x 7 x 1 x 7
81 x 7 = 7 x 1 + 7 x 80 81 x 7 = 10 x 7 x 1 ÷ 7
MATHS7TL002 | Mathematics | Number and Algebra Activity | Rearranging Numbers 5 © Department of Education WA 2015
Activity 3
Calculate the following expressions by firstly creating statements similar to the ones provided
above, thus making the calculations easier. The first two are done for you.
91 x 8 = (90 + 1) x 8 = 90 x 8 + 1 x 8 = 720 + 8 = 728
29 x 7 = (30 – 1) x 7 = 30 x 7 – 1 x 7 = 210 – 7 = 203
21 x 20 = (20 + 1) x 20 = 20 x 20 + 1 x 20 = 400 x 20 = 420
59 x 8 = (60 – 1) x 8 = 60 x 8 – 1 x 8 = 480 – 8 = 472
99 x 4 = (100 – 1) x 4 = 100 x 4 – 1 x 4 = 400 – 4 = 396
399 x 6 = (400 – 1 ) x 6 = 400 x 6 – 1 x 6 = 2400 – 6 = 2394
42 x 8 = (40 + 2) x 8 = 40 x 8 + 2 x 8 = 320 + 16 = 336
121 x 6 = (120 + 1) x 6 = 120 x 6 + 1 x 6 = 720 + 6 = 726
999 x 5 = (1000 – 1) x 5 = 1000 x 5 – 1 x 5 = 5000 – 5 = 4995
1001 x 15 = (1000 + 1) x 15 = 1000 x 15 + 1 x 15 = 15 000 + 15 = 15 015
51 x 11 = (50 + 1) x 11 = 50 x 11 + 1 x 11 = 550 + 11 = 561
599 x 7 =(600 – 1 ) x 7 = 600 x 7 – 1 x 7 = 4200 – 7 = 4193
Activity 4
Write a description of the process you have used in the previous activities and explain how
the calculations are made easier.
The high number to be multiplied is expressed as an addition or subtraction of two numbers.
The numbers are smaller and can be made close to a multiple of 10.
Then the multiplication is applied to each of the numbers in turn.
If the high number to be multiplied is expressed as an addition then the two products are
added. If the high number to be multiplied is expressed as a subtraction then you find the
difference between the two products.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Rearranging Numbers 6 © Department of Education WA 2015
Activity 5
1. If this process works for whole numbers, then should it work for fractions and decimals?
Justify your conclusion.
Yes, it should because they are also numbers which can all be expressed as the addition or
subtraction of two other numbers.
2. Use the process on these decimals and fractions. Two examples are provided.
8 x 1.99 = 8 x (2 – 0.01) = 8 x 2 – 8 x 0.01 = 16 – 0.08 = 15.92
5 x 2 = 5 x (2 + ) = 5 x 2 + 5 x =
9 x 3.01 = 9 x (3 + 0.01) = 9 x 3 + 9 x 0.01 = 27 + 0.09 = 27.09
2 x 3 = 2 x ( 3 + ) = (2 x 3 + 2 x ) = 6 + = 6
3 x 100.02 = 3 x (100 + 0.02) = 3 x 100 + 3 x 0.02 = 300 + 0.06 = 300.06
6 x 10.1 = 6 x (10 + 0.1) = 6 x 10 + 6 x 0.1 = 60 + 0.6 = 60.6
7 x 99.9 = 7 x (100 – 0.1) = 7 x 100 – 7 x 0.1 = 700 – 0.7 = 699.3
8 x 49.9 = 8 x (50 – 0.1) = 8 x 50 – 8 x 0.1 = 400 – 0.8 = 399.2
20 x 8.1 = 20 x (8 + 0.1) = 20 x 8 + 20 x 0.1 = 160 + 2 = 162
20.1 x 8 = (20 + 0.1) x 8 = 20 x 8 + 0.1 x 8 = 160 + 0.8 = 160.8
4 x 10 = (4 + ) x 10 = 4 x 10 + x 10 = 40 + 5 = 45
63.9 x 2 = (64 – 0.1) x 2 = 64 x 2 – 0.1 x 2 = 128 – 0.2 = 127.8
22.002 x 4 = (22 + 0.002) x 4 = 22 x 4 + 0.002 x 4 = 88 + 0.008 = 88.008
11 x 4 = (11 + ) x 4 = 11 x 4 + x 4 = 44 + 2 = 46
39.99 x 6 = (40 – 0.01) x 6 = 40 x 6 – 0.01 x 6 = 240 – 0.06 = 239.94
3 x 89.9 = 3 x (90 – 0.1) = 3 x 90 – 3 x 0.1 = 270 – 0.3 = 269.7
9 x 100.2 = 9 x (100 + 0.2) = 9 x 100 + 9 x 0.2 = 900 + 1.8 = 901.8
Check your expressions and answers.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Rearranging Numbers 7 © Department of Education WA 2015
Activity 6
For each of the following statements, create number sentences to decide if the statement is
(a) possibly true (b) definitely true (c) false. For each example make THREE statements
which are true or find ONE that is false.
1. Adding two numbers together and then subtracting 8 from that total gives the same
answer as subtracting 8 from each original number and then adding the results of these
two subtractions.
40 + 16 = 56 and 56 – 8 = 48
40 – 3 = 32 and 16 – 8 = 8
32 + 8 = 40 and this is not the same as 48 so the statement is FALSE
2. Adding two numbers together and then dividing that total by 2 total gives the same
answer as dividing each original number by 2 and then adding the results of these two
divisions.
10 + 8 = 18 and 18 ÷ 2 = 9 ALSO 10 ÷ 2 = 5 and 8 ÷ 2 = 4 and 5 + 4 = 9
20 + 16 = 36 and 36 ÷ 2 = 18 ALSO 20 ÷ 2 = 10 and 16 ÷ 2 = 8 and 10 + 8 = 18
12 + 14 = 26 and 26 ÷ 2 = 13 ALSO 12 ÷ 2 = 6 and 14 ÷ 2 = 7 and 6 + 7 = 13
Possibly true but we have only checked for a few numbers.
3. Subtracting a small even number from a larger even number and then dividing that total
by 2 total gives the same answer as dividing each original number by 2 and then
subtracting the smaller result from the larger result.
10 - 8 = 2 and 2 ÷ 2 = 1 ALSO 10 ÷ 2 = 5 and 8 ÷ 2 = 4 and 5 - 4 = 1
20 - 16 = 4 and 4 ÷ 2 = 2 ALSO 20 ÷ 2 = 10 and 16 ÷ 2 = 8 and 10 - 8 = 2
12 - 4 = 8 and 8 ÷ 2 = 4 ALSO 12 ÷ 2 = 6 and 4 ÷ 2 = 2 and 6 - 2 = 4
Possibly true, but we have only checked for a few numbers.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Rearranging Numbers 8 © Department of Education WA 2015
STUDENT COPY REARRANGING NUMBERS
This task supports the investigation to learn how numbers can be rearranged to make
calculations easier.
Remember that for addition and multiplication operations the order does not matter if there is
only one type of operation in the number sentence. If addition and multiplication occur in the
same expression (with no brackets) then multiplication is done first.
Examples: 6 + 7 = 7 + 6 5 x 2 = 2 x 5
3 + 8 + 10 = 11 + 10 or 3 + 8 + 10 = 3 + 18 5 x 2 x 4 = 10 x 4 or 5 x 2 x 4 = 5 x 8
Activity 1
1. Consider 61 x 3. Which of the following expressions are equivalent to 61 x 3? Use your
calculator if necessary. Write TRUE or FALSE to show your thinking.
(60 + 1 ) x 3 3 x (60 + 1)
60 x 3 + 1 x 3 60 x 3 x 60 x 3
60 x 3 – 1 x 3 60 x 3 ÷ 3
3 x 60 + 3 x 1 3 x 60 x 1
2. Consider 51 x 4. Which of the following expressions are equivalent to 51 x 4? Use your
calculator if necessary. Write TRUE or FALSE to show your thinking.
(50 + 1 ) x 4 4 x (50 + 1)
50 x 4 + 1 x 4 50 x 4 x 50 x 4
50 x 4 – 1 x 4 50 x 4 ÷ 4
4 x 50 + 4 x 1 4 x 50 x 1
3. Consider 101 x 7. Which of the following expressions are equivalent to 101 x 7? Use your
calculator if necessary. Write TRUE or FALSE to show your thinking.
(100 + 1 ) x 7 7 x (100 + 1)
100 x 7 + 1 x 7 100 x 7 x 100 x 7
100 x 7 – 1 x 7 100 x 7 ÷ 7
7 x 100 + 7 x 1 7 x 100 x 1
MATHS7TL002 | Mathematics | Number and Algebra Activity | Rearranging Numbers 9 © Department of Education WA 2015
Activity 2
1. Consider 99 x 8. Which of the following expressions are equivalent to 99 x 8? Use your
calculator if necessary. Write TRUE or FALSE to show your thinking.
(100 - 1 ) x 8 8 x (100 - 1)
100 x 8 + 1 x 8 100 x 8 x 100 x 8
100 x 8 – 1 x 8 100 x 8 ÷ 8
8 x 100 - 8 x 1 8 x 100 x 1
2. Consider 48 x 2. Which of the following expressions are equivalent to 48 x 2? Use your
calculator if necessary. Write TRUE or FALSE to show your thinking.
(50 - 2 ) x 2 2 x (50 - 2)
50 x 2 + 1 x 2 50 x 2 x 50 x 2
50 x 2 – 1 x 2 50 x 2 ÷ 2
2 x 50 - 2 x 1 2 x 50 x 1
3. Consider 69 x 9. Which of the following expressions are equivalent to 69 x 9? Use your
calculator if necessary. Write TRUE or FALSE to show your thinking.
(70 - 1 ) x 9 9 x (70 - 1)
70 x 9 + 1 x 9 70 x 9 x 70 x 9
70 x 9 – 1 x 9 70 x 9 ÷ 9
9 x 70 - 9 x 1 9 x 70 x 1
4. Consider 81 x 7. Write 4 TRUE statements and 4 FALSE statements similar to the above.
TRUE FALSE
MATHS7TL002 | Mathematics | Number and Algebra Activity | Rearranging Numbers 10 © Department of Education WA 2015
Activity 3
Calculate the following expressions by firstly creating statements similar to the ones provided
above, thus making the calculations easier. The first two are done for you.
91 x 8 = (90 + 1) x 8 = 90 x 8 + 1 x 8 = 720 + 8 = 728
29 x 7 = (30 – 1) x 7 = 30 x 7 – 1 x 7 = 210 – 7 = 203
21 x 20
59 x 8
99 x 4
399 x 6
42 x 8
121 x 6
999 x 5
1001 x 15
51 x 11
599 x 7
Activity 4
Write a description of the process you have used in the previous activities and explain how
the calculations are made easier.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Rearranging Numbers 11 © Department of Education WA 2015
Activity 5
1. If this process works for whole numbers, then should it work for fractions and decimals?
Justify this conclusion.
2. Use the process on these decimals and fractions. Two examples are provided.
8 x 1.99 = 8 x (2 – 0.01) = 8 x 2 – 8 x 0.01 = 16 – 0.08 = 15.92
5 x 2
= 5 x (2 +
) = 5 x 2 + 5 x
=
9 x 3.01
2 x 3
3 x 100.02
6 x 10.1
7 x 99.9
8 x 49.9
20 x 8.1
20.1 x 8
4
x 10
63.9 x 2
22.002 x 4
11
x 4
39.99 x 6
3 x 89.9
9 x 100.2
Check your expressions and answers.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Rearranging Numbers 12 © Department of Education WA 2015
Activity 6
For each of the following statements, create number sentences to decide if the statement is
(a) possibly true (b) definitely true (c) false. For each example make THREE statements
which are true or find ONE that is false.
1. Adding two numbers together and then subtracting 8 from that total gives the same
answer as subtracting 8 from each original number and then adding the results of these
two subtractions.
2. Adding two numbers together and then dividing that total by 2 total gives the same
answer as dividing each original number by 2 and then adding the results of these two
divisions.
3. Subtracting a small even number from a larger even number and then dividing that total
by 2 total gives the same answer as dividing each original number by 2 and then
subtracting the smaller result from the larger result.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Establishing Laws 2 © Department of Education WA 2015
TASK 16: ESTABLISHING LAWS
Overview
In this series of activities students are given the opportunity to identify and test the laws of
arithmetic given in algebraic notation. Students will need to have a sound knowledge and
understanding of operations with whole numbers, decimals and fractions. They also need to
appreciate the role of the variable in an algebraic expression and to understand the values
that variables can assume. Students should be able to calculate all answers without
calculators.
Some students may need
calculators
Relevant content descriptions from the Western Australian Curriculum
Create algebraic expressions and evaluate them by substituting a given value for each
variable (ACMNA176)
Extend and apply the laws and properties of arithmetic to algebraic terms and
expressions (ACMNA177)
Students can demonstrate
fluency when they
o substitute values into algebraic expression and compute the values of the
expressions
understanding when they
o connect the laws and properties of numbers to algebraic terms and expressions
reasoning when they
o justify their conclusions about statements which are true and false
MATHS7TL002 | Mathematics | Number and Algebra Activity | Establishing Laws 3 © Department of Education WA 2015
ESTABLISHING LAWS Solutions and Notes for Teachers
Introduction
In this task, you are given activities which will help you to identify the laws governing
operations with numbers when those laws are written in algebraic terms. In these tasks, the
lower case letters represent variables which can be either positive or negative or zero as well
as being whole numbers, fractions or decimals.
Activity 1: Adding two numbers
When adding two numbers represented by a and b the rule is a + b = b + a
Using any numbers for a and b, write down examples of true statements which follow this
rule. Provide examples as described, and all examples should be different.
The first one is done for you. Answers will vary.
Example a b a + b = b + a
Add two decimals 0.4 1.6 0.4 + 1.6 = 1.6 + 0.4
Add two whole numbers 5 6 5 + 6 = 6 + 5
Add two prime numbers 7 11 7 + 11 = 11 + 7
Add two odd numbers 1 9 1 + 9 = 9 + 1
Add two multiples of 3 6 12 6 + 12 = 12 + 6
Add two fractions +
Add two negative numbers -6 -5 -6 + -5 = -5 + -6
Add two factors of 6 3 2 3 + 2 = 2 + 3
Add a whole number to a decimal 4 2.7 4 + 2.7 = 2.7 + 4
Add a fraction and a decimal 0.3 + 0.3 = 0.3 +
Add a negative and a positive 8 -3 8 + -3 = -3 + 8
Add a fraction to a whole number 11 11 + = + 11
Add two perfect squares 9 25 9 + 25 = 25 + 9
Add a prime and a composite 3 12 3 + 12 = 12 + 3
Add two 4-digit numbers 2315 1436 2315 + 1436 = 1436 + 2315
MATHS7TL002 | Mathematics | Number and Algebra Activity | Establishing Laws 4 © Department of Education WA 2015
Activity 2: Subtracting two numbers
Generally a – b b – a
1. Interpret this statement and explain what it means.
If you subtract a first number from a second number, the answer is not equal to
subtracting the second number from the first. This statement is true: The order in
which the numbers are written and subtracted is important.
2. Give three examples that support this statement.
3 – 5 5 – 3
8 – 4 4 – 8
10 – 5 5 – 10
3. Give an example of a situation when the statement is false.
If a = b then a – b = b – a
e.g., 3 – 3 = 3 – 3
4. Is this statement generally true for fractions? Justify your conclusion.
Yes the statement is true for fractions unless the fractions are equal;
e.g.,
-
=
and this is not equal to
-
, which must be negative.
5. Is this statement generally true for decimals? Justify your conclusion.
Yes the statement is true for decimals unless the decimals are equal,
e.g. 0.7 – 0.3 = 0.4 and 0.3 – 0.7 is negative and less than 0.4
6. Make a true statement by writing an algebraic expression which uses the “=” sign.
a – b = – (b – a )
MATHS7TL002 | Mathematics | Number and Algebra Activity | Establishing Laws 5 © Department of Education WA 2015
Activity 3: Multiplying two numbers
When multiplying two numbers represented by a and b is it true that a b = b a ?
Using any numbers for a and b, write down examples of true statements which follow this
rule. Provide examples as described, and all examples should be different.
The first one is done for you. Answers will vary.
Example a b a b = b a
Multiply two decimals 0.5 1.6 0.5 x 1.6 = 1.6 x 0.5
Multiply two whole numbers 7 8 7 x 8 = 8 x 7
Multiply two composite numbers 100 10 100 x 10 = 10 x 100
Multiply two even numbers 8 12 8 x 12 = 12 x 8
Multiply two multiples of 3 30 3 30 x 3 = 3 x 30
Multiply two fractions x = x
Multiply two negative numbers -3 -1 -3 x -1 = -1 x -3
Multiply two factors of 8 4 8 4 x 8 = 8 x 4
Multiply any number by a decimal 9 0.1 9 x 0.1 = 0.1 x 9
Multiply a fraction by an integer 6 x 6 = 6 x
Multiply any two positive numbers 35 2 35 x 2 = 2 x 35
Multiply zero by a fraction 0 0 x = x 0
Multiply a number by itself 4 4 4 x 4 = 4 x 4
Multiply a prime and a composite 12 3 12 x 3 = 3 x 12
Multiply two 1-digit numbers 7 9 7 x 9 = 9 x 7
Is the statement a b = b a true or false? Write your conclusion.
It is true. When two numbers are multiplied the order in which they are written does not affect
the product of the two numbers.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Establishing Laws 6 © Department of Education WA 2015
Activity 4: Dividing two numbers
Investigate the following statement a ÷ b = b ÷ a
This statement is false, since dividing a first number by a second does not give the same
answer as dividing the second number by the first.
Examples:
30 ÷ 5 5 ÷ 30
8 ÷ 4 4 ÷ 8
100 ÷ 10 10 ÷ 100
Note that you cannot divide by 0, although you can divide 0 by another number.
If a = b then a ÷ b = b ÷ a
Example: 4 ÷ 4 = 4 ÷ 4 = 1
This statement is also false for decimals and fractions (and negative numbers)
a ÷ b = = 1 ÷ and b ÷ a = = 1 ÷
Activity 5: Other Statements
Create statements of equality and provide examples for -
(a) the addition of three variables
a + b + c = a + c + b e.g., 1 + 2 + 3 = 1 + 3 + 2
a + b + c = c + a + b e.g., 2 + 3 + 4 = 4 + 2 + 3
a + b + c = c + b + a e.g., 4 + 5 + 6 = 6 + 5 + 4
a + b + c = b + a + c e.g., 2 + 1 + 4 = 1 + 2 + 4
a + b + c = b + c + a e.g., 5 + 1 + 3 = 1 + 3 + 5
(b) the multiplication of three variables
a x b x c = a x c x b e.g., 2 x 3 x 4 = 2 x 4 x 3
a x b x c = c x a x b e.g., 1 x 5 x 4 = 4 x 1 x 5
a x b x c = c x b x a e.g., 2 x 7 x 3 = 3 x 7 x 2
a x b x c = b x a x c e.g., 1 x 9 x 2 = 9 x 1 x 2
a x b x c = b x c x a e.g., 3 x 6 x 2 = 6 x 2 x 3
MATHS7TL002 | Mathematics | Number and Algebra Activity | Establishing Laws 7 © Department of Education WA 2015
Activity 6:
1. Determine the answers in the table below.
7 + 10 + 1 = 18 (7 + 10) + 1 = 18 7 + (10 + 1) = 18
5 + 8 + 9 = 22 (5 + 8) + 9 = 13 + 9 = 22 5 + (8 + 9) = 5 + 17 = 22
1.1 + 2.7 + 4.8 = 8.6 (1.1 + 2.7) + 4.8 = 8.6 1.1 + (2.7 + 4.8) = 8.6
35 + 29 + 38 = 102 (35 + 29) + 38 = 102 35 + (29 + 38) = 102
9
9
9
2. Describe the pattern of the addition operation in these calculations.
If there are three numbers to add you get the same answer no matter in which order you add
the numbers.
3. Complete this statement in the same way: m + k + w = ( m + k) + w = m + ( k + w)
4. Determine the answers in the table below.
7 x (20 x 10) = 1400 7 x 20 x 10 = 1400 (7 x 20) x 10) = 1400
2 x (8 x 3) = 48 (2 x 8) x 3 = 48 2 x 8 x 3 = 48
0.5 x 8.6 x 4 = 17.2 (0.5 x 8.6) x 4 = 17.2 0.5 x (8.6 x 4) = 17.2
5 x 30 x 3 = 450 (5 x 30) x 3 = 450 5 x (30 x 3) = 450
(
)
5. Describe the pattern of the multiplication operation in these calculations.
When you multiply three numbers the order of multiplication does not affect the answer.
6. Complete this statement in the same manner:
m k w = (m k ) w = m ( k w )
MATHS7TL002 | Mathematics | Number and Algebra Activity | Establishing Laws 8 © Department of Education WA 2015
Activity 7: Reflection
1. Use algebraic statements of equality to summarise the four main rules of operating with
numbers that have been used in this task.
a + b = b + a
a x b = b x a
m + k + w = ( m + k) + w = m + ( k + w)
m k w = (m k) w = m ( k w)
2. Explain, with the use of examples, how knowing such rules can support mental
calculations.
It is particularly useful when there are more than two numbers to add or multiply and you
can split the numbers into smaller numbers.
Example: 546 + 600 = 500 + 46 + 600 = 500 + 600 + 46 = 1100 + 46 = 1146
Example: 25 x 6 = 5 x 5 x 6 = 5 x (5 x 6) = 5 x 30 = 150
3. Write a test of 20 mixed questions based on these rules. Prepare the answers and swap
your test for one written by another student.
Examples: Provide the missing number or variable.
h x w x n = ____ x w x n h + w + n = ____ + w + n
66 + 24 + 55 = 24 + ____ + 66 33 x 22 x 55 = ____ x 22 x 33
6.3 + 4.5 = 3.3 + ____ + 4.5 20 x 6 x 90 = 90 x 6 x 10 x ____
45 x 32 = 5 x _____ x 32 55 x 99 = 99 x 5 x ______
k x m x t = m x ______ x t k + m + t = t + m + ______
2 x h x w = h x ______ x 2 d + 3 + t = t + 3 + ______
0.5 x 444 = 0.5 x 2 x ______ 1.9 + 15 = 10 + 1.9 + ______
7 x 36 = 7 x 6 x _______ 1289 + 99 = 90 + 1289 + ______
(34 + b ) + 5 = 34 + ( _____ + 5) 66 x v x 2 = 2 x ( _____ x 66)
MATHS7TL002 | Mathematics | Number and Algebra Activity | Establishing Laws 9 © Department of Education WA 2015
STUDENT COPY ESTABLISHING LAWS
Introduction
In this task, you are given activities which will help you to identify the laws governing
operations with numbers when those laws are written in algebraic terms. In these tasks, the
lower case letters represent variables which can be either positive or negative or zero as well
as being whole numbers, fractions or decimals.
Activity 1: Adding two numbers
When adding two numbers represented by a and b the rule is a + b = b + a
Using any numbers for a and b, write down examples of true statements which follow this
rule. Provide examples as described, and all examples should be different.
The first one is done for you.
Example a b a + b = b + a
Add two decimals 0.4 1.6 0.4 + 1.6 = 1.6 + 0.4
Add two whole numbers
Add two prime numbers
Add two odd numbers
Add two multiples of 3
Add two fractions
Add two negative numbers
Add two factors of 6
Add a whole number to a decimal
Add a fraction and a decimal
Add a negative and a positive
Add a fraction to a whole number
Add two perfect squares
Add a prime and a composite
Add two 4-digit numbers
MATHS7TL002 | Mathematics | Number and Algebra Activity | Establishing Laws 10 © Department of Education WA 2015
Activity 2: Subtracting two numbers
Generally a – b b – a
1. Interpret this statement and explain what it means.
2. Give three examples that support this statement.
3. Give an example of a situation when the statement is false.
4. Is this statement generally true for fractions? Justify your conclusion.
5. Is this statement generally true for decimals? Justify your conclusion.
6. Make a true statement by writing an algebraic expression which uses the “=” sign.
a – b =
MATHS7TL002 | Mathematics | Number and Algebra Activity | Establishing Laws 11 © Department of Education WA 2015
Activity 3: Multiplying two numbers
When multiplying two numbers represented by a and b is it true that a b = b a ?
Using any numbers for a and b, write down examples of true statements which follow this
rule. Provide examples as described, and all examples should be different.
The first one is done for you.
Example a b a b = b a
Multiply two decimals 0.5 1.6 0.5 x 1.6 = 1.6 x 0.5
Multiply two whole numbers
Multiply two composite numbers
Multiply two even numbers
Multiply two multiples of 3
Multiply two fractions
Multiply two negative numbers
Multiply two factors of 8
Multiply any number by a decimal
Multiply a fraction by an integer
Multiply any two positive numbers
Multiply zero by a fraction
Multiply a number by itself
Multiply a prime and a composite
Multiply two 1-digit numbers
Is the statement a b = b a true or false? Write your conclusion.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Establishing Laws 12 © Department of Education WA 2015
Activity 4: Dividing two numbers
Investigate the following statement a ÷ b = b ÷ a
Activity 5: Other Statements
Create statements of equality and provide examples for
(a) the addition of three variables
(b) the multiplication of three variables
MATHS7TL002 | Mathematics | Number and Algebra Activity | Establishing Laws 13 © Department of Education WA 2015
Activity 6
1. Determine the answers in the table below.
7 + 10 + 1 = (7 + 10) + 1 = 7 + (10 + 1) =
5 + 8 + 9 = (5 + 8) + 9 = 5 + (8 + 9) =
1.1 + 2.7 + 4.8 = (1.1 + 2.7) + 4.8 = 1.1 + (2.7 + 4.8) =
35 + 29 + 38 = (35 + 29) + 38 = 35 + (29 + 38) =
2. Describe the pattern of the addition operation in these calculations.
3. Complete this statement in the same way: m + k + w =
4. Determine the answers in the table below.
7 x (20 x 10) = 7 x 20 x 10 = (7 x 20) x 10) =
2 x (8 x 3) = (2 x 8) x 3 = 2 x 8 x 3 =
0.5 x 8.6 x 4 = (0.5 x 8.6) x 4 = 0.5 x (8.6 x 4) =
5 x 30 x 3 = (5 x 30) x 3 = 5 x (30 x 3) =
5. Describe the pattern of the multiplication operation in these calculations.
Complete this statement in the same manner: m k w =
MATHS7TL002 | Mathematics | Number and Algebra Activity | Establishing Laws 14 © Department of Education WA 2015
Activity 7: Reflection
1. Use algebraic statements of equality to summarise the rules of operating with numbers
that have been used in this task.
2. Explain, with the use of examples, how knowing such rules can support mental
calculations.
3. Write a test of 20 mixed questions based on these rules. Prepare the answers and swap
your test for one written by another student.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Rounding Decimals 2 © Department of Education WA 2015
TASK 18: ROUNDING DECIMALS
Overview
This task provides a basis for considering the size and nature of errors that are made when
numbers are rounded before calculations are performed. It would be useful to provide a
further opportunity for discussing the importance of precision for measuring and to recognise
that there are different levels of precision in different situations. The numbers chosen will
support minimal use of calculators.
Students will need
calculators (for Activity 3 only)
Relevant content descriptions from the Western Australian Curriculum
Round decimals to a specified number of decimal places (ACMNA156)
Multiply and divide fractions and decimals using efficient written strategies and digital
technologies (ACMNA154)
Add and subtract decimals, with and without digital technologies, and use estimation
and rounding to check the reasonableness of answers (Year 6: ACMNA128)
Students can demonstrate
reasoning when they
o explain patterns relating to the size and nature of the error
problem solving when they
o determine a process for completing Activity 5
MATHS7TL002 | Mathematics | Number and Algebra Activity | Rounding Decimals 3 © Department of Education WA 2015
ROUNDING DECIMALS Solutions and Notes for Teachers
Activity 1: Rounding review
(a) Summarise the rules for rounding decimals. Consider the following forms of rounding:
1. Rounding down to the nearest whole number.
2. Truncating a number.
3. Rounding to the nearest tenth, hundredth, thousandth.
1. If the first digit after the decimal point is 5, 6, 7, 8 or 9 then the whole number goes up by
1, otherwise it stays as it is.
2. Truncate means to chop – so the decimal part is removed.
3. To round to the nearest tenth (first decimal point):
If the second digit after the decimal point is 5, 6, 7, 8 or 9 then the first digit goes up by 1,
otherwise it stays as it is and any digits following are removed..
To round to the nearest hundredth (second decimal point):
If the third digit after the decimal point is 5, 6, 7, 8 or 9 then the second digit goes up by
1, otherwise it stays as it is and any digits following are removed. The first digit after the
decimal point is unchanged.
The pattern continues for rounding to the nearest thousandth.
(b) Write a test consisting of 10 items which test the ability to round decimals as described
above. Give your test to another student and then mark their responses.
Example of such a Test:
1. Truncate these to a whole number: 56.3 1093.89
Ans: 56 1093
2. Round these numbers to 2 decimal places: 100.001 43.5623
Ans: 100.00 43.56
3. Round these numbers to the nearest thousandth: 0.09876 1.290545
Ans: 0.098 1.291
4. The answer was 8.79 when rounded to the nearest hundredth.
What might the answer have been originally? Give 3 possibilities
Ans: 8.791 8.788 8.79000005
5. 7.55 is between 7.5 and 7.6. How many other numbers lie between 7.5 and 7.6?
An infinite number of numbers.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Rounding Decimals 4 © Department of Education WA 2015
Activity 2: Investigating the size of “error”
(a) You are given nine values for A and nine values for B. Round these numbers to the
nearest tenth and place them in the columns AR and BR. Add the totals of the original
numbers and enter them in the A + B column. Add the totals for the rounded decimals and
enter them in the AR + BR column. The last error column is the difference between A+B and
AR+BR.
A AR B BR A + B AR + BR Error
1 0.34 0.3 0.43 0.4 0.77 0.7 0.07
2 0.11 0.1 0.54 0.5 0.65 0.6 0.05
3 0.07 0.1 0.21 0.2 0.28 0.3 0.02
4 0.89 0.9 0.59 0.6 1.48 1.5 0.02
5 0.48 0.5 0.67 0.7 1.15 1.2 0.05
6 0.66 0.7 0.75 0.8 1.41 1.5 0.09
7 0.19 0.2 0.24 0.2 0.43 0.4 0.03
8 0.45 0.5 0.33 0.3 0.78 0.8 0.02
9 0.86 0.9 0.92 0.9 1.78 1.8 0.02
Give examples of some situation where this degree of error -
(i) would be important; If the numbers represented litres of fertiliser for a lawn.
(ii) would not be significant. If these were the number of mm on a running track.
(b) As a general rule, when was the error greatest? When both decimals were rounded up?
When both were rounded down? Or when one was rounded up and the other down?
Justify your answer with evidence from the table.
The error was greatest (mean of 0.16 ÷ 3) for both numbers being rounded up but it was only
slightly higher than when both numbers were rounded down (0.14 ÷ 3). Both of these means
were much higher than when one number was rounded up and the other number was
rounded down (0.07 ÷ 3).
(c) What can you do to be more certain of your conclusion?
Need to try many more numbers and a greater variety.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Rounding Decimals 5 © Department of Education WA 2015
Activity 3: Investigating the degree of “error” when multiplying
(a) You are given nine values for A and nine values for B. Round these numbers to the
nearest tenth and place them in the columns AR and BR. Multiply the original numbers ()
and enter the product in the A x B column. Multiply the rounded decimals (without calculator)
and enter the product in the AR x BR column.
For the last error column calculate the difference between A x B and AR x BR.
A AR B BR A x B AR x BR Error
1 0.34 0.3 0.43 0.4 0.1462 0.12 0.0262
2 0.11 0.1 0.54 0.5 0.0594 0.05 0.0094
3 0.07 0.1 0.21 0.2 0.0147 0.02 0.0053
4 0.89 0.9 0.59 0.6 0.5251 0.54 0.0149
5 0.48 0.5 0.67 0.7 0.3216 0.35 0.0284
6 0.66 0.7 0.75 0.8 0.495 0.56 0.065
7 0.19 0.2 0.24 0.2 0.0456 0.04 0.0056
8 0.45 0.5 0.33 0.3 0.1485 0.15 0.0015
9 0.86 0.9 0.92 0.9 0.7912 0.81 0.0188
(b) As a general rule, when was the error greatest? When both decimals were rounded up?
When both were rounded down? Or when one was rounded up and the other down?
Justify your answer with evidence from the table.
The first three examples had numbers which were both rounded down and the error ranged
from 0.0053 to 0.0262. The next three examples had numbers which were both rounded up
and the error ranged from 0.0149 to 0.065 and this is much higher. The last three numbers
had different rounding and the errors were much lower, form 0.0015 to 0.0188.
(c) However, while the size of the error is useful as shown above, it’s the degree of error that
is important. Thus we need to consider what percentage of the actual product is the error;
e.g., the No. 1 percentage error is 0.0262/0.1462 = 0.1792 = 18 hundredths = 18%.
Calculate the percentage errors in the same way for No. 2 to No. 9.
No. 2 percentage error:
0.0094/0.0594 = 0.1582 = 15.8 hundredths = 16% to the nearest per cent.
No,. 3: 26.5%; No. 4: 3% ; No. 5: 9%; No. 6: 13%; No. 7: 12%; No. 8: 1%; No. 9: 2%
MATHS7TL002 | Mathematics | Number and Algebra Activity | Rounding Decimals 6 © Department of Education WA 2015
Activity 4: Investigating the “error” when dividing
(a) You are given nine values for A and nine values for B. Round these numbers to the
nearest whole number and place them in the columns AR and BR. Divide B into A and enter
the result into the A ÷ B column. Divide BR into AR and enter the results into the AR ÷ BR
column. The last column is the difference between A ÷ B and AR ÷ BR; i.e., the error. The
numbers have been determined so that the calculations can be done mentally.
A AR B BR A ÷ B AR ÷ BR Error
1 2.66 3 1.33 1 2 3 1
2 8.88 9 2.22 2 4 4.5 0.5
3 15.5 16 3.1 3 5 5.333 0.333
4 24.5 25 3.5 4 7 6.25 0.75
5 22.8 23 3.8 4 6 5.75 0.25
6 18.5 19 3.7 4 5 4.75 0.25
7 10.05 10 2.01 2 5 5 0
8 100.4 100 25.1 25 4 4 0
9 33.3 33 11.1 11 3 3 0
Compare A ÷ B with AR ÷ BR when -
(i) one number is rounded up and the other is rounded down;
A ÷ B < AR ÷ BR
(ii) both numbers are rounded up;
A ÷ B > AR ÷ BR
(iii) both numbers are rounded down.
A ÷ B = AR ÷ BR
(b) Try some other numbers and see if you can find some examples that contradict these
findings.
If A = 7.2 and B = 2.4 then 7.2 ÷ 2.4 = 3 and 7 ÷ 2 = 3.5. This contradicts (iii)
MATHS7TL002 | Mathematics | Number and Algebra Activity | Rounding Decimals 7 © Department of Education WA 2015
Activity 5
Jon asks his mum who is driving the car, “How much further to go?”
Mum estimates the distance to the nearest kilometre and says “25 km”.
Not long after Jon asks the same question and his mother’s estimate was 20 km.
Determine, to the nearest tenth of a kilometre both the maximum distance and the minimum
distance that the car could have travelled between the two estimates.
When Mum gave the estimate of 25 km, the value could have been from 24.5 km to 25.4 km.
Similarly when Mum gave the estimate of 20 km, the value could have been from 19.5 km to
20.4 km.
The greatest difference between the two estimates is 25.4 – 19.5 = 5.9 km
The least difference between the two estimates is 24.5 – 20.4 = 4.1 km
MATHS7TL002 | Mathematics | Number and Algebra Activity | Rounding Decimals 8 © Department of Education WA 2015
STUDENT COPY ROUNDING DECIMALS
Activity 1: Rounding review
(a) Summarise the rules for rounding decimals. Consider the following forms of rounding:
1. Rounding down to the nearest whole number.
2. Truncating a number.
3. Rounding to the nearest tenth, hundredth, thousandth.
(b) Write a test consisting of 10 items which test the ability to round decimals as described
above. Give your test to another student and then mark their responses.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Rounding Decimals 9 © Department of Education WA 2015
Activity 2: Investigating the size of “error”
(a) You are given nine values for A and nine values for B. Round these numbers to the
nearest tenth and place them in the columns AR and BR. Add the totals of the original
numbers and enter them in the A + B column. Add the totals for the rounded decimals and
enter them in the AR + BR column. The last error column is the difference between A+B and
AR+BR.
A AR B BR A + B AR + BR Error
1 0.34 0.43
2 0.11 0.54
3 0.07 0.21
4 0.89 0.59
5 0.48 0.67
6 0.66 0.75
7 0.19 0.24
8 0.45 0.33
9 0.86 0.92
Give examples of some situation where this degree of error -
(i) would be important;
(ii) would not be significant.
(b) As a general rule, when was the error greatest? When both decimals were rounded up?
When both were rounded down? Or when one was rounded up and the other down?
Justify your answer with evidence from the table.
(c) What can you do to be more certain of your conclusion?
MATHS7TL002 | Mathematics | Number and Algebra Activity | Rounding Decimals 10 © Department of Education WA 2015
Activity 3: Investigating the degree of “error” when multiplying
(a) You are given nine values for A and nine values for B. Round these numbers to the
nearest tenth and place them in the columns AR and BR. Multiply the original numbers ()
and enter the product in the A x B column. Multiply the rounded decimals (without calculator)
and enter the product in the AR x BR column.
For the last error column calculate the difference between A x B and AR x BR.
A AR B BR A x B AR x BR Error
1 0.34 0.43
2 0.11 0.54
3 0.07 0.21
4 0.89 0.59
5 0.48 0.67
6 0.66 0.75
7 0.19 0.24
8 0.45 0.33
9 0.86 0.92
(b) As a general rule, when was the error greatest? When both decimals were rounded up?
When both were rounded down? Or when one was rounded up and the other down?
Justify your answer with evidence from the table.
(c) However, while the size of the error is useful as shown above, it’s the degree of error that
is important. Thus we need to consider what percentage of the actual product is the error;
e.g., the No. 1 percentage error is 0.0263/0.1462 = 0.1792 = 18 hundredths = 18%.
Calculate the percentage errors in the same way for No. 2 to No. 9.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Rounding Decimals 11 © Department of Education WA 2015
Activity 4: Investigating the “error” when dividing
(a) You are given nine values for A and nine values for B. Round these numbers to the
nearest whole number and place them in the columns AR and BR. Divide B into A and enter
the result into the A ÷ B column. Divide BR into AR and enter the results into the AR ÷ BR
column. The last column is the difference between A ÷ B and AR ÷ BR; i.e., the error. The
numbers have been determined so that the calculations can be done mentally.
A AR B BR A ÷ B AR ÷ BR Error
1 2.66 1.33
2 8.88 2.22
3 15.5 3.1
4 24.5 3.5
5 22.8 3.8
6 18.5 3.7
7 10.05 2.01
8 100.4 25.1
9 33.3 11.1
Compare A ÷ B with AR ÷ BR when -
(i) one number is rounded up and the other is rounded down;
(ii) both numbers are rounded up;
(iii) both numbers are rounded down.
(b) Try some other numbers and see if you can find some examples that contradict these
findings.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Rounding Decimals 12 © Department of Education WA 2015
Activity 5
Jon asks his mum who is driving the car, “How much further to go?”
Mum estimates the distance to the nearest kilometre and says, “25 km”.
Not long afterwards Jon asks the same question and his mother’s estimate was 20 km.
Determine, to the nearest tenth of a kilometre both the maximum distance and the minimum
distance that the car could have travelled between the two estimates.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Square Numbers 2 © Department of Education WA 2015
TASK 19: SQUARE NUMBERS
Overview
In this task students are given opportunities to develop further understanding of squares and square roots. They are also invited to review their knowledge of the terminology of types of numbers and investigate the properties of some numbers.
Students will need • calculators
Relevant content descriptions from the Western Australian Curriculum • Investigate and use square roots of perfect square numbers (ACMNA150)• Identify and describe properties of prime, composite, square and triangular numbers
(Year 6:ACMNA122)
Students can demonstrate • fluency when they
o generate mathematical expressions to represent word descriptions• understanding when they
o use the relationships between different types of numbers to make conclusionso describe patterns between sets of numberso generate mathematical expressions to represent word descriptions
• reasoning when theyo generalise the patterns for Activities 4 and 5
MATHS7TL002 | Mathematics | Number and Algebra Activity | Square Numbers 3 © Department of Education WA 2015
SQUARE NUMBERS Solutions and Notes for Teachers
Activity 1: Review
1. Answer these questions and provide evidence/calculations to support your conclusions.
What are square numbers?
Numbers made by multiplying a number by itself; e.g., 9 is a square number as 9 = 3 x 3.
Can square numbers be both odd and even?
Yes. 9 is an odd square number and 4 is an even square number.
Can square numbers be prime numbers?
No. Prime numbers have only two different factors, 1 and the number itself.
How can your calculator be used to determine square numbers?
There is a button to square numbers. x2
What is the name given to the process of “undoing” a square number?
Finding the square root.
2. Determine these numbers:
A palindromic square number between 100 and 200.
121
Two square numbers that add to 100.
64 + 36 = 100
A 5-digit square number ending in 0.
10 000 = 100 x 100
A number which is equal to the square of itself.
1 because 1 x 1 = 1
Four square numbers which are factors of 100.
1, 4, 25, 100
MATHS7TL002 | Mathematics | Number and Algebra Activity | Square Numbers 4 © Department of Education WA 2015
Activity 2 What can be the last digit that a square number can have? Here is a list of the last digits that could exist for a square number. For each digit write Y or N according to its existence as the last digit of a square number. If the last digit can be the ending of a square number, what might have been the ending of the number used to make the square number?
Last digit of number 0 1 2 3 4 5 6 7 8 9
Can it be a square number? Y/N
Y Y N N Y Y Y N N Y
Possible last digit/s of the number?
0 1 9
2
8 5
4
6
3
7
Activity 3
Use 5 different examples to test the truth of each of these statements. 1. The square of any even number is another even number.
22 = 4 82 = 64 202 = 400 142 = 196 162 = 256 True 2. The square of any odd number is another odd number.
32 = 9 52 = 25 212 = 441 132 = 169 192 = 361 True 3. The sum of a number added to its square is always even. 20 + 202 = 420 44 + 442 = 1980 132 + 13 = 182
192 + 19 = 380 632 + 63 = 4032
True
MATHS7TL002 | Mathematics | Number and Algebra Activity | Square Numbers 5 © Department of Education WA 2015
Activity 4
Sums of Odd Numbers.
Write down your answers to the following additions.
1 + 3 = 4
1 + 3 + 5 = 9
1 + 3 + 5 + 7 = 16
1 + 3 + 5 + 7 + 9 = 25
1 + 3 + 5 + 7 + 9 + 11 = 36
2. What will be the sum of the next set of consecutive odd numbers in this pattern?
49
3. Write a rule to find the sum of any number of odd numbers?
The sum of a set of consecutive odd numbers starting at 1 is equal to the square of the number of odd numbers which have been added.
4. How many consecutive odd numbers were added to make a total of -
(a) 100 (b) 441 (c) 1024 (d) 5000
(a) 10 (b) 21 (c) 32 (d) does not work
5. Does your rule apply for the sums of the even numbers? Show the procedure you use todetermine your answer.
2 + 4 = 6
No, two consecutive even numbers are added but the sum is not a square number
MATHS7TL002 | Mathematics | Number and Algebra Activity | Square Numbers 6 © Department of Education WA 2015
Activity 5
One less than a square number.
1. For this activity you are asked to• square a number• subtract 1 from the square• write the above answer as different multiplications of 2 factors. Enter as many as you
can. An example is provided. [Be systematic in determining these factors.]• Complete the table below.
Number Number squared
One less than the number squared
Factor forms of one less than the number squared
N N x N N x N -1
100 10 000 9999 1 x 9999, 3 x 3333, 9 x 1111, 33 x 303, 11 x 909, 99 x 101
11 121 120 1 x 120, 2 x 60, 3 x 40, 4 x 30, 5 x 24, 6 x 20, 8 x 15, 10 x 12
10 100 99 1 x 99, 3 x 33, 9 x 11
9 81 80 1 x 80, 2 x 40, 4 x 20, 5 x 16, 8 x 10
8 64 63 1 x 63, 3 x 21, 7 x 9
7 49 48 1 x 48, 2 x 24, 3 x 16, 4 x 12, 6 x 8
6 36 35 1 x 35, 5 x 7
5 25 24 1 x 24, 2 x 12, 3 x 8, 4 x 6
4 16 15 1 x 15, 3 x 5
3 9 8 1 x 8, 2 x 4
2 4 3 1 x 3
2. Study the numbers in the final column and look for a particular pair of factors that occursin every row. Describe this pair of factors.
In every row one pair of factors is the number one more than N multiplied by the number one less than N.
3. Use this pattern to write an expression for 2202 -1.
221 x 219
MATHS7TL002 | Mathematics | Number and Algebra Activity | Square Numbers 7 © Department of Education WA 2015
Activity 6
1. Investigate the final digits of the squares of numbers ending in 5.Select any 5 numbers ending in 5, square them and record your answers. Can you see apattern in the final digit(s) of these squares? Check your findings with another student’sresults. Record your findings and suggest a reason for this pattern.
Sample of possible answers:
352 = 1225
652 = 4225
852 = 7225
1052 = 11 025
2152 = 46 225
They all end in 25. Could be related to the last digit 5 because 52 = 25
2. Investigate the product of two square numbers; e.g., 4 x 25 = 100. What can you sayabout the product of two square numbers? Provide evidence to support your conclusion.
Sample of possible answers:
32 x 22 = 9 x 4 = 36 which is a square number because 36 = 62
52 x 32 = 25 x 9 = 225 which is a square number because 225 = 152
42 x 22 =16 x 4 = 64 which is a square number because 64 = 82
12 x 72 = 1 x 49 = 49 which is a square number because 49 = 72
32 x 32 = 9 x 9 = 81 which is a square number because 81 = 92
The product of two square numbers is a square number.
3. Does the addition of two square numbers produce another square number? Justify yourconclusion.
72 + 52 = 49 + 25 = 74 and 74 is not a square number, so the sum of two squares is not always a square number.
4. Determine two numbers that multiply to give one million yet neither number ends in 0 andboth numbers are whole numbers.
64 x 15 625 = 1 000 000
MATHS7TL002 | Mathematics | Number and Algebra Activity | Square Numbers 8 © Department of Education WA 2015
STUDENT COPY SQUARE NUMBERS
Activity 1: Review
1. Answer these questions and provide evidence/calculations to support your conclusions.
What are square numbers?
Can square numbers be both odd and even?
Can square numbers be prime numbers?
How can your calculator be used to determine square numbers?
What is the name given to the process of “undoing” a square number?
2. Determine these numbers:
A palindromic square number between 100 and 200.
Two square numbers that add to 100.
A 5-digit square number ending in 0.
A number which is equal to the square of itself.
Four square numbers which are factors of 100.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Square Numbers 9 © Department of Education WA 2015
Activity 2
What can be the last digit that a square number can have?
Here is a list of the last digits that could exist for a square number. For each digit write Y or N according to its existence as the last digit of a square number. If the last digit can be the ending of a square number, what might have been the ending of the number used to make the square number?
Last digit of number 0 1 2 3 4 5 6 7 8 9
Can it be a square number? Y/N Possible last digit/s of the number?
Activity 3
Use 5 different examples to test the truth of each of these statements.
1. The square of any even number is another even number.
2. The square of any odd number is another odd number.
3. The sum of a number added to its square is always even.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Square Numbers 10 © Department of Education WA 2015
Activity 4
Sums of Odd Numbers
1. Write down your answers to the following additions.
1 + 3
1 + 3 + 5
1 + 3 + 5 + 7
1 + 3 + 5 + 7 + 9
1 + 3 + 5 + 7 + 9 + 11
2. What will be the sum of the next set of consecutive odd numbers in this pattern?
3. Write a rule to find the sum of any number of odd numbers?
4. How many consecutive odd numbers were added to make a total of -
(a) 100 (b) 441 (c) 1024 (d) 5000
5. Does your rule apply for the sums of the even numbers? Show the procedure you use todetermine your answer.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Square Numbers 11 © Department of Education WA 2015
Activity 5
One less than a square number.
1. For this activity you are asked to• square a number• subtract 1 from the square• write the above answer as different multiplications of 2 factors. Enter as many as you
can. An example is provided. [Be systematic in determining these factors.]• Complete the table below.
Number Number squared
One less than the number squared
Factor forms of one less than the number squared
N N x N N x N -1
100 10 000 9999 1 x 9999, 3 x 3333, 9 x 1111, 33 x 303 11 x 909, 33 x 303, 99 x 101
11
10
9
8
7
6
5
4
3
2
2. Study the numbers in the final column and look for a particular pair of factors that occursin every row. Describe this pair of factors.
3. Use this pattern to write an expression for 2202 -1.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Square Numbers 12 © Department of Education WA 2015
Activity 6 1. Investigate the final digits of the squares of numbers ending in 5. Select any 5 numbers ending in 5, square them and record your answers. Can you see a pattern in the final digit(s) of these squares? Check your findings with another student’s results. Record your findings and suggest a reason for this pattern.
2. Investigate the product of two square numbers; e.g., 4 x 25 = 100. What can you say about the product of two square numbers? Provide evidence to support your conclusion. 3. Does the addition of two square numbers produce another square number? Justify your conclusion. 4. Determine two numbers that multiply to give one million yet neither number ends in 0 and both numbers are whole numbers.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Ratios 2 © Department of Education WA 2015
TASK 24: RATIOS
Overview
Students would have had the necessary exposure to ratios before Year 7 but need not be
experienced in sharing amounts according to a ratio to engage in this task.
Students will need
calculators
scissors and glue
counters – various colours
Relevant content descriptions from the Western Australian Curriculum
Recognise and solve problems involving simple ratios (ACMNA173)
Students can demonstrate
fluency when they
o manipulate ratios with operations on both numbers
understanding when they
o rank the weedkiller ratios in correct order in Activity 5
reasoning when they
o identify the effect of different operations on proportions
MATHS7TL002 | Mathematics | Number and Algebra Activity | Ratios 3 © Department of Education WA 2015
RATIOS Solutions and Notes for Teachers
Introduction
For the supervision of swimming in open water one lifesaver with a bronze medallion is
needed for every 10 students. This can be written without reference to units as -
one lifesaver for every ten students, or 1:10
Activity 1
Using the following ratios 1:2 1:3 2:3 1:5 2:5 3:4 1:4
Working in groups
Each group is allocated one of the ratios listed above.
Each group collects a handful of counters – two colours only.
Each group makes their ratio as follows, with the first number of counters in one colour
and the second number of counters in the other colour.
o
These would be the counters for the ratio 3:5
Use colours and draw a diagram similar to the one above, to represent your ratio.
Solutions will vary according to the student’s ratio. Some solutions are provided to exemplify
the solutions required.
Now consider changing your ratio. For each change, draw a diagram showing the numbers
of counters of each colour.
1. Doubling the numbers of both colours.
2. Tripling the numbers of both colours.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Ratios 4 © Department of Education WA 2015
3. Quadrupling the numbers of both colours.
4. Doubling the number of counters of the first colour only.
5. Tripling the number of counters of the second colour only.
6. Doubling the number of counters of the first colour and tripling the number of counters
of the second colour.
Changing the numbers in a ratio. The following diagram shows 3 shaded smiley faces to 1
unshaded smiley face.
The diagram below has the same objects but more of them. Are they in the same ratio? Are
there still 3 shaded smiley faces for every 1 unshaded smiley face?
This can be determined by grouping them as above.
Ungrouped Grouped
MATHS7TL002 | Mathematics | Number and Algebra Activity | Ratios 5 © Department of Education WA 2015
7. Use the grouping method (or otherwise) to decide if the ratios were changed when you -
(a) Doubled the numbers of both colours.
NO
(b) Tripled the numbers of both colours.
NO
(c) Quadrupled the numbers of both colours.
NO
(d) Doubled the number of counters of the first colour only.
Yes
(e) Tripled the number of counters of the second colour only.
Yes
(f) Doubled the number of counters of the first colour and tripling the number of counters
of the second colour.
Yes
MATHS7TL002 | Mathematics | Number and Algebra Activity | Ratios 6 © Department of Education WA 2015
Activity 2
Solutions are not provided, as they will vary from student to student.
A. Swap your “ratio” with another student. Draw a coloured diagram of your new ratio.
B. Now consider changing your ratio. For each change, draw a diagram showing the
numbers of counters of each colour. Use the grouping method to present (draw) your
counters and determine the change to the ratio.
1. Add two more counters of each colour.
The numbers in the ratio are no longer in the same proportion; the ratio has been
changed.
2. Add three more counters of each colour.
The numbers in the ratio are no longer in the same proportion; the ratio has been
changed.
3. Add four more counters of each colour.
The numbers in the ratio are no longer in the same proportion; the ratio has been
changed.
4. Add two more counters of the first colour only.
The numbers in the ratio are no longer in the same proportion; the ratio has been
changed.
5. Add three more counters of the second colour only.
The numbers in the ratio are no longer in the same proportion; the ratio has been
changed.
6. Add two more counters of the first colour and three more counters of the second
colour.
The numbers in the ratio are no longer in the same proportion; the ratio has been
changed.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Ratios 7 © Department of Education WA 2015
Activity 3
A. The operations on the numbers in the ratio for Activities 1 and 2 only included
addition and multiplication. Do you think the numbers will still be in proportion (or in
the same ratio) if division or subtraction is used. Provide evidence for your conclusion.
The numbers will still be in proportion for division but not for subtraction.
Example: If there is 1 lifesaver for every 10 swimmers; i.e., 1:10 then subtracting 1
from each ratio is 0:9 and this is not the same as 1:10.
Division is just the multiplication of an inverse so it should apply.
100:10 means 100 for every 10; i.e., 10 for every 1 and this can be achieved by
dividing both the 100 and the 10 by 10, in the same way that we can simplify fractions.
B. Consider the following question to write a summary of the results of these
investigations.
When the numbers in a ratio are changed, are the numbers still in the same proportion
or ratio?
The numbers in the ratio will still be in the same proportion as long as all numbers in
the ratio are either multiplied OR divided by the SAME number. Mostly the ratio will be
changed if the numbers in the ratio have another number added or subtracted; even if
it is the same number.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Ratios 8 © Department of Education WA 2015
Activity 4
The statement below is assumed to be true – the results of the investigation above.
A ratio does not change if both numbers in the ratio are multiplied or divided by the same
number.
Investigate if this statement applies when -
(i) the numbers in the ratio are decimals or fractions; and
(ii) the numbers used to multiply and divide the numbers in the ratio, are decimals or
fractions.
(i)
Ratio 0.1 : 0.4 0.2 : 0.5 0.5 :
Doubled 0.2 : 0.8 0.4 : 1.0 1.0 :
Tripled 0.3 : 1.2 0.6 : 1.5 1.5 :
quadrupled 0.4 : 1.6 0.8 : 2.0 2.0 :
In each column, you can divide the second number by the first and get the same result each
time so the ratio is preserved with multiplication even when there are fractions or decimals in
the ratio.
(ii)
Ratio 10 : 20 2 : 3 12 : 6 100 : 10
x 1.5 15 : 30 3 : 4.5 18 : 9 150 : 15
x 2.5 25 : 50 5 : 7.5 30 : 15 250 : 25
x 5 : 10 1 : 1.5 6 : 3 50 : 5
x 1 : 2 0.2 : 0.3 1.2 : 0.6 10 : 1
In each column, you can divide the second number by the first and get the same result each
time so the ratio is preserved with multiplication even when there are fractions or decimals in
the operating number.
Activity 5
You are given a page of ratios for weedkiller. The ratios are presented as follows.
The first number represents the number of parts of the mixture that are the weed killer and
the second number represents the number of parts of the mixture that are water.
Cut out these ratios and paste them into your workbook in order of strength of the mixture.
1:3
Water Weed
killer
MATHS7TL002 | Mathematics | Number and Algebra Activity | Ratios 9 © Department of Education WA 2015
11 1:3 7 3:4
10 2:5 9 3:5
15 1:8 10 4:10
1 10:1 17 2:40
6 3:2 14 4:28
4 12:4 8 2:3
9 6:10 13 1:4
17 1:20 15 3:24
12 2:7 9 6:15
16 10:90 3 4:1
19 3:100 18 3:50
5 5:3 16 1:9
2 8:1 3 20:5
MATHS7TL002 | Mathematics | Number and Algebra Activity | Ratios 10 © Department of Education WA 2015
STUDENT COPY RATIOS
Introduction
For the supervision of swimming in open water one lifesaver with a bronze medallion is
needed for every 10 students. This can be written without reference to units as -
one lifesaver for every ten students, or 1:10
Activity 1
Using the following ratios 1:2 1:3 2:3 1:5 2:5 3:4 1:4
Working in groups
Each group is allocated one of the ratios listed above.
Each group collects a handful of counters – two colours only.
Each group makes their ratio as follows, with first number of counters in one colour and
the second number of counters in the other colour.
o
These would be the counters for the ratio 3:5
A. Use colours and draw a diagram similar to the one above, to represent your ratio.
B Now consider changing your ratio. For each change, draw a diagram showing the
numbers of counters of each colour.
1. Doubling the numbers of both colours.
2. Tripling the numbers of both colours.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Ratios 11 © Department of Education WA 2015
3. Quadrupling the numbers of both colours.
4. Doubling the number of counters of the first colour only.
5. Tripling the number of counters of the second colour only.
6. Doubling the number of counters of the first colour and tripling the number of counters
of the second colour.
Changing the numbers in a ratio. The following diagram shows 3 shaded smiley faces to 1
unshaded smiley face.
The diagram below has the same objects but more of them. Are they in the same ratio? Are
there still 3 shaded smiley faces for every 1 unshaded smiley face?
This can be determined by grouping them as above.
Ungrouped Grouped
MATHS7TL002 | Mathematics | Number and Algebra Activity | Ratios 12 © Department of Education WA 2015
7. Use the grouping method (or otherwise) to decide if the ratios were changed when you -
(a) Doubled the numbers of both colours.
(b) Tripled the numbers of both colours.
(c) Quadrupled the numbers of both colours.
(d) Doubled the number of counters of the first colour only.
(e) Tripled the number of counters of the second colour only.
(f) Doubled the number of counters of the first colour and tripling the number of counters
of the second colour.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Ratios 13 © Department of Education WA 2015
Activity 2
A. Swap your “ratio” with another student. Draw a coloured diagram of your new ratio.
B. Now consider changing your ratio. For each change, draw a diagram showing the
numbers of counters of each colour. Use the grouping method to present (draw) your
counters and determine the change to the ratio.
1. Add two more counters of each colour.
2. Add three more counters of each colour.
3. Add four more counters of each colour.
4. Add two more counters of the first colour only
5. Add three more counters of the second colour only.
6. Add two more counters of the first colour and three more counters of the second
colour.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Ratios 14 © Department of Education WA 2015
Activity 3
A. The operations on the numbers in the ratio for Activities 1 and 2 only included
addition and multiplication. Do you think the numbers will still be in proportion (or in
the same ratio) if division or subtraction is used. Provide evidence for your conclusion.
B. Consider the following question to write a summary of the results of these
investigations.
When the numbers in a ratio are changed, are the numbers still in the same proportion
or ratio?
MATHS7TL002 | Mathematics | Number and Algebra Activity | Ratios 15 © Department of Education WA 2015
Activity 4
The statement below is assumed to be true – the results of the investigation above.
A ratio does not change if both numbers in the ratio are multiplied or divided by the same
number.
Investigate if this statement applies when -
(i) the numbers in the ratio are decimals or fractions, and
(ii) the numbers used to multiply and divide the numbers in the ratio, are decimals or
fractions.
Activity 5
You are given a page of ratios for weedkiller. The ratios are presented as follows.
The first number represents the number of parts of the mixture that are the weed killer and
the second number represents the number of parts of the mixture that are water.
Cut out these ratios and paste them into your workbook in order of strength of the mixture.
1:3
Water Weed
killer
MATHS7TL002 | Mathematics | Number and Algebra Activity | Ratios 16 © Department of Education WA 2015
1:3 3:4
2:5 3:5
1:8 4:10
10:1 2:40
3:2 4:28
12:4 2:3
6:10 1:4
1:20 3:24
2:7 6:15
10:90 4:1
3:100 3:50
5:3 1:9
8:1 20:5
MATHS7TL002 | Mathematics | Number and Algebra Activity | Equations 2 © Department of Education WA 2015
TASK 25: EQUATIONS
Overview
This focus in this task is on getting ready to interpret, write and solve equations by
establishing values for variables in number sentences. The process for solving the equations
in this task is mainly by guess and check, but more sophisticated methods can still be used.
Students will not need any extra materials
Relevant content descriptions from the Western Australian Curriculum
Introduce the concept of variables as a way of representing numbers using letters
(ACMNA175)
Create algebraic expressions and evaluate them by substituting a given value for each
variable (ACMNA176)
Multiply and divide fractions and decimals using efficient written strategies and digital
technologies (ACMNA154)
Students can demonstrate
fluency when they
o determine the solutions to the equations without using technology
understanding when they
o create number sentences in Activity 2
reasoning when they
o explain their answers in Activities 3 and 4
problem solving when they
o complete Activity 5
MATHS7TL002 | Mathematics | Number and Algebra Activity | Equations 3 © Department of Education WA 2015
EQUATIONS Solutions and Notes for Teachers
Activity 1
Each shape in these number sentences represents one of the digits from 1 to 9.
Which is which?
+ =
+ =
+ =
x =
+ =
x =
– =
x =
The digit represented by is missing from these number sentences. Which digit is
missing?
Start with . It represents 1 because 1 x 1 = 1.
Then + = means = 2
Then = 4 because + =
= 5 because + =
– = means that = 9
x = means that = 3
x = means that = 6
+ = means that = 8
= 7
MATHS7TL002 | Mathematics | Number and Algebra Activity | Equations 4 © Department of Education WA 2015
Activity 2
Use a similar process to the one used in Activity 1 to design and write number sentences
with symbols for the following sets of numbers. Use as many number sentences as you need
to be able to work out the numbers. Ask another student to identify the numbers represented
by the symbols in your number sentences.
(a) The decimals 0.1, 0.2, 0.3, ..., 0.9. Give your number sentences to another student to
solve.
Answers will vary. This is a good opportunity to review the fact that 0.1 x 0.1 0.1
It is also an opportunity to get students to work with simple decimals without using
technology.
(b) The following fractions: , , , , , ,
Answers will vary. This is a good opportunity to review operations with fractions. The
fractions have been chosen so that students can do this exercise without using technology.
Activity 3
The symbols and are used to represent numbers.
Given x = 36:
(a) What numbers might and represent if only whole numbers are allowed?
1 2 3 4 6 9 12 18 36
36 18 12 9 6 4 3 2 1
(b) Give 3 examples of x = 36 when one of the numbers has a value between 0
and 1.
0.5 x 72 = 36
x 144 = 36
360 x 0.1 = 36
(c) Can both numbers be negative? Can one number be negative? Give examples or
explain your answers.
Both numbers can be negative but just one number being negative is not possible:
-9 x -4 = 36 but 9 x -4 = -36
A negative number multiplied by a positive number will give a negative number.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Equations 5 © Department of Education WA 2015
Activity 4
In this activity the letters a, b, k, and m represent numbers. For each equation
give 5 different examples of what the letters could represent; and
determine the number of possible values that each letter could represent and explain
the reasons for your decision.
(a) a + b = 16
a 10 -2 16 0.4 -32
b 6 18 0 15.6 16
The above are examples. The letters a and b can represent any numbers at all as long as
the relationship between them is maintained. Any number can be allocated to a, then b is
determined.
(b) k + 5 = 5 x m
k 10 0 0.5 -8 200
m 3 1 1.1 -0.6 41
The above are examples. The letters k and m can represent any numbers at all as long
as the relationship between them is maintained. Any number can be allocated to k, 5 is
added and m is determined by dividing this result by 5.
(c) k ÷ m = m ÷ k
k 10 -2 16 0.4 -32
m 10 -2 16 0.4 -32
The above are examples. The letters k and m can represent any numbers at all, except
0, as long as the relationship between them is maintained. Any number can be allocated
to k, and m must represent the same number as k.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Equations 6 © Department of Education WA 2015
Activity 5
You are given a set of problems to solve. Use the process described to solve these
problems. An example is provided.
Problem 1: The sum of three consecutive odd numbers is 33. What are the numbers?
Let the first number be represented by k
The next three numbers are k + 2, k + 4
The equation is k + k + 2 + k + 4 = 33
Solving: The numbers must all be about 10 because 3 x 10 = 30
Try 11 + 13 + 15 = 39 (too large)
Try 9 + 11 + 13 = 33 correct The numbers are 9, 11, 13
Problem 2: A number plus its square adds to 650. What is the number?
Let the number be represented by k
Then the square is k2
The equation is k + k2 = 650
The number must be larger than 20 because 20 x 20 + 20 = 420
The number must be less than 30 because 30 x 30 + 30 = 930
The number could end in 5 because 650 is a multiple of 5
Try 25
25 x 25 + 25 = 650 correct The number is 25
Example:
Problem: Four consecutive even numbers add up to 60. What are they?
Let the first number be represented by k
The next three numbers are k + 2, k + 4, k + 6
The equation is k + k + 2 + k + 4 + k + 6 = 60
Solving: The numbers must all be less than 20 because 4 x 20 = 80
Try 10 + 12 + 14 + 16 = 52 (bit short)
Try 12 + 14 + 16 + 18 = 60 correct
MATHS7TL002 | Mathematics | Number and Algebra Activity | Equations 7 © Department of Education WA 2015
Problem 3: The sum of the ages of Granddad’s children is 24. Nickie is the eldest and is
three years older than his sister Lottie. The youngest, Bertie, is 10 years younger than Nickie
and three years younger than Gracie. How old are the children?
Let Nick’s age be represented by m
Then Lottie is m – 3
Bertie is m – 10
Gracie is m – 7
The equation is m + (m – 3) + (m – 10) + (m – 7) = 24.
Nick must be at least 10 (if Bertie is a very small baby), so try 10 for Nick:
10 + 7 + 0 + 3 = 20 (not enough)
Try 11:
11 + 8 + 1 + 4 = 24 correct
Nickie’s age is 11
Lottie is 8
Bertie is 1
Gracie is 4.
Problem 4: Two punnets of strawberries and one punnet of cherry tomatoes together cost $11
and the strawberries were 25c per punnet more expensive than the cherry tomatoes. What
did each punnet cost?
Let the cost of a punnet of strawberries by m.
Then the cost of a punnet of cherry tomatoes is m – $0.25
The equation is m + m + m – $0.25 = $11
The cost must be between $3 and $4 and end in 5c or 10c.
Try $3.50:
$3.50 + $3.50 + $3.25 = $10.25 (too little).
Try $3.75:
$3.75 + $3.75 + $3.50 = $11 correct
The cost of a punnet of strawberries is $3.75.
The cost of a punnet of cherry tomatoes is $3.50.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Equations 8 © Department of Education WA 2015
STUDENT COPY EQUATIONS
Activity 1
Each shape in these number sentences represents one of the digits from 1 to 9.
Which is which?
+ =
+ =
+ =
x =
+ =
x =
– =
x =
The digit represented by is missing from these number sentences. Which digit is
missing?
MATHS7TL002 | Mathematics | Number and Algebra Activity | Equations 9 © Department of Education WA 2015
Activity 2
Use a similar process to the one used in Activity 1 to design and write number sentences
with symbols for the following sets of numbers. Use as many number sentences as you need
to be able to work out the numbers. Ask another student to identify the numbers represented
by the symbols in your number sentences.
(a) The decimals 0.1, 0.2, 0.3, ..., 0.9. Give your number sentences to another student to
solve.
(b) The following fractions: , , , , , ,
MATHS7TL002 | Mathematics | Number and Algebra Activity | Equations 10 © Department of Education WA 2015
Activity 3
The symbols and are used to represent numbers.
Given x = 36:
(a) What numbers might and represent if only whole numbers are allowed?
(b) Give 3 examples of x = 36 when one of the numbers has a value between 0
and 1.
(c) Can both numbers be negative? Can one number be negative? Give examples or
explain your answers.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Equations 11 © Department of Education WA 2015
Activity 4
In this activity the letters a, b, k, and m represent numbers. For each equation
give 5 different examples of what the letters could represent; and
determine the number of possible values that each letter could represent and explain
the reasons for your decision.
(a) a + b = 16
(b) k + 5 = 5 x m
(c) k ÷ m = m ÷ k
MATHS7TL002 | Mathematics | Number and Algebra Activity | Equations 12 © Department of Education WA 2015
Activity 5
You are given a set of problems to solve. Use the process described to solve these
problems. An example is provided.
Problem 1: The sum of three consecutive odd numbers is 33. What are the numbers?
Problem 2: A number plus its square adds to 650. What is the number?
Example:
Problem: Four consecutive even numbers add up to 60. What are they?
Let the first number be represented by k
The next three numbers are k + 2, k + 4, k + 6
The equation is k + k + 2 + k + 4 + k + 6 = 60
Solving: The numbers must all be less than 20 because 4 x 20 = 80
Try 10 + 12 + 14 + 16 = 52 (bit short)
Try 12 + 14 + 16 + 18 = 60 correct
MATHS7TL002 | Mathematics | Number and Algebra Activity | Equations 13 © Department of Education WA 2015
Problem 3: The sum of the ages of Granddad’s children is 24. Nickie is the eldest and is
three years older than his sister Lottie. The youngest, Bertie, is 10 years younger than Nickie
and three years younger than Gracie. How old are the children?
Problem 4: Two punnets of strawberries and one punnet of cherry tomatoes together cost $11
and the strawberries were 25c per punnet more expensive than the cherry tomatoes. What
did each punnet cost?
MATHS7TL002 | Mathematics | Number and Algebra Activity | Scoring Golf 2 © Department of Education WA 2015
TASK 26: SCORING GOLF
Overview
For this task students should have developed an understanding of the number line below
zero and have had some experience with operations on negative integers. In this task
students are provided with an opportunity to consolidate and demonstrate their
understanding of working with negative numbers.
Students will not need any particular equipment
Relevant content descriptions from the Western Australian Curriculum
Compare, order, add and subtract integers (ACMNA280)
Students can demonstrate
fluency when they
o calculate accurately with integers in all activities
o represent integers in flexible ways
understanding when they
o can work backwards to determine operations to produce negative numbers
o interpret and exemplify statements summarising features of operations with
negative integers.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Scoring Golf 3 © Department of Education WA 2015
SCORING GOLF Solutions and Notes for Teachers
Activity 1
Golf courses are designed so that good golfers will usually take the same number of strokes
to hit the ball into the hole. Each hole is usually listed as being one of the following:
Par 3: requiring 3 strokes
Par 4: requiring 4 strokes
Par 5: requiring 5 strokes and so on . . .
When the hole is played the player is said to be either;
Over par: takes more strokes than the par for that hole,
Under par: takes fewer strokes than the par for that hole, or
Par: takes the 'par' number of strokes for that hole.
Scoring for a Par 4 hole
Number of strokes to get
the ball in Description Score Name
7 3 over par 3 Triple Bogey
6 2 over par 2 Double Bogey
5 1 over par 1 Bogey
4 Par 0 Par
3 1 under par – 1 Birdie
2 2 under par – 2 Eagle
1 3 under par – 3 Albatross
1. Complete this table for a Par 3 hole
Number of strokes to get
the ball in Description Score Name
7 4 over par 4
6 3 over par 3 Triple Bogey
5 2 over par 2 Double Bogey
4 1 over par 1 Bogey
3 Par 0 Par
2 1 under par – 1 Birdie
1 2 under par – 2 Eagle
MATHS7TL002 | Mathematics | Number and Algebra Activity | Scoring Golf 4 © Department of Education WA 2015
2. Using positive and negative numbers, determine the Score for the following results.
(a) 5 on a par 3 (d) 4 on a par 4
2 0
(b) 6 on a par 5 (e) 2 on a par 3
1 –1
(c) 7 on a par 4 (f) 3 on a par 5
3 –2
3. What name would be given to each of the above results?
(a) 5 on a par 3 (d) 4 on a par 4
Double Bogey Par
(b) 6 on a par 5 (e) 2 on a par 3
Bogey Birdie
(c) 7 on a par 4 (f) 3 on a par 5
Triple Bogey Eagle
A Birdie is 'one under' and on a par 5 hole, the golfer would succeed in 4 strokes.
4. How many strokes would a golfer take to get an Albatross on a hole that is -
Par 3? Impossible Par 4? 1 Par 5? 2
5. Create a table, similar to Table 1 on the previous page, for a hole of par 5 on a golf
course. Show how you would score and describe the number of strokes (between 1 and 8)
that might be taken to get the ball in.
Number of strokes to get
the ball in Description Score Name
1 Four under par –4 Condor
2 Three under par –3 Albatross
3 Two under par –2 Eagle
4 One under par –1 Birdie
5 Par 0 Par
6 One over par 1 Bogey
7 Two over par 2 Double Bogey
8 Three over par 3 Triple Bogey
MATHS7TL002 | Mathematics | Number and Algebra Activity | Scoring Golf 5 © Department of Education WA 2015
6. The total number of strokes to complete all 18 holes on a golf course is usually 72.
The results of four players for 18 holes are shown in the table. Complete the table to
determine the score for each hole for each player and then identify the winner.
Hole Par
Paul Cath Tom Jill
strokes score strokes score strokes score strokes score
1 3 3 0 2 –1 1 –2 2 –1
2 4 5 1 4 0 3 –1 5 1
3 3 3 0 2 –1 1 –2 3 0
4 4 6 2 5 1 3 –1 4 0
5 3 5 2 6 3 7 4 8 5
6 3 4 1 3 0 6 3 3 0
7 4 4 0 2 –2 5 1 4 0
8 3 7 4 3 0 4 1 5 2
9 5 6 1 4 –1 5 0 4 –1
10 4 5 1 5 1 6 2 5 1
11 4 1 –3 3 –1 7 3 5 1
12 5 3 -2 3 –2 2 –3 4 –1
13 5 2 -3 3 –2 3 –2 4 –1
14 4 3 -1 4 0 4 0 5 1
15 4 3 -1 4 0 4 0 5 1
16 4 4 0 5 1 2 –2 5 1
17 5 5 0 4 –1 5 0 4 –1
18 4 7 3 5 1 4 0 4 0
Final scores: 5 –4 1 8
The winner is Cath.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Scoring Golf 6 © Department of Education WA 2015
Activity 2
2
c) Which player wins?
Investigation Negative Numbers
Use your calculator (if necessary) to investigate the truth of each of the given
statements.
If you think that the statement is false, write one example to support your conclusion.
If you think that the statement is true, write at least three examples to support your
conclusion.
Statements
1. Adding two negative numbers always results in a negative number.
True
–4 + –5 = –9
–6 + –9 = –15
–3 + –7 = –10
2. Doubling a negative number will produce another negative number.
True
2 x –9 = –18
10 x –2 = –20
4.5 x –2 = –9
3. Squaring a negative number will produce another negative number.
False
–9 x –9 = 81
4. Dividing a negative number by a counting number (1, 2, 3, . . . always gives a
quotient that is negative.
True (counting numbers are positive).
–40 ÷ 8 = –5
–100 ÷ 10 = –10
–45 ÷ 5 = –9
5. The product of a negative number and a whole number is always positive.
False (whole numbers, except 0, are positive).
–4 x 8 = – 32
–7 x 0 = 0
MATHS7TL002 | Mathematics | Number and Algebra Activity | Scoring Golf 7 © Department of Education WA 2015
c) Which player wins?
Investigation (cont’d) Negative Numbers
6. Negative numbers are always whole numbers.
False
Decimals can be negative; e.g., –0.5.
7. Adding a negative number to a positive number gives a negative answer.
False
– 4 + 5 = 1
8. Subtracting a positive number from a negative number always gives a negative
answer.
True
-4 – 9 = -13
-6 – 20 = -14
-10 – 1 = -11
9. Subtracting any positive number from a smaller positive number always gives a
negative answer.
True
9 – 12 = -3
10 – 100 = -90
1 – 2 = -1
10. The sum of any positive number and a whole number is another whole
number.
False
1 + 0.5 = 1.5 0.5 is a positive number, but 1.5 is not a whole number.
11. Adding a negative fraction to a positive decimal can result in 0.
True
- 1
2 + 0.5 = 0
0.1 + - 1
10 = 0
- 3
4+ 0.75 = 0
MATHS7TL002 | Mathematics | Number and Algebra Activity | Scoring Golf 8 © Department of Education WA 2015
Activity 3
The card below is part of a game of Neggo where the number is crossed off if it is the result
of the operation that is called out. Your task is to devise three different operations for each
number on the card. The first one is shown as an example.
–30 –2 –42 –16 –24
–10 –8 –9 –18 –60
–12 –36 –1 –20 –100
Examples are given below, but there are many possibilities.
Operation 1 Operation 2 Operation 3
–30 6 x –5 10 – 40 –10 – 20
–2 1 – 3 2 x –1 4 – 6
–42 – 6 x 7 –7 x 6 2 – 44
–16 – 2 x 8 16 x –1 3 – 19
–24 – 6 x 4 3 x –8 2 x –12
–10 2 x – 5 –2 – 8 –3 – 7
–8 8 x –1 1 x –8 – 3 – 5
–9 –3 – 6 –3 x 3 9 x –1
–18 – 36 ÷ 2 –9 x 2 6 – 24
–60 –20 x 3 10 – 70 –3 x 20
–12 24 ÷ –2 3 x –4 6 x –2
–36 9 x – 4 4 x –9 6 x –6
–1 9 ÷ – 9 –4 ÷ 4 6 – 7
–20 –10 x 2 –4 x 5 0.5 x –40
–100 –10 x 10 – 100 ÷ 1 1000 ÷ –10
MATHS7TL002 | Mathematics | Number and Algebra Activity | Scoring Golf 9 © Department of Education WA 2015
STUDENT COPY SCORING GOLF
Activity 1
Golf courses are designed so that good golfers will usually take the same number of strokes
to hit the ball into the hole. Each hole is usually listed as being one of the following:
Par 3: requiring 3 strokes
Par 4: requiring 4 strokes
Par 5: requiring 5 strokes and so on . . .
When the hole is played the player is said to be either;
Over par: takes more strokes than the par for that hole,
Under par: takes fewer strokes than the par for that hole, or
Par: takes the 'par' number of strokes for that hole.
Scoring for a Par 4 hole
Number of strokes to get
the ball in Description Score Name
7 3 over par 3 Triple Bogey
6 2 over par 2 Double Bogey
5 1 over par 1 Bogey
4 Par 0 Par
3 1 under par – 1 Birdie
2 2 under par – 2 Eagle
1 3 under par – 3 Albatross
1. Complete this table for a Par 3 hole
Number of strokes to get
the ball in Description Score Name
7 4
6
5
4
3 Par
2 Birdie
1
MATHS7TL002 | Mathematics | Number and Algebra Activity | Scoring Golf 10 © Department of Education WA 2015
2. Using positive and negative numbers, determine the Score for the following results.
(a) 5 on a par 3 (d) 4 on a par 4
(b) 6 on a par 5 (e) 2 on a par 3
(c) 7 on a par 4 (f) 3 on a par 5
3. What name would be given to each of the above results?
(a) 5 on a par 3 (d) 4 on a par 4
(b) 6 on a par 5 (e) 2 on a par 3
(c) 7 on a par 4 (f) 3 on a par 5
A Birdie is 'one under' and on a par 5 hole, the golfer would succeed in 4 strokes.
4. How many strokes would a golfer take to get an Albatross on a hole that is;
Par 3? Par 4? Par 5?
5. Create a table, similar to Table 1 on the previous page, for a hole of par 5 on a golf
course. Show how you would score and describe the number of strokes (between 1 and 8)
that might be taken to get the ball in.
Number of strokes to get
the ball in Description Score Name
MATHS7TL002 | Mathematics | Number and Algebra Activity | Scoring Golf 11 © Department of Education WA 2015
6. The total number of strokes to complete all 18 holes on a golf course is usually 72.
The results of four players for 18 holes are shown in the table. Complete the table to
determine the score for each hole for each player and then identify the winner.
Hole Par
Paul Cath Tom Jill
strokes score strokes score strokes score strokes score
1 3 3 2 1 2
2 4 5 4 3 5
3 3 3 2 1 3
4 4 6 5 3 4
5 3 5 6 7 8
6 3 4 3 6 3
7 4 4 2 5 4
8 3 7 3 4 5
9 5 6 4 5 4
10 4 5 5 6 5
11 4 1 3 7 5
12 5 3 3 2 4
13 5 2 3 3 4
14 4 3 4 4 5
15 4 3 4 4 5
16 4 4 5 2 5
17 5 5 4 5 4
18 4 7 5 4 4
Final scores:
MATHS7TL002 | Mathematics | Number and Algebra Activity | Scoring Golf 12 © Department of Education WA 2015
Activity 2
c) Which player wins?
Investigation Negative Numbers
Use your calculator (if necessary) to investigate the truth of each of the given
statements.
If you think that the statement is false, write one example to support your conclusion.
If you think that the statement is true, write at least three examples to support your
conclusion.
Statements
1. Adding two negative numbers always results in a negative number.
2. Doubling a negative number will produce another negative number.
3. Squaring a negative number will produce another negative number.
4. Dividing a negative number by a counting number (1, 2, 3, …) always gives a
quotient that is negative.
5. The product of a negative number and a whole number is always positive.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Scoring Golf 13 © Department of Education WA 2015
c) Which player wins?Investigation (cont’d) Negative Numbers
6. Negative numbers are always whole numbers.
7. Adding a negative number to a positive number gives a negative answer.
8. Subtracting a positive number from a negative number always gives a
negative answer.
9. Subtracting any positive number from a smaller positive number always gives a
negative answer.
10. The sum of any positive number and a whole number is another whole
number.
11. Adding a negative fraction to a positive decimal can result in 0.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Scoring Golf 14 © Department of Education WA 2015
Activity 3
The card below is part of a game of Neggo where the number is crossed off if it is the result
of the operation that is called out. Your task is to devise three different operations for each
number on the card. The first one is shown as an example.
–30 –2 –42 –16 –24
–10 –8 –9 –18 –60
–12 –36 –1 –20 –100
Operation 1 Operation 2 Operation 3
–30 6 x –5 10 – 40 –10 – 20
–2
–42
–16
–24
–10
–8
–9
–18
–60
–12
–36
–1
–20
–100
MATHS7TL002 | Mathematics | Number and Algebra Activity | Fraction Graphics 1 © Department of Education WA 2015
MATHS7TL002 | Mathematics | Number and Algebra Activity | Fraction Graphics 2 © Department of Education WA 2015
TASK 27: FRACTION GRAPHICS
Overview
This task requires students to demonstrate an understanding of fractions and operations on
fractions in the creation of a poster highlighting one particular fraction.
Students will need
calculators
access to the internet
printed media; e.g., newspapers
Relevant content descriptions from the Western Australian Curriculum
Compare fractions using equivalence. Locate and represent positive and negative
fractions and mixed numbers on a number line (ACMNA152)
Express one quantity as a fraction of another with and without the use of digital
technologies (ACMNA155)
Connect fractions, decimals and percentages and carry out simple conversions
(ACMNA157)
Students can demonstrate
understanding when they
o represent fractions in various ways
o create accurate statements of operations on their allocated fraction
MATHS7TL002 | Mathematics | Number and Algebra Activity | Fraction Graphics 3 © Department of Education WA 2015
FRACTION GRAPHICS Solutions and Notes for Teachers
Create an information poster for the fraction you have been allocated. For your fraction you
will need to show -
(i) Other number forms for your fraction, including three other equivalent fractions.
(ii) Models showing your fraction as a proportion of the model. These models should
include number lines, continuous area models; e.g., circles, rectangles as well as
discrete models (sets of numbers). Try to show a few examples of each type of model.
(iii) Write questions which require the use of operations with your fraction.
(iv) Write questions for which your fraction is the answer.
(v) Applications of your fraction to situations outside the classroom; e.g., media, news, the
environment etc.
Examine the fraction graphic for on the next page.
Students could work on this task in pairs or small groups.
Any fraction could be considered, such as follows:
The fractions that could be considered are endless in number and could be selected by the
students themselves.
(i) Equivalent number forms that students should consider are percentages, decimals and
equivalent fractions.
(ii) Students should be encouraged to think beyond the traditional representations of
fractions (circles, rectangles, squares) and examine a variety of uncommon
representations.
(iii) Operations should include addition, subtraction, multiplication, division and squaring.
Students may also consider negating and inverting the fraction.
(iv) Students should be encouraged to write questions in a context.
(v) Students could look on the internet or in printed material to determine some applications
of their fraction.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Fraction Graphics 4 © Department of Education WA 2015
is one half
0.5 is one half
50% is one half
4=
50
00 =
33
66=
3
6 =
Half the smiley faces are blue
(discrete model)
Area models of one half
Number line
Operations
x 4 = 2 3.5 + = 4
5 ÷ = 10 of 8.8 = 4.4
+ + = 3 - = 0
x = 4
+ = 1
÷ 2 = 4
5 + 2 x = 6 Questions
1. Roughly what proportion of people
are males?
2. What will a shirt marked at $22 cost
if it is reduced to “half-price”?
3. After round 20 in 2015, the Dockers
had 64 points but 6 teams had less
than half that number. What was the
greatest number of points any of the 6
teams could have had?
4. The half-price of a polar fleece
jacket is $26. What would it normally
cost?
5. Where on the number line is - ?
6. If 10 oranges are cut into halves,
how many halves would there be?
Did you know that -
Dividing by a half is the same as
multiplying by 2.
The chance of a baby being a boy
or a girl is .
The inverse or reciprocal of is 2.
The opposite of is – .
If you toss a normal die, the
chance of the number coming up
being a prime number is .
MATHS7TL002 | Mathematics | Number and Algebra Activity | Fraction Graphics 5 © Department of Education WA 2015
STUDENT COPY FRACTION GRAPHICS
Create an information poster for the fraction you have been allocated. For your fraction you
will need to show -
(i) Other number forms for your fraction, including three other equivalent fractions.
(ii) Models showing your fraction as a proportion of the model. These models should
include number lines, continuous area models; e.g., circles, rectangles as well as
discrete models (sets of numbers). Try to show a few examples of each type of model.
(iii) Write questions which require the use of operations with your fraction.
(iv) Write questions for which your fraction is the answer.
(v) Applications of your fraction to situations outside the classroom; e.g., media, news, the
environment etc.
Examine the fraction graphic for on the next page.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Fraction Graphics 6 © Department of Education WA 2015
FRACTION GRAPHIC
is one half
0.5 is one half
50% is one half
4=
50
00 =
33
66=
3
6 =
Half the smiley faces are blue
(discrete model)
Area models of one half
Number line
Operations
x 4 = 2 3.5 + = 4
5 ÷ = 10 of 8.8 = 4.4
+ + = 3 - = 0
x = 4
+ = 1
÷ 2 = 4
5 + 2 x = 6 Questions
1. Roughly what proportion of people
are males?
2. What will a shirt marked at $22
cost if it is reduced to “half-price”?
3. After round 20 in 2015, the
Dockers had 64 points but 6 teams
had less than half that number. What
was the greatest number of points any
of the 6 teams could have had?
4. The half-price of a polar fleece
jacket is $26. What would it normally
cost?
5. Where on the number line is - ?
6. If 10 oranges are cut into halves,
how many halves would there be?
Did you know that -
Dividing by a half is the same as
multiplying by 2.
The chance of a baby being a boy
or a girl is .
The inverse or reciprocal of is 2.
The opposite of is – .
If you toss a normal dice, the
chance of the number coming up
being a prime number is .
MATHS7TL002 | Mathematics | Number and Algebra Activity | Fraction Graphics 7 © Department of Education WA 2015
MATHS7TL002 | Mathematics | Number and Algebra Activity | Percentages 2 © Department of Education WA 2015
TASK 28: PERCENTAGES
Overview
This task consists of a series of activities designed to encourage thinking about percentages.
Students will need
calculators
access to the internet
Relevant content descriptions from the Western Australian Curriculum
Find percentages of quantities and express one quantity as a percentage of another,
with and without digital technologies. (ACMNA158)
Students can demonstrate
fluency when they
o decide which operations to use in the routine questions and execute these
operations efficiently; e.g., 2(a) and 2(b)
understanding when they
o use 100 as the base for a percentage
reasoning when they
o explain the processes by which they obtained the solutions to questions involving
percentages
problem solving when they
o determine the processes necessary to answer the questions asked in context –
particularly 2(f) and 2(g)
MATHS7TL002 | Mathematics | Number and Algebra Activity | Percentages 3 © Department of Education WA 2015
PERCENTAGES Solutions and Notes for Teachers
Activity 1
1. Here is some of the information printed on two packets of cereal.
Cereal Quantity per 100 g
Carbohydrate Fat Protein
A 69.5 g 1.7 g 8.6 g
B 64.7 g 1.7 g 7.8 g
(a) Complete the table to show the percentages of nutrients in each packet.
Cereal Percentage of nutrients in packets of cereal
Carbohydrate Fat Protein Sugar Fibre
A 69.5% 1.7% 8.6% 28.7% 10.2%
B 69.5% 1.7% 7.8% 24.7% 16.1%
(b) Explain how you determined your answer to part (a).
The figures are give as grams per 100 grams. The units are the same so the
proportions are amounts per 100 and as this is what percentages are, then the amount
in grams per 100 g is the same as the percentage. Per cent = Hundredths.
(c) Percentages are given for sugar and fibre and these are both types of carbohydrates.
Cereals A and B come from different size packets.
Cereal A is in a 460 g packet and Cereal B is in a 775 g packet.
What percentage of fibre would you expect in an 800 g packet of Cereal A?
10.2%
What percentage of sugar would you expect in a 400 g packet of Cereal B?
24.7%
Explain how you determined your answers.
They are the same as the percentages given.
If the product does not vary, then the percentage of its components does not change
as the amount of the product changes.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Percentages 4 © Department of Education WA 2015
2. More data about Cereals A and B.
Cereal Quantity per 100 g Percentage of daily requirements per
average serve (40 g)
Sodium* Riboflavin Niacin Energy Sugar Iron
A 30 mg 1.06 mg 6.2 mg 7% 13% 25%
B 125 mg 1.05 mg 6.2 mg 7% 11% 20%
*Sodium: Not pure sodium, but compounds containing sodium.
(a) Is the percentage of niacin the same in both cereals? Explain.
Yes. There is 6.2 mg per 100 gm in each product. The same amount per 100 g indicates
the percentages are the same.
(b) Which % best represents the percentage of *sodium in Cereal B? Justify your choice.
125% 12.5% 1.25% 0.125% 0.0125%
125 mg per 100 g. These need to be the same units. There are 1000 mg in 1 g.
125 mg = 125 ÷ 1000 g = 0.125 g
0.125 g per 100 g means 0.125%
(c) Is the percentage of *sodium in Cereal A approximately four times the percentage of
*sodium in Cereal B? Explain.
Yes. Comparing 30 mg per 100 g to 125 g per 100 g, they are both out of 100 g. The
*sodium is measured in the same units in each case, so the percentages can be
compared. 125 is about 4 times 30 so the statement is true.
(d) To obtain all your daily iron requirements, how many grams would you need to eat of -
Cereal A? Cereal B?
160 g 200 g
(e) According to information on the packet, 12 mg of *sodium is equivalent to about 0.1 g
salt. Determine the % of salt in Cereal B.
125 mg salt per 100 g cereal is about 10 x 0.1 g salt.
This is about 1 g of salt per 100 g cereal.
So the percentage of salt is about 1%
(f) How many packets of Cereal B would you need to guarantee you have 1 kg sugar?
Cereal B has 24.7% of sugar; so 25% is a good estimate.
Each packet has 25% of sugar so 25% of 775 g = 193.75 g
There are 1000 g in 1 kg
1000 ÷ 193.75 = 5.16 packets
You would need 6 packets of cereal.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Percentages 5 © Department of Education WA 2015
(g) On the packet of Cereal B, a statement is given
More than 20% of your daily fibre needs.
Assuming a standard serve (40 g) of Cereal B has 21% of the recommended daily
intake, determine the number of grams of fibre recommended each day.
40 g of cereal contains 16.1% fibre and this is 21% of the recommended daily intake
Each serve has 16.1% of 40 g = 6.44 g
6.44 g is 21% of the daily need
100 ÷ 21 x 6.44 = 30.7 g of fibre
Activity 2
Search the internet for “nutritional information” about the foods you eat for breakfast or lunch.
Locate three of four different foods and summarise the percentages of carbohydrates,
proteins and fats in each of these foods. Determine the percentages of daily-recommended
intakes in a standard serve for energy, sugar and iron. How do these foods compare with the
cereals presented on the previous pages?
Answers will vary
Activity 3
Write a dot-point summary of features of percentages that you think are important for
students to know, understand and be able to use.
Answers will vary
MATHS7TL002 | Mathematics | Number and Algebra Activity | Percentages 6 © Department of Education WA 2015
STUDENT COPY PERCENTAGES
Activity 1
1. Here is some of the information printed on two packets of cereal.
Cereal Quantity per 100 g
Carbohydrate Fat Protein
A 69.5 g 1.7 g 8.6 g
B 64.7 g 1.7 g 7.8 g
(a) Complete the table to show the percentages of nutrients in each packet.
Cereal Percentage of nutrients in packets of cereal
Carbohydrate Fat Protein Sugar Fibre
A 28.7% 10.2%
B 24.7% 16.1%
(b) Explain how you determined your answer to part (a).
(c) Percentages are given for sugar and fibre and these are both types of carbohydrates.
Cereals A and B come from different size packets.
Cereal A is in a 460 g packet and Cereal B is in a 775 g packet.
What percentage of fibre would you expect in an 800 g packet of Cereal A?
What percentage of sugar would you expect in a 400 g packet of Cereal B?
Explain how you determined your answers.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Percentages 7 © Department of Education WA 2015
2. More data about Cereals A and B.
Cereal Quantity per 100 g Percentage of daily requirements per
average serve (40 g)
Sodium* Riboflavin Niacin Energy Sugar Iron
A 30 mg 1.06 mg 6.2 mg 7% 13% 25%
B 125 mg 1.05 mg 6.2 mg 7% 11% 20%
*Sodium: Not pure sodium, but compounds containing sodium.
(a) Is the percentage of niacin the same in both cereals? Explain.
(b) Which % best represents the percentage of *sodium in Cereal B? Justify your choice.
125% 12.5% 1.25% 0.125% 0.0125%
(c) Is the percentage of *sodium in Cereal A approximately four times the percentage of
*sodium in Cereal B? Explain.
(d) To obtain all your daily iron requirements, how many grams would you need to eat of -
Cereal A? Cereal B?
MATHS7TL002 | Mathematics | Number and Algebra Activity | Percentages 8 © Department of Education WA 2015
(e) According to information on the packet, 12 mg of *sodium is equivalent to about 0.1 g
salt. Determine the % of salt in Cereal B.
(f) How many packets of Cereal B would you need to guarantee you have 1 kg sugar?
(g) On the packet of Cereal B, a statement is given
More than 20% of your daily fibre needs
Assuming a standard serve (40 g) of Cereal B has 21% of the recommended daily
intake, determine the number of grams of fibre recommended each day.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Percentages 9 © Department of Education WA 2015
Activity 2
Search the internet for “nutritional information” about the foods you eat for breakfast or lunch.
Locate three of four different foods and summarise the percentages of carbohydrates,
proteins and fats in each of these foods. Determine the percentages of daily-recommended
intakes in a standard serve for energy, sugar and iron. How do these foods compare with the
cereals presented on the previous pages?
Activity 3
Write a dot-point summary of features of percentages that you think are important for
students to know, understand and be able to use.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Discounts 2 © Department of Education WA 2015
TASK 30: DISCOUNTS
Overview
In this task students investigate discounts. They consider discounting by a fixed amount and
by a percentage. Students are directed to consider applying known facts to ease the
calculation of discounts rather than use an algorithm that may be difficult to remember.
Students will need
calculators
Relevant content descriptions from the Western Australian Curriculum
Find percentages of quantities and express one quantity as a percentage of another,
with and without digital technologies.(ACMNA158)
Students can demonstrate
fluency when they
o determine further percentages given 10%
understanding when they
o distinguish between fixed change and proportional change
o explain their solution process
o recognise equivalences of percentages and fractions
reasoning when they
o explain which is the best discount offer in Activity 3
MATHS7TL002 | Mathematics | Number and Algebra Activity | Discounts 3 © Department of Education WA 2015
DISCOUNTS Solutions and Notes for Teachers
Review activity
1. Changing by a fixed amount. ‘Discount’ means to ‘not count’ or to ‘count off’.
The table below shows the original prices and the discounted prices for various goods.
Item Original
price Discounted
price Value of discount
Bluetooth speaker $144 $118 $26
Big red umbrella $329 $269 $60
Diamond ring (1.25, grade f) $13 210 $11 500 $1710
2 litres milk $2.19 $1.99 20c
Dual cab $22 800 $21 700 $1100
1 ball knitting wool $3.99 $2.99 $1
16 GB iPad mini $298 $258 $40
Vintage Lancia $45 609 $45 069 $540
2. Changing by a fixed percentage. Determine the amount by which these items are
discounted. Show how you determined your answer.
Item Original price
% of discount
Amount of discount
lawnmower $200 10% $20
pool pump $550 10% $55
steel-framed gazebo $150 10% $15
esky $80 20% $16
pyjamas $40 20% $8
sunglasses $250 20% $50
MATHS7TL002 | Mathematics | Number and Algebra Activity | Discounts 4 © Department of Education WA 2015
Activity 1
Determine the amount of the discount when the percentage of the discount varies.
Use your results to complete the table below.
Item Original price
10 % discount
20% discount
30% discount
cot set $100 $10 $20 $30
pillow set $30 $3 $6 $9
sewing machine $120 $12 $24 $36
bed sheets $80 $8 $16 $24
bed quilt $45 $4.50 $9 $13.50
doona $95 $9.50 $19 $28.50
1. How did you calculate 10% of each item?
Divide the original price by 10.
2. How did you calculate 20% of each item?
Double the 10% discount or divide the original price by 5.
3. Can you calculate 20% by multiplying your answer for 10% by 2? Justify your conclusion.
Yes. 10% of an amount is one tenth and 20% of the same amount is one fifth.
One fifth = 2 x one tenth
4. What will the sunglasses cost if the discount is 100%?
Nothing
MATHS7TL002 | Mathematics | Number and Algebra Activity | Discounts 5 © Department of Education WA 2015
5. For these items you are given the value of the 10% discount. Determine the value of the
other discounts and complete the table.
Item Original
price
Discounts
10% 1% 5% 70%
plastic chair $40 $4 40c $2 $28
park bench $150 $15 $1.50 $7.50 $105
barbeque $420 $42 $4.20 $21 $294
ladder $100 $10 $1 $5 $70
rake $4 40c 4c 20c $2.80
Explain how, given 10%, you can determine the other percentages.
To determine 1% of the original amount you can divide the 10% discount by 10.
To determine 5% of the original amount you can -
(a) halve the 10% discount OR
(b) multiply the 1% discount by 5
To determine the 70% discount you can -
(a) multiply the 5% discount by 14 OR
(b) multiply the 1% discount by 70 OR
(c) multiply the 10% discount by 7.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Discounts 6 © Department of Education WA 2015
6. For these items you are given the value of the 30% discount. State the values of the
other discounts and determine the original price.
Item Original
price
Discounts
30% 10% 5% 1%
wheelbarrow $120 $36 $12 $6 $1.20
floodlight $80 $24 $8 $4 80c
tool cabinet $150 $45 $15 $7.50 $1.50
smoke alarm $10 $3 $1 50c 10c
shed $7000 $2100 $700 $350 $70
Explain how, given 30%, you can determine 10% and the original price.
To find 10%, given 30%, you can divide the discount amount by 3.
To determine the original price, you can multiply the 10% discount by 10.
Activity 2
A lounge dining setting is priced at $2000, and for the sale the proposed discount is 35%.
Show FOUR different ways by which you can determine the discount amount.
1. 10% of $2000 is $200.
30% is 3 x 10% so 30% of $2000 = 3 x $200 = $600
5% is half of 10% and half of $200 is $100.
35% = 30% + 5% = $600 + $100 = $700
2. Determine 1%. 1% of $2000 = $20
Determine 35%: 35% = 1% x 35 = $20 x 35 = $700
3. 10% of $2000 is $200
5% of $200 = $2000 ÷ 20 = $100
35% = 10% + 10% + 10% + 5% = $200 + $200 + $200 + $100 = $700
4. 35% = 35 hundredths = 0.35
0.35 x $2000 = $700
MATHS7TL002 | Mathematics | Number and Algebra Activity | Discounts 7 © Department of Education WA 2015
Activity 3
Your grandmother wants to buy a new dress. Which of the following discounts will be better?
Justify your choice.
If the dress costs exactly $100 then 30% off is equal to $30 off.
If the dress costs less than $100; e.g., $90, then 30% off is $27 and this is less than $30.
If the dress costs more than $100; e.g., $120, then 30% off is $36 and this is more than $30.
Conclusion
You get a greater value of discount with 30% if the dress is more than $100.
You get a smaller discount with $30 if the dress is more than $100.
You get the same discount with 30% and $30 off if the dress costs $100.
Activity 4
Jody wanted to buy a pair of shoes. The shop had a week-long sale and claimed that all
shoes were already marked down 10%. On the last day of the sale the owner of the shoe
store marked all shoes down another 10%. Was this the same as a 20% reduction on the
original price?
Use three different starting prices to test this theory and write a conclusion.
Price Price with first 10% off Price with second 10% off Price reduced by 20%
$100 $90 $81 $80
$60 $54 $48.60 $48
$250 $225 $202.50 $200
Conclusion: A one-off discount of 20% gives a greater discount than a 10% discount followed
by another 10% discount. Note that the second 10% is calculated on a smaller amount than
the original price.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Discounts 8 © Department of Education WA 2015
STUDENT COPY DISCOUNTS
Review activity
1. Changing by a fixed amount. ‘Discount’ means to ‘not count’ or to ‘count off’.
The table below shows the original prices and the discounted prices for various goods.
Item Original
price Discounted
price Value of discount
Bluetooth speaker $144 $118
Big red umbrella $329 $269
Diamond ring (1.25, grade F) $13 210 $11 500
2 litres milk $2.19 $1.99
Dual cab $22 800 $21 700
1 ball knitting wool $3.99 $2.99
16 GB iPad mini $298 $258
Vintage Lancia $45 609 $45 069
2. Changing by a fixed percentage. Determine the amount by which these items are
discounted. Show how you determined your answer.
Item Original price
% of discount
Amount of discount
lawnmower $200 10%
2-bedroom apartment $550 000 10%
steel-framed gazebo $150 10%
esky $80 20%
pyjamas $40 20%
sunglasses $250 20%
MATHS7TL002 | Mathematics | Number and Algebra Activity | Discounts 9 © Department of Education WA 2015
Activity 1
Determine the amount of the discount when the percentage of the discount varies.
Use your results to complete the table below.
Item Original price
10 % discount
20% discount
30% discount
cot set $100
pillow set $30
sewing machine $120
bed sheets $80
bed quilt $45
doona $95
1. How did you calculate 10% of each item?
2. How did you calculate 20% of each item?
3. Can you calculate 20% by multiplying your answer for 10% by 2? Justify your conclusion.
4. What will the sunglasses cost if the discount is 100%?
MATHS7TL002 | Mathematics | Number and Algebra Activity | Discounts 10 © Department of Education WA 2015
5. For these items you are given the value of the 10% discount. Determine the value of the
other discounts and complete the table.
Item Original
price
Discounts
10% 1% 5% 70%
plastic chair $40 $4
park bench $150 $15
bbq $420 $42
ladder $100 $10
rake $4 40c
Explain how, given 10%, you can determine the other percentages.
6. For these items you are given the value of the 30% discount. State the values of the
other discounts and determine the original price.
Item Original
price
Discounts
30% 10% 5% 1%
wheelbarrow $36
floodlight $24
tool cabinet $45
smoke alarm $3
shed $2100
Explain how, given 30%, you can determine 10% and the original price.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Discounts 11 © Department of Education WA 2015
Activity 2
A lounge dining setting is priced at $2000, and for the sale the proposed discount is 35%.
Show FOUR different ways by which you can determine the discount amount.
Activity 3
Your grandmother wants to buy a new dress. Which of the following discounts will be better?
Justify your choice.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Discounts 12 © Department of Education WA 2015
Activity 4
Jody wanted to buy a pair of shoes. The shop had a week-long sale and claimed that all
shoes were already marked down 10%. On the last day of the sale the owner of the shoe
store marked all shoes down another 10%. Was this the same as a 20% reduction on the
original price?
Use three different starting prices to test this theory and write a conclusion.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Moving Points 2 © Department of Education WA 2015
TASK 32: MOVING POINTS
Overview
This task provides an opportunity for students to relate specific movements on the Cartesian
plane to particular changes to the points‟ coordinates.
Students will not need any special equipment
Relevant content descriptions from the Western Australian Curriculum
Given coordinates, plot points on the Cartesian plane, and find coordinates for a given
point (ACMNA178)
Compare, order, add and subtract integers (ACMNA280)
Students can demonstrate
fluency when they
o calculate accurately with integers
o read and plot points on the Cartesian plane
understanding when they
o connect the relationship between changes to the coordinates and movement on
the Cartesian plane
o express changes to coordinates as algebraic expressions
MATHS7TL002 | Mathematics | Number and Algebra Activity | Moving Points 3 © Department of Education WA 2015
MOVING POINTS Solutions and Notes for Teachers
Activity 1
1. Review plotting points by plotting the following points and labelling the points with the
letters provided.
A (2,3) B (1,0) C (7,10) D (-4,6) E (-9,0) F (-5,-5) G (-7,8) H (0,-8)
2. Plot any points J K L M N P and ask another student to identify them.
Various recordings & answers.
In the following activities you are asked to “move” a point. The point doesn‟t really move as it
is a position. The new point is positioned away from the original point. When asked to move
a point, label the new point with the same letter but add a dash; e.g., A „moves‟ to A‟.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Moving Points 4 © Department of Education WA 2015
Activity 2
1. Enter the coordinates of the points A to F into the table below. Then “move” the points as
instructed in the table and label the “moved” points.
,
Point Coordinates of given point
„Move‟ the point ... Coordinates of „moved‟ point
A (6,2) Left 4 units (2,2)
B (4,8) Left 5 units (-1,8)
C (-1,3) Left 7 units (-8,3)
D (1,-5) Right 6 units (7,-5)
E (-2,-2) Right 2 units (0,-2)
F (-9,-1) Right 3 units (-6,-1)
2. When a point is moved left or right:
(a) Does the x-value of the point change? YES
(b) Does the y-value of the point change? NO
MATHS7TL002 | Mathematics | Number and Algebra Activity | Moving Points 5 © Department of Education WA 2015
Activity 3
1. Enter the coordinates of the points G to L into the table below. Then “move” the points as
instructed in the table and label the “moved” points.
,
Point Coordinates of given point
„Move‟ the point ... Coordinates of „moved‟ point
G (8,2) Up 6 units (8,8)
H (5,-5) Up 3 units (5,-2)
I (-7,-3) Up 7 units (-7,4)
J (-2,8) Down 4 units (-2,4)
K (-2,-1) Down 3 units (-2,-4)
L (2,5) Down 7 units (2,-2)
2. When a point is „moved‟ up or down:
(a) Does the x-value of the point change? No
(b) Does the y-value of the point change? Yes
MATHS7TL002 | Mathematics | Number and Algebra Activity | Moving Points 6 © Department of Education WA 2015
Activity 4
Enter the coordinates of the points M to T into the table below. Then “move” the points as
instructed in the table and label the “moved” points.
Point Coordinates of given point
„Move‟ the point ... Coordinates of „moved‟ point
M (8,-3) Up 3 units then left 2 units (6,0)
N (2,4) Up 3 units then right 2 units (4,7)
O (-4,6) Down 5 units then left 5 units (-9,1)
P (0,0) Left 9 units then down 9 units (-9,-9)
Q (1,-5) Left 1 unit then up 10 units (0,5)
R (-4,-5) Right 4 units then down 4 units (0,-9)
S (-8,-4) Up 4 units then right 4 units (-4,0)
T (9,9) Down 9 units then right 1 unit (10,0)
MATHS7TL002 | Mathematics | Number and Algebra Activity | Moving Points 7 © Department of Education WA 2015
Activity 5
1. Points W, X, Y and Z were moved to W‟, X‟, Y‟ and Z‟. The movements are given below. Give
the coordinates of the original points W, X, Y and Z.
(a) W‟ is at (4, 2) after W was moved 4 units left. Where was W to start with? (8, 2)
(b) X‟ is at (1, 0) after X was moved 3 units up. Where was X to start with? (1,-3)
(c) Y‟ is at (-6, 7) after Y was moved 1 unit right and 1 unit down.
Where was Y to start with? (-7,8)
(d) Z‟ is at (3, -6) after Z was moved 3 units up and 6 units left. Where was Z to start with?
(0,0)
2. State the coordinates of the new points that will be formed from the following movements.
Point to be „moved‟ Movement New coordinates
(20, 20) Right 4 units (24, 20)
(-10, 30) Left 6 units (-16, 30)
(-20, 0) Up 7 units (-20, 7)
(7, 50) Down 10 units (7, 40)
(20, 25) Left 20 units, down 25 units (0, 0)
(12, 16) Up 2 units, right 3 units (15, 18)
(100, 100) Down 50 units, right 50 units (150, 50)
(20, 50) Right 10 units, up 5 units (30, 55)
(x, y) Up 6 units (x, y + 6)
(x, y) Down 5 units (x, y - 5)
(x, y) Right 3 units (x + 3, y)
(x, y) Left 3 units (x - 3, y)
(x, y) Up m units (x , y + m)
(x, y) Right k units (x + k, y)
(x, y) Down w units then left b units (x - b, y - w)
(x, y) Left k units then up h units (x - k, y + h)
MATHS7TL002 | Mathematics | Number and Algebra Activity | Moving Points 8 © Department of Education WA 2015
3. Create a table with points to be moved and movements listed (similar to the previous table)
and ask another student to determine the new coordinates.
Point to be „moved‟ Movement New coordinates
Various answers
4. Write a summary of what happens to the coordinates of a point when it is “moved”
vertically or horizontally.
If a point is moved horizontally then the first number of the coordinates, the x-value,
increases by the number of units moved if the movement is to the right, and decreases if the
movement is to the left.
If a point is moved vertically then the second number of the coordinates, the y-value
increases by the number of units moved if the movement is up, and decreases if the
movement is down.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Moving Points 9 © Department of Education WA 2015
STUDENT COPY MOVING POINTS
Activity 1
1. Review plotting points by plotting the following points and labelling the points with the
letters provided.
A (2,3) B (1,0) C (7,10) D (-4,6) E (-9,0) F (-5,-5) G (-7,8) H (0,-8)
2. Plot any points J K L M N P and ask another student to identify them.
In the following activities you are asked to “move” a point. The point doesn‟t really move as it
is a position. The new point is positioned away from the original point. When asked to move
a point, label the new point with the same letter but add a dash; e.g., A „moves‟ to A‟.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Moving Points 10 © Department of Education WA 2015
Activity 2
1. Enter the coordinates of the points A to F into the table below. Then “move” the points as
instructed in the table and label the “moved” points.
,
Point Coordinates of given point
„Move‟ the point ... Coordinates of „moved‟ point
A Left 4 units
B Left 5 units
C Left 7 units
D Right 6 units
E Right 2 units
F Right 3 units
2. When a point is moved left or right:
(a) Does the x-value of the point change?
(b) Does the y-value of the point change?
MATHS7TL002 | Mathematics | Number and Algebra Activity | Moving Points 11 © Department of Education WA 2015
Activity 3
1. Enter the coordinates of the points G to L into the table below. Then “move” the points as
instructed in the table and label the “moved” points.
,
Point Coordinates of given point
„Move‟ the point ... Coordinates of „moved‟ point
G Up 6 units
H Up 3 units
I Up 7 units
J Down 4 units
K Down 3 units
L Down 7 units
2. When a point is „moved‟ up or down:
(a) Does the x-value of the point change?
(b) Does the y-value of the point change?
MATHS7TL002 | Mathematics | Number and Algebra Activity | Moving Points 12 © Department of Education WA 2015
Activity 4
Enter the coordinates of the points M to T into the table below. Then “move” the points as
instructed in the table and label the “moved” points.
Point Coordinates of given point
„Move‟ the point ... Coordinates of „moved‟ point
M Up 3 units then left 2 units
N Up 3 units then right 2 units
O Down 5 units then left 5 units
P Left 9 units then down 9 units
Q Left 1 unit then up 10 units
R Right 4 units then down 4 units
S Up 4 units then right 4 units
T Down 9 units then right 1 unit
MATHS7TL002 | Mathematics | Number and Algebra Activity | Moving Points 13 © Department of Education WA 2015
Activity 5
1. Points W, X, Y and Z were moved to W‟, X‟, Y‟ and Z‟. The movements are given below. Give
the coordinates of the original points W, X, Y and Z.
(a) W‟ is at (4, 2) after W was moved 4 units left. Where was W to start with?
(b) X‟ is at (1, 0) after X was moved 3 units up. Where was X to start with?
(c) Y‟ is at (-6, 7) after Y was moved 1 unit right and 1 unit down.
Where was Y to start with?
(d) Z‟ is at (3, -6) after Z was moved 3 units up and 6 units left. Where was Z to start with?
2. State the coordinates of the new points that will be formed from the following movements.
Point to be „moved‟ Movement New coordinates
(20, 20) Right 4 units
(-10, 30) Left 6 units
(-20, 0) Up 7 units
(7, 50) Down 10 units
(20, 25) Left 20 units, down 25 units
(12, 16) Up 2 units, right 3 units
(100, 100) Down 50 units, right 50 units
(20, 50) Right 10 units, up 5 units
(x, y) Up 6 units
(x, y) Down 5 units
(x, y) Right 3 units
(x, y) Left 3 units
(x, y) Up m units
(x, y) Right k units
(x, y) Down w units then left b units
(x, y) Left k units then up h units
MATHS7TL002 | Mathematics | Number and Algebra Activity | Moving Points 14 © Department of Education WA 2015
3. Create a table with points to be moved and movements listed (similar to the previous table)
and ask another student to determine the new coordinates.
Point to be „moved‟ Movement New coordinates
4. Write a summary of what happens to the coordinates of a point when it is “moved”
vertically or horizontally.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Fraction Operations 2 © Department of Education WA 2015
TASK 37: FRACTION OPERATIONS
Overview
The focus of this task is on the addition and subtraction of fractions with unrelated
denominators. Students should attempt to complete these operations using mental
arithmetic.
Students will not need any special equipment
Relevant content descriptions from the Western Australian Curriculum
Solve problems involving addition and subtraction of fractions, including those with
unrelated denominators (ACMNA153)
Compare fractions using equivalence. Locate and represent positive and negative
fractions and mixed numbers on a number line (ACMNA152)
Students can demonstrate
fluency when they
o represent fractions in various ways
o determine fraction addition using different models
understanding when they
o can develop a model to simulate fraction addition
o identify fractions to create matching number sentences
reasoning when they
o explain their use of models for fraction addition
o determine a strategy to locate a fraction halfway between two others
problem solving when they
o determine the processes needed to answer the questions in Activity 1
MATHS7TL002 | Mathematics | Number and Algebra Activity | Fraction Operations 3 © Department of Education WA 2015
FRACTION OPERATIONS Solutions and Notes for Teachers
Activity 1
1. Three family-sized pizzas were shared by five people and the table below shows the
number of pieces eaten by each person and the number left. For any particular pizza all the
pieces were the same size but each pizza was cut into a different number of pieces.
Person Alf Buzz Cody Dani Elly Leftover
Pizza 1 2 3 1 4 2 0
Pizza 2 1 2 2 1 2 0
Pizza 3 1 1 1 0 2 1
This table is reproduced below as a tool to assist with answering the questions. What fraction
of each pizza did each person eat? See table below.
Person Alf Buzz Cody Dani Elly Leftover
Pizza 1
Pizza 2
Pizza 3
2. Buzz and Elly ate the same number or pieces, and so did Alf and Cody. Did each pair eat
the same amount of pizza? Explain your answer mathematically?
Buzz ate = + + =
Elly ate 2 2 2 4 6 8 18
12 8 6 24 24 24 24 Elly ate more than Buzz
Alf ate 2 1 1 4 3 4 11
12 8 6 24 24 24 24
Cody ate 1 2 1 2 6 4 12
12 8 6 24 24 24 24 Cody ate more than Alf
3. What fraction of the three pizzas was eaten by Dani?
Dani ate 4 1 8 3 11
12 8 24 24 24 of 1 pizza. This is equivalent to
11
72 of 3 pizzas.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Fraction Operations 4 © Department of Education WA 2015
Activity 2
I have five fraction tiles as shown below.
Select any of these fractions to create true statements using the number sentences below.
A. + + < 1
B. + + > 1.5
C. _ + < 0
D.
+ _ =
E. _ _ =
55
5
5
5
7
5
MATHS7TL002 | Mathematics | Number and Algebra Activity | Fraction Operations 5 © Department of Education WA 2015
Activity 3
1. Consider the following pairs of fractions.
Determine the fraction that is halfway between each member of the pairs. For each pair, use
a number line to show the positions of the three fractions.
A. and 3
B. and
C. and
D. and
2. Describe a strategy that could be used to determine the fraction halfway between any pair
of fractions.
(1) Add the two fractions and divide the result by 2,
OR
(2) Express the original fractions in an equivalent form – both with the same denominator.
If the numerators add to an even number then halve that sum and express the answer as a
fraction with the denominator.
If the numerators add to an odd number; e.g. 5, then double the denominator. Example:
1 9 2 9
5 10 10 10
2 9 11 20
11
20
and and
so double the denominator is
is
MATHS7TL002 | Mathematics | Number and Algebra Activity | Fraction Operations 6 © Department of Education WA 2015
Activity 4
Use the models provided to show that
+
Explain your use of these models.
First model:
Divide the model vertically into five even columns. Colour 2 columns which is 2 out of 5.
Divide the model horizontally into 4 even rows. Colour in two rows – or the equivalent of two
rows – you cannot colour in parts that are already coloured. Count up the number of parts
coloured. Consider how many parts are coloured (13) and how many parts altogether (20).
So the answer is 13 out of 20 or
Second model:
Colour in 2 out of every 5 smiley faces. Now colour in 1 for every 4 of all of them but do not
colour in one already coloured. So the answer is 13 out of 20 or
MATHS7TL002 | Mathematics | Number and Algebra Activity | Fraction Operations 7 © Department of Education WA 2015
STUDENT COPY FRACTION OPERATIONS
Activity 1
1. Three family-sized pizzas were shared by five people and the table below shows the
number of pieces eaten by each person and the number left. For any particular pizza all the
pieces were the same size but each pizza was cut into a different number of pieces.
Person Alf Buzz Cody Dani Elly Leftover
Pizza 1 2 3 1 4 2 0
Pizza 2 1 2 2 1 2 0
Pizza 3 1 1 1 0 2 1
This table is reproduced below as a tool to assist with answering the questions. What fraction
of each pizza did each person eat?
Person Alf Buzz Cody Dani Elly Leftover
Pizza 1
Pizza 2
Pizza 3
2. Buzz and Elly ate the same number or pieces, and so did Alf and Cody. Did each pair eat
the same amount of pizza? Explain your answer mathematically?
3. What fraction of the three pizzas was eaten by Dani?
MATHS7TL002 | Mathematics | Number and Algebra Activity | Fraction Operations 8 © Department of Education WA 2015
Activity 2
I have five fraction tiles as shown below.
Select any of these fractions to create true statements using the number sentences below.
A. + + < 1
B. + + > 1.5
C. _ + < 0
D.
+ _ =
E. _ _ =
55
7
MATHS7TL002 | Mathematics | Number and Algebra Activity | Fraction Operations 9 © Department of Education WA 2015
Activity 3
1. Consider the following pairs of fractions.
Determine the fraction that is halfway between each member of the pairs. For each pair, use
a number line to show the positions of the three fractions.
A. and 3
B. and
C. and
D. and
2. Describe a strategy that could be used to determine the fraction halfway between any pair
of fractions.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Fraction Operations 10 © Department of Education WA 2015
Activity 4
Use the models provided to show that + Explain your use of these models.
First model:
Second model:
MATHS7TL002 | Mathematics | Number and Algebra Activity | Fraction Action 2 © Department of Education WA 2015
TASK 38: FRACTION ACTION
Overview
This task involves the use of different activities to foster a deeper understanding of fractions
and to provide opportunities to represent fractions in various ways. Use of mental arithmetic
to add and subtract fractions is expected for all questions except the final one in Activity 4.
Students will need
calculators
access to the internet
Relevant content descriptions from the Western Australian Curriculum
Compare fractions using equivalence. Locate and represent positive and negative
fractions and mixed numbers on a number line (ACMNA152)
Solve problems involving addition and subtraction of fractions, including those with
unrelated denominators (ACMNA153)
Express one quantity as a fraction of another with and without the use of digital
technologies (ACMNA155)
Recognise and solve problems involving simple ratios (ACMNA173)
Students can demonstrate
fluency when they
o represent the given fraction in various ways in Activity 1
understanding when they
o develop a fraction wall showing equivalent fractions
reasoning when they
o apply their understanding of ratios
problem solving when they
o determine processes to answer questions in Activity 4
MATHS7TL002 | Mathematics | Number and Algebra Activity | Fraction Action 3 © Department of Education WA 2015
FRACTION ACTION Solutions and Notes for Teachers
Activity 1
The table below shows multiple representations for two thirds. Create a similar table for nine
fifths. Examples shown on next page.
One third plus one third 3
2Two thirds
Four thirds take away two thirds
One third times two
Two oranges pizzas shared
equally between 3 people
3
1
3
1 3
211
3
2
3
4
2 3
1One third of
two 2 ÷ 3
3
1 2 3
11
3
2of 1
3
21
3
11 -
3
2
6
4
3
1
3
1
3
1
3
1
3
1
3
1
One minus one third 30
20
0 1 2 3
MATHS7TL002 | Mathematics | Number and Algebra Activity | Fraction Action 4 © Department of Education WA 2015
10 1
5 5
19
5
Nine fifths
Two minus one fifth
One plus four fifths
Nine oranges shared equally
between 5 people
1.8 8 1
5 5
2 3 4
5 5 5
3 3 3
5 5 5
9
5
18
10
1
9
5÷ 1
51
9
182
5
9
5of 1
33
5
20 11
5 5 Nine divided by 5
1
180%
MATHS7TL002 | Mathematics | Number and Algebra Activity | Fraction Action 5 © Department of Education WA 2015
Activity 2
1. Money has been shared out between two people, but not equally. Determine the ratio of
Ted’s share to Pat’s share as well as the fraction of the whole amount that each person
received. Complete the table provided.
Ted’s amount
Pat’s amount
Ratio Ted: Pat
Total money
Ted’s fraction
Pat’s fraction
$10 $20 1: 2 $30
$40 $10 4:1 $50
$35 $350 1:10 $385
$80 $10 8:1 $90
$75 $5 15:1 $80
$22 $44 1:2 $66
$70 $30 7:3 $100
2. Elia and Ali are sharing different amounts of money according to the ratios given.
Complete the table provided.
Amount of money to share
Ratio Elia: Ali
Elia’s amount Ali’s amount Elia’s
fraction Ali’s
fraction
$100 2:3 $40 $60
$200 1:1 $100 $100
$500 3:7 $150 $350
$350 5:2 $250 $100
$400 5:3 $250 $150
$800 9:1 $720 $80
$135 3:2 $81 $54
3. If the ratio was not known and was written as a : b, what fraction of the money would each
person get?
anda b
a b a b
MATHS7TL002 | Mathematics | Number and Algebra Activity | Fraction Action 6 © Department of Education WA 2015
Activity 3
Design and create a “fraction wall” to show the relationship between the following fractions:
Twentieths, tenths, quarters, eighths, fifths, halves.
1
Activity 4
1. A truck is one-third full of sand. When 45 kg is added it is half full.
How much more sand can be placed in the truck?
1 145
2 3
145
6
45 6 1 270
truckload truckload
truckload
truckload kg
2. A bus starts from the terminal with all seats occupied. At the first stop one third of the
passengers get off and 12 people get on. At the next stop, a half of the new total of
passengers get off and four people get on. There are now 30 passengers on the bus. How many passengers started the trip?
MATHS7TL002 | Mathematics | Number and Algebra Activity | Fraction Action 7 © Department of Education WA 2015
60
1 230 4 (12
2 3
1 2(12
2 3
2(12
3
240
3
of the original number
of the original number
of the original number
of the original number
original number
)
26 = )
52 = )
3. The perimeter of a triangle is 52 cm. All measurements are whole numbers. The shortest
side is half the length of the longest side and the third side is two-thirds of the longest side.
Determine the length of each side.
The longest side must be a multiple of 3 and it must be more than a third of 52. It also must
be even. More than 17, 18 too close, 21 not even. Try 24.
The shortest side would be 12 and the third side must be two-thirds of 24 which is 16.
12 + 16 + 24 = 52 ... good thinking!
4. One Friday afternoon during sport, a quarter of the Year 7 students elected to play
soccer, one-third played frizzball, and that left 40 students who had chosen swimming. How
many students were in Year 7?
1 1 7
3 4 12
5
12
1
12
of Year 7 students is equivalent to 40.
of Year 7 students is equivalent to 8
So 96 students in Year 7
5. A crate half-full of apples has a mass of 130 kg. The same crate had a mass of 90 kg
when it was one-third full of apples. Determine the mass of the empty crate.
Crate + half the apples = 130 kg
Crate + a third of the apples = 90 kg
Difference of 40 kg is mass of
40 kg =
240 kg = mass of the apples
120 kg = mass of half the apples
Crate has a mass of 130 kg – 120 kg = 10 kg
MATHS7TL002 | Mathematics | Number and Algebra Activity | Fraction Action 8 © Department of Education WA 2015
6. When John Isner played Roger Federer at the US Open in 2015 some of their statistics
were as follows:
(i) Federer got 93 of his 133 first serves in and won the point 63 times. When the first
serve was not in he had a second serve (40 second serves) and he won the point 22 times.
(ii) Isner got 65 out of 111 first serves in and won the point 54 times. On the second serve
Federer won the point 22 out 40 times and Isner won the point 33 out of 46 times.
[Data obtained from the US Open tennis official website.]
Compare the statistics of both players and use mathematical arguments to show who won
the highest percentage of points on their serves.
Federer Isner
First serves in = = 0.699 = 70% = 0.586 = 59%
Points won on first serve in = 0.677 = 68% = 0.830 = 83%
Points won on second serve = 0.55 = 55% = 0.717 = 72%
MATHS7TL002 | Mathematics | Number and Algebra Activity | Fraction Action 9 © Department of Education WA 2015
STUDENT COPY FRACTION ACTION
Activity 1
The table below shows multiple representations for two thirds. Create a similar table for nine
fifths.
One third plus one third
Two thirds
Four thirds take away two thirds
One third times two
Two oranges pizzas shared
equally between 3 people
3
211
2
One third of two 2 ÷ 3
2 of 1
-
One minus one third
3
2
3
1
3
1
3
2
3
4
3
1
3
13
11
3
2
3
21
3
11
3
2
6
4
3
1
3
1
3
1
3
1
3
1
3
1
30
20
0 1 2 3
MATHS7TL002 | Mathematics | Number and Algebra Activity | Fraction Action 10 © Department of Education WA 2015
Nine fifths
MATHS7TL002 | Mathematics | Number and Algebra Activity | Fraction Action 11 © Department of Education WA 2015
Activity 2
1. Money has been shared out between two people, but not equally. Determine the ratio of
Ted’s share to Pat’s share as well as the fraction of the whole amount that each person
received. Complete the table provided.
Ted’s amount
Pat’s amount
Ratio Ted: Pat
Total money
Ted’s fraction
Pat’s fraction
$10 $20 1: 2
$40 $10
$35 $350
$80 $10
$75 $5
$22 $44
$70 $30
2. Elia and Ali are sharing different amounts of money according to the ratios given.
Complete the table provided.
Amount of money to share
Ratio Elia: Ali
Elia’s amount Ali’s amount Elia’s
fraction Ali’s
fraction
$100 2:3 $40 $60
$200 1:1
$500 3:7
$350 5:2
$400 5:3
$800 9:1
$135 3:2
3. If the ratio was not known and was written as a : b, what fraction of the money would each
person get?
MATHS7TL002 | Mathematics | Number and Algebra Activity | Fraction Action 12 © Department of Education WA 2015
Activity 3
Design and create a “fraction wall” to show the relationship between the following fractions:
Twentieths, tenths, quarters, eighths, fifths, halves.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Fraction Action 13 © Department of Education WA 2015
Activity 4
1. A truck is one-third full of sand. When 45 kg is added it is half full.
How much more sand can be placed in the truck?
2. A bus starts from the terminal with all seats occupied. At the first stop one third of the
passengers get off and 12 people get on. At the next stop, a half of the new total of
passengers get off and four people get on. There are now 30 passengers on the bus. How many passengers started the trip?
3. The perimeter of a triangle is 52 cm. All measurements are whole numbers. The shortest
side is half the length of the longest side and the third side is two-thirds of the longest side.
Determine the length of each side.
4. One Friday afternoon during sport, half of the Year 7 students elected play soccer, one-
third played frizzball, and that left 40 students who had chosen swimming. How many
students in Year 7?
MATHS7TL002 | Mathematics | Number and Algebra Activity | Fraction Action 14 © Department of Education WA 2015
5. A crate half-full of apples has a mass of 130 kg. The same crate had a mass of 90 kg
when it was one-third full of apples. Determine the mass of the empty crate.
6. When John Isner played Roger Federer at the US Open in 2015 some of their statistics
were as follows:
(i) Federer got 93 of his 133 first serves in and won the point 63 times. When the first
serve was not in he had a second serve (40 second serves) and he won the point 22 times.
(ii) Isner got 65 out of 111 first serves in and won the point 54 times. On the second serve
Federer won the point 22 out 40 times and Isner won the point 33 out of 46 times.
[Data obtained from the US Open tennis official website.]
Compare the statistics of both players and use mathematical arguments to show who won
the highest percentage of points on their first and second serves.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Graphing Relationships 2 © Department of Education WA 2015
TASK 40: GRAPHING RELATIONSHIPS
Overview
For this task students may need further support if they have not previously used letters to
represent variables or if they are lacking in experience with line graphs. For all of these tasks
there is a simple linear relationship between the variables. Each situation can be described
as a rate, though the term is not used in the student activities.
Students will not need any special equipment
Relevant content descriptions from the Western Australian Curriculum
Investigate, interpret and analyse graphs from authentic data (ACMNA180)
Introduce the concept of variables as a way of representing numbers using letters
(ACMNA175)
Students can demonstrate
fluency when they
o complete tables by identifying and continuing patterns
understanding when they
o plot data points
reasoning when they
o interpret graphs of authentic data
o describe the features common to the graphs and tables in this task
problem solving when they
o create formulae to represent relationships between variables
MATHS7TL002 | Mathematics | Number and Algebra Activity | Graphing Relationships 3 © Department of Education WA 2015
GRAPHING RELATIONSHIPS Solutions and Notes for Teachers
Situation 1
Susie works at the local pharmacy store where she is paid $20 per hour.
Examine the table and related graph showing Susie's pay.
Number of hours 0 1 2 3 4 5 6 7
Pay 0 $20 $40 $60 $80 $100 $120 $140
1. How much would Susie get paid for 8 hours’ work? $160
Add this point to the graph above.
2. Does it make sense to join the points? (Can you read the values in between the points?)
Use an example in your explanation.
Yes, you can read values between points. Working for 1.5 hours, Susie could get $30 pay
3. Describe the rule linking number of hours worked and amount paid in -
(i) words
Susie’s pay = number of hours x 20
(ii) symbols (Use P for pay and h for hours worked)
P = 20 x h
0
20
40
60
80
100
120
140
160
180
0 1 2 3 4 5 6 7 8 9
Pay ($)
HOURS WORKED
Susie's Pay
MATHS7TL002 | Mathematics | Number and Algebra Activity | Graphing Relationships 4 © Department of Education WA 2015
Situation 2
Jon is graphing the cost of making apple cakes.
The table and graph show some of these costs.
Number of cakes
0 1 2 3 4 5 6 7
Cost ($) 0 $6 $12 $18 $24 $30 $36 $42
1. Complete both the graph and the table, given that there is a fixed cost per cake.
2. Does it make sense to join the points? (Can you read the values in between the points?)
Use an example in your explanation.
It does not make sense to join the points because you do not make half a cake. There is not
value for 1.5 cakes.
3. Describe the rule linking number of cakes made and the cost using -
(i) words
Cost = number of cakes x 6
(ii) symbols (Use P for cost and c for number of cakes made)
P = c x 6
0
10
20
30
40
50
60
0 1 2 3 4 5 6 7 8 9 10
Cost ($)
Number of cakes
Costing cakes
MATHS7TL002 | Mathematics | Number and Algebra Activity | Graphing Relationships 5 © Department of Education WA 2015
Situation 3
1. There are five tables below showing the costs of different fruits. For each one -
complete the table provided;
plot the graph showing the relationship between number of kilograms and cost;
write a rule linking number of kilograms and cost; and
check that each graph is labelled and points joined if appropriate.
Pears
Number of
kilograms (d) 1 2 3 4 5 6 7 8 9
Total Cost (C) $5 $10 $15 $20 $25 $30 $45 $40 $45
Cost per kg $5 Rule C = d x 5 OR C = 5 d
Peaches
Number of
kilograms (d) 1 2 3 4 5 6 7 8
Total Cost (C) $5.50 $11 $16.50 $22 $27.50 $33 $38.50 $44
Cost per kg $5.50 Rule C = d x 5.5 OR C = 5.5 d
Grapes
Number of
kilograms (d) 1 2 3 4 5 6 7 8 9 10
Total Cost (C) $6 $12 $18 $24 $30 $36 $42 $48 $54 $60
Cost per kg $6 Rule C = d x 6 OR C = 6 d
Nectarines
Number of
kilograms (d) 1 2 3 4 5 6 7 8 9
Total Cost (C) $8 $16 $24 $32 $40 $48 $56 $64 $72
Cost per kg $8 Rule C = d x 8 OR C = 8 d
Oranges
Number of
kilograms (d) 1 2 3 4 5 6 7 8 9
Total Cost (C) $3 $6 $9 $12 $15 $18 $21 $24 $27
Cost per kg $3 Rule C = d x 3 OR C = 3 d
Note: These graphs may be presented by students as connected points.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Graphing Relationships 6 © Department of Education WA 2015
2. Write the fruits in order of cost per kilogram.
A: Oranges B: Pears C: Peaches D: Grapes E: Nectarines
3. Write the names of the graphs in order of steepness.
A: Oranges B: Pears C: Peaches D: Grapes E: Nectarines
4. Comment on your findings.
The more the fruit costs per kilogram, the steeper the graph.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Graphing Relationships 7 © Department of Education WA 2015
Situation 4
For the following vegetable prices
Write the rule linking mass (w) and cost (C)
Create a table of costs.
1. Carrots cost $1.50 per kilogram
Rule: Cost = number of kilogram x $1.50 C = 1.50 x w
Number of
kilograms (w) 0 1 2 3 4 5 6 7 8
Cost (C) 0 $1.50 $3 $4.50 $6 $7.50 $9 $10.50 $12
2. Tomatoes cost $3.85 per kilogram
Rule: Cost = number of kilogram x $3.85 C = 3.85 x w
Number of
kilograms (w) 0 1 2 3 4 5 6 7 8
Cost (C) 0 $3.85 $7.70 $11.55 $15.40 $19.25 $23.10 $26.95 $$30.80
3. Potatoes cost $2.50 per kilogram
Rule: Cost = number of kilogram x $2.50 C = 2.50 x w
Number of
kilograms (w) 0 1 2 3 4 5 6 7 8
Cost (C) 0 $2.50 $5 $7.50 $10 $12.50 $15 $17.50 $20
4. Onions cost $1.20 per kilogram
Rule: Cost = number of kilogram x $1.20 C = 1.20 x w
Number of
kilograms (w) 0 1 2 3 4 5 6 7 8
Cost (C) 0 $1.20 $2.40 $3.60 $4.80 $6.00 $7.20 $8.40 $9.60
5. Plot the costs of these vegetables (up to 10 kg) on the grid on the next page.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Graphing Relationships 8 © Department of Education WA 2015
Reflection
Summarise the features of the tables and graphs that have been used in this task.
All of these graphs are straight lines.
The more the food costs per kilogram the steeper the line for its graph.
All graphs have the point (0,0) because it does not cost money to buy “nothing”.
It makes sense to join the points because you can read values in between the given points.
The changing mass is plotted on the horizontal axis.
The changing total cost is plotted on the vertical axis.
The tables all have the cost increasing by the same amount (as does the mass of the food
considered) which is the cost per kilogram.
The rules are all similar: Total cost = cost per kilogram x number of kilograms.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Graphing Relationships 9 © Department of Education WA 2015
STUDENT COPY GRAPHING RELATIONSHIPS
Situation 1
Susie works at the local pharmacy store where she is paid $20 per hour.
Examine the table and related graph showing Susie's pay.
Number of hours 0 1 2 3 4 5 6 7
Pay 0 $20 $40 $60 $80 $100 $120 $140
1. How much would Susie get paid for 8 hours’ work?
Add this point to the graph above.
2. Does it make sense to join the points? Can you read the values in between the points?
Use an example in your explanation.
3. Describe the rule linking number of hours worked and amount paid using -
(i) words
(ii) symbols (Use P for pay and h for hours worked).
0
20
40
60
80
100
120
140
160
0 1 2 3 4 5 6 7 8
Pay ($)
HOURS WORKED
Susie's Pay
MATHS7TL002 | Mathematics | Number and Algebra Activity | Graphing Relationships 10 © Department of Education WA 2015
Situation 2
Jon is graphing the cost of making apple cakes.
The table and graph show some of these costs.
Number of cakes
0 1 2 3 4 5 6 7
Cost ($) $6 $12 $18 $24
1. Complete both the graph and the table, given that there is a fixed cost per cake.
2. Does it make sense to join the points? (Can you read the values in between the points?)
Use an example in your explanation.
3. Describe the rule linking number of cakes made and the cost using -
(i) words
(ii) symbols (Use P for cost and c for number of cakes made).
0
10
20
30
40
50
60
0 1 2 3 4 5 6 7 8 9 10
Cost ($)
Number of cakes
Costing cakes
MATHS7TL002 | Mathematics | Number and Algebra Activity | Graphing Relationships 11 © Department of Education WA 2015
Situation 3
1. There are five tables below showing the costs of different fruits. For each one -
complete the table provided;
plot the graph showing the relationship between number of kilograms and cost;
write a rule linking number of kilograms and cost; and
check that each graph is labelled and points joined if appropriate.
Pears
Number of
kilograms (d) 1 2 3 4 5
Total Cost (C) $5 $10 $15 $20 $25
Cost per kg Rule
Peaches
Number of
kilograms (d) 1 2 3 4 5
Total Cost (C) $5.50 $11 $16.50 $22 $27.50
Cost per kg Rule
Grapes
Number of
kilograms (d) 1 2 3 4 5
Total Cost (C) $6 $12 $18
Cost per kg Rule
Nectarines
Number of
kilograms (d) 1 2 3 4 5
Total Cost (C) $8 $16 $24 $32 $40
Cost per kg Rule
Oranges
Number of
kilograms (d) 1 2 3 4 5
Total Cost (C) $3 $6 $9 $12 $15
Cost per kg Rule
MATHS7TL002 | Mathematics | Number and Algebra Activity | Graphing Relationships 12 © Department of Education WA 2015
2. Write the fruits in order of cost per kilogram.
3. Write the names of the graphs in order of steepness.
4. Comment on your findings.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Graphing Relationships 13 © Department of Education WA 2015
Situation 4
For the following vegetable prices
Write the rule linking mass (w) and cost (C)
Create a table of costs.
1. Carrots cost $1.50 per kilogram
Rule:
2. Tomatoes cost $3.85 per kilogram
Rule:
3. Potatoes cost $2.50 per kilogram
Rule:
4. Onions cost $1.20 per kilogram
Rule:
5. Plot the costs of these vegetables (up to 10 kg) on the grid on the next page.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Graphing Relationships 14 © Department of Education WA 2015
Reflection
Summarise the features of the tables and graphs that have been used in this task.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Consecutive Numbers 2 © Department of Education WA 2015
TASK 101: CONSECUTIVE NUMBERS
Overview
In this task, students will investigate the addition and subtraction of consecutive numbers.
The activities provide opportunities for students to make connections between consecutive
numbers and the associative law. Students are allowed to choose appropriate methods and
apply their existing strategies to seek solutions. Explaining connections or patterns will help
in the promotion of reasoning mathematically.
No special equipment required
Relevant content descriptions from the Western Australian Curriculum
Apply the associative, commutative and distributive laws to aid mental and written
computation (ACMNA151)
Compare, order, add and subtract integers (ACMNA280)
Students can demonstrate
fluency when they
o identify a way of recording the information
o use their process to record the information and calculate solutions
understanding when they
o identify any connections or patterns in their results
reasoning when they
o show how to identify these connections or patterns
problem solving when they
o explain how they know they have all of the combinations
MATHS7TL002 | Mathematics | Number and Algebra Activity | Consecutive Numbers 3 © Department of Education WA 2015
CONSECUTIVE NUMBERS Solutions and Notes for Teachers
Activity 1
1. Choose any four consecutive numbers between 1 and 9, for example, 1, 2, 3, and 4.
1, 2, 3, 4
2. You want to find all of the different ways of ADDING and SUBTRACTING all of your
chosen numbers. (NOTE: your chosen numbers must always be in numerical order)
How can you do this in a logical way? Can you think of more than one way?
Write a column of each number, with spaces.
Start with all ‘+’ signs.
Change one sign to a ‘-‘, starting from right and moving it left until all signs have been
changed.
Change two signs to a ‘-‘, starting from the right and moving left until all possible
combinations have been changed.
Lastly, all ‘-‘ signs.
3. Using one of your methods from above, record all of the different combinations.
Calculate each answer.
You may want the students to number the operations to help when explaining a rule.
a. 1 + 2 + 3 + 4 = 10
b. 1 + 2 + 3 – 4 = 2
c. 1 + 2 – 3 + 4 = 4
d. 1 – 2 + 3 + 4 = 6
e. 1 + 2 – 3 – 4 = -4
f. 1 – 2 – 3 + 4 = 0
g. 1 – 2 + 3 – 4 = -2
h. 1 – 2 – 3 – 4 = -8
4. How do you know you have all of the combinations?
8 combinations
1 x all +
1 x all –
3! = 6
1 + 1 + 6 = 8
Students may provide a different answer.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Consecutive Numbers 4 © Department of Education WA 2015
Activity 2:
1. Try choosing a different set of numbers and apply your process from above.
Ensure the students calculate the solutions in the same order as in Activity 1.
Ask students show how many combinations should there are.
a. 2 + 3 + 4 + 5 = 14
b. 2 + 3 + 4 – 5 = 4
c. 2 + 3 – 4 + 5 = 6
d. 2 – 3 + 4 + 5 = 8
e. 2 + 3 – 4 – 5 = -4
f. 2 – 3 – 4 + 5 = 0
g. 2 – 3 + 4 – 5 = -2
h. 2 – 3 – 4 – 5 = -10
2. Are there any patterns or connections between this set of solutions and the previous
set of solutions?
The answers to e, f and g are the same in both sets.
They are all even numbers.
Three negative numbers.
Four positive numbers.
Zero in each set.
Any other reasonable connection.
3. Develop a rule to help you find a set of solutions without doing the calculations.
a. The sum of the two middle numbers multiplied by 2.
b. The first number multiplied by 2.
c. The second number multiplied by 2.
d. The third number multiplied by 2.
e. -4
f. 0
g. -2
h. The negative fourth number multiplied by 2.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Consecutive Numbers 5 © Department of Education WA 2015
Activity 3: Extension
1. Investigate how changing the consecutive numbers to a descending order will
affect the results.
Answers will vary
2. Can your previous rule be applied to the process with only three consecutive
numbers?
Answers will vary
3. Can your previous rule be applied to the process with five consecutive numbers?
Answers will vary
4. Consider what might happen if you changed the sign in front of your first number
also.
Answers will vary
MATHS7TL002 | Mathematics | Number and Algebra Activity | Consecutive Numbers 6 © Department of Education WA 2015
STUDENT COPY CONSECUTIVE NUMBERS
Activity 1
2. Choose any four consecutive numbers between 1 and 9, for example, 1, 2, 3, and 4.
3. You want to find all of the different ways of ADDING and SUBTRACTING all of your
chosen numbers. (NOTE: your chosen numbers must always be in numerical order)
How can you do this in a logical way? Can you think of more than one way?
4. Using one of your methods from above, record all of the different combinations.
Calculate each answer.
5. How do you know you have all of the combinations?
MATHS7TL002 | Mathematics | Number and Algebra Activity | Consecutive Numbers 7 © Department of Education WA 2015
Activity 2:
1. Try choosing a different set of numbers and apply your process from above.
2. Are there any patterns or connections between this set of solutions and the previous
set of solutions?
3. Develop a rule to help you find a set of solutions without doing the calculations.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Consecutive Numbers 8 © Department of Education WA 2015
Activity 3: Extension
1. Investigate how changing the consecutive numbers to a descending order will
affect the results.
2. Can your previous rule be applied to the process with only three consecutive
numbers?
3. Can your previous rule be applied to the process with five consecutive numbers?
4. Consider what might happen if you changed the sign in front of your first number
also.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Large Elevens 2 © Department of Education WA 2015
TASK 102: LARGE ELEVENS
Overview
In this task, students will investigate how to mentally multiply 2- and 3-digit numbers by 11.
Students will apply the number laws to develop and test rules to aid mental computation.
They are then required to deduce and justify strategies used and reached when they adapt
the known to the unknown.
Students will need
calculators
Relevant content descriptors from the Western Australian Curriculum
Apply the associative, commutative and distributive laws to aid mental and writtencomputation (ACMNA151)
Students can demonstrate
fluency when they
o use a calculator to check solutions
understanding when they
o identify the connection between problems and solutions
reasoning when they
o develop a rule for multiplying 2-digit numbers by 11
o investigate whether their rule always works
problem solving when they
o investigate whether they can apply their rule to 3-digit numbers
o use previous strategies to develop a rule for multiplying 3-digit numbers by 11
MATHS7TL002 | Mathematics | Number and Algebra Activity | Large Elevens 3 © Department of Education WA 2015
LARGE ELEVENS Solutions and Notes for Teachers
Activity 1
Multiplying large numbers without a calculator can be difficult but there are some tricks that
can help us!!
Consider the following 2-digit numbers multiplied by 11:
21 x 11 = 231
25 x 11 = 275
53 x 11 = 583
62 x 11 = 682
1. Using a calculator, check that the solutions above are correct.
All solutions above are correct.
2. Take the first problem above, write it out in the space below and examine it carefully.
Explain the connection between the problem and its solution.
HINT: Look closely at the digits.
21 x 11 = 231
When multiplying 21 by 11, I place the 2 as the first digit of the answer and the 1 as the last
digit of the answer. The middle digit is the sum of the 2 and 1, which is 3. So the answer is
231.
3. Is this connection between the problem and its solution the same for the other three
problems? Show your reasoning below.
25 x 11 = 275. 2 is the first digit, 5 is the last digit, 2+5=7, which is the middle digit.
53 x 11 = 583. 5 is the first digit, 3 is the last digit, 5+3=8, which is the middle digit.
62 x 11 = 682. 6 is the first digit, 2 is the last digit, 6+2=8, which is the middle digit.
4. Develop a rule for multiplying 2-digit numbers by 11.
When multiplying a 2-digit number by 11, the first digit of the number is the first digit of the
answer. The second digit of the number is the last digit of the answer. The sum of the two
digits becomes the middle digit of the answer.
5. Explain why this rule works in this instance.
The sum of the two digits is less than 10 in these examples.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Large Elevens 4 © Department of Education WA 2015
6. Show how this rule does not always work for 2-digit numbers.
65 x 11 = 715
5 is the last digit but 6 is not the first digit and the sum of the two digits is not the middle digit.
7. Alter your rule from above so that it works for all 2-digit numbers.
When multiplying a 2-digit number by 11, if the sum of the two digits is less than 10, the first
digit of the number is the first digit of the answer. The second digit of the number is the last
digit of the answer. The sum of the two digits becomes the middle digit of the answer.
When multiplying a 2-digit number by 11, if the sum of the two digits is greater than 10, the
second digit of the number is the last digit of the answer. Using the sum of the two digits, the
unit value becomes the middle digit of the answer and the tens value added to the first digit
becomes the first digit of the answer.
Activity 2
Using your work from Activity 1, investigate whether the same rule works for multiplying 3-
digit numbers by 11. If it doesn’t, can you develop a different rule?
Answers will vary.
110 x 11 = 1210
121 x 11 = 1331
123 x 11 = 1353
142 x 11 = 1562
153 x 11 = 1683
1st digit of answer – 1st digit of number
2nd digit of answer – 2nd digit of number +1
3rd digit of answer – sum of the 2nd and 3rd digits of number
4th digit of answer – last digit of number
159 x 11 = 1749
185 x 11 = 2035
247 x 11 = 2717
528 x 11 = 5808
275 x 11 = 3025
MATHS7TL002 | Mathematics | Number and Algebra Activity | Large Elevens 5 © Department of Education WA 2015
4th digit of answer – last digit of number.
3rd digit of answer – unit value of the sum of the 2nd and 3rd digits of number (‘carry’ the tens).
2nd digit of answer – unit value of the sum of the 1st and 2nd digits of number plus tens value
carried over.
1st digit of answer – 1st digit of number plus tens value carried over.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Large Elevens 6 © Department of Education WA 2015
STUDENT COPY LARGE ELEVENS
Activity 1
Multiplying large numbers without a calculator can be difficult, but there are some tricks that
can help us!!
Consider the following 2-digit numbers multiplied by 11:
21 x 11 = 231
25 x 11 = 275
53 x 11 = 583
62 x 11 = 682
1. Using a calculator, check that the solutions above are correct.
2. Take the first problem above, write it out in the space below and examine it carefully.
Explain the connection between the problem and its solution.
HINT: Look closely at the digits.
3. Is this connection between the problem and its solution the same for the other three
problems? Show your reasoning below.
4. Develop a rule for multiplying 2-digit numbers by 11.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Large Elevens 7 © Department of Education WA 2015
5. Explain why this rule works in this instance.
6. Show how this rule does not always work for 2-digit numbers.
7. Alter your rule from above so that it works for all 2-digit numbers.
Activity 2
Using your work from Activity 1, investigate whether the same rule works for multiplying 3-
digit numbers by 11. If it doesn’t, can you develop a different rule?
MATHS7TL002 | Mathematics | Number and Algebra Activity | Ideal Fractions 2 © Department of Education WA 2015
TASK 105: IDEAL FRACTIONS
Overview
In this task, students will investigate the notion of ideal fractions. They will need to carry out
the addition and subtraction of fractions fluently in order to obtain the same solution. They
are required to identify the relationship between these fractions and build their
understanding. Through investigation they will attempt to explain how ideal fractions can be
found.
Students will need
calculators
Relevant content descriptors from the Western Australian Curriculum
Solve problems involving addition and subtraction of fractions, including those with unrelated denominators (ACMNA153)
Multiply and divide fractions and decimals using efficient written strategies and digital technologies (ACMNA154)
Students can demonstrate
fluency when they
o calculate the solution to the addition and multiplication of fractions
o describe, using mathematical language, how the operations where performed
understanding when they
o identify a connection between the solutions of the operations
reasoning when they
o describe, using a rule, how to find ideal fractions
problem solving when they
o attempt to find other examples of ideal fractions
o investigate if this relationship is true for any fraction
o look at the outcome from including subtraction and multiplication
o attempt to source other types of ideal operations
MATHS7TL002 | Mathematics | Number and Algebra Activity | Ideal Fractions 3 © Department of Education WA 2015
IDEAL FRACTIONS Solutions and Notes for Teachers
The fractions 5
3 and
5
2 can be described as Ideal Fractions. Below are two operations that
can be performed with these fractions.
5
3+
5
2
5
3´
5
2
Activity 1
1. Find the solution to the above operations.
Both have a solution of 25
6
2. Use mathematical language to describe how you performed each operation.
When adding fractions, the denominators must be the same. If they are not, they
must be changed. To do this you must find a common multiple of the denominators or
the least common multiple (LCM). This in turn will change the numerators.
To multiply fractions, simply multiply the numerators and multiply the denominators.
3. What is the connection between these two operations?
They both have the same solution.
4. Find 4 other examples that have the connection you have described above.
1. 7
3 and
7
4
2. 9
4 and
9
5
3. 11
5 and
11
6
4. 13
6 and
13
7
5. Is this connection true for any fractions?
No, this does not work with any fractions.
6. Investigate what happens if we take subtraction and division into account?
There is no connection with subtraction or division.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Ideal Fractions 4 © Department of Education WA 2015
7. Is there a rule that you could use to describe how to find ideal fractions?
Improper fractions.The numerators are identical odd numbers.The denominators are two consecutive numbers that add to give the value of thenumerators.
Activity 2: Extension
Are there any other types of ideal operations?
Answers will vary.
MATHS7TL002 | Mathematics | Number and Algebra Activity | Ideal Fractions 5 © Department of Education WA 2015
STUDENT COPY IDEAL FRACTIONS
The fractions 5
3 and
5
2 can be described as Ideal Fractions. Below are two operations that
can be performed with these fractions.
5
3+
5
2
5
3´
5
2
Activity 1
1. Find the solution to the above operations.
2. Use mathematical language to describe how you performed each operation.
3. What is the connection between these two operations?
4. Find 4 other examples that have the connection you have described above.
5. Is this connection true for any fractions?
MATHS7TL002 | Mathematics | Number and Algebra Activity | Ideal Fractions 6 © Department of Education WA 2015
6. Investigate what happens if we take subtraction and division into account?
7. Is there a rule that you could use to describe how to find ideal fractions?
Activity 2: Extension
1. Are there any other types of ideal operations?