Patterns and Algebra in Stages 3 and 4 Judy Anderson The University of Sydney...

Patterns and Algebra in Stages 3 and 4

Judy AndersonThe University of Sydney

[email protected]

AIS Conference 2008

AIS Conference 2008 2

Patterns and Algebra in K–10

Expressing generality (x + y = y + x)

Using and interpreting functions ( y = 3x )

Solving equations ( 3x – 1 = 5)


How can we begin?

Some magic …

A story …

An everyday situation …

A puzzle …

An investigation …


The Magic of Numbers

Think of a number between 1 and 64

Write down everything you can about your number

Share this with a friend – can you think of any other interesting things about these two numbers?

Finding your number …


A long time ago … A mathematician invented the game of chess and presented it to

the king. The king was so pleased with the game that he asked the mathematician to name a reward.

The mathematician looked at the chessboard, consisting of 64 squares, and asked for some rice according to the rule:

“One grain of rice on the first square, 2 grains of rice on the second square, 4 grains on the third square, 8 on the fourth and so on … until the last square.”

The king thought that the mathematician was a bit simple, so he readily agreed and sent for the rice from the royal warehouse.

How much rice did the king need?


I noticed the other morning when hanging some clothes on the line that I hang each item separately with two pegs per item.

How many pegs would I need to hang 1 item, 2 items, 5 items, 30 items, and so on? How would you describe this pattern in words?

Everyday number patternsEveryday number patterns


The clothesline – A peggy problem!

When I was a kid in the country, I used two pegs for the first towel, and then one new peg for each additional towel.

How many pegs are required for 1 item, 2 items, 5 items, 30 items, and so on?

Mum used to overlap the towels but she would put an extra peg in the middle of each towel. What does the pattern look like now?


Think of a number …

Add 2

Multiply by 4

Subtract 4

Divide by 4

Subtract the number your started with

What is the answer?

Use Algebra to show how this works?


Another think of a number …

Think of a number between 2 and 10

Multiply it by nine

Add the two digits of your answer

Subtract five

Choose a letter of the alphabet

corresponding to the number …


Calendar PatternsJune 2008

Sun Mon Tues Wed Thurs Fri Sat

1 2 3 4 5 6 7

8 9 10 11 12 13 14

15 16 17 18 19 20 21

22 23 24 25 26 27 28

29 30 31




1 2 3 4 5 6 7

8 9 10 11 12 13 14

15 16 17 18 19 20 21

22 23 24 25 26 27 28

29 30 31


Algebraic thinking …

Patterns and Algebra

Generating and investigating patterns

Observing, predicting and proving

Describing relationships

Making generalisations and proving results

Using and applying algebraic symbolism to solve problems

Working Mathematically

Questioning

Applying Strategies

Communicating

Reasoning

Reflecting


The developmental sequence

Early Stage 1

Stage 1

Stage 2

Stage 3

Stage 4


Building foundations for Algebrain K – 6 Mathematics

Number Patterns – pattern work leads to expressing generality

eg continue the pattern

3, 6, 9, 12, ….

1, 4, 9, 16, …..

Number Relationships –building understandings of number and operations is also very important


Building foundations includes:

Understanding the properties of numbers and operations

Using all numbers, not just whole numbers

Seeing the operations, not just the answers

125 5 = � � x 5 = 125

4 + 1 = � + 2


What’s my rule?


Activity

Generate three different number patterns that include the number 12.

Try to use different kinds of numbers and different operations.

Look at one of your neighbour’s patterns and find the next three numbers in the pattern.


Questions to pose:

What number comes next?

How do you know?

What number will be 10th?

How do you know?

Can you predict the 20th number?

How could you check if you are correct?

Does the number ‘x’ belong to this pattern?


Consider investigating other patterns

16 - 9=

26 - 9 =

36 - 9 =

continue, predict other cases and explain

The aim of pattern work is to:

develop facility and flexibility with numbers

build intuitive understanding of properties.


Stage 3 Build simple geometric patterns involving multiples

Complete a table of values for geometric and number patterns

Number of Triangles 1

Number of Sides 3

Number of Triangles 1 2 3 4 5 6 7

Number of Sides 3 6 9 12 15 - -


Describe a pattern in words in more than one way

(determining a rule to describe the pattern from the table)

Number of Triangles 1 2 3 4 5 6 7

Number of Sides 3 6 9 12 15 - -

‘It looks like the 3 times tables.’

‘You multiply the top number by three to get the

bottom number.’


Construct, verify and complete number sentences involving the four operations with a variety of numbers

completing number sentences:

5 + � = 12 – 4

7 � = 7.7

constructing number sentences to match a word problem

checking solutions and describing strategies


Learning about using inverse operations

I think of a number, multiply it by 3, take away 9 and then divide by 5. The answer is 3. What was the number I thought of?

( ( � 3 - 9) 5 ) = 3

Answer: 8


Example of “backtracking”

I think of a number, multiply it by 3, take away 9 and then divide by 5. The answer is 3. What was the number I thought of?

3

3 -9 5


Stage 4 - Introducing Pronumerals

K is the number of letters in your name - so always stands for a number

K takes multiple values (unknown or variable)


2 (g + 4) = g + 4 + g + 4= g + g + 8

= 2g + 8


2 (n+n+6)

(2 + n) 2 + 6


Describe this pattern in words?

Describe this pattern using symbols?

What is the value of the 10th, 20th, 50th, 100th terms?

Compare your answers.Discuss any differences to clarify who has predicted correctly.

x 1 2 3 4 5 6 7

y 1 5 9 13 17 21 25


Functional Thinking:Students have difficulty connecting the top number with the bottom number

Common errors:“x goes up by 1 and y goes up by 4”

x+1= y + 4

“ y starts at one and you keep adding 4”x = 1 + 4y

Algebra is not a personal shorthand to jot things down

Algebra cannot be used to express all patterns


Students need to develop mature operations

Counting Adding Multiplying


Algebra resources …

Syllabus and Sample Units of Work

DET Patterns and Algebra

RIC pattern books (Paul Swan)

Origo algebra books (Elizabeth Warren)

AAMT

Others???

Patterns and Algebra in Stages 3 and 4 Judy Anderson The University of Sydney...

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