To be a proportional thinker you need to be able to think multiplicatively

How do you describe the change from 2 to 10?

Additive Thinking: Views the change as an addition of 8

Multiplicative Thinking:Views the change as multiplying by 5

Proportional Thinking

A sample of numerical reasoning test questions as used for the NZ

Police recruitment

½ is to 0.5 as 1/5 is to

a. 0.15

b. 0.1

c. 0.2

d. 0.5

1.24 is to 0.62 as 0.54 is to

a. 1.08b.1.8c. 0.27d.0.48

Travelling constantly at 20kmph, how long will it take to travel 50 kilometres?

a. 1 hour 30 minsb. 2 hoursc. 2 hours 30 minsd. 3 hours

If a man weighing 80kg increased his weight by 20%, what would his weight be now?

a. 96kgb. 89kgc. 88kgd. 100kg

Developing Proportional thinking

Fewer than half the adult population can be viewed as proportional thinkers

And unfortunately…. We do not acquire the habits and skills of proportional reasoning simply by getting older.

What is proportional thinking?

Pre-Stage 7

• What fraction and decimal ideas should you already know about before moving to Stage 7

Fractional Key Ideas Pre-Stage 7

Fractions Teaching Key Ideas

1. Use sets as well as regions from early on and connect different representations

Shapes/Regions(Continuous

models)

Sets(Discrete Models)

€

1

41 quarter

Fractions Key Ideas1. Use sets as well as regions from early on and connect different

representations.

2. Use words first then introduce symbols with care.

How do you explain the top and bottom numbers?

1

2

The number of parts chosen

The number of parts the whole has been divided into

+ = “I ate 1 out of my 2 sandwiches, Kate ate 2 out of her 3 sandwiches so together we ate 3 out of the 5 sandwiches”!!!!!

12

23

35

The problem with “out of”

86

x 24 = 2 out of 3 multiplied by 24!23

= 8 out of 6 parts!

Fractions Key Ideas1. Use sets as well as regions from early on and connect different

representations.

2. Use words first & introduce symbols with care.

3. Go from part-to-whole as well as whole-to-part with both shapes and sets.

6 is one third of what number?

This is one quarter of a shape. What does the whole look like?

Fractions Key Ideas1. Use sets as well as regions from early on and connect different

representations.

2. Use words first & introduce symbols with care.

3. Go from part-to-whole as well as whole-to-part with both shapes and sets.

4. Division is the most common context for fractions when units of one are not accurate enough for measuring and sharing problems.

Initially this is done by halving and halving again.

e.g. 3 ÷ 5

Fractions Key Ideas1. Use sets as well as regions from early on and connect different

representations.

2. Use words first & introduce symbols with care.

3. Go from part-to-whole as well as whole-to-part with both shapes and sets.

4. Division is the most common context for fractions, e.g. 3 ÷ 5

5. Fractions are not always less than 1. Push over 1 early to consolidate the understanding of the top and bottom numbers.

Y7 responses: What is this fraction? 5/2

2 fifths, five lots of halves, tenth, five twoths

Fractions Key Ideas1. Use sets as well as regions from early on and connect different

representations.

2. Use words first & introduce symbols with care.

3. Go from part-to-whole as well as whole-to-part with both shapes and sets.

4. Division is the most common context for fractions, e.g. 3 ÷ 5

5. Fractions are not always less than 1. Push over 1 early to consolidate understanding.

6. Fractions are numbers as well as operators

3/4 is a number between 0 and 1 (number)

Find three quarters of 80 (operator)

Place 3/5 on the number line. (number)

35

0 1

0 100

15

20 60

Find 3/5 of 100. (operator).

Using a double number line or bead string

x3

Fractions Key Ideas1. Use sets as well as regions from early on and connect different

representations.

2. Use words first & introduce symbols with care.

3. Go from part-to-whole as well as whole-to-part with both shapes and sets.

4. Division is the most common context for fractions,

5. Fractions are not always less than 1. Go over 1.

6. Fractions are numbers as well as operators.

7. Fractions are always relative to the whole.

Sam had one half of a cake, Julie had one

quarter of a cake, so Sam had most.

True or False or maybe

What is B?

A A

B B B B

C

D D D D D D D D

What is the whole? (Trains Book 7, p32)

Fractional Key Ideas – Pre Stage 7

1. Use sets as well as regions from early on and connect different representations.

2. Use words first & introduce symbols with care.

3. Go from part-to-whole as well as whole-to-part with both shapes and sets.

4. Division is the most common context for fractions, e.g. 3÷5

5. Fractions are not always less than 1. Go over 1.

6. Fractions are numbers as well as operators

7. Fractions are always relative to the whole.

8. Fractions are really a context for applying add/sub and mult/div strategies.

Stage 7 Decimals and Percentages

Decimals are special cases of equivalent fractions where the denominator is always a

power of ten.

Stage 7 (AM) Level 4 Key IdeasFractions

• Rename improper fractions as mixed numbers, e.g. 17/3 = 52/3

• Find equivalent fractions using multiplicative thinking,, e.g. 2/6 = how many twelfths?

• Order fractions using equivalence and benchmarks like 1 half , e.g. 2/5 < 11/16

• Add and subtract related fractions, e.g. 2/4 + 5/8

• Find fractions of whole numbers using mult’n and div’n e.g.2/3 of 36 and 2/3 of ? = 24

• Multiply fractions by other factions e.g.2/3 x ¼

• Solve measurement problems with related fractions, e.g. 1½ ÷ 1/6 = 9/6 ÷ 1/6 =9

Decimals• Order decimals to 3dp• Round whole numbers and decimals to the nearest whole or tenth

• Solve division problems expressing remainders as decimals, e.g. 8 ÷ 3 = 22/3 or 2.66

• Convert common fractions, i.e. halves, quarters etc. to decimals and percentages• Add and subtract decimals, e.g. 3.6 + 2.89 Percentages• Estimate and solve percentage type problems like ‘What % is 35 out of 60?’, and ‘What is 46% of 90?’ using benchmark amounts like 10%

& 5%

Ratios and Rates

• Find equivalent ratios using multiplication and express them as equivalent fractions, e.g. 16:8 as 8:4 as 4:2 as 2:1 = 2/3

• Begin to compare ratios by finding equivalent fractions, building equivalent ratios or mapping onto 1).

• Solve simple rate problems using multiplication, e.g. Picking 7 boxes of apples in ½ hour is equivalent to 21 boxes in 1½ hours.

Misconceptions with Decimal Place Value:

How do these children view decimals?

1. Bernie says that 0.657 is bigger than 0.7

2. Sam thinks that 0.27 is bigger than 0.395

3. James thinks that 0 is bigger than 0.5

4. Adey thinks that 0.2 is bigger than 0.4

5. Claire thinks that 10 x 4.5 is 4.50

Developing understanding of decimal tenths and hundredths

place value

The CANON law in our place value system is that

1 unit must be split into TEN of the next smallest unit AND NO OTHER!

Developing Decimal Place Value Understanding

1. Use decipipes, candy bars, or decimats to

understand how tenths and hundredths

arise and what decimal numbers ‘look like’

2. Make and compare decimal numbers, e.g.

Which is bigger? 0.6 and 0.47

3. How much more make.. e.g. 0.47 + ? = 0.6

1. establish the whole, half, quarter rods then tenths 2. 1 half = ? tenths3. 1 quarter = ? tenths + 4. 1 eighth = ? tenths? +

Using Decipipes

View children’s response to this task

Now compare:0.4 0.38 0.275

3 chocolate bars shared between 5 children.

30 tenths ÷ 5 =

0 wholes + 6 tenths each = 0.6

Using candy bars

3 ÷ 5

Using decimats and arrow cards

2. Make and compare decimals

•Which is bigger: 0.6 or 0.43?

•How much more make…

Add and subtract decimals

Rank these questions in order of difficulty.

a)0.8 + 0.3,

b)b) 0.6 + 0.23

c)c) 0.06 + 0.23,

Exchanging ten for 1

Mixed decimal values

Same decimal values

Add and subtract decimals (Stage 7) using decipipes or candy bars

1.6 - 0.98

Tidy Numbers Place Value

Equal Additions Reversibility

Standard written form (algorithm)

When you multiply the answer always gets bigger.

True False

0.4 x 0.3Which is the correct answer?

0.12 1.2 0.012

Decimals multiplied by a whole number(Stage 8) Using candy bars or

decipipes

7 x 0.2

Tidy Numbers Place Value

Proportional Adjustment

Written form

Ww

w

Decimal Multiplied by a Decimal (Stage 8)

1. Convert to a fraction, i.e.

0.25 x 0.8 is the same as 1 quarter of 0.6

2. Use Arrays e.g. 0.4 x 0.3

Ww

w

0 1

1

0.3

0.4

Using Arrays0.4 x 0.3 = 0.12

Division of decimals by a decimal

Sue had 2.5 kg of fruit, if it takes 0.5 kg of fruit to make 1 jar of jam, how many jars can Sue make?2.5 ÷ 0.5

Division of decimals by a whole number4.2 metres of string is cut int 7 equal lengths, how long is each length? 4.2 ÷ 7

‘Target Time’ (from FIO Number L3 Book 2 page 16)

Target Number is 6

+ =

• Roll a dice and place the number thrown.

• Try and make the number sentence as close to the target number as possible.

• Score = the difference between your total and the target number.

Decimal Keyboard

Decimal Games and Activities

• Digital learning Objects: http://digistore.tki.org.nz/ec/viewMetadata.action?id=L1079

1. Decimal Sort2. First to the Draw3. Four in a Row Decimals4. Beat the Basics5. Decimal Keyboard6. FIO N3–4:2 Fraction Distraction

Thought for the day

A DECIMAL POINT  

When you rearrange the letters becomes

I'M A DOT IN PLACE

It is a method of comparing fractions by giving both fractions a common denominator i.e. hundredths. So it is useful to view percentages as hundredths.

Why calculate percentages?

=

Applying PercentagesTypes of Percentage Calculations at Level 4 (stage 7)

• Estimate and find percentages of amounts,

e.g. 25% of $80

• Expressing quantities as a percentage

(Using equivalence – Jo’s workshop)

e.g. What percent is 18 out of 24?

Estimate and find percentages of whole number amounts.

25% of $80

35% of $80

Using benchmarks like 10%, and ratio tablesFIO: Pondering Percentages NS&AT 3-4.1(p12-13)

Using common conversions halves, thirds, quarters, fifths, tenths

Book 8:21 (MM4-28) , Decimats. Bead strings, slavonic abacus

Practising instant recall of conversionsBingo, Memory, I have, Who has, Dominoes,

100%

Find __________ (using benchmarks and ratio tables)

100%

$80

Find 35% of $80

$80

100%

$80

Find 35% of $80

$80

100%

$80

Find 35% of $80

100%

$80

Find 35% of $80

$8

10%

$8

35%

$28

$4

5%

$4

$8$8

30%

$24

Now try this…

46% of $90

46% of $90100% 10% 40% 5% 1% 6% 46%

$90 $9 $36 $4.50 $0.90 $5.40 $41.40

Is there an easier way to find 46%?

46% of 90

Estimating Percentages

16% of 3961 TVs are found to be faulty at the factory and need repairs before they are sent for sale. About how many sets is that?

(Book 8 p.26 - Number Sense)

Using Number Properties:

Explain how you would estimate 61% of a number?

About 600

Dice: 20% 25% 40% 50% 60% 75%

Roll then move to chosen spot60% MM (7-5)

What do you know now that you didn’t know before?

What parts of this workshop would you include in a short after-school workshop to share with your staff.

Useful PD resources available on nzmaths•Equipment animations, •Digital Learning Objects, •Online PD Modules•Figure It Out activities

Misconceptions with Decimal Place Value:

How do these children view decimals?

1. Bernie says that 0.657 is bigger than 0.7

(decimals are 2 separate whole number systems separated by a decimal point, so 657 is bigger than 7)

2. Sam thinks that 0.27 is bigger than 0.395

(the more decimal places, the tinier the number becomes because thousandths are really small)

3. James thinks that 0 is bigger than 0.5

(decimals are negative numbers)

4. Adey thinks that 0.2 is bigger than 0.4 (direct link to fraction numbers , i.e. ½ = 0.2, ¼ = 0.4)

5. Claire thinks that 10 x 4.5 is 4.50 (when you multiply by 10, just add a zero)