Proportions & Ratios Workshop

Lisa Heap and Alison Howard

Mathematics Facilitators

Objectives• Understand the progressive strategy stages of

proportions and ratios

• Understand common misconceptions and key ideas when teaching fractions and decimals

• Explore equipment and activities used to teach fraction knowledge and strategy

Revising the Framework:

• Sort the addition/subtraction framework.• Align the multiplication/division framework.

• How do they fit?

Solving a Division Problem:

A sheep station has eight

paddocks and 296 sheep. How many sheep are

there in each paddock?

296 ÷ 8 Reversibility

8 x 30 = 240

8 x 7 = 56

Place Value

240 ÷ 8 = 3056 ÷ 8 = 7 30 + 7 = 37

Tidy Numbers

4000 ÷ 8 = 500

500 - (320 ÷ 8)=

500 - 40 = 460

Rounding and Compensating

296 ÷ 8 =

148 ÷ 4 =

74 ÷ 2 = 37

Proportional Adjustment:

Algorithm320 ÷ 8 =40

40 - (24 ÷ 8)=

40 - 3= 37

Multiplication Division Share Back:

Did you try…– A knowledge check?– Diagnostic snapshot?– Recording in your modelling book?– Some Equal grouping/Sharing?

Importance of Place Value

• What is place value?• Where does place value

start?• What place value equipment

have you currently got in your school?

• Order the equipment from least abstract to most abstract.

Place Value Hats:

Place Value Ideas:

• 100 Day Party.• Place Value Hats.• Large Numbers Roll Over Page 43.

Place Value

Read, Say, Do x2

• Write the number as 63• Write the number as sixty-three• Say the numeral one way, 63 is sixty-three• Say the numeral another way, 63 is six tens and three ones• Model the number as ones, 63 individual ones• Model the number in the PV form as 6 tens and 3

ones

The Rope Game

Find 2/5

FractionRope Game

Fraction Knowledge Test:• Write the symbols for one half, one eighth, one

quarter, one third and one fifth

• Put those fractions in order (smallest first)

• Draw 3 pictures to represent one quarter

• 7 is one third of what number?

• 12 is three fifths of what number?

• What is 3 ÷ 5?

• On a number line from 1 – 5 show where five halves live.

• Show me one half as a ratio.

5 children share three chocolate bars evenly. How much chocolate does each child receive?

3 ÷ 5

1/2 1/

2 1/2 1/

2 1/2

What are these pieces called?

1/2 +

1/10 =

2/12 !!

What do you think they have done?

A more sophisticated method for 3 ÷ 5

1/5+1/5+1/5 =3/5

Y7 response: “3 fifteenths!” Why?

Which letter shows 5 halves as a number?

0 1 2 3

A B C D E F

Ratios (Introduced at Stage 6)

Write 1/2 as a ratio

3: 4 is the ratio of red to blue beans.

What fraction of the beans are red?

Think of some contexts when ratios are used.

1:1

3/7

Choose your share of chocolate!

Framework Practice

Match the strategy stages to their definitions and assessment task(s) from GloSS.

What does this mean?

€

3

73 over 73 : 7

3 out of 7 3 ÷ 7

3 sevenths

+ = “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!

The Problem with Language

Use words first before using the symbols

e.g. one fifth not 1/5

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

Models of Fractions:

• Complete the activity on discrete and continuous models of fractions.

• In your thinking groups discuss the meaning of continuous and discrete.

10

Continuous Model:• Models where the object can be divided in any

way that is chosen.e.g. ¾ of this line and this square are blue.

Discrete Model:• Discrete: Made up of individual objects.

e.g. ¾ of this set is blue

Whole to Part:• Most fraction problems are about giving

students the whole and asking them to find parts.

• Show me one quarter ofthis circle?

Part to Whole:

• We also need to give them part to whole problems, like:

• 5 is a quarter of thisnumber.

What is the number?

Missing Number Decimal Fraction Bingo

1.4 1.5

2.4 2.6

2.9 3.3

The Strategy Teaching Model

Using Number Properties

Using Imaging

Using Material

s

New Knowledge & Strategies

Existing Knowledge & Strategies

Using Materials

What equipment do you use to teach decimals?

Developing Decimal Place Value Understanding

Decipipes, candy bars, or decimats can be used

to demonstrate how tenths and

hundredths arise and what decimal

numbers ‘look like’ and to compare decimals

numbers.

Explore the Decipipes. What is each piece called? How did you know?

• Build 0.365 and 0.37• Which is bigger? Why?

• Add 0.4 and 0.25 on your decipipes. Discuss what you did and what you found out.

Using Decipipes

Book 7: Pipe Music with Decimals, p.38

3 chocolate bars shared between 5 children.

30 tenths ÷ 5 =

0 wholes + 6 tenths each = 0.6

Using candy bars (and expressing remainders as decimals)

3 ÷ 5

Solve this problem…

I had 1.6m of ribbon. I used 0.98m for my gift wrapping. How much

ribbon do I have left?

Place Value

Equal Additions Reversibility

Standard written form (algorithm)

Ratios

• In the rectangle below, what is the ratio of blue cubes to green cubes?

• What is the fraction of blue and green cubes?

• Can you make another structure with the same ratio? What would it look like?

• What confusions may children have?

More on Ratios• Divide a rectangle up so that the

ratio of its blue to green parts is 7:3.

• What is the fraction of blue and green?

• If I had 60 cubes how many of them will be of each colour?

A Ratio Problem to Solve• There are 27 pieces of fruit. The ratio of

fruit that I get to the fruit that you get is 2:7. How many pieces do I get?

• How many pieces would there have to be for me to get 8 pieces of fruit?

• What key mathematical knowledge is required?

The 4 Stages of the P.D Journey:Organisation

Organising routines, resources etc.

Focus on ContentFamiliarisation with books, teaching model etc.

Focus on the StudentMove away from what you are doing to noticing what

the student is doing

Reacting to the StudentInterpret and respond to what the student is doing

Final Evaluation

Complete your initial evaluation and mark on your progress.

Thank you!