Dynamic Representations - Core concept 3.1: Understanding multiplicative relationships

Dynamic Representations

Core concept 3.1:

Understanding multiplicative relationships

3.1.1.1* Appreciate that any two numbers can be connected via a multiplicative relationship

3.1.1.2 Understand that a multiplicative relationship can be expressed as a ratio and as a fraction

3.1.1.3 Be able to calculate the multiplier for any given two numbers

3.1.1.4 Appreciate that there are an infinite number of pairs of numbers for any given multiplicative relationship (equivalence)

Appreciate that any two numbers can be connected via a multiplicative relationship

This activity can be modelled with Cuisenaire rods (below) and the dynamic representation (left).

Ask students to describe how to get from 9 to 1 to 5 using division and multiplication.

9 ÷ 9 x 5 = 5

When they are comfortable with this ask how to get from a to a to b.

a ÷ a x b = b

Can you get from 9 to 5 without division?

9 x ⅑ x 5 = 5

9 x 5/9 = 5

Can you get from a to b without division?

a x ⅟a x b = b

a x b/a = b

secmm-cc-theme-3-key-idea-23.pptx

3.1.1.1 Appreciate that any two numbers can be connected via a multiplicative relationship

●Interpret multiplication as scaling.

●Interpret multiplication as a rate.

●Understand that any number can be transformed into any other number by multiplying.

Multiplication as Scaling

This representation models stretching a springs and compares this with a table and graph of two points on the spring. See https://www.ncetm.org.uk/media/mqfp3xb3/ncetm_ks3_cc_3_1.pdf page 9

a) Look at the 4s row. What does 4 need to be multiplied by:

(i) to move to 8?

(ii) to move to 28?

(iii) to move to 16?

(iv) to move to 20?

b) Still within the 4s row, what does the 8 need to be multiplied by:

(i) to move to 16?

(ii) to move to 24?

(iii) to move to 20?

(iv) to move to 4?

c) Consider corresponding entries in the 4s row and the 6s row.

(i) What is the relationship between 8 and 12; 20 and 30; 40 and 60, etc.?

(ii) Picture where 10 might be in the 4s row. What number would be in the corresponding position in the 6s row?

a)This double number line can be used to convert between pounds and dollars. £3 is equivalent to $4.

(i) Describe how you would use the double number line to roughly convert $4.50 to pounds.

(ii) Describe how you would use the double number line to accurately convert $4.50 to pounds.

b) This graph can be used to convert between pounds and dollars.

(i) What features of the graph show that the rate of exchange is £3 for every $4?

(ii) What features of the double number line in part a) show that the rate of exchange is £3 for every $4?

(iii) Describe how the graph would change if the rate of exchange changed to £3 for every $5.

(iv) Describe how the double number line would change if the rate of exchange changed to £3 for every $5.

3.1.2.1* Use a double number line to represent a multiplicative relationship and connect to other known representations

3.1.2.2 Understand the language and notation of ratio and use a ratio table to represent a multiplicative relationship and connect to other known representations

3.1.2.3 Use a graph to represent a multiplicative relationship and connect to other known representations

3.1.2.4 Use a scaling diagram to represent a multiplicative relationship and connect to other known representations

secmm-cc-theme-3-key-idea-24.pptx

3.1.2.1 Use a double number line to represent a multiplicative relationship and connect to other known representations

●Use the multiplicative nature of the double number line to find pairs of values using scalar (along the line) multipliers.

●Use the multiplicative nature of the double number line to find pairs of values.

●Use the double number line to represent multiplicative relationships, efficiently using both scalar and functional multipliers.

●Use the multiplicative nature of the double number line to solve problems efficiently.

Ellie and her dad walk side by side along a straight path. The number of steps they take is represented here:

a) Which line represents Ellie’s steps and which represents her dad’s? Explain how you know.

b) When Ellie had taken 12 steps, her dad had taken 8 steps. When else were Ellie and her dad stepping at precisely the same time?

c) If they continued walking like this, find more points when they would be stepping at the same time. What is the connection between the number of Ellie’s steps and the number of her dad’s steps?

Ali buys eight identical packets of cakes for a birthday party. The total cost is £12.

a) Which line in this double number line represents the number of packets of cakes Ali buys and which represents the total cost? Explain how you know.

b) Use this diagram to write down the cost of some other quantities of packets of cakes.

c) Can you write down roughly how much nine packets of cakes would cost? Can you be precise? Explain how.

Which of these representations do you find most useful to answer question c?

Why?

secmm-cc-theme-3-key-idea-25.pptx

3.1.2.2. Understand the language and notation of ratio and use a ratio table to represent a multiplicative relationship and connect to other known representations

●Identify the multipliers for information arranged in a ratio table

●Understand ratio tables as an abstraction of the double number line

●Identify situations in which the ratio table is a suitable model

Teaching whole-number multiplication to promote children’s proportional reasoning: a practice-based perspective from Japan - ZDM – Mathematics EducationWhole-number arithmetic is a core content area of primary mathematics, which lays the foundation for children’s later conceptual development. This paper focuses on teaching whole-number multiplication (WNM) to build a stepping stone for children’s proportional reasoning. Our intention in writing this paper is to obtain a practice-based perspective on the teaching of WNM, given the growing attention being given to significant connections between children’s understanding of multiplication and their proportional reasoning. To this end, we review studies on teaching WNM that were conducted in Japan. Our review highlights several aspects of teaching WNM that are integral to building an initial understanding of proportional relationships. Linear representations such as tape diagrams and number lines are also found to help make explicit the important properties of proportional relationships that can be used progressively across grade levels.

Here the same section coloured differently. In this diagram there is also a constant multiplier to move from the numbers in blue to those in red.

a) Find the constant multiplier to move from the blue to red.

b) Shift the group of four numbers to a different position on the table-square. Is there always a constant multiplier to move from left to right? From top to bottom?

Here is a section from a times-table square. A set of four numbers is highlighted.

In this diagram there is a constant multiplier to move from the numbers in blue to those in red.

a)Find the constant multiplier to move from the blue to red.

b)Find the constant multiplier to move from the red to blue.

c)Shift the group of four numbers to a different position on the table-square. Is there always a constant multiplier to move from left to right? From top to bottom?

Write down the multipliers and find the missing values in these ratio tables.

For a birthday party, Pete buys 8 identical packets of cakes. The total cost is £12.

a) Use the double number line to estimate the cost of 9 packets of cakes.

b) Which of these ratio tables could be used to calculate the cost of 9 packets of cakes?

3.1.3.1 Find a fraction of a given amount

3.1.3.2 Given a fraction and the result, find the original amount

3.1.3.3 Express one number as a fraction of another

3.1.4.1 Be able to divide a quantity into a given ratio

3.1.4.2 Be able to determine the whole, given one part and the ratio

3.1.4.3* Be able to determine one part, given the other part and the ratio

3.1.4.4 Use ratio to describe rates (e.g. exchange rates, conversions, cogs, etc.)

secmm-cc-theme-3-key-idea-26.pptx

3.1.4.3 Be able to determine one part, given the other part and the ratio

●Interpret mathematically a situation involving unequal sharing, using correct notation and relevant diagrams.

●Recognise the multiplicative relationship between a and b in the ratio a : b.

●Find missing parts when a quantity is divided into more than two unequal parts.

Oscar and Pietro share some money in the ratio 2 : 3.

Oscar’s share is £9.

Explain how each of these diagrams can help to calculate Pietro’s share of the money.

3.1.5.1 Describe one number as a percentage of another

3.1.5.2 Find a percentage of a quantity using a multiplier

3.1.5.3 Calculate percentage changes (increases and decreases)

3.1.5.4 Calculate the original value, given the final value after a stated percentage increase or decrease

3.1.5.5 Find the percentage increase or decrease, given start and finish quantities

secmm-cc-theme-3-key-idea-27 (1).pptx

3.1.5.3 Calculate percentage changes (increases and decreases)

●Find a percentage increase or decrease using an additive method.

●Use multipliers to calculate percentage increase or decrease.

●Solve familiar and unfamiliar problems, including real-life applications.

Bar models, double number lines and ratio tables are all powerful representations that can help students work ‘beyond 100%’ and identify the both the whole, and the multiplier linked to the percentage.