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KS3 Multiplicative reasoning project

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Mathematics Subject knowledge supplement

The activities in the KS3 multiplicative reasoning workshops provided opportunities for pedagogical and subject knowledge development. The main focus for this was provided by the lessons from the teaching units and feedback on the lessons through lesson study. This supplement focusses on teacher’s responses to the subject knowledge activities designed as part of the workshops and linked in most cases to the lessons themselves and the key wider mathematical knowledge associated with them.

The activities are described with teacher’s responses and suggestions for improvement or adaptations to the workshops. Findings and recommendations are summarised in the main report.

This document will complement the project materials and support professional development leaders in designing and running workshops on multiplicative reasoning.

Contents:

Summary of Subject knowledge areas focussed on in each workshop

Details of the subject knowledge activities and teachers responses:

Workshop1 (introductory workshop)

Workshop 2

Workshop 3

Workshop 4

Workshop 5 (final workshop)

NCETM Report on the KS3 Multiplicative Reasoning Project


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KS3 multiplicative reasoning project Mathematics subject knowledge

Workshop teaching focus

Workshop Subject knowledge focus

Workshop 1

(2 days)

Project introduction

The mathematics of

mltiplicative reasoning

Unit 0 assessment

tasks

A. Recognising areas of the curriculum connected by proportionality

Recognising connections where the underlying mathematics is multiplicative Considering common approaches to solving proportional problems (unitary and

scale factor) I. Identifying the meaning of the operations involved

II. Showing their equivalence mathematically B. Considering the mathematics of multiplicative reasoning (this was optional, see

appendix 1) Transforming one number into another

via two operations - multiplication and division Reducing this to a single operation of either multiplication or division.

Moving towards a broad definition of the term multiplicative reasoning used for the project

Ratios and fractions Definition of a proportion

C. Considering different interpretations of multiplication and division

Division as sharing (partitive) or grouping (quotitive) and multiplication as repeated addition or scale factor enlargement

Interpreting the meaning of calculations from a given context D. Generating equivalent expressions involving fractions

Workshop 2 Teaching unit 1:

Deepening understanding of

fractions

Identifying the whole in solving problems involving fractions

Applications of the distributive law for fractions.

Geometric images to support the visualisation of infinite power series

Consideration of a fraction as an operator

Workshop 3

Teaching unit 2: Understanding and

identifying

proportional contexts

Partition and its relation to area partition - partition of brie segment in appendix

Fractions in relation to ratios

The effect of proportional changes and their inverses

Definition of a reciprocal as an inverse operator

Equivalence of division to multiplication by reciprocal, hence their role in relation to division of fractions

Exploring the properties of different relationships including directly proportional relationships and linear relationships

Exploring the representation of functions through mapping diagrams and double number lines

Workshop 4

Teaching unit 3: Application to a wider

range of contexts

Making sense of procedures involving calculating with fractions Interpreting the meaning of multipliers and chains of multipliers in fraction and

decimal form and simplifying such expressions to show equivalence

Models for multiplication and division including decimals and fractions

Workshop 5

Plenary workshop

Necessary and sufficient conditions for proportionality

Exploring a proportional problem across representations

Combining intermediate steps (operations) to a single operation

Proving the distributive law for proportional functions


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NCETM KS3 Multiplicative reasoning project report

Subject knowledge supplement Reporting on the responses to the subject knowledge elements of the KS3 project

Workshop 1 (Introductory workshop)

A. Recognising areas of the curriculum connected by proportionality and considering the underlying structure of the mathematics

Subject knowledge focus: Making connections where the underlying mathematics is multiplicative Considering common solution approaches: Identifying the meaning of the operations involved Showing their equivalence mathematically.

Considering interpretations of division as sharing or grouping

Activity 1: Making connections where the underlying mathematics is multiplicative Teachers look through a booklet comprising a variety of GCSE exam questions ranging across many strands of the curriculum. – Q: what connects them?

Response:

Teachers stated that they recognised these questions were all connected by their underlying proportionality and hence would involve essentially the same maths. They also noted that they had never made this explicit in their teaching.

However they did feel that teaching that did reflect this, in particular schemes of work that systematically developed pupils learning with this in mind could potentially be transformative.

They also clearly stated that this should happen from early Primary and continue right through Secondary teaching.

Some teachers were surprised and interested in the range of topics that were underpinned by the same multiplicative structures e.g. similarity, trigonometry, pie charts and some algebra questions.

Suggested further task:

In what ways does the teaching of enlargement, similarity and trigonometry relate to multiplicative reasoning? How might a progression in the scheme of work highlight and develop understanding in this way?

Activity 2: Comparing two key calculation strategies, identifying their meaning and demonstrating their equivalence.

Part 1: Teachers answer Q8 from the above GCSE question booklet and discuss solution in pairs.

Q8 Holiday: In 1976 the average yearly wage was £3275. On average, people spent 17% of £3275 on their family holiday. How much is 17% of £3275? Show your working.

Response: Most teachers used a multiplier approach (3275 x 0.17). The majority of the teachers indicated this was

the method they also used to teach percentages i.e. using a decimal equivalent to the % as an operator. However some used what they described as a ’unitary’ method (3275 ÷100 x 17) i.e. finding the amount for 1% and scaling up.


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Part 2: They then look at two alternative solutions to Q8 (right) presented on a slide and are asked in pairs to describe the approach in each case and show how the calculations are equivalent. Response: Teachers described the left hand solution as finding the

amount for 1% and then ‘multiplying up’. They referred to this as the ‘unitary method’. The right hand solution they described as a ‘multiplier method’.

Working in pairs teachers mainly demonstrated equivalence along the following lines:-

× 17 = × = 3275 × = 3275 × 0.17 = 0.17 x 3275 = 556.75

Teachers recognised this was based on the commutative property of multiplication and accepting the rule for multiplying fractions- potential further discussion on which other operations of +, -, ×, ÷ were commutative or non-commutative.

Some teachers indicated they had not reflected on the mathematical link between the use of a multiplier to find a percentage and what was referred to as a unitary approach.

Discussion also included some consideration in pairs on the nature of the two methods in terms of a rate – an ‘amount for one’ compared to a ‘scale factor’ of enlargement. In some cases this anticipated later discussions on the different interpretations for division and multiplication.

Some discussion focussed briefly on how the divisions were also expressed in ‘fraction form’

Part 3: As a further activity teachers were asked to choose another question of their own choice from the booklet and apply the two approaches. Response:

In some cases the nature of the question suggested more strongly to teachers one method over the other. This was often due to the numbers involved indicating one approach as easier to calculate or sometimes the context made more ‘sense’ with a particular method.

Teachers noted that, where a question involved different units e.g. litres of oil cost £, then it was easier to ‘make sense’ of the unitary method in the context of the question compared to use of a scale factor method. However where units are the same e.g. lengths in a geometry question, then a scale factor (multiplier) made more sense e.g. similarity, trig etc.

B. Considering the mathematics of multiplicative reasoning (See appendix ) Activities relating to this item were included in the original project scripts for workshop 1, however many workshops gave only a brief treatment or left out much of this due to time constraints. Following evaluation of the project, this section has been revised and is now considered an important SK element to the project for the early workshops. Hence a revised treatment is given here in appendix 1. It includes work on constructing multipliers between numbers and how they relate to fraction, decimal and percentage changes, in particular when they are applied to sets of numbers in the same ratio (proportional sets). This allows an exploration of the multiplicative reasoning nature of proportional problems and supports the definition given in the project description sent to schools.

C. Interpretations of division and multiplication SK focus: division as sharing or grouping and multiplication as repeated addition or scaling

Activity 1: Models of multiplication and division – How are these developed in school? Teachers consider a series of slides introducing multiplication as repeated addition or scaling and division as sharing or grouping, they then look through a worksheet (see appendix 2) and consider the models and discuss in pairs which suggest multiplication as repeated addition or multiplication as scaling.


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Response: evaluation feedback suggested that that some teachers appeared not to consider this an important distinction at Secondary (though later in the project this belief appears to change, suggesting the use of ways of assessing changes in beliefs be employed more formally in the project). Suggestions for improvement: Streamline activity and communicate more clearly the purpose of the session to focus on the development of pupil understanding of multiplication and division at KS1 and KS2 in order to better appreciate Teaching and Learning difficulties. Challenge teachers to:

Offer contexts or images that suggest multiplication as repeated addition or as scale factor enlargement.

Offer examples and proofs of the commutative properties of multiplication and non commutivity of division.

Activity 2: Devising questions from a context reflecting division as either sharing or grouping Teachers are shown the context below from a KS1 SAT question:

Task: Devise examples of questions for this context, stating in each case whether they involve division as sharing or grouping. Give examples of questions involving multiplication as repeated addition or scale factor enlargement.

Response:

Many teachers were able to devise examples that implied division as sharing (partitive) e.g. if Sita cut it into 5 equal lengths how long would each length be? or grouping (quotitive) e.g. Sita needs 9cm lengths for hemming, how many can she cut from the ribbon?

Discussion took place where different units were involved e.g. if 45 cm cost 90p how much would 1 cm cost, 10 cm cost? What length would cost 1p, £1? Here discussion centred on the meaning of partitioning as a rate (unit per unit) and its connection with unitary method.

Generating examples for multiplication as repeated addition presented no problems but discussion centred on difficulty in constructing examples involving multiplication as scale factor enlargement. Examples suggested included considering the ribbon as elastic – how long would it be if the ribbon was stretched by 50%? Or – How wide apart would two points 4 cm apart be after a 1 ½ times stretch? This provoked a discussion on the nature of stretching compared to repeated addition – e.g. isn’t stretching still repeated addition?

Activity 4: Interpreting the meaning of calculations related to a given context

Part 1: For a given context interpret the meaning of different calculations. Task: Teachers are shown an image of a bag of flour with some information relating to weight, cost etc. They are asked to look at a set of calculations and in each case identify and justify a possible question relating to the bag of flour context that the calculation represents. Response:

This was a popular activity in relation to thinking of the questions the calculations could represent and how this would be a good activity for students.

Identifying the meaning of the calculation in terms of the context was useful, in particular identifying the meaning of the unitary method in each case and the resulting division.

Some of the questions were challenging with the last one being finally solved by only one or two.

Considerable pedagogic discussions were also stimulated here.


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Part 2: A follow up activity asked teachers to identify what interpretation of division each calculation involved e.g. sharing (partitive) or grouping (quotitive). Response:

A key recognition by teachers related to division (in these calculations most often in the form of a fraction) – hence numerator and denominator where the dividend and divisor were of different units. In this case the division suggested a rate (which teachers identified as a unitary method) and division as sharing (partitive). In the case where the units were the same, this suggested a dimensionless scale factor of enlargement which was more easily interpreted as division as grouping (quotitive).

Suggested follow up: Give time for teachers to consider the importance of pupils making sense of these classifications more explicitly (this is dealt with further in Lesson 3cd). Include as part of pre project a subject knowledge survey/test?

D. Generating equivalent expressions involving fractions Activity 1: Teachers are shown a slide centred on a fraction expression with some equivalent expressions. Teachers are asked to suggest further equivalent expressions to place on the diagram. The activity focussed on the properties of multiplication and division of fractions exploiting different interpretations of a fraction as an operator. Response – This activity was not completed at all workshops due to time considerations, however where it did take place teachers commented on the usefulness of the activity for manipulating expressions and understanding and applying the arithmetic of fractions or fraction reasoning. Many teachers felt this would be a useful activity with pupils. Suggested further activity: complete a similar activity for a given fraction (e.g. starting with a vulgar fraction- see slide below right) and construct multiple interpretations including fractions, decimals, ratio, proportion and percentages.

Core 0: Pupil assessment questions SK focus: Considering multiplication as scale factor of enlargement as different to repeated addition

Teachers discussed the Core 0 pupil assessment questions. In particular question 2 (left) involving calculating the length of a segment of a shape given a similar shape Response: teachers recognised the question required a scale factor of enlargement calculated from two known corresponding sides and applied to find the missing length on the enlarged shape. However many were surprised by the number of pupils who incorrectly applied an incorrect additive approach. Subsequent to a discussion on this, teachers also went on to discuss the difference between multiplication considered as repeated addition or a scale factor of enlargement.

Where this discussion took place teachers noted how it was ‘easier to see enlargement of a linear segment as the same as repeated addition of its original length even when the length and scale factor were decimals, however it was much harder to see the length of an enlargement of a curved segment as repeated addition of the original shape’ suggesting an important distinction in these two definitions for multiplication. Suggested further task: How might you describe links between multiplication as scale factor enlargement (multiplier) or repeated addition?


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Workshops and Teaching units These subject knowledge activities led by the workshop presenters were designed to link to the pupil activities in the lessons from the teaching unit the workshop was focussing on.

Workshop 2

Unit 1: Deepening understanding of fractions

Subject knowledge focus:

Part whole interpretation of fractions

Applications of the distributive law for fractions

Geometric images to support the visualisation of infinite power series

Consideration of a fraction as an operator

Lesson 1c: Fair shares Teacher activity 1 SK focus: The importance of identifying the whole when using fractions Teachers undertook this activity as it is presented in the lesson where a number of pictures of students holding sandwiches are shown. The activity requires a justification of the answer using a diagram (RME). The teacher SK activity focussed on picture A below.

Response: A number of approaches were used including an approach where each of the three sandwiches was divided into 5 equal bits from which each person would get three. In some justifications (example above right) the ‘equal bits’

were described as ‘s giving: + + = = as the fraction of a sandwich each person got – conflicting

with the answer ⅗ which they knew would be the ‘correct’ answer. The ensuing debate allowed teachers to clarify their thinking and to realise the significance of identifying the whole correctly. That the same fraction can represent different quantities lead to a discussion of the fraction as an operator on an identified whole. The different solutions generated by teachers also in some cases led to discussion of the interpretation of a fraction of a fraction.

Lesson 1a: Parts of a shape SK focus: Applications of the distributive law for fractions.

The distributive law of multiplication applied to fractions

Fractions considered as a measure and as an operator

Geometric images can provide a model for visualising infinite series and their convergence

Fraction of a fraction and the multiplication of fractions Teachers undertake the activities from the lesson with key activities below expanded on by the PD leads


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Lesson 1a Stage 3. An important subject knowledge point arose while discussing solutions to the stage 3 task - shading ⅜ of rectangle Z below left.

Response: In discussing approaches taken to shading 3/8’s of rectangle Z, many teachers extended the partition of the ‘bottom right’ 1/9 to all the other 1/9’s creating 72 equal sections. Many shaded 27 of them knowing that 27/72 = 3/8, some teachers had used this partition to shade 3/8’s of each of the 1/9’s (example shown above). Discussion identified this as the distributive law for multiplication.

A further geometric example of the distributive law in action was demonstrated by the shading of Z exhibited in the example below:

Response: Here a number of teachers were surprised or unsure at first that the shading was equivalent to ⅜ (In some cases the insight came following working

the lesson with pupils and the subsequent discussion). All teachers recognised

× (a + b) = × a + × b as an expression of the distributive law for

multiplication but had not straight away linked this geometrically to the areas within a rectangle. Discussion revealed teachers felt that it was less obvious as the partition in this case was of different sizes, though teachers on reflection

agreed it was equivalent to the algebraic expression where in the example above a and b represent the partitioned areas 8/9 and 1/9 respectively.

Lesson 1a stage 3 extension activity This idea was further explored in the lesson stage 3 extension activity where the lesson gives the partition below.

Response: This suggested to many teachers taking the given partition approach to its limit and thus creating an infinite series. Many teachers were able to derive and justify from the diagram the infinite

series ( + x + x x + ..…. ) to represent the limit for this partition

process which they recognised would be equivalent to ⅗ from the distributive law. Some were able to show and discuss how it converges to ⅜. Many

commented that this approach would particularly be useful in their A level teaching as an alternative image for an infinite series and its convergence. Further activities: - construct geometric images for other converging series.

Lesson 1a stage 4 activities The activity required deriving a fraction to represent the shaded area of a number of given examples in lesson 1a.

Response: This resulted in insightful discussion on the multiplication of fractions. In particular, looking at shape X (left) the discussion explored the meaning of ⅓ of ¼ and a justification of this as multiplication of the two fractions; and this being the product of the numerators over the product of the denominators. In some cases teachers had only considered the multiplication of fractions as a procedure without considering its justification.

Example solution:


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Workshop 3 Teaching unit 2: Understanding and identifying proportional contexts SK focus:

Partition and its relation to area partition – partition of brie segment in appendix

Fractions in relation to ratios

The effect of proportional changes and their inverses

Definition of a reciprocal as an inverse operator

Equivalence of division to multiplication by reciprocal, hence their role in relation to division of fractions

Exploring the properties of different relationships including directly proportional relationships and linear relationships

Exploring the representation of functions through mapping diagrams and double number lines

Lesson 2a: Using a bar to represent and solve a variety of proportional problems Lesson 2a Stage 4 rectangular pizza Q4 and 5 SK focus: Fractions in relation to ratios

SK Q: what would be the meaning of 5/9 and 4/9 in this context? What would be the meaning of 4/5 and 5/4? Response: While many related 5/9 and 4/9 as the respective fractions of the whole when the pizza was shared in the ratio 5:4 others were unsure of the meaning of 4/5 and 5/4 which discussion illuminated. Further activity: Others were challenged as in the lesson: ‘Where students offer strategies to answer one of these questions not using the bar, it is important to ask them to describe where they can see their own method

on the bar. This can help students to make sense of the rationale behind their strategy’. e.g. many teachers used this approach to justify, and hence ‘make sense of’ for themselves, the unitary method.

Lesson 2b: Percentages on the bar model SK focus:

The effect of proportional increases and decreases on a given quantity

Definition of reciprocal and its role in relation to multiplication and division as inverse operators Teachers worked on problems involving the use of a bar to model percentage increase and decrease Lesson 2b presentation slides posed the following question (Q11 from lesson 2b) A football stadium has increased its seating by 20%. The stadium can now hold up to 30 000 supporters.

Describe how the bar drawn below represents this information. How many supporters could the stadium hold

before the increase?

Response: Here the bar allowed teachers to visualise how a proportional increase followed by the same proportional decrease does not return to the original quantity. Many teachers had often

approached this problem through constructing a multiplier (scale factor) to represent the increase and then dividing by the multiplier to reverse the increase (emphasising the inverse relations of × and ÷). Discussion allowed the relationship to be more fully explored. In some cases teachers generalised so: an increase by a fraction

of can be decreased by leading to scale factor increase of 1 + = can be reversed by scale factor

which is its reciprocal. Hence leading to the definition: the product of a number and its reciprocal is 1 or, when considered as operators, the product gives the identity operation. This operator approach highlighted how division by a number is equivalent to multiplying by its reciprocal and hence justifying the often used rule for dividing fractions as ‘turning upside down and multiplying’. These ideas were also simply extended to percentage increases and decrease. Test questions: a 25 % increase is reversed with a ? % decrease. Why?


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Lesson 2c: Identifying proportional scenarios Lesson 2d: Directly or inversely proportional Activity: teachers complete the lesson activities relating to identifying the properties of a problem that

determine the presence or absence of direct proportionality (one example below), others included multiple representations including directly proportional graphs and linear graphs representing different problems.

Response: Teachers participation in this activity revealed that in some cases there was a lack of clarity on the difference between linear and directly proportional relationships. In particular this was revealed through the graphical representation activity and the ‘doubling one quantity doubles the other’ true / false statements. The resulting discussion clarified for all participants the link between proportionality and a straight line graph through the origin, and its difference to a context represented by a straight line graph not passing through the origin. Also how the straight line graph through the origin supported the statement ‘double one quantity doubles the other’ and this was not true for straight line graphs that did not pass through the origin (however the reverse is not always true -i.e. ‘doubling one quantity doubles the other’ may not imply proportionality and hence a straight line through origin). Counterexamples explored in final workshop can be found, though they are of a sophisticated nature. Key SK Q: What is the difference between linear and directly proportional relationships?

Lesson 2e Double number line. Activity: Comparing representations

Before introducing the double number line teachers are asked to consider a number of different representations of a taxis fare problem (left). The properties of the representations are considered (mapping diagram, Cartesian graph and ratio table). This is followed by animated slides demonstrating how a mapping diagram can be transformed into a Cartesian graph and into a double number line (left) hence revealing the nature of the representations and their connection. This includes the extension of the mapping lines which appear to intersect at a common point. Teachers are asked to consider the significance of this intersection point. Response: Many teachers were less familiar with mapping diagrams as a representation and the use of congruent scales gave significance to the visual pattern of the mapping lines. It was suggested that the point of

intersection of the mapping lines appeared to act as a centre of enlargement and the lines as enlargement rays. This gave rise to discussion on the model for multiplication as enlargement and different to repeated addition. The animation generates interesting but difficult insights into the geometric relationships between a mapping diagram and double number line i.e. the change between the geometry of the rays and the geometry of the scales.

Lesson 2e activity 2 Having thus introduced the double number line (DNL) this is now used to model a number of proportional and non-proportional problems, two examples of which are given below.


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Response: Discussion around completing entries on the double number line supported discussion on the nature of proportionality. Key observations were:

Significance of the zeros being lined up for quantities to be in direct proportion (links were made with linear sequences and their nth term expression) along with corresponding numbers on the DNL being in a common ratio.

Teachers compared the nature of using along the line calculations to that of between the line calculations (scale factor enlargement) compared to use of a common conversion rate, their respective advantages and meaning in the context of the question.

Some question situations could be categorised through recognition of the underlying structure behind their DNL’s e.g. time to boil in a pan different numbers of eggs and time for different numbers of items to dry on a clothes line.

Activity 3 (optional) Teachers were asked to consider the statement below taken from a campaign for better transport poster.

The potential of this particular question was unexplored due to time constraints Suggestion: to set as gap task.

Workshop 4 Teaching Unit 3: Application to a wider range of problems SK focus:

Making sense of procedures involving calculating with fractions Making sense of reciprocals and their role in calculations with fractions

Interpreting the meaning of multipliers and chains of multipliers in fraction and decimal form

Simplifying such expressions to show equivalence

Interpretations of multiplication and division

Lesson 3ab SK focus: Making sense of procedures involving calculating with fractions Teachers considered Q25 from lesson 3ab and the advantages and limitations of each of the three approaches given by pupils. Q25. How many sequins ⅛ of an inch wide can be cut from a strip of plastic 5½ inches in length? Here are three approaches to the problem:

Response: Method 3 surprised many teachers as they had not considered such an approach involving equivalent fractions. They mentioned how it built on pupils existing understanding of division and equivalent fractions. Method 1 provoked consideration of finding how many times a fraction went into the multiplicand and the ease of this where the divisor was a unit fraction. Some linked this to the idea of the reciprocal. Further suggested challenges:

Explain division of fractions in as many ways as you can

Justify the procedure for dividing fractions


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The bar allowed teachers to justify approaches and make sense of procedures involving fractions. A key discussion was the role of the reciprocal of the divisor in the procedure for dividing fractions. This different approach to the previous workshop activity (focussing on reciprocal as an inverse multiplier and its role in the division of fractions) reveals a (quotitive) interpretation of the reciprocal of a number as: ‘how times the number goes into 1’. In some cases teachers noted that previously they had not been convinced of the value or need to know why the procedure works, though the use of the bar here and the challenge to make sense of the procedure had in many cases changed their minds.

Lesson 3cd: Exploring multiplicative structures SK focus:

Interpretation of division and multiplication as grouping or sharing Different models for representing multiplication and division

Lesson 3cd pre task activity: Division as grouping (quotitive) and division as sharing (partitive) Here the subject knowledge discussion for teachers returns to interpretations of division encountered in earlier workshops. Teachers complete an activity using the rod diagrams to divide 100 by different numbers in different ways (two examples of the tasks are given below).

As before many teachers were able to reflect on and distinguish between interpretations of division as either sharing or grouping in each case. In working through the pre lesson activity teachers had particularly rich discussions related to making sense of the interpretations when the numbers (divisors) were changed to decimals – e.g. partitive interpretation: how do you divide something into 2.5 pieces?

Lesson 3cd Stage 1 activity Teachers complete the lesson activity below where they discuss how well each representation for 3 x 8 fits the story.

Response: By working across representations the discussion gave insight to interpretations of multiplication. The key to this activity occurred when they were challenged to consider the suitability of each representation when the numbers were changed e.g. 3.2 x 8 or 3.2 x 8.6 etc. Here the discrete models became increasingly more difficult to interpret and again the significance of models for multiplication as scale factor enlargement was noted by the teachers.


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Lesson 3cd Stage 2 activity Teachers were asked to draw diagrams to represent ⅝ x ¼ then ⅝ ÷ ¼; this was a challenging and rich activity. Many teachers were able to justify a representation for a quotitive (grouping) interpretation of ⅝ ÷ ¼ but many struggled to provide a partitive (sharing) interpretation – how can you share something into a ¼ number of bits? Here discussion in some cases centred on extrapolating from sharing into increasingly smaller whole numbers, allowing ⅝ to be seen as the ¼, so what is the whole?

Lesson 3ef SK focus:

Interpreting the meaning of multipliers and chains of multipliers in fraction and decimal form

Simplifying such expressions to show equivalence

Activity: Working on the making orange juice activity

Teachers engage in the main activity of the lesson via a pupil slide show. This introduces the context of a citrus farm growing orange trees for the production of orange juice (above left). The key task is to use this to complete a data table (above right) which has titled columns relating to key variables from the context. This involves inputting key information correctly then using ‘between column’ and ‘between row’ relationships to complete the empty spaces. Response: This was a popular activity and lesson with the teachers. Teachers found the table (which was recognised as a ratio table - due to the constant ratio between data in a particular pair of columns) illuminating. Generating horizontal multipliers and vertical multipliers to calculate empty spaces and particularly comparing and interpreting them in terms of the context i.e. a rate or a dimensionless scale factor; the importance and relevance of units in interpreting meaning of multipliers in solving the problems; and the power of generating alternative solutions and showing their equivalence. Many had not appreciated the versatility of the mathematics here. Points included:

Simplifying chains of multipliers involving fractions.

Not all teachers at first were sure how to construct a multiplier from one number to another, some thinking in terms of ‘the number you’re going to, divided by the number you have come from’. Some were able to consider the problem here as scaling down to 1 (multiply by the reciprocal), then scale up to the required number (multiplying reciprocal by this number) giving the multiplier as a fraction.

Final workshop (workshop 5) For the final workshop there are no remaining teaching units to be introduced. So once the lesson study feedback on the teaching of unit 3 has been completed the workshop focusses on reflecting on both pedagogical and subject knowledge professional development progress by teachers from the project. This is done through specifically designed evaluation processes and consideration of next steps. However opportunities are provided on subject knowledge through two further key sessions.


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These included an activity focussed on defining ‘necessary and sufficient conditions’ for a direct proportion relationship between two variables to exist and an activity considering how a particular problem is modelled across a number of different representations that have been considered in the project. SK focus:

Necessary and sufficient conditions for proportionality

Exploring a proportional problem across representations

Combining intermediate steps (operations) to a single operation

Proving the distributive law for proportional functions

Activity 1: Necessary and sufficient conditions for proportionality:

Here teachers are given that y α x and asked (working in pairs on A3 paper) to construct as many resultant statements regarding y and x as they can. The task is followed by considering which of them is necessary but also sufficient i.e. equivalent.

Response: Many teachers included ‘doubling x doubles y’ as a necessary but also sufficient condition for proportionality but this is in fact not true as there exist counterexamples (though fairly sophisticated as functions and an example can be provided from the materials). In fact discussion here engaged significant interest, with many teachers wanting to know the function, having attempted some possible speculation as to what it might be (it is possible to generate –geometrically – many graphs with the property of ‘doubling x doubles y’ that are not straight-line graphs through the origin and thus not a proportional relationship). This led teachers to refine this particular condition to one that was necessary and sufficient e.g. ‘a proportional

change in x results in an equivalent proportional change in y’.

Activity 2: Exploring a proportional problem across representations

Again teachers worked in pairs on A3 paper centred on a typical GCSE proportional problem (see slide below). Each diagram on the paper represented a visual model which would support a solution from a different mathematical perspective (e.g. algebraic, geometric, etc.) with teachers tasked to discuss how the key calculations occur in each representation and in some cases complete the model and calculation for the answer.

Response: This is potentially a very powerful activity however this received a mixed response where it was used, suggesting the task should be adapted or introduced differently. Many teachers responded that the multitude of representations revealed the interconnectedness of mathematics. It was clear that some approaches seemed more natural/suitable for the particular problem being looked at; however the different mathematical perspectives shed light on the structure of the problem leading potentially to greater insight and deepening understanding, not just of the problem but also the different

approaches e.g. graphs, algebra, etc. This challenged previous views (first broached in workshop 1) that there, ‘… should only be one common approach to solving a particular type of problem so as not to confuse children’ compared to the point of view where actively asking pupils to solve and justify a given problem from a number of different approaches and compare solutions might deepen understanding. Here the task prompted discussion on teaching for procedural fluency versus teaching for conceptual understanding. This also included reference to the 2014 national curriculum where procedural and conceptual understanding should be taught in parallel. Possible extension

Consider which particular representations best suit the given problem

Construct a similar multi representation for a different problem


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Further considerations

Could this activity be used with your department?

What purpose or insights does it bring?

Could such an activity be useful to use with pupils? Who? When?

Other subject knowledge activities provided In many cases there was insufficient time in the final workshop for their inclusion or completion but these might be considered as post workshop tasks or scheduled to ensure they do take place during the workshop as perhaps important subject knowledge assessment tasks for teachers’ subject knowledge.

Activity sheet 1 – Combining intermediate steps (operations) to a single operation Often pupils will solve a proportional problem by finding a number of intermediate steps e.g. for 15% of 60 they might find 10% of 60 = 6, then 5% of 60 = 6/2 = 3 and hence combine to give 15% of 60 = 6 + 3 = 9. How can you prove this is correct and can you show this as a single operation e.g. 0.15x 60 etc.?

Activity sheet 2 – Proving the distributive law for proportional functions

mathematics subject knowledge supplement · mathematics subject knowledge supplement the activities...

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