1 Multiplicative Relationships and Ways of Working 1 Professional Development Session 3

PD Session Outline / Structure Reflection Planning Preparation Reflection on teaching of a lesson dealt with during a previous PD session and student responses to the lesson. May include analysis of a video extract to provide a common point of reflection and discussion. Engagement with another lesson or collection of lessons (to be taught post this PD session). Focus on: lesson intentions; targeted misconceptions; deliberate lesson strategies to overcome these. Planning for teaching the lesson. Planning for cascading the PD session/materials to other members of the maths department.

Guiding principles behind the PD materials / sessions PD Investigate mathematical structures and contents Identify student methods and misconceptions Identify and engage strategies for dealing with these through engagement with ICCAMS lessons through engagement with sample student work and video extracts Elaboration of the ICCAMS lesson design principles use of models and multiple representations

Investigating different ways of working with multiplicative relationships 4

5 Multiplication Lesson 4A: Shadow Lengths Direct link to MR Lesson 4A

6 ACTIVITY 6: MR Lesson 4A - Post Shadows Direct link to MR Lesson 4A

7 Now focus specifically on the different problem scenarios presented on each of the slides shown on Page 6:Page 6  Identify and discuss how the problem scenarios are different and the specific aspect of understanding that is targeted in each scenario.  Also identify the different methods that can be used to solve each of these problems and the challenges and possible errors which students might encounter. ACTIVITY 6: MR Lesson 4A - Post Shadows Direct link to MR Lesson 4A

8  To prompt engagement with a scenario involving a proportional (multiplicative) relationship [one-way stretch].  To prompt extrapolation from a given situation to a similar situation involving an unknown. Lesson 4A – Post Shadows: Discussion - Slide 1a Direct link to MR Lesson 4A Slide 1a – Intention

9 Students find it very difficult to recognise ratio relationships in geometric settings, particularly when the relationship involves a fractional multiplier. × 2½ Lesson 4A – Post Shadows: Discussion - Slide 1a Direct link to MR Lesson 4A Slide 1a – Intention

10 Slide 1a – Targeted Misconceptions Lesson 4A – Post Shadows: Discussion - Slide 1a + 2 Direct link to MR Lesson 4A Do students see this as a multiplicative or an additive relationship? (Incorrect) additive approaches:  12 m increases to 14 m, so 30 m shadow must increase to 32 m

11 Do students see this as a multiplicative or an additive relationship? (Incorrect) additive approaches:  12 m increases to 14 m, so 30 m shadow must increase to 32 m  The shadow length on P is 18 m longer than the pole length. Using the same principle on pole Q, the shadow length must be 14 m + 18 m = 32 m Lesson 4A – Post Shadows: Discussion - Slide 1a +18 m Direct link to MR Lesson 4A Slide 1a – Targeted Misconceptions

12 This slide is designed to help students who see the scenario in Slide 1a as involving an additive relationship.  Students who use an additive strategy will make the short segment of the shadow 2 m.  However, ‘visually’ students should be able to see that the length of this shadow segment is longer than the length of the 2 m portion on the pole, thus conflicting with their additive approach. ≠ Lesson 4A – Post Shadows: Discussion - Slide 1b Direct link to MR Lesson 4A Slide 1b – Intention

13 What questions might we ask students here to help them to move their thinking forward or reflect on a (previous or current) incorrect additive strategy? Lesson 4A – Post Shadows: Discussion - Slide 1b Direct link to MR Lesson 4A Slide 1b – Targeted Misconceptions

14 This slide reinforces in a more explicit and directed way the fact that the shadow length of the 2 m section of pole must be longer than 2 m. This will hopefully prompt students who have used an addition strategy in Slide 1b (to get an answer of 32 m for the shadow length by using a length of 2 m for the short shadow portion) to rethink this method. Lesson 4A – Post Shadows: Discussion - Slide 1c ≠ 2m Direct link to MR Lesson 4A Slide 1c – Intention

15 What questions might we ask students here to help them to move their thinking forward or reflect on a (previous or current) incorrect additive strategy? Lesson 4A – Post Shadows: Discussion - Slide 1b Direct link to MR Lesson 4A Slide 1c – Targeted Misconceptions

16 This slide makes use of even more complex multiplier values for the relationship between post and shadow length to further test understanding of the multiplicative (ratio) nature of this relationship (for those who completed Slide 1a easily). Lesson 4A – Post Shadows: Discussion - Slide 2 Direct link to MR Lesson 4A Slide 2 – Intention

17 The more complex multipliers may prompt some students to resort to an (incorrect) addition strategy:  Pole Q is 2 m taller than P, so the shadow for Q must be 31 m + 2 = 33 m Lesson 4A – Post Shadows: Discussion - Slide Direct link to MR Lesson 4A Slide 2 – Targeted Misconceptions

18 The more complex multipliers may prompt some students to resort to an (incorrect) addition strategy:  Pole Q is 2 m taller than P, so the shadow for Q must be 31 m + 2 = 33 m Lesson 4A – Post Shadows: Discussion - Slide 2  From Slide 1a the length of the original shadow for Pole Q is 35 m. Since the shadow for pole P increases by 1 m the shadow for pole Q must also increase by 1 m to 36 m. 35 m +1 m Direct link to MR Lesson 4A Slide 2 – Targeted Misconceptions

19 The more complex multipliers may prompt some students to resort to an (incorrect) addition strategy: Lesson 4A – Post Shadows: Discussion - Slide 2  The shadow for Pole P is 19 m (i.e. 31 m – 12 m) longer than the pole. So, the shadow for Pole Q must also be 19 m longer, which gives 14 m + 19 m = 33 m. +19 m Direct link to MR Lesson 4A Slide 2 – Targeted Misconceptions

20 What questions might we ask students here to help them to move their thinking forward or reflect on a (previous or current) incorrect additive strategy? Lesson 4A – Post Shadows: Discussion - Slide 2 Direct link to MR Lesson 4A Slide 2 – Targeted Misconceptions

21 How might the (deliberate) design of this slide prompt students to reconsider an additive approach? Lesson 4A – Post Shadows: Discussion - Slide 3a The significant difference in the heights of the poles facilitates that a ‘visual’ check should confirm that an additive approach does not give the correct answer. Direct link to MR Lesson 4A Slide 3a – Intention 6 m 20 m + 6 m

22 Two further deliberate design features in this scenario facilitate the use of ‘rated addition’ as a strategy: Lesson 4A – Post Shadows: Discussion - Slide 3a  The longer Pole V is again 2½ times the length of the shorter Pole U.  The difference in heights between the poles is much bigger than in the scenarios on Slides 1 and 2. Direct link to MR Lesson 4A Slide 3a – Intention

23 ‘Rated Addition’ approach: 1.Working with the scalar relation (post to post): Every 4 m of pole casts a 10 m shadow. The 14 m pole can be divided into the following lengths: 4 m + 4 m + 4 m + half of 4 m Lesson 4A – Post Shadows: Discussion - Slide 3a This means that the shadow for the 14 m pole is 10 m + 10 m + 10 m + half of 10 m, which gives 35 m. Direct link to MR Lesson 4A Slide 3a – Intention 4 m 2 m 10 m 5 m

24 ‘Rated Addition’ approach: 2.Working with the functional relation (post to post): Every 4 m of pole casts a 10 m shadow. Lesson 4A – Post Shadows: Discussion - Slide 3a On the second pole: 4 m pole section → 10 m shadow + 4 m pole section → 10 m shadow + 2 m pole section → 5 m shadow Direct link to MR Lesson 4A Slide 3a – Intention 14 m pole → 35 m shadow 4 m 2 m 10 m 5 m

25 It is essential that students understand the difference between an (incorrect) additive approach and a rated (or repeated) addition approach. Lesson 4A – Post Shadows: Discussion - Slide 3a Direct link to MR Lesson 4A

26 ‘Rated Addition’ Lesson 4A – Post Shadows: Discussion - Slide 3a Direct link to MR Lesson 4A Slide 3a – Intention Incorrect Additive Approach (V1) 4 m 2 m 10 m 5 m 4 m 10 m 2 m 4 m 2 m

27 ‘Rated Addition’ Lesson 4A – Post Shadows: Discussion - Slide 3a Direct link to MR Lesson 4A Slide 3a – Intention Incorrect Additive Approach (V2) 4 m 10 m 4 m 2 m 4 m 2 m 4 m 2 m 10 m 5 m

28 How might comparing these two slides with students help to demonstrate the multiplicative (and non-additive) relationship between post height and shadow length? Lesson 4A – Post Shadows: Discussion - Slides 3a & 1a Direct link to MR Lesson 4A (The importance and usefulness of) Comparing Slides 3a and 1a

29 Slide 3a is the first instance where the length of the ‘shorter’ pole changes from 12 m (P) to 4 m (U). As such, comparing these two slides might reinforce for students the non-additive (multiplicative) nature of the relationship between pole height and shadow length. Lesson 4A – Post Shadows: Discussion - Slides 3a & 1a Direct link to MR Lesson 4A (The importance and usefulness of) Comparing Slides 3a and 1a

30 Lesson 4A – Post Shadows: Discussion - Slides 3a & 1a Shadow and height difference = 6 m Ratio of shadow to height = 10 : 4 = 2.5 : 1 i.e. Shadow is 2.5 times longer than pole Shadow and height difference = 18 m Ratio of shadow to height = 30 : 12 = 2.5 : 1 i.e. Shadow is 2.5 times longer than pole Direct link to MR Lesson 4A (The importance and usefulness of) Comparing Slides 3a and 1a

31 As with slide 1C, the inclusion of extra ‘partitions’ are intended to help students to visualise that each segment of shadow length (and, hence, the total shadow length) is longer than each corresponding section of pole height. Lesson 4A – Post Shadows: Discussion - Slide 3b ≠ Direct link to MR Lesson 4A Slide 3b – Intention

32 What questions might we ask students here to help them to move their thinking forward or reflect on a (previous or current) incorrect strategy? Lesson 4A – Post Shadows: Discussion - Slide 3b Direct link to MR Lesson 4A Slide 3b – Targeted Misconceptions

33 Slides 4a & b – Intention Lesson 4A – Post Shadows: Discussion - Slides 4a & b These slides prompt students to extrapolate and generalise their understanding of the (multiplicative) relationship between pole height and shadow length to another (unrepresented) scenario. Direct link to MR Lesson 4A

34 Slides 4a & b – Intention Lesson 4A – Post Shadows: Discussion - Slides 4a & b Here the focus is on the functional relationship between pole height and shadow length (i.e. between two different systems). Here the focus is on the scalar relationship between the heights of two poles and later their shadows (i.e. between quantities in the same system). Direct link to MR Lesson 4A

35 Slides 4a & b – Targeted Misconceptions Lesson 4A – Post Shadows: Discussion - Slides 4a & b  What methods might students make use of in each of these scenarios?  Likely errors or inappropriate strategies? Direct link to MR Lesson 4A