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Introduction to Lesson 1:

Multiplicative reasoning

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Aims of professional development

Teachers should

be prepared to teach Lesson 1: Multiplicative reasoning;
understand how the lesson is developed and brought to a close in ways that values and builds upon what students already know;
understand how to work with students’ prior learning about multiplicative reasoning;
know what they need to do next.

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Reminder: Teaching for Mastery key principles

https://www.masteringmaths.org/mm-approach/key-principles

Visit the Mastering Maths website to read more about the Key Principles

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Overview of Lesson 1:

Multiplicative reasoning

When solving proportional problems, students often have just one strategy, most commonly an additive approach. This lesson uses the context of hotel stays and a double number line to explore different ways of solving these problems using both additive and multiplicative approaches. Students are asked to think about the approaches in terms of efficiency for solving proportion problems.

Using representations to provide access to the mathematical structure of a problem is a key principle of teaching for mastery (Key Principle 1). In this lesson, the double number line is used to illustrate and compare the thinking that relies on addition with that which relies on multiplication.

Proportion problems can crop up in many places in the curriculum, e.g. exchange rates, distance–time, rates of pay, conversion between units and similar triangles. This lesson could be used effectively in any of these different places in the curriculum, emphasising that, while the context may change, the approach stays the same. This helps to support students in developing a connected understanding of mathematical topics, which is an important aspect of teaching for mastery.

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Objectives of Lesson 1: Multiplicative reasoning ��

Students attempt a multiplicative reasoning problem and look at four different approaches

Students solve the problem in their own way and then share approaches

Students use their prior knowledge and understanding

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Lesson 1: Outline plan

Where does the lesson value and build on prior learning?

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Lesson 1: Research questions

Pedagogic focus

How is the lesson developed and brought to a close in ways that values and builds upon what students already know?

Maths focus

What evidence do you observe of students’ prior learning and how do they work with or modify this?

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Lesson 1: Multiplicative reasoning

Introduction and Discuss 1

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Teacher activity: think about this

How would you do this question?

How might your students do it?

Note as many different approaches as you can think of.

What is your response to these questions?

Tell the teachers that the lesson begins by introducing the context of the owner of a hotel charging the same price per night, regardless of how long someone wants to stay (this is an important assumption in this task). They tell students that the charge for a 2-night stay is £50. Now teachers ask half the students to find the cost of a 5-night stay, and the other half of the class to find the cost of an 11-night stay. Students are given a few minutes on their own to think about how to tackle the problem, and then they discuss what they did with a partner. As students work through the problem, teachers should identify any students who use one of the four approaches described on Slide 4. These students can be called on when these methods are covered in Discuss 1.

Give the teachers Handout Proportional Reasoning Example. Ask them to respond to the questions on the handout and follow this up with a short discussion.

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Handout: Exploring student thinking

Work out what the student has done

Represent the thinking on a double number line

Attempt to classify the approach as additive, scale factor, unitary or rate

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Exploring student thinking

Using a strip diagram to find the cost of one night - Value their thinking and build on it.

A process to be followed – What are students understanding?

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Exploring student thinking

Mixed thinking – Multiplier method to get £250 – Additive method to get £25

Dropped units - Confusion

A process to follow – What are students understanding?

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5 nights and 11 nights: Watch the video

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Now watch1521 Lesson 1 Introducing the DNL ONLINE.mp4 (about 6 minutes) showing students working and then the teacher discussion. The video demonstrates how one teacher approached the activity and the discussion, and, if you feel that teachers understand the design of this part of the lesson well enough, you might like to skip Slides 14 to 17. Depending on time, consider showing the video more than once.

Ask the teachers to consider the research questions as they watch the video:

How is the lesson developed and brought to a close in ways that values and builds upon what students already know?

What evidence do you observe of students’ prior learning and how do they work with or modify this?

Also ask them to make a note of anything they find interesting, and in particular, some of the approaches used by the students.

You might like to have some questions ready to provoke discussion, for example:

What different approaches did you observe?

In which ways did the teacher value students’ different approaches?

How did students explain their approaches?

How did the teacher use the double number line to explain students’ thinking?

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Video, 5 nights and 11 nights: discussion

What did you notice?
Did anything surprise you?

What different approaches did you observe?
In which ways did the teacher value students’ different approaches?
How did students explain their approaches?
How did the teacher use the double number line to explain students’ thinking?

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Class methods for 11- night stay

This section of the lesson discusses the proportion problem that students worked on in the previous section, as seen on the video.

A double number line is used to illustrate students’ own approaches , as the teacher did in the video, and to examine alternative approaches that may not have been used within the class. Mathematical representations, such as double number lines, help students to develop an understanding of multiplicative structure (Key Principle 1).

The teacher asks students to explain their method for finding the cost of an 11-night stay, capturing the different methods on the double number line provided on this slide. Emphasise to the teachers that the double number line is used to capture the students’ thinking and is not something to ‘be taught’.

It is common for students to use an additive approach. For example, students might multiply the cost of a 2-night stay by 5 (£50 × 5 = £250), find the cost of 1 night (£50 ÷ 2 = £25) and then add these two values together (£250 + £25 = £275).

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Four approaches for discussion

These are four slides, but there are also two summary slides. Would you go straight to the summary slides?

The four slides shown here are used to explain the four different approaches. Discuss with teachers how useful they would find them, and how their students would respond to them. Note that all four are also provided on one slide and it may be that teachers decide to skip these four slides and go straight to the summary slide.

Ava uses an additive approach. Ava has added the cost of 2 nights twice to find the cost of 4 nights (£100). She has then halved the cost of 2 nights to find the cost of 1 night (£25). Adding these together gives the cost of 5 nights (£125). �To find the cost for 11 nights, she has used 5 nights + 5 nights + 1 night �(£125 + £125 + £25 = £275).

Bahir uses the unitary approach. Bahir has worked out the cost of one night and then multiplied by 11.

Catia: Rather than working along the number lines, this method looks at the relationship between the lines and gives the rate of £25 per night, a multiplier that is true for all corresponding points on the number lines.

Danny has worked out that 2 is multiplied by 5.5 to get to 11. He has multiplied £50 by 5.5.

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Different approaches for discussion

Summary slides.

How would you use them?

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Direct proportion

What is the purpose of this slide?

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The role of class discussions in Mastering Maths

Valuing and building on prior learning (KP2)

A focus on development of understanding of mathematical structure (KP1) and the role of representations to provide insight (see Handout Models of structure)

Fluency and understanding of key ideas (KP4)

Everyone believes everyone can succeed (KP5)

Give it enough time (it’s an important part of the Teaching for Mastery approach)

Conclude this phase of the cluster meeting by discussing how the class discussion contributes to the Mastering Maths approach in FE.

It values and builds on the prior learning of students (KP2), by drawing out and developing their thinking. It is important that teachers allow students to use approaches with which they are comfortable, and to avoid teaching them a ‘new’ method.
It uses representations (double number lines in this case) to provide insight into mathematical structure (KP1). As the discussion progresses, students will distinguish between additive and multiplicative approaches, for example. (You could distribute Handout Models of structure as you discuss this. Ask the teachers to read it in their own time and keep it for reference.)
Related to this, it promotes an understanding of key ideas. Students need to be fluent with their methods, but they also need to understand what they are doing and why (KP4).
It provides opportunities to develop a culture in which students’ voices are valued and everyone is made to feel that they can succeed (KP5).��Emphasise that it is important to allow enough time for the discussion.

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Lesson 1: Multiplicative reasoning �Explore 1 and Discuss 2

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Lesson 1: Multiplicative reasoning �Teachers working in pairs

Discuss:

working along the rows and between the rows
why a table is used
why the order of the questions matters
what the point of question f is
what students might struggle with

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Take the role of students

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Lesson 1: Design of the task

Order of the questions

The role of the table

Mathematical structure

Class discussion

What is the point of Question f?

This slide is optional but you might like to use it to structure your whole-group discussion which follows the teacher pairwork.

Ask the group if they think students might work between the rows of the table. When the lesson was observed in the trials, most – or even all – students worked along the rows, treating the cost in Euros and the cost in South African Rand separately. However, the way the questions are ordered intends to encourage working between the rows. Discuss whether this is the point of question f.

When discussing question f, also ask the teachers what might confuse the students about this question. There is nowhere to write an answer!

Explain that the table is used because the transition to the double number line is easy.

Ask them for their thoughts about

The ordering of the questions (They have been designed, using conceptual variation, to guide students to complete the table in a way that explores different relationships in a sequential way.)

The role of the table (It is designed with the feel of a double number line, to emphasise the relationships both along each row and between the rows.)

How mathematical structure is emphasised (through the ordering of the questions and the use of the table)

Their role when the students are working in pairs (They should observe students as they complete this task but should not intervene unless necessary. As students work on the task, they should notice how they illustrate their strategy and identify those with different approaches could be called upon to explain in the next part of the lesson.)

How they would bring out the key ideas in a class discussion. (The focus of this discussion is not on whether students have arrived at the correct answers to the questions, but rather on the approaches and the relationships used to answer the questions. Asking students with different approaches to a question to share their thinking gives value to their strategies and encourages a collaborative community in which everyone believes everyone can succeed (Key Principle 5). Teachers could ask pairs of students (identified earlier) with different approaches to share their approaches to the task. The teacher might reinforce their explanations using double number lines on the board or use those provided).

Point out to teachers that two pdf versions of this task are provided: one portrait and one landscape. Tell them that in the trials, some teachers stuck the task sheets in the middle of a sheet of A3, which gave the students space for working. This can be seen on the video you are about to watch.

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Lesson 1: Explore 1

Students work in pairs

Explicit guidance about how they should work (developing a collaborative classroom culture)

Try not to intervene – but when should you?

This slide introduces the pair-work task to the students.

Teachers ask students to work in pairs and give each pair a copy of the ‘Cost of stay’ handout. It is important that students answer the questions in order because they have been carefully designed to guide them to complete the table in a way that explores different relationships in a sequential way.

Encourage teachers to make sure that students explain their approach to their partner, working together to agree a solution and develop their own understanding. If they notice that one student is answering all the questions, or they are not working collaboratively, they should ask a student to explain a question that has been answered by their partner.

Teachers tend to find it difficult to observe students as they complete this task without intervening. Ask the teachers in your group when they should intervene.

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Students work on the pair-work task: watch the video

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Video: discussion

What did you notice?
Did anything surprise you?

How does the teacher encourage collaboration?
How do the students work together?
How did the students show their working?
How and why did the students struggle with questions e and f?

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Capturing thinking: ways of working

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Would you discuss this third double number line?

The slide shown here is provided for use in a whole class discussion. Encourage teachers to use at least some of the double number lines to illustrate students’ thinking. It is not necessary to use all three double number lines. Some teachers like to draw their own number lines.

Point out to the teachers that the top pair of lines replicate the top two rows of the table, the middle pair replicates the top and the second row and the third pair of lines replicates the bottom two rows. If the top two pairs have been completed, then the answers could simply be copied to the third pair. The third pair compares the currencies and the multiplier between the lines is the exchange rate, but point out the the teachers that the exchange rate is explored in more detail in the next part of the lesson, so they might not want to discuss this at this stage.

Students are likely to want to check that they have answered the questions correctly but ask teachers to encourage them also to share their approaches to the task.

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Video: Students work on the DNLs

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Now watch this video: 1541 Lesson 1 Using the DNL ONLINE.mp4 (about 2½ minutes). The teacher in this video explains that the double number lines can be used to capture the information in the table, and asks the students to draw out the three sets of lines and fill in the missing boxes.

Ask the teachers to consider the research questions as they watch the video:

How is the lesson developed and brought to a close in ways that values and builds upon what students already know?

What evidence do you observe of students’ prior learning and how do they work with or modify this?

Also ask them to make a note of anything they find interesting, and in particular, anything they think exemplifies a mastery approach in FE.

You might like to have some questions ready to provoke discussion, for example:

How do students use the double number lines to explain their own thinking?
What different approaches did you notice?
What do students appear to find difficult, and what would you do to help them?

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Students work on the DNLs: discussion

What did you notice?
Did anything surprise you?
How do students use the double number lines to explain their own thinking?
What different approaches did you notice?
What do students appear to find difficult, and what would you do to help them?

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Lesson 1: Multiplicative reasoning

Explore 2 and Discuss 3

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Exchange rates: Direct proportionality

What questions would you ask about direct proportionality?

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Exchange rates: Converting currencies

Would you ask the students to draw out the double number line?

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Along the lines or between the lines or not at all?

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Explore 2

Discussion of approaches – promotes coherence (KP3)

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Lesson 1: Multiplicative reasoning

Explore 3 and review

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Explore 3

How would YOU do this?

How would your students do it?

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Thinking about the activity

How does this task develop fluency?
Will students work out the speed? Is this necessary?
What is the between-lines multiplier? What does it represent?
How were the numbers chosen?

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Sample student responses

What have these students done?

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Closure

How important is this slide?

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Practice question

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Lesson 1: Multiplicative reasoning Closing thoughts

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Lesson plan and PowerPoint

What can you change in the lesson and what should NOT be changed?

What do you need to do to prepare for teaching the lesson?

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Lesson 1: Research questions

Pedagogic focus

How is the lesson developed and brought to a close in ways that values and builds upon what students already know?

Maths focus

What evidence do you observe of students’ prior learning and how do they work with or modify this?

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Aims of professional development

Teachers should

be prepared to teach Lesson 1: Multiplicative reasoning;
understand how the lesson is developed and brought to a close in ways that values and builds upon what students already know;
understand how to work with students’ prior learning about multiplicative reasoning;
know what they need to do next.

For a reminder of what has been discussed today, please have a look at the self-study materials on Desmos

https://www.masteringmaths.org/mm-approach/self-study-materials

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Thank you