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Lesson 4: �Algebraic thinking in problem solving

Objectives

Represent contextual problems mathematically
Use diagrams to represent mathematical structure
Determine the value of an unknown in a problem
Solve problems involving angles

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Making walls out of blocks

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Em

Finch

Twice as long as Em’s wall

Gill

Three times as long as Em’s wall

What are possible lengths for the three walls?

Tell students that Em, Finch and Gill are making walls out of blocks. �Finch builds a wall that is twice as long as Em’s wall and Gill builds a wall that is three times as long as Em’s wall. Ask students to show possible lengths for Em’s, Finch’s and Gill’s walls on their mini whiteboards.

Tell students to hold up their mini whiteboards showing the number of blocks in each wall. If all students have identified the same three walls (for example, the simplest case: 1 (Em), 2 (Finch) and 3 (Gill)), ask a couple of different students to explain their reasoning. If there are different walls identified within the class, ask students to explain their thinking.

If necessary, check your students’ understanding by asking further questions, for example:

A wall is 4 blocks long. Whose wall could this be? �(It could belong to Em or Finch, but not Gill)
Suggest a wall that could be made by Em and Gill, but not Finch. �(e.g. 3, 9, 15, 21, which are the odd multiples of 3)

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Em’s, Finch’s and Gill’s walls

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Em

Finch

Twice as�long as�Em’s wall

Gill

Three times as long as�Em’s wall

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How many blocks?

4

Em

Finch

Twice as long as Em’s wall

Gill

Three times as long as Em’s wall

How many blocks will Finch need?
How many blocks will Gill need?
How many blocks will Em, Finch and Gill use altogether?

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Total number of blocks

5

Em

4 blocks

Finch

Twice as long as Em’s wall

8 blocks

Gill

Three times as long as Em’s wall

12 blocks

Total� 24 blocks

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Finding the number of blocks

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DISCUSS

Handout�available

Lara, Meena and Noah are answering this question.

Em, Finch and Gill build a wall each.

Altogether they use 60 blocks.
Finch’s wall is twice as long as Em’s wall.
Gill’s wall is three times as long as Em’s wall.

How many blocks are in each of their walls?

Show the question given on the slide and allow the students a few minutes to think about it, either individually or in pairs. Then explains that the ‘Approaches’ handout shows three different approaches to working out how many blocks Em, Finch and Gill will each use, given that there are 60 blocks. Tell students to work in pairs and give each pair a copy of the handout. Ask students to look at, and discuss the three different approaches and to answer the questions below each approach.

After a few minutes, bring the class together. Ask students which of the three approaches they found easiest to understand. Would they have used one of the approaches described, or taken a different approach? Encourage students to listen to one another as they share their thinking with the class and consider other viewpoints.

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Lara’s approach

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REVIEW

What has Lara done?

What could Lara have done to save some time?

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Meena’s approach

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REVIEW

What has Meena done?

What does Meena still need to do to answer the question?

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Noah’s approach

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REVIEW

What has Noah done?

How did Noah use his diagram to find the number of blocks that Em used?

Noah’s method introduces a bar model diagram. When discussing Noah’s approach, spend some time discussing the bar model first, before looking at the answers.

Many students have no experience of creating and using bar models, so you may need to spend some time unpicking exactly how Noah constructed his bar model. Make sure that students understand how Noah knew to draw 6 parts in his diagram, and where the 60 came from. Emphasise that the 6 parts and the 60 end at the same place: the 6 parts make 60 altogether. (He used 1 part to represent the number of blocks in Em’s wall, 2 parts to represent the number of blocks in Finch’s wall and 3 parts to represent the number of blocks in Gill’s wall, resulting in a total of 6 parts. There are 60 blocks in total.)

Ask students how Noah used his bar model to find out the number of blocks that Em, Finch and Gill each used. (Noah divided the total of 60 blocks by the number of parts (6) to get the number of blocks in Em’s walls (10). He then calculated the sizes of 2 and 3 parts to find the corresponding number of blocks in Finch’s and Gill’s walls.)

When all three approaches have been discussed with the class, ask students what is the same and what is different about these methods. Draw out that all three approaches identify the base value (10) and relate the lengths of Em’s and Gill’s walls to this.

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Building train tracks

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DISCUSS

Em, Finch and Gill use 25 pieces of track.
Finch’s track is twice as long as Em’s.
Gill’s track is 5 pieces longer than Finch’s.

How long is each of their tracks?

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Modelling relationships

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DISCUSS

Em, Finch and Gill use 25 pieces of track.
Finch’s track is twice as long as Em’s.
Gill’s track is 5 pieces longer than Finch’s.

Lara’s bar model

Meena’s bar model

Whose bar model�is correct?

After a minute or so, tell students that Lara and Meena have tried to draw bar models to represent the relationships between the numbers of pieces in the three tracks. In their pairs, can they decide whose bar model is correct?

After a couple of minutes, ask for a show of hands for those that think Lara is correct and for those who think that Meena is correct. If there is agreement, ask a couple of students to explain their reasoning. If there is disagreement, ask pairs of students with differing views to share their thinking.

Emphasise the importance of considering problems carefully to ensure that the structure of the relationships between values is correctly identified. If students are struggling, encourage them to consider whose model correctly shows that Gill’s track is 5 pieces longer than Finch’s track. Confirm that Meena’s model is correct, and that Lara’s is not.

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Modelling relationships

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DISCUSS

Ollie’s bar model

Pippa’s bar model

What have Ollie and Pippa done wrong?

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Using a bar model

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REVIEW

Em, Finch and Gill use 25 pieces of track.
Finch’s track is twice as long as Em’s.
Gill’s track is 5 pieces longer than Finch’s.

Meena’s bar model

How long is each of their tracks?

Ask students to work in pairs to use Meena’s bar model to answer the question. After a few minutes, ask a couple of pairs of students to describe their approach.

Check whether students have correctly identified that ? represents 4 pieces, and check whether students have correctly identified the number of pieces in each of the three tracks (Em: 4 pieces long, Finch: 8 pieces long, Gill: 13 pieces long).

Students often lose sight of the original problem during the solution process and it is important to remind them of the need to relate any solution that they obtain back to the question posed. In this case, the length of each of the three tracks needs to be worked out and stated.

It is important that students understand how the bar model is used to solve the problem because they will depend on this understanding later in the lesson.

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Working in pairs

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YOUR

TURN

Take turns to answer questions.
Explain your reasoning to your partner.
Your partner must agree, disagree or ask for more explanation.

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Describing diagrams

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YOUR TURN

Handout�available

Look at the geometry diagrams in rows �A and B and on cards C1 and C2.
Write down what you know about the angles in each diagram.

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Geometry diagram (row A)

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Geometry diagram (row B)

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Geometry diagram (card C1)

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Geometry diagram (card C2)

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Placing the cards

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YOUR TURN

Handout�available

Place the cards �C1 to C4 in the table.
Draw bar models to represent the maths.
Find the value of the unknowns.

Explain that the task also includes four word questions with the same relationships between the unknowns as those in the geometry diagrams. Explain that this means that the same bar model can be used to represent the geometry question and the word question.

Ask students to:

place cards C1 to C4 in the table
draw bar models that could be used to represent the maths.

Some students may prefer to do the card matching first, and others might choose to draw the bar models first. Either way, they should take turns to fill a cell in the table, explaining their reasons to their partner.

Students may not complete all four rows of the table and it is not essential that they do so. If students are struggling, ask them to focus on rows A and B as the relationships between the unknowns in these two rows mirror those explored earlier in the lesson. Also explain that they need to find a ‘base value’ that all the other values relate to (one question mark in Meena’s bar model, for example). It may be helpful to fold the table in half if students are finding working with all four rows overwhelming.

As you circulate around the class, observe what the students are doing, but try not to intervene, other than reminding them that, in each row, the word problem and the geometry diagram are essentially the same. Notice how different pairs approach the task and use your observations to inform the discussion in the next part of the lesson.

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Completing row A

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REVIEW

A	Word problem	Geometry problem	Bar model
	C4 Ingrid is twice as old as Jay. Hugh is 3 times as old as Jay. The sum of Hugh’s, Ingrid’s and Jay’s ages is 180. How old is Jay? Jay is 30 years old.	y = 30

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Completing row B

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REVIEW

B	Word problem	Geometry problem	Bar model
	C3 A ruler costs twice as much as a pencil. A pen costs 5p more than a ruler. The total cost of a ruler, pen and pencil is £1.80. Find the cost of a pencil. A pencil costs 35 pence.	x = 35

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Completing row C

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REVIEW

C	Word problem	Geometry problem	Bar model
	Lee gets 10 more marks than Kia in a science test. Kia and Lee score a total of �90 marks. Calculate Kia’s score. Kia scored 40 marks.	C2 z = 40

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Completing row D

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REVIEW

D	Word problem	Geometry problem	Bar model
	Abdul sells 360 bags of sweets. He sells twice as many jellies as chews, 5 times as many fruit drops as chews and 45 more wine gums than chews. How many bags of chews does he sell? 35 bags of chews	C1 w = 35

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Mathematical structure

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?	?	?	?	?	?
180

Row A

?	?	?	?	?	5
180

Row B

?	?	10
90

?	?	?	?	?	?	?	?	?	45
360

Row C

Row D

Different problems with the same mathematical structure can be represented using the same bar model.

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Practice questions

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YOUR TURN

Handout�available

Find the value of x.

………………. (3)

54

Hilary, Imogen and Jeeha are playing a game with cards.

Imogen has 3 cards more than Hilary.

Jeeha has twice as many cards as Imogen.

They have a total of 53 cards.

Work out how many cards Hilary has.

………………. (4)

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Q9 from June 2017, 1MA1/2F

Q10 from June 2015, 5MB3H/01

Distribute a copy of the practice questions to each student and give them a couple of minutes to come up with a solution to either one or both of the problems. Make sure that students understand that these are two separate problems, not a pair of problems with the same underlying mathematical structure.

Notice whether students draw a bar model to represent the problems. If they do, do they use 360 as the total for the bars in the first question, or do they simplify the problem straight away by considering the two unknown angles adding up to 270? Once students have found a value for x, encourage them to check that 2x° + 3x° + 90° = 360°.

Notice whether students use ‘Hilary’ as the base value for the second question. It is possible to draw a bar model using ‘Imogen’ as the base value, but students may run into problems representing the fact that Hilary has three fewer cards.

Deepening understanding The first practice question involves a 2x unknown and a 3x unknown, but the mathematical structure of the problem is different to the problems in row A of the task / Em’s, Finch’s and Gill’s walls in the introductory activities (which also involve a 2x and a 3x). Can students explain why this is? Do students recognise that the base value (x) is calculated but does not actually feature in the bar model?

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Representing the problem – 1

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Why are all the diagrams correct?

How can they be used to find x?

Lara

Meena

Noah

?	?	?	?	?	90
360

3x

2x

x	x	x	x	x
270

2x	3x	90
360

Begin by asking students how they answered the questions. It is important that you spend a bit of time on what they did as well as what the correct answers are. Ask students that drew a bar model to describe their thinking and explain in what ways their bar model helped them to make sense of the problem.

Show students possible bar models to represent the practice questions and discuss as a class the questions related to these possible bar models. For this first question, three different bar models are given, and students should understand that although they are different, they are all correct. Make sure that they know how to use the bar models to find the value of x.

Meena’s simplifies the first problem by subtracting the right angle from the total, so that angle sum is 270° rather than 360°. Both Lara’s and Meena’s bar models make the need to divide by 5 clear and Noah’s bar model makes the unknown value of x explicit.

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Representing the problem – 2

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Hilary, Imogen and Jeeha are playing a game with cards.

Imogen has 3 cards more than Hilary.

Jeeha has twice as many cards as Imogen.

They have a total of 53 cards.

Work out how many cards Hilary has.

Noah’s first diagram

H	H	3	H	3	H	3
53

Jeeha

Hilary

Imogen

Noah’s second diagram

H	H	H	H	9
53

Why did Noah re-draw his diagram?

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Lesson review: �Algebraic thinking in problem solving

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Objectives

Represent contextual problems mathematically
Use diagrams to represent mathematical structure
Determine the value of an unknown in a problem
Solve problems involving angles

Suggested further steps/areas to work on

Solve linear equations involving negative terms

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Lesson 4: �Credits

Text acknowledgements

Pearson Education Ltd: Pearson Edexcel GCSE (9 – 1) In Mathematics (1MA1) Foundation (Calculator) Paper 2F (June 2017)

Pearson Education Ltd: Pearson Edexcel GCSE (9 – 1) In Mathematics (5MB3H/01) Foundation (Calculator) Paper 3 (June 2015)

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