1. Materials developed by Paul Dickinson, Steve Gough & Sue Hough at MMU
A Question of Balance
Materials developed by Paul Dickinson, Steve Gough & Sue Hough at MMU

2. Thank you
Sue, Steve and Paul would like to thank all the teachers and students who have been involved in the trials of these materials
Some of the materials are closely linked to the ‘Making Sense of Maths’ series of books and are reproduced by the kind permission of Hodder Education

3. Note to teacher
This PowerPoint uses the context of a see saw and traditional weighing scales to encourage students to develop an appreciation of what it means if scales are balanced, what happens if items are removed from one side of balanced scales and what strategies can be employed to bring the scales back in balance. The leaners are asked to draw their own representations of the balance scale pictures and to consider how shorthand notation such as 3a + 12 = 8a + 2 can be thought of in the context of weighing scales. Students are encouraged to operate on their balance pictures by removing items (or adding items) from both side of the scales. (See slide 19 for Ellie’s method) The weighing scales context is not a new idea for teaching solving equations with ‘x’ on both sides, but here the context and the images associated with it are embraced in more detail. In trials some students were aware that this was giving them a fresh approach to tackling solving equations with x on both sides, which is a topic they had previously experienced little success with.

4. Who is heavier?
A gentle introduction. Students can be encouraged to embrace this context by additional questions such as ‘When is the last time that you went on a see saw?’ ‘Who was it with?’ ‘What’s the most amount of people you have ever been on a see saw with?’ …… and so on.

5. How could you balance the see saw?
Students make various suggestions as to who they would sit with who. In previous trials the see saw context was found to be important as a way of enabling students to tap into the concept of balance.

6. The human balance
Invite a student out to the front to act as a human balance. Place various objects on their ‘scales’ (their hands) and ask them to tilt according to the weight of the objects, as though they are a see saw or a set of balance scales.
Have a scenario where equal numbers of say, books are placed on both sides of their ‘scales’. Remove items from one side of their ‘scales’ and ask them to tilt accordingly. Then ask the question ‘ How could we make the scales balance again?’ The first response to this question is to say ‘Put the books back on’. But it is worth asking them to think of an alternative way to get the scales to balance. Eventually a student may suggest ‘removing the same amount of books from the other side of the scales. This is an important idea for what is to come.
In trials of this activity, sometimes when their empty scales are presented with a set of books placed on one side, people can have a tendency to raise the scales on the side where the books have been placed. Almost as though the heavier item needs to be higher up the scales. This was a surprise initially, but would seem to indicate how the concept of weighing scales is not necessarily well understood by the students.

7. How can you tell these scales are balanced?
This photograph may require some close scrutiny. For students unfamiliar with the context of traditional weighing scales, some discussion around the context may be helpful. Questions like ‘Where might you see weighing scales like these?’ and How do they work?’ can be useful. Or ‘what do the weighing scales we use now look like?’ may encourage students to think about weighing fruit in the supermarket and today’s electronic scales. Initially students may think that the scales are not balanced because they do not look the same on both sides. However close inspection of the bar which runs across underneath the trays would suggest that this is horizontally placed and therefore the scales are balanced.

8. What can you say about the flour?
In this picture the scales are lower down on the side of the weights which indicates that the weights are heavier than the flour. The next slide shows how much each weight weighs in ounces. Therefore it is possible to say that the flour weighs a little less than 16 oz + 1 oz + ½ oz

9. What can you say about the flour?
This is a very old set of weights. The 4 oz weight is missing. It is possible to weigh many many quantities using a complete set of weights.

10. What can you say about the flour?
In this picture the scales are lower down on the side of the flour which indicates that the flour is slightly heavier than 16 oz + 1 oz + ¼ oz
Therefore the weight of flour is between 17 ¼ amd 17 ½ oz

11. Describe what you see in the picture below
This picture would be described as a model or representation of the context of the traditional weighing scales seen in the previous photographs. It is interesting to note whether the students can make the transition from the actual photograph to this picture version. Is it obvious to them from the picture that the flour must weigh 25 oz?

12. What can you say about this bag of flour?
At a glance students may be tempted to say that this bag of flour weighs 22 oz. Follow up questions such as ‘How do you know?’ may encourage them to scrutinise the picture further and notice that the scales are not actually balanced. Therefore it is only possible to say that the flour weighs more than 22 oz. (i.e flour > 22 oz)

13. What can you say about this bag of flour?
Again with close scrutiny students should notice that the scales are lower down on the right hand side. This suggests that the flour weighs less than 12 oz . (flour < 12 oz, or 12 oz > flour.)

14. What about this bag of flour?
This image represents two shifts from the previous pictures. Now we have a hand drawn version of the traditional weighing scales, with the trays represented by a horizontal line.
Secondly the left hand ‘tray’ now contains weights as well as flour .
2 oz + 4 oz + flour balances with 16 oz.
This implies that the flour must weigh 10 oz, otherwise the scales would not be drawn in balance.
Worksheet A8 consists of more problems of this type.
Now try the problems on Worksheet A8

15. Describe what has happened from one picture to the next
In trials students were quick to point out that a pepper weighs 6 ounces. The focus here however, is what has happened from one scales picture to another. From the first picture to the second, ounces have been removed from both sides of the scales. A useful question is to ask whether the scales would remain balanced like they are in the picture if you had removed ounces from both sides of the scales. ‘What would happen to the scales when you removed 5 ounces from the left hand tray? Would that side of the scales go up or down? What happens to your friend on the other side of the see saw when you get off one side?
In other words, ask supplementary questions which help the students synthesise the idea about how to get the scales balanced again, once you start taking things off the scales.

16. Draw your own version of this picture
Students’ drawing will vary. Here is one possibility: (make sure you are in ‘Notes page’ view in order to view this image)
Notice how this student has drawn the ounces as one block of weight. Some students will draw individual ounces. It is worth comparing student’s pictures, using a visualiser or mini whiteboards and asking questions like ‘Which picture shows the most detail?’ ‘Which picture was quickest to draw?’ and so on. Sharing pictures may help some students to refine the way they draw their scales pictures.

17. Describe how you can find the weight of one orange
Invite a student to the board so that they can describe and point to the objects they are referring to. Keep alive the image of the scales tipping as objects are removed from one side of the scales and ask how they can be put back in balance. Interestingly, students do not always approach this problem by removing objects from both sides of the scales. Two possible student approaches are shown on the next 2 slides.

18. Describe Gary’s thinking
Gary’s method
This is the most common type of approach where students remove equivalent objects from both sides of the scales to reveal the weight of (in this case) one orange.
Describe Gary’s thinking

19. Now try the problems on Worksheet A9
Ellie’s method
Describe Ellie’s thinking
This approach was one student’s response to solving the problem. They saw it as a need to make the scales look exactly the same on both sides, which could be achieved by adding (not removing objects). Because the scales start off balanced and must be balanced once they have exactly the same objects on both sides, then it seemed obvious to this student that what they have added to both sides must be equivalent in weight. This is indeed a valid strategy and can be applied to all scales problems.
Worksheet A9 consists of more problems of this type.
Now try the problems on Worksheet A9

20. Explain the short hand
In answering Worksheet A9, some students may naturally start to make jottings where they use letters to stand for the fruit shown in the pictures. Their jottings could be shared with the class, and used as a lead into this slide.
Questions such as ‘Where is the equals in the scales picture?, ‘What does the ‘b’ stand for?’ are useful prompts.
NOTE: It would seem natural to choose the starting letter for the fruit to represent the actual fruit. Actually the letter in this case stands for the weight of the fruit, and therefore is a variable. This point can be made in response to what the students say for this slide.

21. What other short hand could you use to represent this picture?
Explain the short hand
Explanations will vary.
Students will recognise that it is possible to group the ounces together to give a short and version of 11a + 40 = 15a + 20.
It is probably worth asking again ‘Where they think the equals sign is in the picture?’
What other short hand could you use to represent this picture?

22. What’s the shorthand for this situation?
There may be some debate about what type of fruit is shown on the scales. This may lead to different shorthand situations.
It was designed to represent a grapefruit.
A possible shorthand is:
9G + 12 = 6G +48
or 9G = 6G …..and so on.
There are various ways to find the weight of a grapefruit, either by removing objects, or by adding on objects.
The weight of a grapefruit G = 12 ounces
Describe how to find the weight of a grapefruit.

23. How does the weight of a tomato compare with the weight of a banana?
Still working on the ideas of finding the weight of one item in terms of (in this case,) another item.
Removing tomatoes and bananas reduces to 3 tomatoes balancing with 2 bananas
Using Ellie’s method of adding items to both sides in order to make the scales have exactly the same items on each side gives: 2 bananas must be equivalent to 3 tomatoes.
Either can be reduced to the idea that 1 banana is equivalent in weight to 1 ½ totatoes.

24. Draw a scales picture for this shorthand
5g + 17 = 9g + 5
Pictures will vary. Some students assume that ‘g’ stands for grapefruit, or grape. Some students want to write the letter ‘g’ on the scales. Some group the ounces altogether. Some want to draw individual ounces or even make up the weight using the standard weights from before i.e. make up 17 ounces from a 16 ounce weight and a 1 ounce weight. Here is one possible representation: (make sure that you are in ‘Notes page’ view if you wish to see this drawing)
Students can then work on their picture or write jottings next to it to deduce that
12 ounces balances with 4g
3 ounces balances with 1g
1g balances with 3 ounces g = 3
Now find the value of the letter g

25. Draw a scales picture for this shorthand
7p + 17 = 10p + 2
A possible scales picture is shown below: (make sure that you are in ‘Notes page’ view if you wish to see this drawing)
By removing (or adding on p’s and ounces) it is possible to see that
15 balances with 3p
p will balance with 5 ounces p = 5
Worksheet A10 contains similar problems. Some students may be able to move away from the scales picture and produce jottings which may look quite similar to the traditional way of solving equations. The formality is not necessary, it is much more important that the students develop a sense of ownership and understanding of how to solve these problems.
Find the value of the letter p
Now try the problems on Worksheet A10

26. Draw a scales picture for this shorthand
3(t + 2) = 11t + 2
In the trials several students were independently able to access problems containing bracket notation. For some students it seemed obvious that they needed to represent a ‘t ’ (tomato) and 2 ounces, 3 ‘lots of’.
Below is a possible scales picture: (make sure that you are in ‘Notes page’ view if you wish to see this drawing)
Removing objects it can be seen that
4 = 8t
Students can struggle with this idea.
Returning to the scales picture showing 4 ounces balancing with 8 tomatoes is a way of helping them make sense of it.
Or it can be helpful to draw a bar diagram with 4 blocks labelled as 8 across the top
i.e. each block is worth ½
Now find the value of the letter t

27. Draw a scales picture for this shorthand
4(x + 3) = 3(x + 5)
A possible scales picture is shown below: (make sure that you are in ‘Notes page’ view if you wish to see this drawing)
Note the weighing scales have been reduced to just a tray on either side. The repetitive weights have been grouped into a total of 12 on the LHS and 15 on the RHS.
Worksheet A11 provides similar problems. Again some students may start to become more efficient in the way they record their solutions. However they can always return to the drawing and the context of the weighing scales if they come unstuck.
Find the value of the letter x
Now try the problems on Worksheet A11

28. Draw a scales picture for this shorthand
4t + 1 = 6t – 5 3y – 4 = 5y – 6 10 – p = 2p - 8
This slide requires the students to consider what happens when the ‘weights’ are negative amounts. Some students are able to have a go at representing this on a scales picture but it is challenging.
Ideally by this stage you would want the students who are able to go this far to be operating less with the scales and more with the idea that as long as you do the same to both sides of the ‘scales’ be that taking off, or adding on items / weights then the equivalence still holds.
These problems are not recommended for students who are just about coping with the positive versions. It is enough for them to develop a concrete understanding when the scales context is applicable.
Answers: t = 3 y = 5 p = 6

29. Summary
You have used balanced weighing scales as a way of making sense of solving linear equations like those shown below. Check to see which of these equations you can now solve: 1)
4x + 3 = ) 13x + 1 = 11x + 15
4) 3( t + 4) = ) 5y – 4 = 3y + 6
Strategies will vary from one student to another.
Answers:
1) A = 4 2) x = 6 3) x = 7 4) t = 3 5) y = 5

30. Blank template slide Information Question 1 Question 2
Available if you wish to add to this presentation for your particular group.
Question 2