1. Charity Events and School Trips
Photo
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
Please note that the intention is for no calculator use for any of the sessions in this Number module. This session is aimed at giving students a way of developing methods for adding and subtracting fractions using a segmented bar. This session builds on the ideas introduced in The Sweet Shop lesson, of using contexts which are ‘bar like’ when you draw them. Here the context is about displaying the results of various surveys on a segmented bar. On slide 6 students are shown how to turn their segmented bar into a pie chart and so make connections between these two representations. As the context develops students are asked to compare the survey results of a Year 8 class of 30 students with those of a Year 12 class of 20 students (slide 9). The idea is to choose a segmented bar which allows you to represent both 30 students and 20 students. In the trials most learners were able to see that choosing a bar of 60 segments (from Worksheet N3b) allows for this. However, deciding how many segments to use when asked to compare the results of 45 students with the results of 60 students (slide 14) proved more challenging and exposed a lack of security with numbers in several learners. The strategy of listing what amounts of segments are possible for the 60 students i.e 60, 120, 180, 240, and so on. Along with listing what amounts are possible for 45 students i.e. 35, 90, 135, 180, … proved to be helpful for many learners. These problems are designed to encourage the learners to work on informal ideas associated with common denominator through the context of choosing (or imagining) a segmented bar which would allow both survey results to be displayed. The context is important as it gives a visualisation and some purpose to this process. By the end the learners are set traditional bare fractions questions such as 𝟐 𝟓 + 𝟏 𝟑 (slide 25, part b) , but the idea is that many learners will still be thinking in the context of the segmented bar, either by drawing one, or by thinking along the lines of what size bar should I draw to be able to represent a group of 5 people and to be able to represent a group of 3 people. Standard methods for teaching learners to add and subtract fractions rely heavily on memory rather than understanding or making sense of what is actually going on. Hence the reason that many learners forget these methods very quickly. Although the approach used here takes some time to develop, it encourages learners to develop their own ways of thinking about how to add fractions based on meaning and understanding. This session also requires learners to interpret and compare data presented in tables and charts and so exposes them to GCSE type problems traditionally associated with the Handling data aspects of the curriculum.

4. School charity events
City college has appointed a new bursar Jake. Part of his role is to oversee the running of trips and charity events. The college has recently taken part in a sponsored walk and Jake decides to survey the students to see which charities students would like the money to go to out of Barnardos, Cancer Research World Wildlife fund or Help the Aged.
Jake made these pictures from the results he got for Year 9 and Year 12 students
Students will have various opinions as to which is which. For example some students will argue that younger children will be more interested in caring for wild animals and therefore the top strip must be Year 9. It is worth hearing various ideas. (See next slide for which is which.)
The purpose of this question is to encourage students to engage with the actual context. They may also talk about any charities that their own college supports, or that they themselves support.
Try to decide which bar represents the Year 9 student votes and which bar represents the Year 12 student votes.

5. Year 12 charity preferences
The Year 12 students raised close to £1200.
According to the survey roughly how much should be given to Barnardos?
The Barnardo’s strip is close to one fifth of the whole of the Year 12 bar. Students will have various strategies for deciding on this proportion, such as seeing how many blue strips can be ‘fitted in’ to the whole bar. The gesture of marking along the whole bar in gaps the size of the Barnardo’s strip is an important one and it can be demonstrated at the board by both the teacher and a student. It is important to insist on some precision when this happens as paying attention to scale is an important feature of many RME models.
An alternative strategy is to mark on half the bar and then a quarter (by halving the half) and note that the Barnardo’s strip is less than this. Therefore may be closer to a fifth than a quarter.

6. Favourite charity
a) Which charity would you have voted for? b) Ask everyone in your group to choose their favourite charity from Barnardos, Cancer Research, World Wildlife and Help the Aged. c) Cut a bar from Worksheet N4a and shade it to show your results. (E.g. If 3 people choose Barnardos shade 3 segments one colour and label/colour the block.)
This activity is worth doing, since the use of a segmented bar is fundamental to the development of ideas and meaning in this resource. At this stage the segmented bar is used to represent the results of a survey. It is likely that students will use a one to one mapping of segments to people. Although in the trials some students tried to make use of the bar in its entirety, and consequently may have given two or even three segments per person.
Scissors are needed, shading can be done with stripes / spots etc if no colours are available

7. Favourite charity You can make a pie chart from your bar as follows:
Stick the ends together
Draw a circle around the outside of your ring
Mark inside where the colours change
Remove the ring and mark where you think the centre is. Join the centre to your markers
Colour in the sections and make a key
Cellotape or paper clips are needed to join the ends of the strip together. This can be fiddly but makes a really powerful connection between the strip/ bar representation and a circle representation.
Strategies for estimating how much should be given to Barnardo’s will very: Some students see what fraction of the whole strip is taken up by the Barnardo’s block and portion out the £1200 pounds accordingly.
Some students use the circular representation and think what fraction is taken up by the Barnardo’s piece of pie.
Labelling the start and end of the bar with £0 and £1200 may help prompt students.
The emphasis here is on hearing their informal methods for estimating.
According to your group’s result, roughly how much of the £1200 should be donated to Barnardos ?

8. Swim/ Keep fit /Fasting/ 5 a-side or netball
Which charity event?
The college do a sponsored walk for charity every other year. The years in between they try out different sponsored events. Jake has the job of choosing the sponsored event for next year.
Again he asks the students what they would prefer out of four choices:
Swim/ Keep fit /Fasting/ 5 a-side or netball
The results for Year 12 are shown on the bar below:
The next two slides provide another opportunity to survey the group and represent their results on both a segmented bar and a pie chart. It may not be necessary for your group to do this.
Spend a little time embracing this context. i.e. What charity activities do your students take part in? On the next slide, they will be asked to choose from the four activities listed above

9. Which charity event?
Which charity sponsored event would your group choose from
Swim/ Keep fit /Fasting/ 5 a-side or netball ?
Cut another bar from worksheet N4a and use it to display the results
Use the bar to make a pie chart
How do your group compare with the results of Jake’s survey?
Part c) comparing their groups strip with Jake’s is challenging. The strips are not of the same length and so it is necessary to consider the proportion or fraction of each part. Some students may suggest turning Jake’s bar into a pie chart to help with comparisons.
Students may comment on which is the most / least popular choice for their group compared with Jake’s.

10. The college trip to London
For several years the college has run a weekend trip to London for students in Year 8 and Year 12. This has included a visit to the Tower of London and a morning of sporting activity.
Jake decides it is time to review what places and activities are most popular with the students. Firstly he asks a class in Year 8 and a class in Year 12 which places they would most like to visit.
Here are the results:
Attraction
Year 8 class
(30 students)
Year 12 class
(20 students)
Harry Potter World 10 5
Madam Tussauds 7
Tower of London 3 2
London Zoo 6
London Dungeons 4
Again some time is spent engaging with the context. Students are likely to volunteer their own experiences of visiting any of these London attractions.
Students will start to make informal comparisons like ‘Harry Potter is much more popular with the younger students.’ Some students may reveal that they are making absolute comparisons (i.e. 10 students is twice as many) as opposed to making relative comparisons . ( 1/3 of year 8 said Harry Potter whereas 1/4 of Year 12 said Harry Potter) and so on. There is no need to explore the formal cancelling down process.
The purpose of this slide and the next few is to start to push a need for comparison when the class sizes are different.
Are you surprised by any of these results?

11. The college trip to London
Attraction
Year 8 class
(30 students)
Year 12 class
(20 students)
Harry Potter World 10 5
Madam Tussauds 7
Tower of London 3 2
London Zoo 6
London Dungeons 4
a) Choose a bar from Worksheet N4b for each class.
b) Colour in the bars and make a key to display the results for each class.
c) Write down what fraction of each class prefer each attraction.
In the trials students used different length bars to answer parts a) & b) which is to be expected at this stage.
Part c) answers will vary, some students will naturally ‘cancel down’ by recognising that say 10 is 1/3 of 30 students (as opposed to using the cancelling down formal process.) Whereas others will refer to 10 out of 30 for Harry Potter in Year 8 and so on. There is no need to explore the formal cancelling down process here.

12. Worksheet N4b – Segmented bars
This is the second of the segmented bar worksheets which includes more segments per bar. This allows students to choose bars where one person may be represented by 1 or 2 or even 3 or more segments . However in the case of the previous slide questions part a students are most likely to have used one segment to represent one person.

13. The college trip to London
Attraction
Year 8 class
(30 students)
Year 12 class
(20 students)
Harry Potter World 10 5
Madam Tussauds 7
Tower of London 3 2
London Zoo 6
London Dungeons 4
d) Is Madame Tussauds more popular with the Year 8 or the Year 12’s?
e) What about London Zoo? Is it more popular with the Year 8’s or the Year 12’s?
Part d) often causes a difference of opinion. Some students will argue it’s the same because it is 7 students in each case. Others will recognise that 7 out of 30 students represents a smaller chunk than 7 out of 20 students. Therefore Madame Tussaud’s is more popular with the Year 12 class. It is not necessary to resolve this issue at this stage.
Part e) To compare 6 people out of 30 with 5 people out of 20. The number of people has gone down by one but the amount of people has gone down by 10 which is a lot compared with the one. Sketching (to scale) 6 on a bar of length 30, then knocking it down by 1 and marking this in a bar of length 20 can help to show that proportionally 5 out of 20 is more than 6 out of 30.#
Alternatively students may talk about comparing 6/30 or 1/5 with 1/4 and be able to argue why 1/4 is bigger than 1/5.
This slide is about hearing informal reasoning which can be a very powerful way of making sense of how to draw comparisons when the total amounts are not the same.

14. The college trip to London
Look at the bars you made for this survey. It can be tricky to compare them because they do not have the same number of segments.
Cut out 2 more bars from Worksheet N4b which you could use to show the classes results, so that each bar has the same number of segments
Use you new bars to help you write some statements comparing the results of this survey.
Attraction
Year 8 class
(30 students)
Year 12 class
(20 students)
Harry Potter World 10 5
Madam Tussauds 7
Tower of London 3 2
London Zoo 6
London Dungeons 4
After discussions most students recognise that it would be good to use a bar of 60 segments. What is interesting is the way in which students then construct their bars. For example in the case of representing the 30 Year 8 students, some choose to draw the 30 bar twice, whereas others double the amount in each category and then draw the bar. This is a distinction which may have some bearing on the way they see equivalence in fractions.
Students now find it much easier to compare the bars. The can now use absolute quantities i.e compare the number of pieces there are for each category. However they can also talk about relative quantities such as fractions.

15. Year 8 (60 students responded)
Activity centres
As part of the London trip students spend time at an activity centre. Jake wants to review students’ preferences for this. He s all the students in Year 8 and in Year 12 to find out.
Here are the results:
Activity
Year 8 (60 students responded)
Year 12 (45 students responded)
High ropes 11 8
Climbing 17
Forest adventures 23
Team building 9
In the trials this appeared to be quite a leap for some students. Particularly since doubling one of the totals does not give a total which will work for both Year groups.
Teachers found it useful to ask what amount of segments could you use for the 60 people i.e. 60, 120, 180, 240 and so on.
Then to ask what amount of segments could be used for the 45 people i.e. 45, 90, 135, 180,… and so on and to see if there are any matches. Students comment on how long these segmented bars will be which is a good point and not very practical. The teacher can start to draw empty bars to scale (see below) to provide students with a way of still retaining the context of the segmented bars and hence keeping a sense of what the list of numbers means.
Part b) Forest adventures scaled up becomes 69 out of 180 for the Year 8’s and 68 out of 180 for the Year 12’s so it’s very close but slightly more popular with the Year 8’s.
a) What size of segmented bar could you use to show these results if you wanted each bar to have the same number of pieces?
b) Are the Forest adventures more popular with the Year 8 or the Year 12 students?

16. Favourite type of sandwich
The College provides a packed lunch for all the students on the trip down to London. Jake s some students to find out which flavours they prefer. 𝟏 𝟓 said cheese, 𝟏 𝟐 said tuna, 𝟏 𝟒 said ham, 𝟑 𝟐𝟎 said egg.
a) How many of the 40 Students who responded said ham?
Strategies will vary. Some students may wish to think on a segmented bar.
Sketching a bar marked as a length of 40 can be helpful, particularly when finding 1/5. Some students may not make the connection to see how may 5’ there are in 40. Some may need to guess an amount for 1/5 and scale it up 5 times to check if it gives 40.
Part a) 8 said cheese, 20 said tuna, 10 said ham, 6 said egg.
Part b) 14 said cheese or ham.
b) How many said cheese or egg?

17. Evening Entertainment – Year 12 students
On one of the evenings of the trip the Year 12 students take part in a dance class. Jake asks a sample of Year 12 students which type of class they would prefer said Salsa, said Zumba, said Ballroom
How many people do you think Jake asked?
How many people said none of the above choices?
c) What other choices might they have preferred?
Part a) Suggestions will vary. Some students suggest 10 people were asked but soon realise this would not be possible. After rejecting various numbers, because they do not fit with out of 3, out of 5 and out of 10, students suggest 30 people were asked.
It is important to note that many students will still be thinking and talking in the context of Jake’s survey which is to be encouraged.
It is worth asking if Jake might have asked a different number of people other than 30.
Part b) Working with 30 people asked then 10 said Salsa, 6 said Zumba and 3 said Ballroom. That leaves 11 saying none of these 3 choices. But really we can only give this as 11 out of 30, because if 60 people were asked then the amount of people would be different. Again the ideas linked to absolute comparisons and relative comparisons are around.
Part c) various suggestions: Disco, Latin, Jive and so on….

18. Evening Entertainment – Year 8 students
The Year 8 students spend one evening of the trip playing games. Jake surveys the Year 8 students about their favourite type of game said board games, said table football, said playing cards.
In the trials some students used 3 bars marking them up as follows:
First bar marked in groups of 3
Second bar marked underneath first bar in groups of 4
Third bar marked in groups of 8.
Using this strategy they found the end of the bars matched after 24 segments.
Although GCSE resit students will no doubt have met ‘common denominator’ many times they will have tended to focus only of the ‘number aspects of that, as opposed to what it means in a pictorial sense.
a) Use a segmented bar from worksheet N3b to display this information.

19. Favourite film type – Year 8 & Year 12
On the last night of the trip the students watch a film. Jake is unsure whether to put all the students in together or if the Year 8 would prefer a different type of film to the Year 12’s. He surveys some of the students to find out their favourite type of film
a) Students may still need segmented bars in front of them. Or it may be enough to list:
5, 10, 15, 20, 25, …
4, 8, 12, 16, 20, 24, ….
So you could use 20 segments or 40 segments or 60 segments and so on.
b) Out of 20 students, 4/20 in year 8 compared with 5/20 in Year 12. This is a difference of 1/20, which is hard to make sense of when you give it as a fraction. It’s easier to say if 20 people were asked in each Year group then one more person voted for Action in Year 12 than in Year 8.
Give three sizes of segmented bar could you use to represent both and 1 4
What is the difference between the fractions of students in year 8 and those in year 12 whose favourite film type is Action?

20. Favourite film type – Year 8 & Year 12
Part c) Strategies will vary.
If 8 students were asked altogether then 4/8 said comedy in Year 8 and 1/8 said Comedy in Year 12. This would mean 3 more students in Year 8 compared with Year 12 said Comedy. Or 3/8.
Students may suggest that 8 is not many students to survey, so they must have asked more i.e. 16, 24, 32, 40. Some students may realise that it must match the total number of students asked about Action and Fantasy.
Part d) comparing ‘out of 6’ with ‘out of 5’ must mean either 30 or 60 or 90 ….. Students were asked.
Using 30: 1/6 becomes 5/30 compared with 2/5 becomes 12/30.
So 7/30 more preferred Fantasy in Year 12 than in Year 8.
c) How does the fraction of Year 8 students who prefer Comedy compare with the fraction of Year 12 students?
d) What about for fantasy?

21. Favourite film type – Year 8 & Year 12
e) There are some categories missing from this table. How can you tell? Suggest what they might be.
f) If Jake was to choose a film type for both the year 8 and the Year 12 students to watch, what type do you think he would go for?
Students may need support by sketching a bar and marking on the ½ , the 1/5 and the 1/6 to see that it does not make up a whole bar..
Alternatively 1/5 out of 30 is 6, 1/2 out of 30 is 15 and 1/6 out of 30 is 5, making 26 out of 30 in total,.
Likewise for the Year 12.
Possible other categories include horror, romance, thriller etc.
Part f) Using a total of 120 in order to be able to express all fractions gives:
Action 1/5 + 1/4 = 24/ /120 = 54/120
Comedy 1/2 + 1/8 = 60/ /120 = 75/120
Fantasy 1/6 + 2/5 = 20/ /120 = 68/120
According to these survey results, Comedy is the most popular.

22. Drawing bars to compare fractions
Draw a bar like this:
a) How many segments should you use in your bar if you want to show and ? b) Which is more, or ? c) What is the difference between than ?
Part a) It might help students to mark up the top of their bar in jumps of 2 and the bottom of their bar in jumps of 7, until they find a match. This happens after 14 segments.
1/2 = 7/14 1/7 = 2/14
b) 1/2 is more/
c) The difference is 5 segments which is 5/14.

23. a) Which is larger, 3 5 or 4 7 ? How much larger?
b) Which is larger, or ?
It is anticipated that students will be drawing a segmented bar to answer these questons, or using the method of listing possible sizes of bar
i.e. 5, 10, 15, 20, 25, 30, …
7, 14, 21, 28, 35, …. And so on.
3/5 = 21/35 4/7 = 20/ /5 is bigger by 1/35
Part b) Students may use informal reasoning such as 5/6 is larger because its only 1/6 away froma whole one. Whereas 2/3 is 1/3 away.
Or 2/3 = 4/6 which is smaller than 5/6. So 5/6 is larger.

24. Drawing bars to add fractions
a) Draw a segmented bar to help you show that =
A possible approach is shown below: (make sure that you are in ‘Notes Page’ view if you want to view this image)
b) How much bigger is than ?

25. Drawing bars to add and subtract fractions
Start by drawing a segmented bar to work out the following:
a) 𝟏 𝟒 + 𝟏 𝟏𝟐 b) 𝟐 𝟓 + 𝟏 𝟑 c) 𝟏 𝟐 + 𝟑 𝟒
d) 𝟏 𝟒 + 𝟑 𝟕 e) 𝟔 𝟏𝟓 + 𝟕 𝟏𝟎 f) 𝟏 𝟐 + 𝟓 𝟗
g) 𝟐 𝟑 𝟐 𝟗 h) 3 𝟒 𝟕 𝟏 𝟓 i) 1 𝟏 𝟐 𝟐 𝟑
It is anticipated that some students will still be drawing segmented bars, or maybe doing jottings representative of that thinking.
Some students will want to use a formal method they have been previously taught, but they should be encouraged to make sense of how this actually works and why it give them the correct answer.
a) = b) c) = 1 1 4
d) e) = f) =
g) h) i) 5 6

26. Our survey said …….
In a survey, 100 people were asked what is 𝟏 𝟒 + 𝟏 𝟐 ?
52 people said 2 6
26 people said 1 4
22 people said don’t know
This is a question we have trialled on many school students. The most common response is to add the tops and add the bottoms and hence come up with the answer 2 6 , even though this is smaller than one of the original fractions.
Students now have various strategies they can use to convince each other and themselves why adding the tops and adding the bottoms, although an understandable strategy does not work.
In particular drawing a segmented bar shading and 1 4 , provides a good rationale for why the answer has to be
Which group of people do you think are correct and why?

27. Dividing with fractions
Challenge
Experiment with using a segmented bar to help you work out
2 3 ÷ 1 4
This is available as a challenge to those students who would benefit from taking these ideas further.
By drawing a segmented bar with 12 segments this problem can be thought of as how many = pieces can be fitted into =
This amounts to seeing how many 3’s you can fit into 8 =
This can be seen more clearly on the picture.
Hence ÷ = ÷ = 8 ÷ 3 =

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