Arsenal collection: match the matches
Decide which charts and graphs represent the number of goals Arsenal and Manchester United scored in fifteen matches.
Age
7 to 14
Challenge level
Problem
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The data below show how many goals the teams scored in their matches.
There are six different collections of data, three show the results for Arsenal and three show Manchester United's goals. Can you match the data to the teams?
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Getting Started
What is the total number of goals each team scored over the fifteen
matches?
You could try comparing any two of the charts to start with. For example, do you think the pie chart and the pictogram represent the same team's goals? Why or why not?
How about comparing the pictogram and the tally chart?
You could try comparing any two of the charts to start with. For example, do you think the pie chart and the pictogram represent the same team's goals? Why or why not?
How about comparing the pictogram and the tally chart?
Student Solutions
Brenchley and Matfield School sent in this excellently reasoned solution:
Arsenal and Manchester United played 15 games each.
- The pie chart and the bar chart had the same amount of goals.
- The tally chart and the pictogram had the same amount of goals.
- I then did this to find the mean of the pictogram: 0x3=0 1x3=3 2x2=4 3x5=15 4x2=8 0+3+4+15+8 = 30 Divided by 15 =2.
- I then found the mode which is 3 because 3 goals was scored mostly
- Finally I found the median which is 2. 000111223333344
- The mode is 1 higher than the mean and the median is the same as the mean.
- This matches the data of the writing about Arsenal.
- This means the pictogram and the tally chart are Arsenal.
- This means the pie chart and the bar chart are Manchester United.
- Arsenal scored 30 goals.
- Manchester United scored 30 goals.
George and Dominic from St Nicholas' C of E School, Newbury, also sent in a very clear solution:
We looked at the pictogram and worked out how many goals were scored altogether - that was 30.We divided it by 15 (the number of games) to find the average or mean - that was 2.
The tally chart goes with the pictogram as both have a mode of 3 scored goals. These two are both Arsenal because Arsenal's mode of the number of goals scored (3) is one more than the mean number of goals scored (2).
On the frequency chart, the most goals scored was one. When we added all of them together (30) and divided it by 15, we got 2. This is the mean.
For Man U, the mean number is one more than the mode number so the mode is one. This means the frequency chart must be Man U.
The pie chart shows that one goal was scored the most often and that 0 goals was scored the least. These facts also apply to the frequency chart. So the pie chart is also Beta Rovers.
So the tally chart and the pictogram were Arsenal, and the pie chart and frequency chart were Man U.
Here are some more solutions.
We started by looking at the pie chart, the mode was 1 goal, which was the same as the bar line graph. We knew that the pictograph and the tally chart matched because the mode is 3 on both. The mean is 2 for all 4 graphs. Arsenal has its mode one more than the mean, so Arsenal has to be grouped with the tally chart and the pictograph. Therefore Man U has to be with the pie chart and the bar
line graph.
(Cirby, Isabella, George and Hannah, St Helen's C of E Primary School)
We started by ordering the numbers from the pictogram which were 0,0,0,1,1,1,2,2,3,3,3,3,3,4,4 and then calculated the median which was 2. We then calculated the mean which was 2. This was equal to the median which was one of the clues in the second box. We then used the chart to help calculate the mode by doing the reverse of the clue and adding 1 to 2 which was the mean and median.This
gave us our mode which matched up to the clue we then repeated the process to calculate the other mean, median and mode this gave us our answer. Arsenal's data was the pictogram and the tally. Man U's was the pie chart and bar chart.
(Patrick and James, Redlands Primary School)
The first thing we did was pair the cards that had the same number of goals. We paired the tally chart with the pictogram and the pie chart with the bar line graph. Then we found out the mode of both pairs by looking at the tally and finding the most tally marks. We also looked at the pie chart and noticed the largest section. After that we found the mean. We had to time the number of goals
by the number of times they came up, added the answers together and divided by 15 because they'd played 15 matches. We found the median by putting the numbers in order (eg for Arsenal it was 000111223333344) and finding the middle number. Finally, we read the description and found which team fitted it. Arsenal was the team that had scored 3 goals 5 times and Manchester United was the team that
had scored 1 goal 6 times.
(Grace and Katie, Greenacre School for Girls)
Teachers' Resources
Why do this problem?
This problem could be used at the start of a series of lessons on data handling, or as an assessment opportunity at the end of the unit. It will get children talking meaningfully about mathematics, presenting and justifying arguments.
Possible approach
As an introduction to this task, you may choose to ask general questions about the different forms of data. This might be most helpful in the case of the pie chart if the class is not so familiar with this method of representation. For example, you could ask questions such as:
- Looking at the pie chart, in approximately what fraction of the total number of games did the team score one goal?
- What does the tally chart show us?
This activity would be ideal to tackle in pairs or threes. You could print off this Word document or this pdf containing the six different forms of data which could be cut up to create six cards. In this way, children would be encouraged to talk to each other as they interpret the data and
the richness of their discussion will allow you to assess their understanding.
In the plenary, you can focus on how pupils knew which forms of data go together.
Key questions
What is the total number of goals each team scored over the fifteen matches?
Have you tried comparing two of the charts with each other?
Do you think they represent the same team's goals? Why or why not?
Possible extension
You could challenge children to make their own version of this problem in pairs.
Possible support
It might be helpful for children to be encouraged to make jottings on the cards as they work on this task.