Top Shoot

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Purpose

This is a level 5 number activity from the Figure It Out series. It relates to Stage 8 of the Number Framework.
A PDF of the student activity is included.

Achievement Objectives
Student Activity

Click on the image to enlarge it. Click again to close. Download PDF (568 KB)

Specific Learning Outcomes

compare ratios

Description of Mathematics

Number Framework Links
Use this activity to:
• help students consolidate and apply their knowledge of fractions (stages 7 and 8)
• develop confidence in students who are beginning to use advanced proportional strategies (stage 8).
 

Required Resource Materials
A calculator

FIO, Level 3-4+, Proportional Reasoning, Book One, Top Shoot, page 24

Computer (optional)

Activity

Use this high-interest context to highlight the contribution good mathematical thinking can make to everyday situations.
Students who understand how to compare fractions by means of common denominators should be able to attempt this activity independently or in small groups. Others will need to do it in a guided teaching situation.
The key to this activity is found in question 1 and highlighted in the thought bubble. Students should discuss the issue (How can the coach compare ...?) in small problem-solving groups and then report back before going any further.
Students need to understand how the goals:attempts ratio works as a measure of accuracy. Useful questions for discussion could be:
• Why might the person who scores the most goals not necessarily be the most accurate?
• Is it always the fault of the goal shoot if a team does not score many goals?
• How can you tell from the goals:attempts ratio which goal shoot is more accurate?
Make sure that discussion raises and answers the question “Does the number of attempts include the successful shots?” The answer is yes. Students get used to the idea that a ratio shows the parts that make up the whole, but this is clearly not the case in this context.
You could take this opportunity to emphasise that ratios are a kind of mathematical shorthand: they can never be understood without a contextual explanation. Students may realise that a goals:attempts ratio can equally well be written as a goals:misses ratio. 7:12 (goals:attempts) is the same as 7:5 (goals:misses).
Use some simple goals:attempts ratios, for example 10:20 and 5:10, to show how equivalent fractions are a useful strategy for comparing accuracy. Both 10:20 and 5:10 describe situations in which half of the attempts made were successful, because 10/20 = 1/2 = 5/10.
Question 2 presents a challenge because there are four ratios to be ordered by size and, if expressed as fractions, they all have difficult, unrelated denominators. There are at least three ways that the students could approach the problem:
1. Try to find the common denominator, even if it is very big. The fractions are 7/12, 9/15, 12/21, and 9/13, and the denominator is 5 460 (13 x 7 x 5 x 3 x 2 x 2). Students using this method will find that a calculator is indispensable!
2. Avoid comparing the four girls’ records simultaneously and focus instead on comparing pairs of girls and finding the best performer in each pair. Here is one line of reasoning:
• Although Toni makes 6 more attempts than Mere, she scores only 3 more goals. Clearly, Mere has the better result, so Toni is eliminated.
• Peti scores 9 from 13 attempts, while Mere scores 9 from 15. Peti clearly has the better result, so Mere is eliminated.
• Peti makes just one more attempt than Rowena but scores 2 more goals. This gives her the better result and eliminates Rowena.
3. Use a variation on the strip diagram theme, as illustrated in the student book. Strip diagrams normally model situations in which there is a small standard unit, and the strips vary in length depending on how many units they represent. In this variation, it is the length of the strip itself that is the unit. The difference is the way in which it is divided up. Diagrams like these can’t realistically be created by hand, because the computer program’s stretch function is needed to shorten or lengthen one strip to match another. Once this has been done, two or more ratios can be compared at a glance. A diagram of this kind provides an excellent visual model of what is meant by comparing ratios.
All three approaches can be used when answering question 3, to compare the girls’ records at the end of both games. Here is a line of reasoning using pair comparisons:
• If the students combine the results from both halves, they will get the ratios 18:26, 22:33, 20:35, and 15:20. These can be simplified to give the equivalent ratios 9:13, 2:3, 4:7, and 3:4.
• Rowena (18:26) scores the equivalent of 9 goals from 13 attempts while Peti (15:20) scores the equivalent of 9 goals from 12 attempts. This gives Peti the better result and eliminates Rowena.
• Mere (22:33) scores the equivalent of 4 goals from 6 attempts and Toni (20:35) the
equivalent of 4 goals from 7 attempts. This gives Mere the better result and eliminates
Toni.
• Mere scores the equivalent of 2 goals from 3 (a success rate of 2/3) while Peti scores the equivalent of 3 goals from 4 (a success rate of 3/4). 3/4 is greater than 2/3 so Peti has the better result and Mere is eliminated.
It may be worth raising with your students the fact that mathematically identical ratios may not be so identical in real-life situations. You could provoke this discussion by asking “Are the skills of a girl who makes 4 attempts and scores 3 goals in the same league as those of a girl who attempts 40 and scores 30?” Mathematically, the goals:attempts ratio is identical for each, but in practice, the second girl has proven a level of skill that could in the case of the first be just luck. A player who, in a season, attempts 400 and scores 300 provides even greater proof of skill.

Answers to Activity

1. Accuracy is measured by comparing the number of goals with the number of attempts. It can be expressed using a fraction, decimal, percentage, ratio, or using a diagram (for example, a strip diagram).
2. It depends on what is meant by “best shooting record”. Toni scored most goals, 3 more than her nearest rivals. But Peti was the most accurate shoot. The record for the four girls at the end of game 1 can be summarised like this:

strip diagram.

3. Again, it depends on what the coach is looking for. If it is accuracy, Peti is the clear winner: 15 out of her 20 attempts scored goals. The other three girls couldn’t match this level of accuracy, but all three of them scored more goals. Mere
scored most goals (22) but came only third as far as accuracy goes.
The record for the four girls at the end of both trial games can be summarised like this:

strip diagram.

Attachments
TopShoot.pdf567.82 KB
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Level Five