Problem
In the first quarter of last Friday’s netball game, Katie and Sarah scored all the goals for the Gold team. Sarah shot ¼ of all the goals. Katie shot 12 goals. How many points did the Gold team score?

In the second quarter of the game, Katie scored a fifth of the team’s 20 points while again Sarah scored the rest. How many points did Sarah score?

What is this problem about?
Fractions are important numbers in their own right and should be known and used because they occur in everyday use. Children need to know what various small fractions are and how they fit together. For instance, an important part of this question is knowing that one quarter plus three-quarters is one.

But fractions too develop into more complicated objects after children have done some algebra. To be able to divide confidently by expressions such as x and 2x + 3, it is important that they have a strong grounding in fractions and dividing by simple numbers like four and five.

Finally, like many of the problems on this website, the question requires children to do more than one step while holding in their heads, information about both the problem itself and the direction they are going in order to solve it.

Achievement Objectives
Number (Level 2)
- write and solve story problems which involve halves, quarters, thirds and fifths.

Mathematical Processes
- devise and use problem solving strategies (guess and check, draw a diagram, use equipment)

Resources
Counters
Blackline master of the problem

Specific Learning Outcomes
The children will be able to:
- find a ¼ and ¾ of a set.

1. Introduce the problem by discussing netball. Ask:
   How many in the team can shoot? How many points for each goal?
2. Read the problem with the class.
3. Ask for ideas about how they might solve the problem.
4. Let the children work on the problem in pairs. As they work ask questions that focus their thinking on the fractions.
   How do you work out a ¼ of a set?
   If Sarah shot ¼ how much did Katie shoot?
   Convince me that your answer is correct.
5. Encourage the children to think about how they could convince others in the class that their answer was correct.
6. Share explanations.

Extension Problem
What happened in the last two quarters? Get the children to finish off the story of the netball game by writing two more stories involving fractions. They can use the first two quarters as a model. Encourage them to use fractions other than quarters and fifths.

What fraction of the match total did Sarah score?

Other Contexts for the Problem
Various other games could be used
Shopping, where points are changed for money, and quarters for different shops

Solution
Let’s do the first quarter first. As only 2 players shoot for the team, Katie must have scored ¾ of the goals. We know that Katie scored 12 points. If 12 points is ¾, a ¼ is 4 points. Therefore Sarah scored 4 points. This means that between them, Sarah and Katie scored 12 + 4 = 16 points. So the team scored 12 points in the first quarter.

In the second quarter, 20 points were scored. Katie scored a fifth of these. A fifth of 20 is 4. So Katie scored 4 points and Sarah scored 20 – 4 = 16 points.

It is likely that children will use other approaches than we have used here. What we have done may well be the most sophisticated way to solve the problem.

If guess and check is used, help the children to see how to improve their guesses so that their next guess is better than their first. For instance, they might guess 20 points for the first quarter answer. By dividing the 20 sticks into four groups they will see that a quarter is 5. Then three-quarters is 15. But this is more than Katie actually scored. So the original guess of 20 was too high.