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Rationals Between

What fractions can you find between the square roots of 56 and 58?

At the corner of the cube circular arcs are drawn and the area enclosed shaded. What fraction of the surface area of the cube is shaded? Try working out the answer without recourse to pencil and paper.

Do Unto Caesar

At the beginning of the night three poker players; Alan, Bernie and Craig had money in the ratios 7 : 6 : 5. At the end of the night the ratio was 6 : 5 : 4. One of them won $1 200. What were the assets of the players at the beginning of the evening?

Peaches Today, Peaches Tomorrow....

Stage: 3 and 4 Challenge Level:

Why do this problem?

This problem could replace repetitive textbook work on calculating fractions of integers. It offers plenty of practice of these calculations while requiring students to come up with problem-solving strategies. It offers a good context for thinking about factors of numbers.

Possible approach

Introduce the first part of the problem to the class, and give them a little bit of time to work in pairs to solve it. All three parts of this problem can be displayed on these PowerPoint slides.

Once students have had a chance to work on the first challenge, share strategies and discuss any difficulties that arose.

The second task is much more demanding and interesting, requiring students to work quite systematically and record the steps they take clearly. Challenge students to find the solution and to convince themselves that there is no other way of ordering the fractions.

Finally, the last challenge would work very well as a 'simmering activity' set for students to think about beyond a single lesson - perhaps this could be set as a homework. The best solution could be displayed on the classroom wall, and students could be challenged to improve on it.

Key questions

How will you record your work efficiently so that you can keep track of what is happening?

Possible extension

In part (iii) a solution can be found where the peaches last for more than a fortnight! Finding a solution of this magnitude should provide a suitable extension challenge.

Ben's Game and Fair Shares? are both good follow-up fractions problems.

Possible support

For part (ii), suggest to students who are struggling to record their thinking effectively that they could use a tree diagram: at each stage, branch off the fractions it would be possible to try next so that all possibilities are checked.

Divisibility. Fractions. Factors and multiples. Working systematically. Networks/Graph Theory. Calculating with fractions. Mathematical reasoning & proof. Visualising. Modulus arithmetic. Simultaneous equations.