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Tweedle Dum and Tweedle Dee

Two brothers were left some money, amounting to an exact number of pounds, to divide between them. DEE undertook the division. "But your heap is larger than mine!" cried DUM...

Sum Equals Product

The sum of the numbers 4 and 1 [1/3] is the same as the product of 4 and 1 [1/3]; that is to say 4 + 1 [1/3] = 4 � 1 [1/3]. What other numbers have the sum equal to the product and can this be so for any whole numbers?

Special Sums and Products

Find some examples of pairs of numbers such that their sum is a factor of their product. eg. 4 + 12 = 16 and 4 × 12 = 48 and 16 is a factor of 48.

A Chance to Win?

Age 11 to 14
Challenge Level

Why do this problem?

This problem improves students’ fluency in calculating proportion/percentage increases and decreases, and could be used to encourage students to use multiplicative methods rather than additive methods. It is also an opportunity for students to develop reasoning and mathematical arguments which explore and exploit the commutative property of multiplication.

There is the possibility of investigating novel equations with unknowns as powers.


Possible approach

We might begin by asking students to guess solutions, possibly prompted by questions such as ‘is it better to win early on?’ We could note any conjectures on the main whiteboard.

Next, students to try ordering the cards and calculate their winnings in pairs or small groups. Encourage them to record their findings logically. You could hand out manipulatives to represent the red and black cards.

Ask students to present their best sequence of cards. This is a chance for them to describe their thinking, including how it was guided by the initial conjectures. It will also ensure that everyone has understood the task and could showcase a variety of methods for calculating the final winnings.

The final winnings should be the same for every sequence presented, so you could challenge the students to find an order which gives a different total. If they can’t, why not? Once they’ve had a chance to develop their reasoning in small groups, you could have a class discussion about why the order of the cards will not affect the final winnings.

Next introduce the idea of additional winning cards. Some students might want some time to check that their conclusions still hold true. Others may be ready to immediately tackle the questions ‘how many winning cards do I need to make the game profitable?’, or ‘what happens it there is 1 losing card and $n$ winning cards?’. You could suggest that the students use a table to record their results.

You could bring the class together at the end to present and discuss their conclusions.


Key questions

Can you calculate your winnings for a particular sequence?

Do you notice anything about your results? Can you explain why this is?

How many winning cards do you need to make the game profitable?

How many winning cards do you need to ‘undo’ the effect of a losing card?


Possible support

To lead students towards seeing that the order of the cards does not matter, students might be provided with a table of suggested orders, and space to write their winnings. Calculators could be used if students are struggling numerically.

To help students express finding proportions/fractions/percentages as multiplication, you could use and/or

It is possible students will struggle with the concept of a ‘stake’; this might be illustrated through an example sequence of cards, or by first introducing a simpler game.


Possible extension

Win or Lose uses similar concepts. Students could also consider what happens if they are allowed to choose the amount they bet, or if they are given different odds on winning.