Problem bits one

“Bits” are designed to be focus statements for group discussion. As such, they could be used as short starting points during a mathematics lesson. It is important that the results of the group discussion are shared with the whole class.

Bit 1

Although the aim is to swap the 10c with the 50c or 10c with the 20c, other coins might also end up in a different position.

Students’ answers will vary, but the minimum number of moves is:

a. 10 cents swaps with 50 cents (13 moves)
b. 10 cents swaps with 20 cents (22 moves)

It is important for students to have a way to record the moves they make. This may involve drawing the whole frame each time or just recording the coins that move in each situation.

For example, the 10 cent and 20 cent coin movements could be recorded as:

• 5, 10, 20, 5, $1, 50, 10, $1, 5, 20, $1, 5, 50, 10, 5, 50, 10, 5, 50, $1, 20, 10 (the 5 c + $1 have swapped as well)

The first move, “5”, indicates that the 5 cent coin moves to the only vacant space, followed by the 10 cent coin moving to the space created by moving the 5 cent coin.

Bit 2

The circle is the limiting case of a polygon with an infinite number of sides and corners. Students are likely to answer along these lines, with responses such as,

• There are so many sides/corners that they are too small to see.

Students may like to look at the number of lines of reflective symmetry that each polygon has:

The circle has an infinite number of lines of reflective symmetry.

Bit 3

The “addition makes bigger” belief is one that students frequently adopt as an informal rule. This becomes an obstacle when they add negative integers. For example,

• ⁺3 + ^-2 = ⁺1

So the answer is less than the starting addend.

Students should note that adding zero is an exception to the rule because the answer and first addend are the same. For example,

• 4 + 0 = 4.

1.

Answers will vary.

The minimum number of moves is:

a. 10 cents swaps with 50 cents: 13 moves

b. 10 cents swaps with 20 cents: 22 moves

2.

Answers will vary but may include:

• There are so many sides and corners that they are too small to see.

3.

Answers will vary. If you add 0, the answer isn’t larger, and if you add a negative integer (for example, ⁺3 + ^-2 = ⁺1), the answer is less than the starting addend.