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← The Maths They Teach Now

The Maths They Teach Now

Four short lessons for grown-ups who learned maths another way: number bonds, ten-frames, bar diagrams and the area model, each with a game for the kitchen table.

Spark & Whetstone · sparkwhetstone.org/learn/maths-they-teach-now · CC BY-NC-SA 4.0. Answers are at the end, so you can try each question first.

Lesson 1. Number bonds

A number bond shows a whole and two parts that make it: 10 at the top, and 7 and 3 below. Children learn the bonds to 10 by heart, the way many of us learned times tables.

The point is flexibility. A child who knows that 7 and 3 make 10 can work out 7 + 5 as 7 + 3 + 2: make ten, then add what is left.

If your child says "I made ten first", that is not a shortcut they invented to cheat. It is exactly the method.

Question. Your child works out 8 + 6 by saying "8 and 2 make 10, then 4 more". What are they doing?

  1. Guessing and getting lucky.
  2. Making ten first, using a number bond.
  3. Counting on their fingers.

Say it back. In your own words: what is a number bond, and how does "make ten" help with adding?

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Lesson 2. Ten-frames

A ten-frame is a box of two rows of five. Counters fill it from left to right, top row first.

It lets a young child see an amount without counting: a full top row and two more is seven, at a glance. It also shows how far a number is from ten.

At home, an egg box with ten cups and a handful of buttons does the same job.

Question. A ten-frame has its top row full and three counters on the bottom row. How many counters?

  1. Seven
  2. Eight
  3. Three

Say it back. Say it back: what does a ten-frame let a child see that a pile of counters does not?

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Lesson 3. Bar diagrams

A bar diagram draws a word problem as rectangles. The whole is one long bar; the parts are pieces of it. "Sam had some sweets, gave away 5, and has 12 left" becomes a bar split into 5 and 12.

Drawing it first stops children from grabbing the numbers and adding them because the word "left" appeared. The picture shows whether you add or subtract.

You can ask "can you draw it?" about almost any word problem, including ones in your own day.

Question. A ribbon is cut into two pieces. One is 30 cm and the whole was 80 cm. How does a bar diagram help?

  1. It shows the whole bar of 80 with a 30 piece, so the other piece is 80 take away 30.
  2. It shows you should add 30 and 80.
  3. It does not help with lengths.

Say it back. In your own words: why does drawing the bar first help a child choose the right sum?

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Lesson 4. The area model

To multiply 23 by 14, children draw a rectangle 23 wide and 14 tall, split it into 20 and 3 one way, 10 and 4 the other, and find the four small areas: 200, 30, 80 and 12. Then they add them: 322.

It looks longer than the column method many of us learned. Its strength is that every number means something: you can see where each part comes from.

It is also the same picture they will use later for algebra, when (a + b)(c + d) becomes four rectangles.

Question. In the area model for 23 × 14, which four parts are added?

  1. 20 × 10, 20 × 4, 3 × 10 and 3 × 4
  2. 23 + 14
  3. 2 × 1, 3 × 4

Say it back. Say it back: what does the area model show that the column method hides?

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At the table: Make ten at the table

Players: 2 or more, any ages from about 5. You'll need: An egg box with ten cups and a handful of buttons or beans.

  1. One player puts some buttons in the egg box without counting aloud.
  2. The other says how many there are, and how many more would make ten.
  3. Check together by filling the empty cups.
  4. Swap, and when it gets easy, try making twenty with two boxes.

Answers

  1. Making ten first, using a number bond. — They split 6 into 2 and 4 so that 8 could make 10. That is the number-bond method, and it scales to bigger numbers.
  2. Eight — A full row is five, and three more makes eight. The empty spaces show it is two short of ten.
  3. It shows the whole bar of 80 with a 30 piece, so the other piece is 80 take away 30. — The bar makes the whole and the known part visible, so the unknown part is clearly what is left: 50 cm.
  4. 20 × 10, 20 × 4, 3 × 10 and 3 × 4 — Splitting 23 into 20 and 3 and 14 into 10 and 4 gives four rectangles: 200, 80, 30 and 12, which add to 322.