Patterns in the addition table

Patterns and sequences · level 2, lesson 4 of 6

A question to start

The addition table below holds the answer to every sum from 0 + 0 up to 9 + 9. That is 100 sums.

Here is a puzzle. How many of the 100 squares hold the number 9? How many hold 18? Before you look, make a guess. Then, by the end of this lesson, you will be able to say not just how many, but where they are and why, without counting square by square.

The idea

How the table works

The bold numbers along the top and down the left side are the numbers you add. To find 3 + 5, go along the row for 3 (blue) and down the column for 5 (green). Where they meet (orange) is the answer: 8.

A row goes across. A column goes down. (A column holds up a roof, standing straight up and down. That may help you remember.)

Pattern 1: along a row, one more each time

Read along the blue row for 3: 3, 4, 5, 6, 7, 8, 9, 10, 11, 12. Each number is 1 more than the one before.

Why? Moving one square right means the top number goes up by 1 (from 5 to 6, say), while the row number stays 3. If one of the two numbers you add goes up by 1, the answer goes up by 1. So 3 + 6 is one more than 3 + 5.

The same is true going down a column: each step down, the left number goes up by 1, so the answer goes up by 1.

Pattern 2: the slanting lines of the same number

Now colour every square that holds 7.

They make a straight slanting line from the top right, going down to the left. Why?

Start at 0 + 7 = 7 (top row). Now take one step down and one step left. Down makes the first number 1 bigger. Left makes the second number 1 smaller. One more and one less balance out: 1 + 6 = 7. Again: 2 + 5 = 7. And again: 3 + 4 = 7. Every step down-and-left keeps the answer the same.

It is like moving one block from one pile to the other. The piles change, but the total stays the same:

0 + 7, 1 + 6, 2 + 5, 3 + 4, 4 + 3, 5 + 2, 6 + 1, 7 + 0

These 8 sums are all the ways to make 7 from two numbers 0 to 9. So 7 appears 8 times.

Now we can answer the starting puzzle. The line of 9s goes from 0 + 9 (top right) to 9 + 0 (bottom left). It crosses the whole table, one square in every row: 10 squares. It is the longest slanting line.

18 appears only once: 9 + 9, in the bottom right corner. To make 18 from two numbers that are 9 or less, both must be 9. And 0 appears only once too: 0 + 0, top left.

Pattern 3: the doubles

The slanting line the other way, from the top left to the bottom right (orange), holds 0, 2, 4, 6, 8, 10, 12, 14, 16, 18. These are the doubles: 0 + 0, 1 + 1, 2 + 2, ... 9 + 9. They are where a number is added to itself.

They go up by 2 each time. Why? A step down and a step right makes both numbers 1 bigger, so the answer gets 2 bigger.

Every double is an even number. Doubling means making two equal piles, and an even number is exactly one you can split into two equal piles. So of course every double is even.

The green squares just to the right of the doubles are the near doubles: 0 + 1, 1 + 2, 2 + 3, ... They are 1 more than the doubles: 1, 3, 5, 7, ..., 17, all odd. This gives a fast mental trick: 6 + 7 is double 6 and one more, 12 + 1 = 13.

Pattern 4: the table is the same flipped

Think of the blue doubles line as a mirror. 3 + 5 is in row 3, column 5. Its mirror square is row 5, column 3: that is 5 + 3. Both hold 8. The green pair, 2 + 7 and 7 + 2, both hold 9.

Every square has a twin on the other side of the mirror, with the same number in it. If you folded the table along the mirror line, every number would land on the same number.

Why? Because 3 + 5 and 5 + 3 always give the same answer. Here is a picture that shows it.

Put 3 orange blocks and 5 blue blocks in a row: that is 3 + 5. Now walk round and look at the same row from the other side. You see 5 blue blocks, then 3 orange ones: 5 + 3. It is the same row of blocks. Nothing was added or taken away. So 3 + 5 = 5 + 3. The same thing works for any two numbers. This is called the turn-around rule.

The turn-around rule saves you work: the table has 100 squares, but you only need to learn about half of them. The other half are the same sums, turned around.

Pattern 5: a chessboard of odds and evens

Colour the even answers. You get a chessboard! Along a row the answers go up by 1, so they take turns: even, odd, even, odd. And each row starts with the other kind from the row above. That is the same as the chessboard quilt in Level 1.

The chessboard tells you a rule: even + even is even, odd + odd is even, even + odd is odd. Check: 4 + 6 = 10 (even), 3 + 5 = 8 (even), 3 + 4 = 7 (odd).

Worked examples

Example 1. Using the slanting line. Write all the ways to make 5 by adding two numbers from 0 to 9.

1. Start at the top: 0 + 5. Then step down and left: one more, one less.
2. 0 + 5, 1 + 4, 2 + 3, 3 + 2, 4 + 1, 5 + 0.
3. That is 6 ways. (5 appears 6 times in the table.)

Example 2. Near doubles. Work out 8 + 9 using a double.

1. 8 + 9 is one more than 8 + 8.
2. 8 + 8 = 16, so 8 + 9 = 17.
3. (Or: 8 + 9 is one less than 9 + 9 = 18. Also 17. ✓)

Example 3. A missing number. In the row for 4, one square is hidden: 4, 5, 6, ?, 8, 9. What is it, and what sum is it?

1. Along a row the numbers go up by 1, so it is 7.
2. It is the 4th square (column 3), so the sum is 4 + 3 = 7. ✓

Example 4. How many times does 12 appear?

1. List the ways with numbers 0 to 9: 3 + 9, 4 + 8, 5 + 7, 6 + 6, 7 + 5, 8 + 4, 9 + 3.
2. We cannot use 2 + 10, because 10 is not on the table.
3. 7 times. Notice that 12 is 3 more than 9, and appears 3 times fewer than 9 (10 − 3 = 7). The slanting lines get shorter as you go past the middle.

Try it

1. Use the table to find 6 + 7 and 7 + 6. Why do they match?
2. Read down the column for 2: what are the first five numbers? What is the rule?
3. Write all the ways to make 4 with two numbers from 0 to 9. How many squares hold 4?
4. Use a near double to work out 7 + 8.
5. How many squares in the table hold 15? Write the sums.
6. Is 5 + 6 odd or even? Is 7 + 9 odd or even? Use the chessboard rule to explain.
7. Which number appears most often in the 0 to 9 table? Which numbers appear only once?
8. Lily says: "The doubles line has only even numbers, so the line next to it must have only odd numbers." Is she right? Explain.

Answers

1. Both are 13. They are mirror squares across the doubles line: the same two numbers in the other order. By the turn-around rule, 6 + 7 = 7 + 6.
2. 2, 3, 4, 5, 6. Start at 2 and add 1 each time.
3. 0 + 4, 1 + 3, 2 + 2, 3 + 1, 4 + 0: 5 squares.
4. 7 + 8 is one more than 7 + 7 = 14, so 15.
5. 6 + 9, 7 + 8, 8 + 7, 9 + 6: 4 squares.
6. 5 + 6 is odd + even, so odd (11). 7 + 9 is odd + odd, so even (16).
7. 9 appears most often (10 times). 0 and 18 appear only once each.
8. Yes. Each square next to a double is that double plus 1 (one step right adds 1). An even number plus 1 is always odd. So every number on that line is odd.

Common mistake

Mixing up rows and columns. A row goes across and a column goes down. With the turn-around rule it does not change the answer, but it matters when you are finding a square or describing a pattern.

Thinking every number appears the same number of times. The slanting lines are long in the middle and short at the corners: 9 appears 10 times, but 0 and 18 only once.

Where it leads

• Previous lesson: The hundred chart. Next lesson: Growing shapes.
• This level: Level 2: Pattern rules. The whole topic: Patterns and sequences.
• Doubles, near doubles and the turn-around rule are mental-maths tools: Fact families and mental tricks.
• Odd and even sums: Odd and even sums.
• Folding along a mirror line is symmetry: Lines of symmetry.
• Adding two dice gives the same slanting lines, and that is why 7 is the most common total of two dice: Probability.