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The hundred chart

Patterns and sequences · level 2, lesson 3 of 6

A question to start

Colour the numbers you say when you count by 9s on a hundred chart: 9, 18, 27, 36, ...

Nine of them make a perfect slanting line, from 9 down to 81, going down and to the left, like a slide. Counting by 2s makes straight stripes going down. Counting by 3s makes slanting stripes. Why does each count make its own picture? And why does counting by 9 slide one down and one left, every single time?

The idea

The hundred chart

A hundred chart shows the numbers 1 to 100 in rows of ten. The first row is 1 to 10, the second row is 11 to 20, and so on, down to 91 to 100.

Two moves on the chart are worth knowing by heart:

  • One step right is +1. (From 23, one right is 24.)
  • One step down is +10. (From 23, one down is 33.) That is because every row is 10 numbers long, so the number below is a whole row, 10, further on.

And so one step left is −1 and one step up is −10. Almost every pattern on the chart comes from these four moves.

Counting by 2s: stripes

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100

The numbers in the count by 2s (2, 4, 6, 8, 10, 12, ...) are called the even numbers. On the chart they make five straight stripes going down: every second column.

Why straight down? Because each row is 10 long, and 10 is five 2s. So after counting by 2s along a whole row, you start the next row in exactly the same place: on the 2nd square, then the 4th, the 6th... The coloured squares in each row sit right under the ones above.

Another way to say it: each coloured number is a step down (+10) from the one above it, and if a number is even, then 10 more is even too.

Counting by 5s and 10s

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100

The count by 10s (green) is the last column: 10, 20, 30, ..., 100. Each one is exactly one step down from the one before, because one step down is +10.

The count by 5s is the green column and the column of 5, 15, 25, ... (orange). Two stripes. That is because 10 is two 5s: every row holds exactly two hops of 5, so the pattern lines up from row to row, just like the 2s.

Counting by 9s: the slide

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100

Here is the puzzle from the start. Why does each number in the count by 9s sit one down and one left of the one before?

Think about what +9 does on the chart. Adding 9 is the same as adding 10, then taking away 1:

  • 18 + 9: first +10 takes you to 28 (one step down). Then −1 takes you to 27 (one step left).
  • 27 + 9: first +10 takes you to 37 (one down). Then −1 takes you to 36 (one left).

That happens every time, because 9 is always 10 take away 1. So every hop of 9 is the same move on the chart: one down, one left. Repeating the same move makes a straight line, and that is the slide.

What about 90? From 81, one step down is 91. One step left of 91 falls off the left edge of the chart, so it wraps round to the end of the row above: 90. The move is still "+10, then −1"; the picture just has to jump to the other side because the chart is cut into rows. That is why 90 is not on the slide.

There is a hidden pattern in the digits too. Look at 9, 18, 27, 36, 45, 54, 63, 72, 81, 90. The tens digit goes up by one (the step down) and the ones digit goes down by one (the step left). And the two digits always add to 9: 1 + 8, 2 + 7, 3 + 6, ... because one goes up exactly as much as the other goes down.

Counting by 11s: the other slide

Now try 11. Adding 11 is adding 10, then adding 1: one step down and one step right.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100

11, 22, 33, 44, 55, 66, 77, 88, 99: a slide going down to the right. (And the two digits are always the same!)

Counting by 3s: slanting stripes

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100

Counting by 3s does not make straight stripes, because 10 is not a whole number of 3s (three 3s make 9, and four 3s make 12). So each row starts the pattern in a different place, and the stripes lean.

But look at one stripe, such as 6, 15, 24, 33, 42, 51. From 6 to 15 is +9. From 15 to 24 is +9, and so on. Each stripe is a slide of 9s! That is because three hops of 3 make 9, so every third coloured square is one down and one left of the last.

So the picture of the 3s is made of 9-slides, side by side.

What the stripes tell you

The pictures are not only pretty. They help you check and predict.

  • Is 74 in the count by 2s? It sits in the column of 4, 14, 24 ..., a coloured stripe. Yes.
  • Is 74 in the count by 3s? Find it on the 3s picture: 72 is coloured and 75 is coloured, but 74 is not. No.
  • What comes after 63 in the count by 9s? One down (73), one left: 72.

Worked examples

Example 1. Moves on the chart. Start at 46. Where do you land after one step down, then two steps right?

  1. One step down: 46 + 10 = 56.
  2. Two steps right: 56 + 1 + 1 = 58.
  3. Check: altogether we added 10 + 2 = 12, and 46 + 12 = 58. ✓

Example 2. Adding 9 the quick way. Work out 57 + 9 using the chart.

  1. +9 is one down, one left.
  2. One down from 57 is 67. One left is 66.
  3. Check: 57 + 10 = 67, then take 1: 66. ✓

Example 3. Adding 19. Work out 34 + 19.

  1. 19 is 20 take away 1. So: two steps down, one step left.
  2. Two down from 34: 44, then 54. One left: 53.
  3. Check: 34 + 20 = 54, then take 1: 53. ✓

Example 4. Why 7s make no simple picture. Look at the count by 7s: 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, ...

  1. 7 is in column 7, 14 in column 4, 21 in column 1, 28 in column 8.
  2. Each hop of 7 is "one down, three left" or "seven right", depending on where you are. It does not keep doing the same move.
  3. The pattern only lines up again after 70, because 70 is the first number that is both in the count by 7s and in the count by 10s. 70 is at the end of a row, so 77 sits right below 7, 84 below 14, and so on. The 7s make a pattern that repeats every 7 rows!

Try it

  1. On a hundred chart, what number is one step down from 38? One step up from 38?
  2. Start at 25. Go one step down and one step left. Where are you? What did you add?
  3. Use the "one down, one left" move to work out 45 + 9 and 72 + 9.
  4. Use the chart to work out 26 + 11 and 63 + 11.
  5. Which of these are in the count by 9s: 36, 46, 54, 64, 81?
  6. Count by 4s on a hundred chart: 4, 8, 12, 16, 20, 24, ... Which columns are coloured in the first row? In the second row? Do they make straight stripes?
  7. What number is two steps down and one step left of 41? What did you add altogether?
  8. Explain why counting by 10s makes a straight stripe going down.
Answers
  1. One down: 48. One up: 28.
  2. One down: 35. One left: 34. You added 10 and took away 1, so you added 9.
  3. 45 + 9: one down is 55, one left is 54. 72 + 9: one down is 82, one left is 81.
  4. 26 + 11: one down is 36, one right is 37. 63 + 11: one down is 73, one right is 74.
  5. 36, 54 and 81. (The count by 9s is 9, 18, 27, 36, 45, 54, 63, 72, 81, 90. The digits of 36, 54 and 81 add to 9; 46 and 64 add to 10.)
  6. First row: 4 and 8 (columns 4 and 8). Second row: 12, 16, 20 (columns 2, 6 and 10). Third row: 24 and 28 (columns 4 and 8 again). They do not make straight stripes; the rows take turns. That is because 10 is not a whole number of 4s, but 20 is: so the pattern repeats every two rows.
  7. Two down from 41: 51, then 61. One left: 60. You added 20 and took away 1: 19.
  8. One step down is always +10. So each number in the count by 10s is exactly one step below the last one, in the same column.

Common mistake

Going the wrong way for 9. Adding 9 is one down and one left (10 take away 1), not one right. One down and one right is +11.

Forgetting the edge. At the end of a row, "one step right" jumps to the start of the next row: one right from 30 is 31, which is at the far left of the next row. The moves still work as numbers, but the picture wraps round.

Where it leads