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Repeating patterns

Patterns and sequences · level 1, lesson 1 of 6

A question to start

Mia is making a necklace. Here are the beads she has put on the string so far.

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Which bead goes on next?

You probably said "orange" very fast. Nobody told you the rule. Your eyes saw it all by themselves: orange, blue, orange, blue, again and again.

Now a harder question. Mia wants her necklace to have 20 beads. What colour will the 20th bead be? Can you tell without drawing all 20? By the end of this lesson you can.

The idea

What a pattern is

A pattern is something that follows a rule, so you can tell what comes next.

Some patterns repeat. The same few things come again, and again, and again, in the same order, like the beads on Mia's necklace. This lesson is all about repeating patterns.

Other patterns grow: each step is a bit bigger than the one before. You will meet those in lesson 5.

The core: the part that repeats

Look at Mia's beads again. Put a bracket under each part that repeats.

The part that repeats is orange, blue. We call it the core of the pattern. The whole necklace is just the core, then the core again, then the core again.

The core is the most important thing to find. Once you know the core, you know the whole pattern, even the parts nobody has made yet.

Here is a good way to say it out loud:

"The core is orange, blue. It is 2 beads long."

Different cores make different patterns

The same two colours can make many different patterns. It depends on the core.

Orange, orange, blue. The core is 3 beads long.

Orange, blue, blue. This core is also 3 beads long, and it uses the same colours. But it is a different pattern! Look: here the orange bead is on its own, and the blue beads come in pairs.

A pattern can also use three different things. Here the core is square, triangle, circle.

Giving patterns letter names

Grown-ups who study patterns give them short names with letters. They use a new letter for each new thing in the core.

  • orange, blue is called an AB pattern. A stands for orange and B stands for blue.
  • orange, orange, blue is an AAB pattern.
  • orange, blue, blue is an ABB pattern.
  • square, triangle, circle is an ABC pattern.

The letters do not care about colours or shapes. Red, green is an AB pattern too. So is cat, dog, cat, dog. The letter name tells you the shape of the pattern: what repeats, and how.

Here is a trick for finding the letter name. Start with the first thing and call it A. Every time you see something new, give it the next letter: B, then C. Every time you see something you have seen before, use its old letter.

For the beads orange, orange, blue, orange, orange, blue: orange is A. Orange again is A. Blue is new, so it is B. Then orange is A again... So it is A A B, A A B. An AAB pattern.

How long must a pattern be before you can trust it?

Here is something important. If you only see orange, blue, you cannot be sure what comes next. Maybe it is orange again. Maybe it is blue (orange, blue, blue). Maybe it is green!

To be sure about a repeating pattern, you need to see the core at least twice, and better three times. That is why patterns on worksheets usually show the core three times. When you see the core again and again, you can trust the rule.

Why the core lets you see far ahead

Now back to the necklace question: what colour is the 20th bead?

Let us number the beads.

1 2 3 4 5 6 7 8 9 10

Look at the numbers under the orange beads: 1, 3, 5, 7, 9. Look at the numbers under the blue beads: 2, 4, 6, 8, 10. Every blue bead has a number you say when you count by 2s: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20.

Why? Because the core is 2 beads long, and blue is the last bead of the core. Each time you finish one core, you have used 2 more beads. So the end of each core lands on 2, then 4, then 6, and so on.

20 is in the count by 2s. So the 20th bead is the end of a core, and it is blue.

The same idea works for any core. Here is the square, triangle, circle pattern with numbers.

1 2 3 4 5 6 7 8 9 10 11 12

The core is 3 long, and the green circle is last in the core. So the green circles sit at 3, 6, 9, 12: the count by 3s. The 15th shape is a green circle too, because 15 is in the count by 3s (3, 6, 9, 12, 15). So the 16th shape starts a new core: an orange square.

You do not need to draw the whole pattern. You only need the core, and some counting.

Patterns with blocks, beads and stickers

You can make a repeating pattern out of anything: blocks, beads, stickers, buttons, crayons, toy cars. Here is one made from blocks.

The core is orange, orange, blue, green. That is 4 blocks long, an AABC pattern. The next block after the orange one at the end is another orange.

Worked examples

Example 1. Find the core. Sam lines up his toy cars: red, red, yellow, red, red, yellow, red, red, yellow. What is the core, and what is the pattern's letter name?

  1. Look for where it starts again. Red, red, yellow... then red again. So it starts again after yellow.
  2. The core is red, red, yellow. It is 3 long.
  3. Check: red, red, yellow / red, red, yellow / red, red, yellow. Yes, every part matches.
  4. Red is A. Red again is A. Yellow is new, so it is B. The letter name is AAB.

Example 2. Fill the gaps. What goes in the two empty spaces?

  1. Find the core: green, green, orange. Then green, green, orange again. The core is green, green, orange.
  2. After the last orange comes the core again: green, green, orange.
  3. The picture already shows one green after the second orange. So the two empty spaces are the rest of the core: green, then orange.

Example 3. Look far ahead. A pattern of stickers goes star, moon, star, moon, star, moon... What is the 9th sticker? What is the 14th?

  1. The core is star, moon. It is 2 long. The moon is last in the core.
  2. So the moons sit at 2, 4, 6, 8, 10, 12, 14 (counting by 2s). The stars sit at 1, 3, 5, 7, 9, 11, 13.
  3. 9 is not in the count by 2s, so the 9th sticker is a star.
  4. 14 is in the count by 2s, so the 14th sticker is a moon.

Example 4. Same letters, different things. Which of these is an ABB pattern, like orange, blue, blue?

  • (a) dog, cat, cat, dog, cat, cat
  • (b) dog, dog, cat, dog, dog, cat
  • (c) dog, cat, dog, cat

Give letters. (a) dog is A, cat is B, cat again B: A B B. Yes. (b) A A B: that is AAB. (c) A B: that is AB. So the answer is (a). Dogs and cats are not beads, but the pattern has the same shape.

Try it

Questions 1 to 3 use these three rows.

Row 1.

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Row 2.

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Row 3.

  1. What comes next in row 1?
  2. What comes next in row 2?
  3. What goes in the two empty spaces in row 3?
  4. Give the letter name of this pattern: jump, jump, sit, jump, jump, sit.
  5. Make an AB pattern using the words "up" and "down". Make an ABB pattern using the same two words.
  6. A pattern of beads goes red, yellow, red, yellow, ... What colour is the 12th bead? What colour is the 15th?
  7. A pattern goes apple, pear, banana, apple, pear, banana, ... What is the 9th fruit? What is the 10th?
  8. Ben says: "Red, blue, red, blue is a different pattern from blue, red, blue, red." Is he right?
Answers
  1. Blue. The core is blue, orange, 2 long. The row ends with orange, which is the end of a core, so the next bead starts a new core: blue.
  2. Orange triangle. The core is orange triangle, orange triangle, green circle. The row ends with the first orange triangle of a new core, so the next one is the second orange triangle.
  3. Green triangle, then orange circle. The core is blue square, green triangle, orange circle. The row ends with the blue square that starts a new core, so the rest of the core comes next.
  4. AAB. Jump is A, jump again is A, sit is new so it is B.
  5. AB: up, down, up, down, up, down. ABB: up, down, down, up, down, down, up, down, down. (Other answers are right too, as long as the core repeats: for example down, up, up is also ABB.)
  6. The core red, yellow is 2 long, and yellow is last. Yellow sits at 2, 4, 6, 8, 10, 12, 14... So the 12th is yellow. 15 is not in the count by 2s, so the 15th is red.
  7. The core is 3 long, and banana is last. Bananas sit at 3, 6, 9, 12 (count by 3s). So the 9th fruit is a banana, and the 10th starts a new core: an apple.
  8. Both are right in a way. The two rows start differently, so as rows of beads they are different. But both are AB patterns: two things taking turns. If you cover the first bead of red, blue, red, blue, red, the rest is blue, red, blue, red. Same pattern shape, different starting place.

Common mistake

Stopping the core too early. In orange, orange, blue, orange, orange, blue, some people say the core is orange, orange. But then the next would be orange, orange again, with no blue. The core must be the whole part that repeats. Test your core: say it again and again and check that it matches every bead you can see.

Forgetting where the row ends. To find what comes next, look at the last bead and ask: where am I in the core? If the row ends in the middle of a core, the next bead finishes that core. It does not start a new one.

Where it leads