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Pattern rules in words

Patterns and sequences · level 2, lesson 1 of 6

A question to start

Kai writes the start of a number pattern on the board: 2, 4, ... and asks the class what comes next.

Ana says 6. "You add 2 each time: 2, 4, 6, 8."

Ben says 8. "You double each time: 2, 4, 8, 16."

Who is right?

The surprising answer is: both of them, or rather, nobody can tell yet. Two numbers are not enough to know the rule. Kai has to say his rule, or show more numbers. In this lesson you will learn to say pattern rules in clear words, so that everyone gets the same pattern.

The idea

Number patterns

In Level 1 the patterns were made of beads, blocks and claps. A number pattern is a list of numbers that follows a rule, like

3, 5, 7, 9, 11, 13, ...

The three dots at the end mean "and it keeps going the same way".

Each number in the list is called a term. In 3, 5, 7, 9, ... the first term is 3, the second term is 5, the third term is 7, and so on.

A rule has two parts

To tell someone a number pattern, you need to tell them two things:

  1. Where it starts. This is the first term.
  2. What to do each time to get the next term.

For 3, 5, 7, 9, ... the rule in words is:

Start at 3 and add 2 each time.

Here it is on a number line. Each hop is 2 long.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 +2 +2 +2 +2 +2

With those few words, anyone can write the pattern, as far as they like. They do not need to see it. That is what makes a rule so useful: a short sentence holds a pattern that goes on forever.

Finding the rule from the numbers

When someone gives you the numbers and you have to find the rule, work like a detective.

  1. The start is easy: it is the first number.
  2. Look at the gaps. Find how much each number changes to get to the next one.
  3. Check every gap. The rule must work for all of them, not only the first one.

For 4, 9, 14, 19, 24:

  • From 4 to 9 is +5. From 9 to 14 is +5. From 14 to 19 is +5. From 19 to 24 is +5.
  • Every gap is the same, so the rule is: start at 4 and add 5 each time.

A helpful way to write it is to put the gaps between the numbers:

4  (+5)  9  (+5)  14  (+5)  19  (+5)  24

The same start, a different rule

Two patterns can start at the same number and still be very different, because the rule says what to do each time. Here are two patterns that both start at 3.

Start at 3 and add 2 each time: 3, 5, 7, 9, 11, 13, ... (the number line above).

Start at 3 and add 4 each time: 3, 7, 11, 15, 19, ...

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 +4 +4 +4 +4

A rule can also take away. Taking away from 3 would run out almost at once, so this one starts higher.

Start at 20 and take away 3 each time: 20, 17, 14, 11, 8, 5, 2.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 −3 −3 −3 −3 −3 −3

The hops go backwards this time, so the numbers get smaller.

So the start alone does not tell you the pattern, and the hop alone does not tell you the pattern. You need both.

Why two numbers are not enough

Back to Kai's puzzle: 2, 4, ... Ana's rule is start at 2 and add 2 each time:

0 1 2 3 4 5 6 7 8 9 10 +2 +2 +2

Ben's rule is start at 2 and double each time. Doubling means adding a number to itself: 2 doubled is 4, 4 doubled is 8, 8 doubled is 16.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Both rules give 2, then 4. They only disagree at the third term: 6 or 8. With only two numbers, there is just one gap to look at, and many rules could make that one gap. That is why you should always look at at least three numbers (two gaps or more) before you trust a rule. And even then, it is best when the person who made the pattern tells you the rule in words.

This is a big idea in mathematics: a few examples are not a proof. You will meet it again and again, all the way up to Level 3, where a famous circle puzzle fools almost everybody.

Rules that repeat

Not every number pattern grows. Some repeat, just like the bead patterns:

1, 2, 3, 1, 2, 3, 1, 2, 3, ...

The rule in words is: repeat 1, 2, 3 again and again. The core is 1, 2, 3. If you look at the gaps, they are +1, +1, then −2 (from 3 back down to 1), and then the gaps repeat too.

Saying rules clearly

A rule must be clear enough that two people who follow it get exactly the same numbers. Compare:

  • "The numbers go up." Not clear! Up by how much? From where?
  • "Add 5." Better, but where does it start?
  • "Start at 10 and add 5 each time." Clear. Everyone gets 10, 15, 20, 25, 30, ...

When you write a rule, read it back and ask: could someone follow this without asking me any questions?

Worked examples

Example 1. Find the rule. 6, 9, 12, 15, 18, ...

  1. The start: 6.
  2. The gaps: 6 to 9 is +3, 9 to 12 is +3, 12 to 15 is +3, 15 to 18 is +3.
  3. All the same. The rule: start at 6 and add 3 each time.
  4. The next two terms: 18 + 3 = 21, 21 + 3 = 24.

Example 2. Use a rule. Write the first six terms of: start at 7 and add 10 each time.

  1. Start: 7.
  2. 7 + 10 = 17. 17 + 10 = 27. 27 + 10 = 37. 37 + 10 = 47. 47 + 10 = 57.
  3. The pattern: 7, 17, 27, 37, 47, 57.
  4. Notice: the ones digit stays 7, and the tens digit goes up by one each time. Adding 10 always does that.

Example 3. A gap that does not fit. Mo says the rule for 5, 10, 15, 21, 25 is "add 5". Is he right?

  1. Gaps: 5 to 10 is +5. 10 to 15 is +5. 15 to 21 is +6. 21 to 25 is +4.
  2. Two gaps do not fit "add 5". But if the 4th term were 20, every gap would be +5: 5, 10, 15, 20, 25.
  3. So the pattern has a mistake: 21 should be 20. The rule is "start at 5 and add 5 each time". (Just like finding the wrong bead in Level 1, we look for the smallest fix: one number changed.)

Example 4. Three rules, same start. Write the first five terms of each rule, all starting at 1: (a) add 1 each time, (b) add 3 each time, (c) double each time.

  1. (a) 1, 2, 3, 4, 5.
  2. (b) 1, 4, 7, 10, 13.
  3. (c) 1, 2, 4, 8, 16.
  4. Rules (a) and (c) agree for the first two terms (1, 2) and then split apart. By the fifth term, doubling is far ahead: 16 against 5.

Try it

  1. Write the rule in words: 2, 4, 6, 8, 10, ...
  2. Write the rule in words: 11, 21, 31, 41, ...
  3. Write the first five terms: start at 5 and add 4 each time.
  4. Write the first five terms: start at 30 and take away 5 each time.
  5. Two patterns start at 4. One adds 3 each time, one adds 6 each time. Write the first four terms of each. Which numbers are in both patterns?
  6. Find the mistake and fix it: 3, 7, 11, 14, 19, 23.
  7. A pattern starts 5, 10, ... Write two different rules that both start this way, and the third term for each.
  8. Write the rule in words: 4, 8, 4, 8, 4, 8, ...
Answers
  1. Start at 2 and add 2 each time. (Every gap is +2.)
  2. Start at 11 and add 10 each time. (Every gap is +10.)
  3. 5, 9, 13, 17, 21. (5 + 4 = 9, 9 + 4 = 13, 13 + 4 = 17, 17 + 4 = 21.)
  4. 30, 25, 20, 15, 10.
  5. Add 3: 4, 7, 10, 13. Add 6: 4, 10, 16, 22. In both: 4 and 10. (If you keep going, 16 and 22 also turn up in the add-3 pattern: 4, 7, 10, 13, 16, 19, 22. Every number in the add-6 pattern is in the add-3 pattern, because adding 6 is the same as adding 3 twice!)
  6. Gaps: +4, +4, +3, +5, +4. The 4th term should be 15, not 14. Fixed: 3, 7, 11, 15, 19, 23. The rule is "start at 3 and add 4 each time".
  7. For example start at 5 and add 5 each time (third term 15), and start at 5 and double each time (third term 20). Other answers are right too, such as start at 5, add 5, then add 10, then add 15...
  8. Repeat 4, 8 again and again. (The gaps go +4, −4, +4, −4.)

Common mistake

Giving only half the rule. "Add 3" is not a whole rule, because 1, 4, 7, 10 and 5, 8, 11, 14 both "add 3". Say where it starts too.

Checking only the first gap. In 5, 10, 15, 21, 25 the first gap is +5, but not every gap is. Check every gap before you write the rule.

Where it leads