Patterns and sequences
A pattern question asks for two things and only marks one of them: work out the rule, then use it. Most mistakes are in the first half, and a child who says the rule out loud before answering — "it goes up in sevens" — gets far more of them right than a child who jumps to the next number.
Start the levels Print a worksheet
Finding the rule from the gaps
3, 10, 17, 24, ___
- Find the gaps: 10 take 3 is 7, 17 take 10 is 7, 24 take 17 is 7.
- Every gap is 7, so the rule is add 7.
- 24 and 7 more is 31.
- The answer is 31.
Worth knowing. Checking every gap rather than just the first is what stops a wrong rule. Two numbers agree by luck all the time; four in a row do not.
When the gaps grow
2, 6, 18, 54, ___
- The gaps are 4, 12 and 36 — not the same, so it is not adding.
- Try dividing instead: 6 divided by 2 is 3, 18 divided by 6 is 3, 54 divided by 18 is 3.
- Every step multiplies by 3.
- 54 times 3 is 162.
Worth knowing. Gaps that grow are the signal to try multiplying. It is worth trying the subtraction first anyway, because ruling it out is what makes the multiplying obvious.
Where it usually goes wrong
- Checking only the first gap, so 2, 6, 10, 14 and 2, 6, 18, 54 look like the same pattern at the start.
- Continuing the pattern backwards by mistake when it counts down.
- Giving the rule instead of the next number — "add 7" is the working, not the answer.
The Professor writes the gaps between the numbers underneath before he does anything else. If the gaps are all the same, the rule is right there.
Practise it
All level pathsQuestions teachers and parents ask
How many terms does it take to be sure of a rule?
Strictly, no number is enough — any four numbers can be continued in infinitely many ways. In practice three or four with a constant gap or a constant multiplier fixes what is being asked, and every pattern here shows at least three so the question has one fair answer.
When are patterns taught?
Generating a pattern from a rule is grade 4 (4.OA.C.5), and comparing two patterns arrives in grade 5 (5.OA.B.3). Skip counting in earlier grades is the same idea before it has the name.
Why not use patterns like 1, 1, 2, 3, 5?
Fibonacci and its relations are lovely and they are not fair as a test, because there is no way to work them out — you either recognise them or you do not. The patterns here follow a constant step or a constant multiplier, which a child can always find from the numbers in front of them.
Where this shows up
Standards
- K.OA · Operations and algebraic thinking
- 4.OA · Operations and algebraic thinking
- 5.OA · Operations and algebraic thinking
- 8.F · Functions
- CCSS.MATH.CONTENT.K.CC.A.1 — Count to 100 by ones and by tens.
- CCSS.MATH.CONTENT.2.NBT.A.2 — Count within 1000, and skip-count by 5s, 10s and 100s.
- CCSS.MATH.CONTENT.4.OA.C.5 — Generate a number or shape pattern that follows a given rule, and notice features of the pattern the rule did not state.
- CCSS.MATH.CONTENT.5.OA.B.3 — Generate two numerical patterns from two rules and identify the relationship between corresponding terms.
- CCSS.MATH.PRACTICE.MP7 — Look for and make use of structure: find the rule or relationship underneath a set of facts, and use it to settle what has to follow.