Patterns and Algebra

The rule, not the next number

Ask a child what comes next in 4, 7, 10, 13 and most will answer sixteen correctly and learn nothing at all. The mathematics is in the sentence you add three each time, because a rule can be run forward, run backward, and started from anywhere. A child who only produces the next term is guessing one step ahead of you.

So change what you ask for. Ask for the rule in words before the number. Ask what the tenth term would be without writing out the ones in between. Ask them to break the pattern on purpose and see whether you can catch it. Each of those forces the rule into the open where it can actually be checked.

The equal sign is the misreading that costs the most later

Try this before reading further. Write 8 + 4 = __ + 5 in front of a child who has done a year of arithmetic pages and watch what goes in the blank. If it is twelve, they have read the equal sign as an instruction meaning write the answer here, which is exactly what a page of 8 + 4 = __ teaches. Every equation in algebra depends on it meaning something else: both sides name the same amount.

The cure is true-or-false sentences, said aloud, thirty seconds a day. Is 7 = 7 true. Is 5 + 3 = 8 true. Is 9 = 4 + 5 true. Is 6 + 2 = 3 + 5 true. Children object loudly to the backward ones at first, and the objection is the lesson. Keep going until the objection stops coming.

Eight minutes with cups, counters and a balance

This strand needs very little time and benefits from being short and frequent. Use counters your child can move, a plastic cup to hide an unknown underneath, and a balance scale if you own one, because balance is the physical picture the equal sign has been trying to convey all along.

1. Lay out a repeating pattern with buttons or silverware and ask for the rule in a full sentence before anything else.
2. Remove an item from the middle of the pattern rather than the end, and ask what is missing. The middle forces the rule; the end can be guessed.
3. Put seven counters out, cover some with a cup, and say: there were seven, now I can see four, how many are hidden?
4. Write that same situation as 4 + __ = 7, then again as 7 = 4 + __, and read both out loud.
5. Finish with three true-or-false sentences, at least one of them written backward.

You will hear the rule before you hear the answer

When this is working, the rule arrives first: add three each time, double it, add one. Your child solves 8 + __ = 13 by counting up from eight rather than staring at the page. They accept 9 = 4 + 5 without complaint. And they invent patterns for you to solve, which is the clearest sign the structure has become theirs rather than yours.

When it is not working, the next term appears with no explanation behind it, a missing addend is answered by adding the two visible numbers, an equation with the answer on the left provokes a protest, and the letter in 3 + n = 7 is treated as a label for something rather than a number that has a value.

When the cup has to come back out

Go back to objects your child can pick up. Build the pattern physically, say the rule out loud, then break it and ask for the missing piece from the middle. For missing addends, return to the cup and counters until your child stops adding the two numbers they can see and starts counting up to the total instead.

Repeating patterns belong in pre-K and kindergarten, true-or-false equations and missing addends across first and second grade, rules and input-output tables in third and fourth, letters standing for numbers in fifth. The equal-sign misreading takes several weeks of daily true-or-false sentences to dislodge, and it tends to come back after a long stretch of one-sided computation, so keep the thirty seconds going.

First, your child should already…

This skill builds directly on:

• Numbers and Counting — master this first.
• Addition — master this first.

What your child is learning

Patterns and algebra help children describe what stays the same and what changes. This begins with repeating patterns and grows into equations and variables.

Signs your child is ready

• Can continue a simple pattern
• Can explain a rule in words
• Understands equality as balance

Practice this skill

Worksheets and decodable practice for Patterns and Algebra are in the library.

• Kindergarten — Patterns and Early Algebra 12 worksheets
• Grade 3 — Number Patterns 9 worksheets
• Kindergarten — Shapes and Patterns 80 worksheets
• Kindergarten — Logic and Patterns 48 worksheets

Every set opens its first three worksheets to everyone; the rest are included with All Access.

How to know it’s mastered

The child completes, describes, or writes rules and equations for grade-level patterns.