Finding and Explaining Patterns
A 45-minute Grade 3 lesson on spotting arithmetic patterns and, more importantly, on saying why they happen — because a pattern nobody can explain is a coincidence.
Finding and Explaining Patterns is a free 45-minute Grade 3 maths lesson plan for Common Core standard 3.OA.D.9 — spot and explain number patterns. It is timed across 5 sections, gives the exact wording to use where the wording carries the mathematics, names 3 misconceptions with the teaching move that addresses each, and ends with an exit ticket and a printable worksheet with its answer key.
- Grade:
- Grade 3
- Length:
- 45 minutes
- Standard:
- 3.OA.D.9
Grade 3 · Operations & Algebraic Thinking
3.OA.D.9
Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations.
Objective
Students will be able to identify a pattern in a number sequence or table and explain it using properties of operations.
Students are successful when they can
- Describes a pattern in words, not just by continuing it.
- Explains why the pattern happens.
- Predicts a term further along without listing every term.
What you need
- A completed multiplication grid per pair
- Coloured pencils
- Counters for building small cases
- The printable practice sheet linked at the end of this plan
The lesson
Warm-up — what do you notice?
5 min- Show the 5 times table written out. Take everything the class notices.
- Accept all observations, including the obvious ones. Sorting them comes next.
Teach — noticing, then explaining
14 min- Shade the multiples of 4 on the grid. They are all even. Ask why.
- Push past 'they just are' to the reason: four is even, so any number of fours is a number of pairs.
- Now the 9s: the digits of every multiple sum to 9. Notice it, then check it on 9 × 7 and 9 × 12.
- Explaining that one is genuinely hard, and saying so is honest. Some patterns are noticed long before they are explained.
- Take an easier one: adding two odd numbers always gives an even. Build it with counters — each odd has one left over, and the two leftovers pair up.
- Say the standard: noticing is the first half, explaining is the second, and the second is where the mathematics is.
“Anyone can see the pattern. The question is why it has to be that way — and 'because it always is' is not yet an answer.”
Guided practice — notice and justify
12 min- Pairs find three patterns on the multiplication grid and write each as a sentence.
- For each, they must attempt a reason. An honest 'we could not explain this one' is an acceptable outcome for one of the three.
- Pairs swap and try to explain each other's unexplained pattern.
Independent practice
11 min- Students complete a patterns page.
- A continued sequence with no description means the pattern was matched rather than understood — ask for the rule in words.
Close
3 min- Ask whether an odd number times an odd number is odd or even, and why.
- Take a prediction and one reason, and leave the checking for tomorrow.
What goes wrong, and why
Each of these is a reasoning error rather than carelessness. The fix addresses the reasoning.
Continues a sequence correctly but cannot say the rule.
Why: The next term has been found by matching the visual gap rather than by identifying an operation. On a simple sequence this always works.
Fix: Ask for the twentieth term. Matching gaps requires listing everything; a rule does not, and the demand exposes which one the pupil has.
Says a pattern is true because it worked for three examples.
Why: Checking cases is how patterns are found, and nothing has yet distinguished evidence from proof.
Fix: Find a pattern that holds for several cases and then fails. One counterexample teaches the difference permanently.
Describes a pattern by its appearance — 'the numbers go down the diagonal' — without any arithmetic.
Why: The grid is being read as a picture, which is a legitimate first observation but not yet a mathematical statement.
Fix: Ask what the numbers on that diagonal are, and what they have in common. The picture points at the pattern; the numbers are the pattern.
If they are not there yet
- Work with the 2s, 5s and 10s where the patterns are strong and the explanations are within reach.
- Build small cases with counters so an explanation can be seen rather than argued.
If they finish early
- Explain why the multiplication grid is symmetric about its diagonal.
- Investigate what happens to the digit sum of multiples of 3.
- Find a pattern in the addition table and explain it.
Exit ticket
Every multiple of 6 is even. Why must that be true?
What to look for: A reason involving 6 being even, or 6 being made of pairs. 'Because they all end in 0, 2, 4, 6 or 8' is an observation restated, not an explanation — worth pushing on.
Printable practice for 3.OA.D.9
Free to print and copy, answer key included. Each sheet can generate a fresh set of problems for a retake.
- Patterns — AddingWrite the missing numbers.
- Patterns — Counting by NinesWrite the missing numbers.
Questions about Finding and Explaining Patterns
How long does the Finding and Explaining Patterns lesson take?
45 minutes, split across 5 timed sections: warm-up 5 min, teach 14 min, guided practice 12 min, independent practice 11 min, close 3 min. The minutes are stated per section and add up to the stated length, so the plan can be cut or extended at a section boundary rather than abandoned halfway.
Which standard does Finding and Explaining Patterns teach?
CCSS.MATH.CONTENT.3.OA.D.9 — 3.OA.D.9, Operations & Algebraic Thinking for Grade 3: “Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations.” In plain terms, spot and explain number patterns.
What do I need to teach this lesson?
A completed multiplication grid per pair, coloured pencils, counters for building small cases and the printable practice sheet linked at the end of this plan. Everything listed is either ordinary classroom equipment or a free printable from this site — there is nothing to buy and nothing to prepare beyond photocopying.
What mistakes should I expect in this lesson?
Three, each a reasoning error rather than carelessness: continues a sequence correctly but cannot say the rule; says a pattern is true because it worked for three examples and describes a pattern by its appearance — 'the numbers go down the diagonal' — without any arithmetic. The plan gives the thinking behind each one and the teaching move that addresses the thinking rather than the symptom.
How do I know whether the lesson worked?
The exit ticket asks: “Every multiple of 6 is even. Why must that be true?.” A reason involving 6 being even, or 6 being made of pairs. 'Because they all end in 0, 2, 4, 6 or 8' is an observation restated, not an explanation — worth pushing on. That tells you whether the class needs a reteach or a thirty-second conversation before you plan tomorrow.
Is there a worksheet to go with Finding and Explaining Patterns?
Yes — 2 printable 3.OA.D.9 worksheets, linked at the end of the plan and free like the rest of the site. Each one comes with an answer key and 25 versions, so the practice after the lesson can be a different paper on every desk.
Is Finding and Explaining Patterns free to use?
Yes. The plan, its printables and their answer keys are free to read, print and photocopy for your own class, with no account and no email capture. It was last reviewed on 5 September 2026. Please do not resell it or republish it as your own.
Last reviewed 2026-09-05. Aligned to 3.OA.D.9 of the Common Core State Standards for mathematics.