3.OA.9: Find and Explain Arithmetic Patterns
Also written as 3.OA.D.9 and CCSS.MATH.CONTENT.3.OA.D.9 · Cluster: Solve problems involving the four operations, and identify and explain patterns in arithmetic.
The Official Wording
Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations. For example, observe that 4 times a number is always even, and explain why 4 times a number can be decomposed into two equal addends.
Common Core State Standards, Grade 3, Operations & Algebraic Thinking · Read it at the source · Checked October 14, 2026
What 3.OA.9 Means in Plain English
Third graders hunt for patterns in the addition and multiplication tables, then explain why each one works. Every product in the 5s row ends in 0 or 5, and every number in the 6s row is double the one above it in the 3s row.
The explaining is the real work: “6 is two 3s, so 6 × 7 is two 3 × 7s” is a reason, while “it just goes up” is not. In 4th grade students build patterns from a rule and explain features the rule never mentioned (4.OA.5).
3.OA.9 I Can Statements
- I can find a pattern in the multiplication or addition table.
- I can explain a pattern using what I know about multiplying.
- I can tell whether a sum or product will be even or odd.
3.OA.9 Worked Example
Two rows of the multiplication chart side by side, and the reason one is always double the other.
In the multiplication chart, every number in the 6s row is double the number above it in the 3s row. Why?
- Line up the two rows and compare them column by column.3, 6, 9, 12, 15 and 6, 12, 18, 24, 30
- Test one column, the 4s. Three fours is 12 and six fours is 24, which is double 12.3 × 4 = 12, 6 × 4 = 24
- Six is two threes, so 6 groups of 4 is 3 groups of 4, twice.6 × 4 = 2 × (3 × 4)
- That works in every column, because 6 is always 2 × 3, whatever it multiplies.6 × n = 2 × (3 × n)
Check it: Test a column you didn’t use: 3 × 9 = 27, double 27 is 54, and 6 × 9 = 54.
Common Mistakes with 3.OA.9
Describing instead of explaining
“The numbers go up by 6” given as the reason the 6s row doubles the 3s row
Try this: Ask “why?” a second time and push for a reason built from groups, like 6 groups is 3 groups twice.
Stretching a pattern to another operation
“Odd times odd is odd, so odd plus odd is odd too”
Try this: Build odd numbers from pairs plus one leftover. Two odds bring two leftovers, which make a new pair, so the sum is even.
Trusting a pattern past the chart
“The digits of a nines product add to 9,” then stuck at 9 × 11 = 99
Try this: Test patterns past the edge of the chart and say exactly where they hold, like 9 × 1 through 9 × 10.
Where 3.OA.9 Fits
Practice 3.OA.9
Quick Check
- Is 7 × 6 even or odd? Explain without multiplying.
- Each number in the 10s row is double the number in which row? Why?
- What happens to the sums 5 + 4, 6 + 4, 7 + 4 and 8 + 4? Why?