Patterns in addition and multiplication tables
Spotting and explaining patterns in the addition and multiplication tables using properties of operations
About two lessons of 45 to 60 minutes
The multiplication table is full of hidden patterns
A multiplication table looks like a grid of a hundred separate facts to memorise, but it is really full of patterns that make those facts predictable, and even easier to check. Once you know the pattern in a row or a column, you can find a missing fact without having memorised it.
This unit is about noticing those patterns, and then explaining WHY they happen using what you already know about addition and multiplication, not just spotting them by chance.
- The 4-times row: 4, 8, 12, 16, 20each number is 4 more than the one before it
- 3 x 4 and 4 x 3 land in mirror-image cellsthe table is symmetric because multiplication can be done in either order
- Every number in the 5-times table ends in 0 or 5a pattern you can explain, not just notice
What students will be able to do
Students will identify patterns in the addition table and the multiplication table, and explain why those patterns happen using the properties of operations, such as the commutative property.
- I can describe the pattern in a row or column of the multiplication table.
- I can use a pattern in the times table to find a missing fact without recalculating from scratch.
- I can explain why a x b and b x a land on the same value, using the shape of the table.
- I can explain, not just state, why a pattern happens (for example, why the 5-times table always ends in 0 or 5).
Standards this unit teaches
- 3.OA.D.9Common Core (US)Patterns in arithmetic
Identify and explain patterns in addition and multiplication tables using properties of operations.
- AC9M3N07Australian Curriculum v9 (ACARA)Create number algorithms
Follow and create step-by-step procedures that generate sets of numbers, then look for and describe any patterns.
Prior knowledge
This unit builds on skills students should already have met. Revisit any that are shaky first.
Words to teach and display
- Pattern
- a rule that repeats and lets you predict what comes next
- Commutative property
- the rule that multiplying two numbers in either order gives the same answer, such as 3 x 4 = 4 x 3
- Row
- the numbers reading across a table
- Column
- the numbers reading down a table
Teach it: concrete, pictorial, abstract
The lesson moves from things students can hold, to pictures and diagrams, to the written maths. The diagrams below are drawn from data, so they are accurate and print cleanly. Teach straight from them.
1. Patterns in the addition table
PictorialIn an addition table, reading along the row for 6 gives 6+1=7, 6+2=8, 6+3=9, 6+4=10: each answer is exactly 1 more than the one before it, because the second addend went up by 1 each time. That is the whole pattern for every row of an addition table: it grows by the same step as the addend that is changing.
Describe the pattern in the row of the addition table for 6, and use it to predict 6 + 7 without adding from scratch.
- Read the known part of the row: 6+1=7, 6+2=8, 6+3=9, 6+4=10.
- Notice the pattern: each answer is exactly 1 more than the last, matching the second number going up by 1 each time.
- Extend the pattern: 6+5=11, 6+6=12, 6+7=13.
Answer: 6 + 7 = 13, following the +1 pattern.
- What would the row for 9 look like, and why does it grow by the same step as the row for 6?
2. Patterns in the multiplication table
PictorialA multiplication table's rows do not grow by a fixed +1 step, they grow by the row's own number. The 4-times row goes 4, 8, 12, 16, 20, growing by 4 each time, which is really just skip counting by 4. The table also has a mirror-image pattern: the cell for 3 x 4 and the cell for 4 x 3 both hold 12, because multiplying two numbers in either order gives the same product.
This same idea explains why every number in the 5-times table ends in 0 or 5: multiplying 5 by an even number always lands on a multiple of 10 (ending in 0), and multiplying 5 by an odd number always lands halfway between two multiples of 10 (ending in 5).
Describe the pattern in the 4-times row of the multiplication table, then use it to find 4 x 5, and check it against 5 x 4.
- Read the known part of the row: 4x1=4, 4x2=8, 4x3=12, 4x4=16.
- Notice the pattern: each answer is 4 more than the last, since the row is skip counting by 4.
- Extend the pattern: 4x5=20.
- Check the mirror cell: 5x4 also equals 20, because multiplying in either order gives the same product.
Answer: 4 x 5 = 20, and it matches 5 x 4 = 20 exactly.
- Without calculating from scratch, use the row pattern to find 4 x 6.
- Why does every entry in the 2-times table come out even?
Common misconceptions and how to address them
MisconceptionTrusting a pattern rule after checking only the first two entries, without confirming it against a third or fourth pair.
Why it happens: Two matching examples feel convincing, but some near-patterns break after a couple of terms, especially when a student mis-happened to pick coincidental first steps.
How to address it: Make checking at least three consecutive pairs a fixed habit before trusting any rule, and celebrate when a student catches a rule that breaks.
MisconceptionTreating the table's mirror symmetry (3x4 and 4x3 both equal 12) as a coincidence to memorise separately, rather than connecting it to the commutative property.
Why it happens: The symmetry is usually noticed visually before the property behind it is named, so the two ideas sit unconnected in memory.
How to address it: Point directly at the two mirrored cells and say the rule together: multiplying in either order gives the same answer, that is why the table is symmetric across its diagonal.
MisconceptionReading a multiplication table cell as if it were an addition table cell, for example reading the cell where row 3 meets column 4 as 3+4=7 instead of 3x4=12.
Why it happens: Addition tables are usually taught first, so the habit of adding the row and column labels can carry over into reading a multiplication table.
How to address it: Label every table clearly with a x or + symbol in the corner, and have students say 'row times column' or 'row plus column' out loud before reading any cell.
MisconceptionAssuming every pattern in a table must be increasing, missing patterns that stay constant or repeat, such as the x0 row (always 0) or the x1 row (matches the column headers exactly).
Why it happens: Growing patterns get most of the classroom attention, so a constant or repeating pattern does not register as a 'real' pattern.
How to address it: Explicitly explore the x0 and x1 rows as patterns in their own right and describe them in words: the x0 row never changes, the x1 row copies the column header.
Guided practice (with answers)
1. Describe the pattern in the 7-times row (7, 14, 21, 28...) and use it to find 7 x 5.
Answer: 35. The row grows by 7 each time (skip counting by 7): 28 + 7 = 35.
2. Explain why 6 x 8 and 8 x 6 give the same answer.
Answer: Because multiplying two numbers in either order gives the same product (the commutative property); both equal 48.
3. Why does every number in the 5-times table end in 0 or 5?
Answer: Multiplying 5 by an even number always lands on a multiple of 10 (ends in 0); multiplying 5 by an odd number lands halfway between two multiples of 10 (ends in 5).
4. What is the pattern in the addition table's row for 8, and what is 8 + 6?
Answer: Each answer is 1 more than the last as the second addend goes up by 1; 8 + 6 = 14.
5. What number fills every cell of the multiplication table's x0 row?
Answer: 0, since any number multiplied by 0 is always 0.
Independent practice worksheets
Set the matching ChalkBee worksheets for independent work. The answer keys are computed in code, so they are never wrong. A printable multiplication chart is the best first resource for spotting these patterns directly.
Differentiation
- Stay with a single row at a time (such as the 2s or 5s) and describe its pattern in words before moving to a new row.
- Use a hundred-square style highlighted chart so a pattern's visual shape (a diagonal, a column) is easy to see.
- Pair every 'what' question (what is the pattern) with a 'why' question (why does it happen) so explaining becomes routine, not an extra step.
- Explore the diagonal of square numbers (1, 4, 9, 16...) running through the table and describe its own pattern.
- Predict a fact two rows beyond the memorised times tables (such as the 12-times row) using only the row-growth pattern.
- Write a 'true or false' pattern claim for a partner to test against the actual table, such as 'every number in the 6-times table is even'.
Assessment: exit ticket
A short exit ticket sampling addition-table and multiplication-table patterns.
1. The 3-times row is 3, 6, 9, 12. What is the pattern, and what is 3 x 5?
Answer: Each answer is 3 more than the last; 3 x 5 = 15 (12 + 3).
2. Why do 7 x 2 and 2 x 7 land on the same value in the table?
Answer: Because multiplying in either order gives the same product; both equal 14.
3. Every entry in the 4-times table is what kind of number, even or odd? Why?
Answer: Even, because 4 is even, and an even number multiplied by anything always gives an even product.
Teacher notes and timings
- Rough timing across two lessons: Lesson 1 the addition table (section 1), Lesson 2 the multiplication table plus the exit ticket (section 2 and assessment).
- Language to keep saying: check at least three pairs before trusting a rule, and always ask why as well as what.
- The mirror-symmetry pattern in section 2 is the clearest classroom link between this unit and the commutative property first met in the multiplication unit; make that connection explicit rather than leaving it implicit.
- Curriculum note: ACARA's AC9M3N07 covers creating and describing patterns in generated number sets at the same Year 3 level, a close match to this unit's explain-the-pattern focus.
- Present mode and print both work: use Present to build a multiplication chart live and highlight rows, columns, and the diagonal, then print the worksheets for independent practice.