Problem
Complete the multiplication table
A times table has 1-6 across the top and 1-5 down the side. Each cell holds its row number times its column number. Some cells on the torn strips are blank. Fill every blank with the right product.
Reasoning · Grade 2-2 The Multiplication Table
Your answer
Watch out Counting on from a neighbour goes wrong across a torn gap — multiplying the two headers always works.
How to solve
Strategy Look for a Pattern — Each cell is just row x column, so the fastest way to fill any blank is to read off its row and column headers and multiply. The row and column patterns (each row counts up by the row number, each column by the column number) let us double-check every entry by skip-counting.
1STEP 1
Read the rule of the table
Every cell is just row × column.
cell = (row number) × (column number)
The whole table is built from one rule, so every blank is answered by the same row-times-column multiplication.
3.OA.D.9Look For A Pattern2STEP 2
Fill rows 1 and 2
Rows 1 and 2 give 1,2,3,4,5,6 and 2,4,6,8,10,12.
1{×}1=1, …, 1{×}6=6; 2{×}1=2, …, 2{×}6=12
Counting by 1s then by 2s gives each row, matching multiplication facts you already know.
3.OA.C.7Look For A Pattern3STEP 3
Fill rows 3, 4, and 5
Rows 3, 4, 5 skip-count to end at 18, 24, 30.
3,6,9,12,15,18; 4,8,12,16,20,24; 5,10,15,20,25,30
Skip-counting by 3, by 4, and by 5 fills each row, and you can check a column the same way (e.g. column 6 reads 6, 12, 18, 24, 30, going up by 6).
3.OA.C.7Draw A DiagramSkip-counting by 3, by 4 and by 5 fills those rows, and reading the same cells down a column checks them.
Why?
Each cell is a count of equal groups, so moving one step along a row simply adds one more group of that row's number.
🧱Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.
Why?
A cell read as a row times a column gives the same number as the column times the row, so the column sweep must agree with the row sweep.
🧱Commutativity of multiplicationa rows of b and b rows of a count the same array of dots — rows and columns just swap.
Answer
row 1: 1,2,3,4,5,6 / row 2: 2,4,6,8,10,12 / row 3: 3,6,9,12,15,18 / row 4: 4,8,12,16,20,24 / row 5: 5,10,15,20,25,30
Every entry equals its row number times its column number, and both directions check out: each row goes up by its row number and each column goes up by its column number (e.g. column 4 reads 4, 8, 12, 16, 20). The largest entry, in row 5 column 6, is 5 x 6 = 30, which is the biggest product possible in this 5-by-6 body.
Takeaway
This only needs the Grade 3 times tables you already know -- every blank is just its row number times its column number!
- Read the rule of the table
- Fill rows 1 and 2
- Fill rows 3, 4, and 5