Multiplication table folds to equal cells

3.OA.B.53.OA.D.9 · take · grade 3

Archetype: Generalize a Growing Pattern into a Rule · step in a 12-type progression

In the multiplication table, fold along the green dashed line. Find the two numbers that meet cell ㉠ and cell ㉡, then find the sum of those two numbers.

The multiplication table is a $4 \times 4$ table with $5$ , $6$ , $7$ , $8$ written across the top and $5$ , $6$ , $7$ , $8$ written down the left side. Each cell holds the product of its top number and its left number.

The green dashed line is the diagonal through the cells where the left number equals the top number ( $5{\times}5$ , $6{\times}6$ , $7{\times}7$ , $8{\times}8$ ). When the table is folded along this line, the two cells that land on each other hold the same number.

㉠ is the cell with left number $8$ and top number $6$ .
㉡ is the cell with left number $6$ and top number $7$ .

Fold along the dashed line, find the number in the cell that meets ㉠ and the number in the cell that meets ㉡, and add those two numbers together.

Show solution

Understand

In a 4x4 multiplication table with heads 5, 6, 7, 8 across and down, fold along the diagonal where left equals top. Find the numbers in the cells that land on ㉠ and ㉡ after folding, then add those two numbers.

Givens

Each cell holds (left number) x (top number).
The fold line is the diagonal through 5x5, 6x6, 7x7, 8x8.
When folded, two mirror cells land on each other and hold the same number.
㉠ is the cell with left 8, top 6.
㉡ is the cell with left 6, top 7.

Unknowns

The number in the cell that meets ㉠ after folding.
The number in the cell that meets ㉡ after folding.
The sum of those two numbers.

Constraints

Folding along the left=top diagonal swaps a cell at (left a, top b) with the cell at (left b, top a).
Because a x b = b x a, the meeting cell holds the same number.

Plan

#1 Draw a Diagram · also uses: #5 Look for a Pattern

Folding along the diagonal is a spatial mirror that swaps the row and column heads of a cell, so I picture the mirror move, use the commutative pattern a x b = b x a to find each meeting value, then add.

Execute

#1 Draw a Diagram 3.OA.B.5

㉠ is at left 8, top 6, so its mirror across the diagonal is at left 6, top 8. That meeting cell holds 6 x 8 = 48 (and ㉠ itself is 8 x 6 = 48, the same number).

6 \times 8 = 48

Folding swaps the two heads of a cell, and a x b equals b x a, so the meeting cell shows the same product.

#5 Look for a Pattern 3.OA.B.5

㉡ is at left 6, top 7, so its mirror is at left 7, top 6. That meeting cell holds 7 x 6 = 42.

7 \times 6 = 42

Same idea: swapping the heads keeps the product, so the mirror cell is also 42.

#1 Draw a Diagram 3.OA.C.7

The cell meeting ㉠ holds 48 and the cell meeting ㉡ holds 42, so their sum is 48 + 42 = 90.

48 + 42 = 90

Once both meeting values are known, the answer is a simple addition.

Answer: 90

Review

Both meeting cells lie in the 5-to-8 multiplication range, where products run from 25 to 64; 48 and 42 sit comfortably in that range, and their sum 90 is reasonable.

You could fill in the whole 4x4 product table, mark the diagonal, and physically check that the cells across from ㉠ and ㉡ read 48 and 42 before adding.

Standards · min grade 3

3.OA.B.5 Apply properties of operations as strategies to multiply and divide — Using the commutative property a x b = b x a to find the mirror cells across the fold.
3.OA.C.7 Fluently multiply and divide within 100 — Computing the products 6x8 and 7x6 and adding the results.

💡 Folding the table just swaps the row and column numbers, and since 6x8 = 8x6, the mirror cell shows the very same product!