FOUNDATIONAL MATHEMATICS

Tile Workshop

Rearrange coloured tiles to discover that an amount can be described in different ways.

Turn rectangular arrays to see the commutative property, and split arrays into colour groups to master the distributive property.

Interactive Tile Workshop Challenge 1 of 4 0 XP

Level 1: Commutative Property Total: 18 tiles 3 rows × 6 cols

Build 3 rows with 4 blue and 2 gold tiles in each row. What do you notice when you turn or split the array?

Discovered Mathematical Form

Rearrange or turn the tiles above to reveal the equation!

Observation Mode

Warm-Up Challenge

The Distributive Misconception Check

A board holds 3 rows with 4 blue tiles and 2 gold tiles in each row. Why is splitting it NOT equal to \(3 \times 4 + 2\)?

The Mathematics Behind the Board

How Physical Moves Create Algebraic Equations

When children physically arrange tiles into rows and columns, abstract arithmetic rules transform into obvious visual facts:

Move 1: Turn the array — same total, swapped dimensions: Rearrange the 3 rows of 6 tiles into 6 rows of 3 tiles. The number of tiles stays 18. This shows the commutative property of multiplication: \(3 \times 6 = 6 \times 3\). The same principle applies to addition inside rows: \(4 + 2 = 2 + 4\).
Move 2: Split the colours — multiply both parts: Separate the blue and gold tiles into two distinct arrays. Now you see \(3 \times 4\) blue tiles and \(3 \times 2\) gold tiles. This shows the distributive property: \(3 \times (4 + 2) = (3 \times 4) + (3 \times 2) = 18\).

Sandbox Workbench

Custom Array Playground

Change row and tile counts freely. Turn, split, join, and swap to explore any array.

Rows

Blue Tiles

Gold Tiles

3 rows × 6 columns Total: 18 tiles

Discovered Mathematical Form

Rearrange or turn the tiles to reveal the equation!

Sandbox Mode

Step-by-step lesson

Tile arrays: see why multiplication properties work

Before you start: Count equal rows and columns and add small groups.

Read an array as equal groups

An array with 3 rows and 6 tiles in each row contains 6 + 6 + 6 = 18 tiles, written 3 × 6. Counting the same 18 tiles one by one works, but multiplication records the equal-group structure. The row lengths must match for a rectangular array.

Turn the array: the commutative property

Rotate the rectangle so that its 6 columns become 6 rows of 3. No tile is added or removed, so 3 × 6 = 6 × 3. This illustrates ab = ba for whole-number arrays. The result applies to multiplication, not every operation: 6 − 3 = 3, while 3 − 6 = −3.

Split the array: the distributive property

Keep all 3 rows, but split each row into 4 blue and 2 gold tiles. There are 3 × 4 = 12 blue tiles and 3 × 2 = 6 gold tiles. Therefore 3 × (4 + 2) = 12 + 6 = 18. The same three rows occur in both smaller rectangles, which explains why the 3 multiplies both parts.

The expression 3 × 4 + 2 counts only 12 blue tiles and 2 gold tiles, a total of 14. It misses four gold tiles. The difference is visible on the board, so the correction follows from counting rather than memorising a slogan.

Use the picture for mental calculation and algebra

To find 7 × 14, split 14 into 10 + 4: 7 × 10 + 7 × 4 = 70 + 28 = 98. For a symbolic extension, (x + 2)(x + 3) splits a rectangle into four areas: x², 3x, 2x and 6. Their sum is x² + 5x + 6. Positive side lengths suit this physical picture; the algebraic identity has a wider domain.

Try it yourself

Split a 4-row, 7-column array into 5 columns and 2 columns. Write the two products and their sum.

Show answer and explanation

The smaller arrays contain 4 × 5 = 20 and 4 × 2 = 8 tiles. Thus 4 × (5 + 2) = 20 + 8 = 28. Both parts retain all four rows.

Use the activity to check your understanding

After a turn or split, explain what changed and what stayed the same. Use the visible tile counts to justify the equation.

Try the next activity: Order of Operations →

Sources and further reading

OpenStax: Use the Language of Algebra