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Year 5 Maths Multiplication

Multiplication with arrays & the area model

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Theory

An array sets objects out in equal rows and columns, so the total is rows \(\times\) columns. The area model draws a multiplication as a rectangle and splits it by place value into easier partial products. This Year 5 maths page shows arrays, turning arrays round, using arrays to divide, and using the area model to multiply two-digit numbers by one-digit numbers.

An array is an arrangement of objects in equal rows. An array with \(6\) rows of \(7\) shows \(6 \times 7 = 42\).

Turning an array on its side swaps the rows and columns but keeps the same number of objects, so \(6 \times 7 = 7 \times 6\). This is the commutative law: the order of multiplying does not matter.

Arrays also show division: \(56\) counters in \(7\) equal rows gives \(56 \div 7 = 8\) in each row.

The area model draws \(6 \times 37\) as a rectangle \(6\) high and \(37\) wide, split into \(6 \times 30\) and \(6 \times 7\). Each part is a partial product; adding them gives the answer.

Array of 6 rows of 7 dots An array of 42 dots in 6 rows with 7 dots in each row.
\(6\) rows of \(7\): \(6 \times 7 = 42\).
Area model for 6 times 37 A rectangle 37 wide and 6 tall split into parts: 6 times 30 is 180, 6 times 7 is 42. Together they make 222. 6 180 30 42 7
Area model: \(6 \times 37 = 180 + 42 = 222\).
Array of 4 rows of 9 split after 5 columns An array of 4 rows of 9 dots. A dashed red line splits each row into 5 blue dots and 4 orange dots, showing 4 times 9 equals 4 times 5 plus 4 times 4.
Splitting an array: \(4 \times 9 = 4 \times 5 + 4 \times 4 = 20 + 16 = 36\).

Arrays and area models both use rows times columns:

\[ \text{total} = \text{rows} \times \text{columns} \]
total = rows × columns
\[ 6 \times 7 = 7 \times 6 = 42 \]
6×7=7×6=42
\[ 6 \times 37 = 6 \times 30 + 6 \times 7 \]
6×37=6×30+6×7
Area of a rectangle. The area of a rectangle is length \(\times\) width, so a garden \(9\) m by \(23\) m has area \(9 \times 23 = 207\) square metres.
Split by place value. Split the two-digit number into tens and ones: \(58 = 50 + 8\). Multiply each part, then add.

How to use the area model

  1. Draw a rectangle with the one-digit number down the side.
  2. Split the two-digit number into tens and ones along the top.
  3. Multiply to find the area of each part.
  4. Add the partial products.

How to use an array to divide

  1. Share the total into the given number of equal rows.
  2. Count how many are in each row.
Example 1 — Array
Chairs are set out in \(7\) rows of \(13\). How many chairs are there?
Solution
\(7 \times 10\)\(=\)\(70\)
\(7 \times 3\)\(=\)\(21\)
\(70 + 21\)\(=\)\(91\)
7×13=91

There are \(91\) chairs.

Example 2 — Turned array
An array of \(3\) rows of \(9\) is turned on its side. How many objects are in the new array, and how is it arranged?
Solution
\(3 \times 9\)\(=\)\(27\)
\(9 \times 3\)\(=\)\(27\)
3×9=9×3=27

Still \(27\) objects, now in \(9\) rows of \(3\).

Example 3 — Array for division
\(56\) tiles are laid in \(7\) equal rows. How many tiles are in each row?
Solution
\(56 \div 7\)\(=\)\(8\)
56÷7=8

\(8\) tiles in each row, because \(7 \times 8 = 56\).

Example 4 — Area model
Use an area model to work out \(4 \times 58\).
Area model for 4 times 58 A rectangle 58 wide and 4 tall split into parts: 4 times 50 is 200, 4 times 8 is 32. Together they make 232. 4 200 50 32 8
Solution
\(4 \times 50\)\(=\)\(200\)
\(4 \times 8\)\(=\)\(32\)
\(200 + 32\)\(=\)\(232\)
4×58=232

\(4 \times 58 = 232\).

Example 5 — Area of a garden
A garden bed is \(9\) m wide and \(23\) m long. What is its area?
Solution
\(9 \times 20\)\(=\)\(180\)
\(9 \times 3\)\(=\)\(27\)
\(180 + 27\)\(=\)\(207\)
9×23=207

The area is \(207\) square metres.

Example 6 — Missing part
An area model splits \(6 \times 37\) into \(6 \times 30\) and \(6 \times \square\). Find the box and the answer.
Solution
\(37 - 30\)\(=\)\(7\)
\(180 + 42\)\(=\)\(222\)
6×37=222

The box is \(7\), and \(6 \times 37 = 222\).

Example 7 — Find the mistake
Tom split \(47\) into \(4\) and \(7\) for \(3 \times 47\), and got \(12 + 21 = 33\). What went wrong?
Solution
\(3 \times 40\)\(=\)\(120\)
\(3 \times 7\)\(=\)\(21\)
\(120 + 21\)\(=\)\(141\)
3×47=141

The \(4\) means \(40\), so the parts are \(40\) and \(7\). The answer is \(141\).

Common pitfalls

Splitting into digits. \(47\) splits into \(40 + 7\), not \(4 + 7\).
Forgetting to add the parts. The answer is the sum of all the partial products.
Mixing up rows and columns. Rows go across, columns go down. Either order gives the same total, so it does not change the answer.
Adding instead of multiplying. \(7\) rows of \(13\) is \(7 \times 13\), not \(7 + 13\).

Frequently asked questions

What is an array in maths?

An array is a set of objects in equal rows and columns. \(6\) rows of \(7\) show \(6 \times 7 = 42\).

Why does turning an array not change the total?

The same objects are just viewed the other way, so \(3 \times 9 = 9 \times 3\). This is the commutative law.

What is the area model for multiplication?

It draws the multiplication as a rectangle, splits one side by place value and adds the areas of the parts: \(4 \times 58 = 200 + 32 = 232\).

What is a partial product?

One of the smaller products in a split multiplication, such as \(4 \times 50\) and \(4 \times 8\) in \(4 \times 58\).

How can an array show division?

Share the objects into equal rows. \(56\) in \(7\) rows gives \(8\) per row, so \(56 \div 7 = 8\).

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