☰ Year plan · Mathematics

Breaking arrays and perimeter

Week 10 · Breaking arrays and perimeter · 3 lessons
1

Many ways to break an array

Discover Mathematics P3: Term 1, Lessons 31–40
GoalBreak an array apart in different ways and see that every way gives the same product.
New words
break apart · split a number or an array into smaller partsdistributive property · a rule: a product can be found by adding the products of its partsstrategy · a way of working something out
Explanation

There is more than one way to break an array apart, and every way gives the same product. Take 6 × 7 = 42. We can break the 7 columns into 5 and 2: (6 × 5) + (6 × 2) = 30 + 12 = 42. We can break them into 3 and 4: (6 × 3) + (6 × 4) = 18 + 24 = 42. We can also break the 6 rows into 3 and 3: (3 × 7) + (3 × 7) = 21 + 21 = 42. This rule is called the distributive property. Choose the split that uses the facts you know best, such as the 2s, 5s and 10s.

6 × 5 = 306 × 2 = 12
Worked examples
An 8 by 6 array is broken into two equal parts by a vertical line. Each part is 8 × 3 = 24. The total is 24 + 24 = 48, so 8 × 6 = 48.
The same array can be broken by a line across the rows: 5 rows and 3 rows. The parts are 5 × 6 = 30 and 3 × 6 = 18. The total is 30 + 18 = 48. The product is the same.
We can use three parts. A 5 by 9 array is broken into three 5 by 3 arrays: 5 × 3 = 15 each. The total is 15 + 15 + 15 = 45, so 5 × 9 = 45.
Class activity

Colourful breaks. Draw a 7 by 8 array on squared paper. Break it in three different ways, using a different coloured pencil each time. For each way write the equation and the total. Is the total 56 every time? Which break was easiest for you?

Practice
1
This 7 by 8 array is broken after the 5th column. Write the equation and find the product.
2
Break 9 × 4 into two equal parts and find the product.
3
Show that 6 × 8 = 48 by breaking the 8 into 4 and 4.
4
Youssef breaks 5 × 7 into (5 × 4) + (5 × 4) and gets 40. What is wrong?
5
Find two different ways to break 4 × 9 and check that both give 36.