A times table fact is a rectangle you can see
Equal rows make a rectangle
Put 4 groups of 6 dots in rows, one group to a row, and the dots make a rectangle: 4 rows with 6 dots in each. A rectangle of dots like this is called an array. Count it a row at a time, in sixes: 6, 12, 18, 24. So 4 × 6 = 24.
Every times table fact can be drawn this way. The first number is how many rows there are, the second is how many dots are in each row, and the answer is the number of dots in the whole rectangle.
4 rows of 6: 4 × 6 = 24.
Turn it on its side
Now turn the rectangle a quarter turn. The same 24 dots stand in 6 rows of 4, so 6 × 4 = 24 as well. No dot was added and none was taken away, so the answer cannot change. Multiplying two numbers in either order gives the same answer, and that rule is called the commutative property.
It halves the work of learning the times tables: know 4 × 6 and you know 6 × 4. Only a fact whose two numbers are the same, like 7 × 7 = 49, has no partner.
Make 24 as a rectangle, then find another factor pair
multiplication is commutative: 6 × 3 = 3 × 6
Every rectangle of 24 dots is a times table fact for 24: 3 × 8, 4 × 6, 6 × 4 and 8 × 3.
Multiplication and division · Rows of Chairs Seen from the Front and from the Side
QuestionThe chairs in a hall stand in 3 rows. There are 4 chairs in each row. (a) How many chairs are there? (b) Mei stands at the side of the hall. From there she sees the same chairs in 4 rows. How many chairs are in each of her rows?
1.Draw the chairs as dots. Make 3 rows and put 4 dots in each row.
One dot is one chair. There are 3 rows with 4 chairs in each row. 2.Count by rows. Each row has 4 chairs: 4, 8, 12.
Count by rows: 4, 8, 12. 3.(a) 3 rows of 4 is 3 × 4 = 12. There are 12 chairs.
(a) 3 rows of 4 is 3 × 4 = 12 chairs. 4.Now look from the side. Each column is a row for Mei. There are 4 columns, so she sees 4 rows.
From the side, each column is a row for Mei. There are 4 of them. 5.(b) Each column has 3 dots, so each of her rows has 3 chairs. Check: 3, 6, 9, 12, so 4 × 3 = 12, the same 12 chairs.
(b) Each of her rows has 3 chairs. 4 × 3 = 12, the same 12 chairs.
Answer: (a) 12 chairs; (b) 3 chairs
Watch out for
- Saying that Mei's rows also have 4 chairs. From the side, the 4 is the number of rows. The number in each row is the number of dots in one column, which is 3.
- Thinking that 4 rows of 3 is a different number of chairs from 3 rows of 4. No chair has moved, so both are 12: 4 × 3 = 3 × 4.
Multiplication and division · Buns Packed in Rows of 3 or Rows of 4
QuestionA baker has 24 buns to put on a tray. (a) If he puts them in rows of 3, how many rows are there? (b) If he puts them in rows of 4 instead, how many rows are there?
1.Put the buns in rows of 3 and count in threes until 24 is reached: 3, 6, 9, 12, 15, 18, 21, 24.
The 24 buns in rows of 3. Count in threes: 3, 6, 9, 12, 15, 18, 21, 24. 2.(a) That is 8 threes, so 24 ÷ 3 = 8. There are 8 rows of 3.
(a) 24 ÷ 3 = 8 rows of 3. 3.Now put the same 24 buns in rows of 4 and count in fours: 4, 8, 12, 16, 20, 24.
The same 24 buns in rows of 4. Count in fours: 4, 8, 12, 16, 20, 24. 4.(b) That is 6 fours, so 24 ÷ 4 = 6. There are 6 rows of 4. Check: 8 × 3 = 24 and 6 × 4 = 24.
(b) 24 ÷ 4 = 6 rows of 4. Each row is bigger, so there are fewer rows.
Answer: (a) 8 rows; (b) 6 rows
Watch out for
- Thinking that rows of 4 must give more rows because 4 is more than 3. Each row of 4 uses up more buns, so there are fewer rows: 6 instead of 8.
- Finding 8 rows in part (a) and then answering 8 + 1 = 9 rows in part (b) because each row has 1 more bun. The buns must be packed again from the start: 24 ÷ 4 = 6.
Your turn
1.Now the rectangle is turned on its side. How many squares?