A day with Grandpa
Grandpa was measuring a rug using yards, feet and inches. Can you help William to work out its area?
Problem
William went to spend the day with Grandpa. Grandpa was busy measuring a rug to put on the hall floor. He was using yards, feet and inches to measure with. Grandpa always did.
William was learning about area at school. "I wonder what the area is?" he pondered.
Grandpa wrote down his measurements and drew this sketch.
"It's two yards and one foot by one yard and one foot. There's three feet in a yard, so that's seven foot by four foot. Let me see, that's twenty-eight square feet and that's .... hmm ..... three square yards and one square foot."
Was his calculation correct?
"Let's measure the doormat," he said, "You do it." William used Grandpa's measuring tape.
"The doormat's two feet and six inches by one foot three inches," he announced.
Grandpa did another sketch.
He thought for a while.
"There's twelve inches in a foot, so it's three square feet and eighteen square inches," he announced at last.
Was this calculation correct?
Grandpa was getting into his stride. "Let's measure the rug in front of the fire, too," he said excitedly.
It was two yards, one foot and five inches long and one yard, two feet and seven inches wide.
"How do you work it out?" inquired William. "You multiply them together, of course," said Grandpa airily.
He wrote some figures on his paper.
"Wow! How do you do that?" asked William peering at it.
Grandpa looked at his paper for a long time.
"I can't remember," he hesitated, "Can you think of a way to do it?"
Can you think of a way to do it?
Getting Started
Remember there are 12 inches in a foot and three feet in a yard.
Might it help to draw a sketch?
Perhaps you can use your diagram to think about how many square inches there are in a square foot and how many square feet there are in a square yard?
Student Solutions
It is easy to get confused by these sorts of calculations. Fortunately Felipe from Quarry Bay School and Toby, Liam, Abi and Livvy from Shute Community Primary School didn't get confused, and all came up with the correct solutions.
The first rug measures $2$ yards $1$ foot by $1$ yards $1$ foot.
$2$ yards $1$ foot = $(2\times 3) + 1 = 7$ feet and $1$ yard $1$ foot = $(1\times 3) + 1 = 4$ feet. Therefore the area of the rug is $7\times 4 = 28$ square feet.
Many of you correctly realised that if $1$ yard is $3$ feet then $1$ square yard is $3\times 3 = 9$ square feet. $28 = 3\times 9 + 1 = 3$ square yards $1$ square foot. So Grandad was correct.
Grandad was also correct the second time:
$2$ feet $6$ inches $\times 1$ foot $3$ inches
$= (2\times 12) + 6$ inches $\times (1\times 12) + 3$ inches
$= 30\times 15$ square inches
$= 450$ square inches
$= (3\times 144) + 18$ square inches
$= (3\times 12^2) + 18$ square inches
$= 3$ square feet $18$ square inches.
We work out the last one like this:
$2$ yards $1$ foot $5$ inches $\times 1$ yard $2$ feet $7$ inches
$= (2\times 3) + 1$ feet $5$ inches $\times (1\times 3) + 2$ feet $7$ inches
$= 7$ feet $5$ inches $\times 5$ feet $7$ inches
$= (7\times 12) + 5$ inches $\times (5\times 12) + 7$ inches
$= 89$ inches $\times 67$ inches
$= 89\times 67 = 5963$ square inches
$= (41\times 144) + 59$ square inches
$= 41$ square feet $59$ square inches
$= (4\times 9) + 5$ square feet $59$ square inches
$= 4$ square yards $5$ square feet $59$ square inches
Teachers' Resources
A Day with Grandpa
William went to spend the day with Grandpa. Grandpa was busy measuring a rug to put on the hall floor. He was using yards, feet and inches to measure with. Grandpa always did.
William was learning about area at school. "I wonder what the area is?" he pondered.
Grandpa wrote down his measurements and drew this sketch.
"It's two yards and one foot by one yard and one foot. There's three feet in a yard, so that's seven foot by four foot. Let me see, that's twenty-eight square feet and that's .... hmm ..... three square yards and one square foot."
Was his calculation correct?
"Let's measure the doormat," he said, "You do it." William used Grandpa's measuring tape.
"The doormat's two feet and six inches by one foot three inches," he announced.
Grandpa did another sketch.
He thought for a while.
"There's twelve inches in a foot, so it's three square feet and eighteen square inches," he announced at last.
Was this calculation correct?
Grandpa was getting into his stride. "Let's measure the rug in front of the fire, too," he said excitedly.
It was two yards, one foot and five inches long and one yard, two feet and seven inches wide.
"How do you work it out?" inquired William. "You multiply them together, of course," said Grandpa airily.
He wrote some figures on his paper.
"Wow! How do you do that?" asked William peering at it.
Grandpa looked at his paper for a long time.
"I can't remember," he hesitated, "Can you think of a way to do it?"
Can you think of a way to do it?
Why do this problem?
Possible approach
Key questions
Possible extension
Possible support