Mental Strategies for Multi-Digit Whole-Number Multiplication
Without using the standard algorithm, solve:
48 × 250
Explain a mental strategy that makes this multiplication easier.
Answer:
12,000
One strategy:
48 × 250 = 48 × 1,000 ÷ 4 = 12,000
Use the standard algorithm to solve:
327 × 46
15,042
A school orders 24 boxes of pencils. Each box contains 36 pencils. The pencils are divided equally among 8 classrooms.
How many pencils does each classroom receive?
Answer:
24 × 36 = 864
864 ÷ 8 = 108 pencils
Solve:
4.8 × 25
120
A rectangular garden is 12.5 meters long and 8 meters wide.
What is the area?
12.5 × 8 = 100 square meters
Find the product mentally:
399 × 25
Explain how you used a nearby friendly number.
Answer:
9,975
399 × 25
= 400 × 25 − 1 × 25
= 10,000 − 25
= 9,975
Solve using the standard algorithm:
2,406 × 318
765,108
A theater has 28 rows with 36 seats in each row. For a special event, 125 seats are reserved for guests.
How many seats are available for everyone else?
28 × 36 = 1,008
1,008 − 125 = 883 seats
Solve:
3.75 × 24
90
A runner completes 6 laps around a track. Each lap is 0.75 mile.
How many miles does the runner travel?
6 × 0.75 = 4.5 miles
Which is easier to calculate mentally:
125 × 48 or 125 × 52?
Calculate both mentally and explain what makes the strategy efficient.
125 × 48 = 6,000
125 × 52 = 6,500
Possible strategy:
125 × 48 = 125 × 4 × 12
= 500 × 12
= 6,000
125 × 52 = 125 × 4 × 13
= 500 × 13
= 6,500
A student calculates:
3,204 × 207 = 663,228
Without completely redoing the problem, explain why the answer wouldn't be correct.
The answer is unreasonable because:
3,204 × 200 is already 640,800.
The correct answer is:
3,204 × 207
= 3,204 × 200 + 3,204 × 7
= 640,800 + 22,428
= 663,228
Important: The student's answer actually is correct.
A school is buying notebooks for 18 classrooms. Each classroom needs 27 packs, and each pack contains 24 notebooks.
The school already has 1,500 notebooks.
How many additional notebooks must the school purchase?
18 × 27 = 486 packs
486 × 24 = 11,664 notebooks
11,664 − 1,500 = 10,164 notebooks
A student says:
“2.4 × 3.5 must be greater than 10 because 24 × 35 is 840.”
Explain the student's mistake and find the correct answer.
The student forgot to account for the decimal places.
2.4 × 3.5 = 8.4
Since both factors are greater than 1 but less than 4, an answer of 8.4 is reasonable.
A rectangular swimming pool is 18.5 meters long and 7 meters wide.
A safety mat covers 12.5 square meters of the surrounding area.
How many square meters are not covered by the safety mat?
18.5 × 7 = 129.5 square meters
129.5 − 12.5 = 117 square meters
DAILY DOUBLE!!!!!
A student says:
“I can find 2,498 × 40 by doing 2,500 × 40 and then subtracting 2.”
Is the student correct? If not, explain the mistake and find the correct answer mentally.
The student is not correct.
They need to subtract 2 × 40, not just 2.
2,498 × 40
= 2,500 × 40 − 2 × 40
= 100,000 − 80
= 99,920
DAILY DOUBLE!!!!
Find the missing digit:
□,426 × 307 = 1,052,802
What digit belongs in the box?
TRY: 3
3,426 × 307
= 3,426 × 300 + 3,426 × 7
= 1,027,800 + 23,982
= 1,051,782
So actually 3 does not work.
Let's solve correctly.
Test 3,429:
3,429 × 307 = 1,052,703
Test 3,? We need:
1,052,802 ÷ 307 = 3,429.322...
Therefore, there is no whole-number digit that makes the equation true.
A company packs 48 boxes. Each box contains 36 packages, and each package contains 125 stickers.
The company gives away 18,500 stickers.
How many stickers remain?
48 × 36 = 1,728 packages
1,728 × 125 = 216,000 stickers
216,000 − 18,500 = 197,500 stickers
A store sells fabric for $7.25 per yard.
A customer buys 18.4 yards.
How much does the fabric cost?
7.25 × 18.4
= 7.25 × 18 + 7.25 × 0.4
= 130.50 + 2.90
= $133.40
A construction crew needs 24 pieces of wood, each measuring 2.75 meters.
They have already purchased 18.5 meters of wood.
How many more meters of wood do they need?
24 × 2.75 = 66 meters
66 − 18.5 = 47.5 meters
They need 47.5 more meters.
Without using the standard algorithm, find:
4,999 × 125
Then explain two different mental strategies that could be used.
624,875
Strategy 1:
5,000 × 125 = 625,000
1 × 125 = 125
625,000 − 125 = 624,875
Strategy 2:
4,999 × 100 = 499,900
4,999 × 25 = 124,975
499,900 + 124,975 = 624,875
A student multiplies:
4,208 × 306
and gets 1,287,648.
Without doing the entire standard algorithm first, determine whether the answer is reasonable. Then calculate the correct answer.
Estimate:
4,208 × 300 ≈ 1,262,400
So 1,287,648 is reasonable.
Exact:
4,208 × 306
= 4,208 × 300 + 4,208 × 6
= 1,262,400 + 25,248
= 1,287,648
The student's answer is correct.
A school is planning a field trip for 126 students.
Each bus can hold 48 students, and each bus costs $725 to rent.
The school also needs to purchase lunch for every student at $9 per student.
The school has a budget of $5,000.
Will the school stay within its budget? How much money will be left over or how much more will be needed?
126 ÷ 48 = 2 remainder 30, so 3 buses are needed.
3 × $725 = $2,175
126 × $9 = $1,134
Total:
$2,175 + $1,134 = $3,309
$5,000 − $3,309 = $1,691
The school stays within its budget and has $1,691 left.
Determine which product is greater:
24.8 × 3.75
or
25 × 3.7
Then find the difference between the two products.
24.8 × 3.75 = 93
25 × 3.7 = 92.5
Therefore, 24.8 × 3.75 is greater.
Difference:
93 − 92.5 = 0.5
A rectangular playground is 24.5 meters long and 18 meters wide.
A rubber surface costs $12.75 per square meter.
The school receives a $1,500 discount on the total cost.
What is the final cost of the rubber surface?
First find the area:
24.5 × 18 = 441 square meters
Then multiply by the cost:
441 × $12.75 = $5,622.75
Subtract the discount:
$5,622.75 − $1,500 = $4,122.75
Final answer: $4,122.75