3A · Chapter 4 · Lesson 2
Strategies for Finding the Product
Break one factor into friendly parts using the distributive property.
Today's challenge
Calculate 59 × 6 by splitting 59 into tens and ones. Try your own method, then compare it with a model.
What you'll learn
- Break one factor into friendly parts using the distributive property.
- Explain the connection between a diagram and a number sentence.
- Solve a fresh application without step-by-step hints.
Before this lesson
Arrays and place value.
Prerequisite competency: 3A-C04-L01
Entry · 2 questions
Warm-up: prerequisite check
Try on paper before continuing. If the questions are hard, review the prerequisite concepts.
- E1
There are 4 equal rows with 5 counters each. How many counters?
Show answer and reasoning
Answer: 20
There are 4 groups of 5; multiply 4×5=20. Turning the array does not change the total.
- E2
There are 5 equal rows with 3 counters each. How many counters?
Show answer and reasoning
Answer: 15
There are 5 groups of 3; multiply 5×3=15. Turning the array does not change the total.
Learn the concept
Words to know
- distributive property
- multiply each part and then combine the products
- partial product
- the product of one decomposed part
Break one factor into friendly parts using the distributive property.
Split a factor into tens and ones. Make two smaller rectangles of the same height, find their areas, and combine the results.
Why this works
An array can be divided into smaller rectangles. For 17 × 4, split 17 into 10 and 7. Four rows of 10 contain 40; four rows of 7 contain 28. Add the partial products: 40 + 28 = 68. Both pieces must be multiplied by four because both pieces occur in each row.
Check the meaning
Both parts must be multiplied by the other factor. Draw an array if you are unsure.
Concrete → Pictorial → Abstract
- Build: Use counters, grid paper, or labeled bundles to represent the story or quantity.
- Draw: Sketch an array, place-value exchange, or equal-unit bar; label what is known.
- Calculate: Write an equation matching the representation and solve.
- Verify: Use an inverse operation or a second strategy to check.
For advanced students: if you can already explain this concept, start with the independent or challenge set.
Worked examples and immediate checks
Explain each step aloud or in writing. Work the fresh question before revealing its solution.
Worked example 1
Reason and verify
Calculate 59 × 6 by splitting 59 into tens and ones.
- Split a factor into tens and ones. Make two smaller rectangles of the same height, find their areas, and combine the results.
- (50+9)×6=(50×6)+(9×6)=354.
- Both parts must be multiplied by the other factor. Draw an array if you are unsure.
Result: 354
Understanding check 1 — your turn
Calculate 22 × 6 by splitting 22 into tens and ones.
Reveal reasoning
Answer: 132 — (20+2)×6=(20×6)+(2×6)=132.
Worked example 2
Reason and verify
Calculate 66 × 7 by splitting 66 into tens and ones.
- Split a factor into tens and ones. Make two smaller rectangles of the same height, find their areas, and combine the results.
- (60+6)×7=(60×7)+(6×7)=462.
- Both parts must be multiplied by the other factor. Draw an array if you are unsure.
Result: 462
Understanding check 2 — your turn
Calculate 29 × 7 by splitting 29 into tens and ones.
Reveal reasoning
Answer: 203 — (20+9)×7=(20×7)+(9×7)=203.
Worked example 3
Reason and verify
Calculate 73 × 8 by splitting 73 into tens and ones.
- Split a factor into tens and ones. Make two smaller rectangles of the same height, find their areas, and combine the results.
- (70+3)×8=(70×8)+(3×8)=584.
- Both parts must be multiplied by the other factor. Draw an array if you are unsure.
Result: 584
Understanding check 3 — your turn
Calculate 36 × 8 by splitting 36 into tens and ones.
Reveal reasoning
Answer: 288 — (30+6)×8=(30×8)+(6×8)=288.
Guided · 4 questions
Guided practice
Practice with the model visible. Explain the units at each step.
- G1
Calculate 17 × 2 by splitting 17 into tens and ones.
Show answer and reasoning
Answer: 34
(10+7)×2=(10×2)+(7×2)=34.
- G2
Calculate 24 × 3 by splitting 24 into tens and ones.
Show answer and reasoning
Answer: 72
(20+4)×3=(20×3)+(4×3)=72.
- G3
Calculate 31 × 4 by splitting 31 into tens and ones.
Show answer and reasoning
Answer: 124
(30+1)×4=(30×4)+(1×4)=124.
- G4
Calculate 38 × 5 by splitting 38 into tens and ones.
Show answer and reasoning
Answer: 190
(30+8)×5=(30×5)+(8×5)=190.
Independent · 8 questions
Independent practice
Work without relying on the model or checking answers prematurely.
- I1
Calculate 67 × 7 by splitting 67 into tens and ones.
Show answer and reasoning
Answer: 469
(60+7)×7=(60×7)+(7×7)=469.
- I2
Calculate 74 × 8 by splitting 74 into tens and ones.
Show answer and reasoning
Answer: 592
(70+4)×8=(70×8)+(4×8)=592.
- I3
Calculate 81 × 2 by splitting 81 into tens and ones.
Show answer and reasoning
Answer: 162
(80+1)×2=(80×2)+(1×2)=162.
- I4
Calculate 13 × 3 by splitting 13 into tens and ones.
Show answer and reasoning
Answer: 39
(10+3)×3=(10×3)+(3×3)=39.
- I5
Calculate 20 × 4 by splitting 20 into tens and ones.
Show answer and reasoning
Answer: 80
(20+0)×4=(20×4)+(0×4)=80.
- I6
Calculate 27 × 5 by splitting 27 into tens and ones.
Show answer and reasoning
Answer: 135
(20+7)×5=(20×5)+(7×5)=135.
- I7
Calculate 34 × 6 by splitting 34 into tens and ones.
Show answer and reasoning
Answer: 204
(30+4)×6=(30×6)+(4×6)=204.
- I8
Calculate 41 × 7 by splitting 41 into tens and ones.
Show answer and reasoning
Answer: 287
(40+1)×7=(40×7)+(1×7)=287.
Challenge · 2 questions
Challenge practice
Solve original application and explanation problems. Draw and defend a useful model.
- C1
Calculate 19 × 6 without 19 repeated additions.
Show answer and reasoning
Answer: 114
Think (20−1)×6 = 20×6 − 1×6 = 120−6=114.
- C2
A student says 18×5 = 10×5 + 8. Explain the missing operation and correct answer.
Show answer and reasoning
Answer: 90
Both parts are multiplied by five: (10×5)+(8×5)=50+40=90.
Exit · 3 questions
Exit check
Solve independently; revisit the skill if an important explanation is missing.
- E1
Calculate 17 × 3 by splitting 17 into tens and ones.
Show answer and reasoning
Answer: 51
(10+7)×3=(10×3)+(7×3)=51.
- E2
Calculate 24 × 4 by splitting 24 into tens and ones.
Show answer and reasoning
Answer: 96
(20+4)×4=(20×4)+(4×4)=96.
- E3
Calculate 31 × 5 by splitting 31 into tens and ones.
Show answer and reasoning
Answer: 155
(30+1)×5=(30×5)+(1×5)=155.