3A · Chapter 4 · Lesson 1
Looking Back at Multiplication
Model products as equal groups or rectangular arrays.
Today's challenge
There are 3 equal rows with 7 counters each. How many counters? Try your own method, then compare it with a model.
What you'll learn
- Model products as equal groups or rectangular arrays.
- Explain the connection between a diagram and a number sentence.
- Solve a fresh application without step-by-step hints.
Before this lesson
Addition, skip counting.
Prerequisite competency: PREREQ-PRIOR-EQUAL-GROUPS
Entry · 2 questions
Warm-up: prerequisite check
Try on paper before continuing. If the questions are hard, review the prerequisite concepts.
- E1
Draw 4 groups of 3 counters and find the total.
Show answer and reasoning
Answer: 12
4 equal groups with 3 in each group make 4 × 3 = 12. This is prerequisite equal-group reasoning.
- E2
Draw 5 groups of 6 counters and find the total.
Show answer and reasoning
Answer: 30
5 equal groups with 6 in each group make 5 × 6 = 30. This is prerequisite equal-group reasoning.
Learn the concept
Words to know
- array
- objects arranged in equal rows and columns
- factor
- a number multiplied by another
Model products as equal groups or rectangular arrays.
Use an array of equal-size rows. The total can be counted by rows or by columns; this shows commutativity without relying on a slogan.
Why this works
An array has equal rows and columns. Four rows with six counters each make 24 counters. Turn the array so it has six rows with four counters each; you still have the same 24 counters. This shows why 4 × 6 = 6 × 4 even though the rows and columns swap names.
Check the meaning
Explain which factor counts groups and which counts items per group in the story.
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
There are 3 equal rows with 7 counters each. How many counters?
- Use an array of equal-size rows. The total can be counted by rows or by columns; this shows commutativity without relying on a slogan.
- There are 3 groups of 7; multiply 3×7=21. Turning the array does not change the total.
- Explain which factor counts groups and which counts items per group in the story.
Result: 21
Understanding check 1 — your turn
There are 7 equal rows with 3 counters each. How many counters?
Reveal reasoning
Answer: 21 — There are 7 groups of 3; multiply 7×3=21. Turning the array does not change the total.
Worked example 2
Reason and verify
There are 4 equal rows with 5 counters each. How many counters?
- Use an array of equal-size rows. The total can be counted by rows or by columns; this shows commutativity without relying on a slogan.
- There are 4 groups of 5; multiply 4×5=20. Turning the array does not change the total.
- Explain which factor counts groups and which counts items per group in the story.
Result: 20
Understanding check 2 — your turn
There are 8 equal rows with 10 counters each. How many counters?
Reveal reasoning
Answer: 80 — There are 8 groups of 10; multiply 8×10=80. Turning the array does not change the total.
Worked example 3
Reason and verify
There are 5 equal rows with 3 counters each. How many counters?
- Use an array of equal-size rows. The total can be counted by rows or by columns; this shows commutativity without relying on a slogan.
- There are 5 groups of 3; multiply 5×3=15. Turning the array does not change the total.
- Explain which factor counts groups and which counts items per group in the story.
Result: 15
Understanding check 3 — your turn
There are 2 equal rows with 9 counters each. How many counters?
Reveal reasoning
Answer: 18 — There are 2 groups of 9; multiply 2×9=18. Turning the array does not change the total.
Guided · 4 questions
Guided practice
Practice with the model visible. Explain the units at each step.
- G1
There are 3 equal rows with 11 counters each. How many counters?
Show answer and reasoning
Answer: 33
There are 3 groups of 11; multiply 3×11=33. Turning the array does not change the total.
- G2
There are 4 equal rows with 9 counters each. How many counters?
Show answer and reasoning
Answer: 36
There are 4 groups of 9; multiply 4×9=36. Turning the array does not change the total.
- G3
There are 5 equal rows with 7 counters each. How many counters?
Show answer and reasoning
Answer: 35
There are 5 groups of 7; multiply 5×7=35. Turning the array does not change the total.
- G4
There are 6 equal rows with 5 counters each. How many counters?
Show answer and reasoning
Answer: 30
There are 6 groups of 5; multiply 6×5=30. Turning the array does not change the total.
Independent · 8 questions
Independent practice
Work without relying on the model or checking answers prematurely.
- I1
There are 8 equal rows with 4 counters each. How many counters?
Show answer and reasoning
Answer: 32
There are 8 groups of 4; multiply 8×4=32. Turning the array does not change the total.
- I2
There are 2 equal rows with 3 counters each. How many counters?
Show answer and reasoning
Answer: 6
There are 2 groups of 3; multiply 2×3=6. Turning the array does not change the total.
- I3
There are 3 equal rows with 10 counters each. How many counters?
Show answer and reasoning
Answer: 30
There are 3 groups of 10; multiply 3×10=30. Turning the array does not change the total.
- I4
There are 4 equal rows with 8 counters each. How many counters?
Show answer and reasoning
Answer: 32
There are 4 groups of 8; multiply 4×8=32. Turning the array does not change the total.
- I5
There are 5 equal rows with 6 counters each. How many counters?
Show answer and reasoning
Answer: 30
There are 5 groups of 6; multiply 5×6=30. Turning the array does not change the total.
- I6
There are 6 equal rows with 4 counters each. How many counters?
Show answer and reasoning
Answer: 24
There are 6 groups of 4; multiply 6×4=24. Turning the array does not change the total.
- I7
There are 7 equal rows with 11 counters each. How many counters?
Show answer and reasoning
Answer: 77
There are 7 groups of 11; multiply 7×11=77. Turning the array does not change the total.
- I8
There are 8 equal rows with 9 counters each. How many counters?
Show answer and reasoning
Answer: 72
There are 8 groups of 9; multiply 8×9=72. Turning the array does not change the total.
Challenge · 2 questions
Challenge practice
Solve original application and explanation problems. Draw and defend a useful model.
- C1
Explain why 4 rows of 6 objects and 6 rows of 4 objects contain the same number.
Show answer and reasoning
Answer: 24 objects
Rotating the rectangular array changes which side is called rows, but the set of 24 positions is unchanged.
- C2
Describe two different rectangles containing 24 counters, and explain why both give 24.
Show answer and reasoning
Answer: 3×8 and 4×6
Three rows of eight and four rows of six each count 24 equally arranged counters; factor pairs can differ without changing total.
Exit · 3 questions
Exit check
Solve independently; revisit the skill if an important explanation is missing.
- E1
There are 4 equal rows with 10 counters each. How many counters?
Show answer and reasoning
Answer: 40
There are 4 groups of 10; multiply 4×10=40. Turning the array does not change the total.
- E2
There are 5 equal rows with 8 counters each. How many counters?
Show answer and reasoning
Answer: 40
There are 5 groups of 8; multiply 5×8=40. Turning the array does not change the total.
- E3
There are 6 equal rows with 6 counters each. How many counters?
Show answer and reasoning
Answer: 36
There are 6 groups of 6; multiply 6×6=36. Turning the array does not change the total.