← 2-2 · Smaller unit, more measured counts · Length as Sum of Parts with Unit Matching

Smaller unit, more measured counts · 12 practice problems

2.MD.A.1

Generated variants — 12

Freshly produced from the archetype’s parameters — problem, figure, and solution derived together.

Variant 1 answer: 6 times

A wooden stick is $3\ \text{ft}$ long, and a wire is $4\ \text{ft}$ long. Find how many times you would lay the wire end to end to cover the same length as $8$ lengths of the wooden stick.

Show solution

Understand

A wooden stick is 3 ft long and a wire is 4 ft long. We want to cover the length of 8 sticks laid end to end. How many wire lengths laid end to end cover that same total?

Givens

The wooden stick is 3 ft long.
The wire is 4 ft long.
The target length is 8 sticks laid end to end.

Unknowns

How many wire lengths laid end to end equal the length of 8 sticks.

Constraints

Both units (stick, wire) measure the same total length; the count must come out whole.

Plan

#8 Analyze the Units · also uses: #7 Identify Subproblems

First find the total length to cover (8 sticks), then see how many 4 ft wires fit into it. Because the wire is longer than the stick, fewer wires are needed than sticks - the smaller unit needs more counts.

Execute

#7 Identify Subproblems 2.MD.A.1

Each stick is 3 ft, and 8 are laid end to end, so multiply.

8 \times 3 = 24 \text{ ft}

Laying equal sticks end to end adds their lengths, giving the total to cover.

#8 Analyze the Units 2.MD.A.1

Each wire is 4 ft, so see how many 4 ft pieces make 24 ft by dividing.

24 \div 4 = 6

Counting how many copies of the unit fit into the total is measuring with that unit.

Answer: 6 times

Review

Check: 6 wires x 4 ft = 24 ft, which equals 8 sticks x 3 ft = 24 ft. The wire (longer unit) needs fewer counts than 8 sticks, as expected.

Look for a Pattern / repeated addition (tool 5): add 4 + 4 + 4 + 4 + 4 + 4 = 24 ft, reaching the target in 6 wire lengths.

Standards · min grade 2

2.MD.A.1 Measure the length of an object by selecting and using appropriate tools — Measuring the same total length with two different unit lengths (stick, wire).

💡 A longer measuring unit needs fewer counts to reach the same length -- Grade 2 measuring sense!

Variant 2 answer: 3 times

A wooden stick is $9\ \text{ft}$ long, and a wire is $12\ \text{ft}$ long. Find how many times you would lay the wire end to end to cover the same length as $4$ lengths of the wooden stick.

Show solution

Understand

A wooden stick is 9 ft long and a wire is 12 ft long. We want to cover the length of 4 sticks laid end to end. How many wire lengths laid end to end cover that same total?

Givens

The wooden stick is 9 ft long.
The wire is 12 ft long.
The target length is 4 sticks laid end to end.

Unknowns

How many wire lengths laid end to end equal the length of 4 sticks.

Constraints

Both units (stick, wire) measure the same total length; the count must come out whole.

Plan

#8 Analyze the Units · also uses: #7 Identify Subproblems

First find the total length to cover (4 sticks), then see how many 12 ft wires fit into it. Because the wire is longer than the stick, fewer wires are needed than sticks - the smaller unit needs more counts.

Execute

#7 Identify Subproblems 2.MD.A.1

Each stick is 9 ft, and 4 are laid end to end, so multiply.

4 \times 9 = 36 \text{ ft}

Laying equal sticks end to end adds their lengths, giving the total to cover.

#8 Analyze the Units 2.MD.A.1

Each wire is 12 ft, so see how many 12 ft pieces make 36 ft by dividing.

36 \div 12 = 3

Counting how many copies of the unit fit into the total is measuring with that unit.

Answer: 3 times

Review

Check: 3 wires x 12 ft = 36 ft, which equals 4 sticks x 9 ft = 36 ft. The wire (longer unit) needs fewer counts than 4 sticks, as expected.

Look for a Pattern / repeated addition (tool 5): add 12 + 12 + 12 = 36 ft, reaching the target in 3 wire lengths.

Standards · min grade 2

2.MD.A.1 Measure the length of an object by selecting and using appropriate tools — Measuring the same total length with two different unit lengths (stick, wire).

💡 A longer measuring unit needs fewer counts to reach the same length -- Grade 2 measuring sense!

Variant 3 answer: 6 times

A wooden stick is $6\ \text{ft}$ long, and a wire is $7\ \text{ft}$ long. Find how many times you would lay the wire end to end to cover the same length as $7$ lengths of the wooden stick.

Show solution

Understand

A wooden stick is 6 ft long and a wire is 7 ft long. We want to cover the length of 7 sticks laid end to end. How many wire lengths laid end to end cover that same total?

Givens

The wooden stick is 6 ft long.
The wire is 7 ft long.
The target length is 7 sticks laid end to end.

Unknowns

How many wire lengths laid end to end equal the length of 7 sticks.

Constraints

Both units (stick, wire) measure the same total length; the count must come out whole.

Plan

#8 Analyze the Units · also uses: #7 Identify Subproblems

First find the total length to cover (7 sticks), then see how many 7 ft wires fit into it. Because the wire is longer than the stick, fewer wires are needed than sticks - the smaller unit needs more counts.

Execute

#7 Identify Subproblems 2.MD.A.1

Each stick is 6 ft, and 7 are laid end to end, so multiply.

7 \times 6 = 42 \text{ ft}

Laying equal sticks end to end adds their lengths, giving the total to cover.

#8 Analyze the Units 2.MD.A.1

Each wire is 7 ft, so see how many 7 ft pieces make 42 ft by dividing.

42 \div 7 = 6

Counting how many copies of the unit fit into the total is measuring with that unit.

Answer: 6 times

Review

Check: 6 wires x 7 ft = 42 ft, which equals 7 sticks x 6 ft = 42 ft. The wire (longer unit) needs fewer counts than 7 sticks, as expected.

Look for a Pattern / repeated addition (tool 5): add 7 + 7 + 7 + 7 + 7 + 7 = 42 ft, reaching the target in 6 wire lengths.

Standards · min grade 2

2.MD.A.1 Measure the length of an object by selecting and using appropriate tools — Measuring the same total length with two different unit lengths (stick, wire).

💡 A longer measuring unit needs fewer counts to reach the same length -- Grade 2 measuring sense!

Variant 4 answer: 3 times

A wooden stick is $6\ \text{ft}$ long, and a wire is $8\ \text{ft}$ long. Find how many times you would lay the wire end to end to cover the same length as $4$ lengths of the wooden stick.

Show solution

Understand

A wooden stick is 6 ft long and a wire is 8 ft long. We want to cover the length of 4 sticks laid end to end. How many wire lengths laid end to end cover that same total?

Givens

The wooden stick is 6 ft long.
The wire is 8 ft long.
The target length is 4 sticks laid end to end.

Unknowns

How many wire lengths laid end to end equal the length of 4 sticks.

Constraints

Both units (stick, wire) measure the same total length; the count must come out whole.

Plan

#8 Analyze the Units · also uses: #7 Identify Subproblems

First find the total length to cover (4 sticks), then see how many 8 ft wires fit into it. Because the wire is longer than the stick, fewer wires are needed than sticks - the smaller unit needs more counts.

Execute

#7 Identify Subproblems 2.MD.A.1

Each stick is 6 ft, and 4 are laid end to end, so multiply.

4 \times 6 = 24 \text{ ft}

Laying equal sticks end to end adds their lengths, giving the total to cover.

#8 Analyze the Units 2.MD.A.1

Each wire is 8 ft, so see how many 8 ft pieces make 24 ft by dividing.

24 \div 8 = 3

Counting how many copies of the unit fit into the total is measuring with that unit.

Answer: 3 times

Review

Check: 3 wires x 8 ft = 24 ft, which equals 4 sticks x 6 ft = 24 ft. The wire (longer unit) needs fewer counts than 4 sticks, as expected.

Look for a Pattern / repeated addition (tool 5): add 8 + 8 + 8 = 24 ft, reaching the target in 3 wire lengths.

Standards · min grade 2

2.MD.A.1 Measure the length of an object by selecting and using appropriate tools — Measuring the same total length with two different unit lengths (stick, wire).

💡 A longer measuring unit needs fewer counts to reach the same length -- Grade 2 measuring sense!

Variant 5 answer: 2 times

A wooden stick is $6\ \text{ft}$ long, and a wire is $9\ \text{ft}$ long. Find how many times you would lay the wire end to end to cover the same length as $3$ lengths of the wooden stick.

Show solution

Understand

A wooden stick is 6 ft long and a wire is 9 ft long. We want to cover the length of 3 sticks laid end to end. How many wire lengths laid end to end cover that same total?

Givens

The wooden stick is 6 ft long.
The wire is 9 ft long.
The target length is 3 sticks laid end to end.

Unknowns

How many wire lengths laid end to end equal the length of 3 sticks.

Constraints

Both units (stick, wire) measure the same total length; the count must come out whole.

Plan

#8 Analyze the Units · also uses: #7 Identify Subproblems

First find the total length to cover (3 sticks), then see how many 9 ft wires fit into it. Because the wire is longer than the stick, fewer wires are needed than sticks - the smaller unit needs more counts.

Execute

#7 Identify Subproblems 2.MD.A.1

Each stick is 6 ft, and 3 are laid end to end, so multiply.

3 \times 6 = 18 \text{ ft}

Laying equal sticks end to end adds their lengths, giving the total to cover.

#8 Analyze the Units 2.MD.A.1

Each wire is 9 ft, so see how many 9 ft pieces make 18 ft by dividing.

18 \div 9 = 2

Counting how many copies of the unit fit into the total is measuring with that unit.

Answer: 2 times

Review

Check: 2 wires x 9 ft = 18 ft, which equals 3 sticks x 6 ft = 18 ft. The wire (longer unit) needs fewer counts than 3 sticks, as expected.

Look for a Pattern / repeated addition (tool 5): add 9 + 9 = 18 ft, reaching the target in 2 wire lengths.

Standards · min grade 2

2.MD.A.1 Measure the length of an object by selecting and using appropriate tools — Measuring the same total length with two different unit lengths (stick, wire).

💡 A longer measuring unit needs fewer counts to reach the same length -- Grade 2 measuring sense!

Variant 6 answer: 5 times

A wooden stick is $5\ \text{ft}$ long, and a wire is $6\ \text{ft}$ long. Find how many times you would lay the wire end to end to cover the same length as $6$ lengths of the wooden stick.

Show solution

Understand

A wooden stick is 5 ft long and a wire is 6 ft long. We want to cover the length of 6 sticks laid end to end. How many wire lengths laid end to end cover that same total?

Givens

The wooden stick is 5 ft long.
The wire is 6 ft long.
The target length is 6 sticks laid end to end.

Unknowns

How many wire lengths laid end to end equal the length of 6 sticks.

Constraints

Both units (stick, wire) measure the same total length; the count must come out whole.

Plan

#8 Analyze the Units · also uses: #7 Identify Subproblems

First find the total length to cover (6 sticks), then see how many 6 ft wires fit into it. Because the wire is longer than the stick, fewer wires are needed than sticks - the smaller unit needs more counts.

Execute

#7 Identify Subproblems 2.MD.A.1

Each stick is 5 ft, and 6 are laid end to end, so multiply.

6 \times 5 = 30 \text{ ft}

Laying equal sticks end to end adds their lengths, giving the total to cover.

#8 Analyze the Units 2.MD.A.1

Each wire is 6 ft, so see how many 6 ft pieces make 30 ft by dividing.

30 \div 6 = 5

Counting how many copies of the unit fit into the total is measuring with that unit.

Answer: 5 times

Review

Check: 5 wires x 6 ft = 30 ft, which equals 6 sticks x 5 ft = 30 ft. The wire (longer unit) needs fewer counts than 6 sticks, as expected.

Look for a Pattern / repeated addition (tool 5): add 6 + 6 + 6 + 6 + 6 = 30 ft, reaching the target in 5 wire lengths.

Standards · min grade 2

2.MD.A.1 Measure the length of an object by selecting and using appropriate tools — Measuring the same total length with two different unit lengths (stick, wire).

💡 A longer measuring unit needs fewer counts to reach the same length -- Grade 2 measuring sense!

Variant 7 answer: 4 times

A wooden stick is $8\ \text{ft}$ long, and a wire is $10\ \text{ft}$ long. Find how many times you would lay the wire end to end to cover the same length as $5$ lengths of the wooden stick.

Show solution

Understand

A wooden stick is 8 ft long and a wire is 10 ft long. We want to cover the length of 5 sticks laid end to end. How many wire lengths laid end to end cover that same total?

Givens

The wooden stick is 8 ft long.
The wire is 10 ft long.
The target length is 5 sticks laid end to end.

Unknowns

How many wire lengths laid end to end equal the length of 5 sticks.

Constraints

Both units (stick, wire) measure the same total length; the count must come out whole.

Plan

#8 Analyze the Units · also uses: #7 Identify Subproblems

First find the total length to cover (5 sticks), then see how many 10 ft wires fit into it. Because the wire is longer than the stick, fewer wires are needed than sticks - the smaller unit needs more counts.

Execute

#7 Identify Subproblems 2.MD.A.1

Each stick is 8 ft, and 5 are laid end to end, so multiply.

5 \times 8 = 40 \text{ ft}

Laying equal sticks end to end adds their lengths, giving the total to cover.

#8 Analyze the Units 2.MD.A.1

Each wire is 10 ft, so see how many 10 ft pieces make 40 ft by dividing.

40 \div 10 = 4

Counting how many copies of the unit fit into the total is measuring with that unit.

Answer: 4 times

Review

Check: 4 wires x 10 ft = 40 ft, which equals 5 sticks x 8 ft = 40 ft. The wire (longer unit) needs fewer counts than 5 sticks, as expected.

Look for a Pattern / repeated addition (tool 5): add 10 + 10 + 10 + 10 = 40 ft, reaching the target in 4 wire lengths.

Standards · min grade 2

2.MD.A.1 Measure the length of an object by selecting and using appropriate tools — Measuring the same total length with two different unit lengths (stick, wire).

💡 A longer measuring unit needs fewer counts to reach the same length -- Grade 2 measuring sense!

Variant 8 answer: 6 times

A wooden stick is $8\ \text{ft}$ long, and a wire is $12\ \text{ft}$ long. Find how many times you would lay the wire end to end to cover the same length as $9$ lengths of the wooden stick.

Show solution

Understand

A wooden stick is 8 ft long and a wire is 12 ft long. We want to cover the length of 9 sticks laid end to end. How many wire lengths laid end to end cover that same total?

Givens

The wooden stick is 8 ft long.
The wire is 12 ft long.
The target length is 9 sticks laid end to end.

Unknowns

How many wire lengths laid end to end equal the length of 9 sticks.

Constraints

Both units (stick, wire) measure the same total length; the count must come out whole.

Plan

#8 Analyze the Units · also uses: #7 Identify Subproblems

First find the total length to cover (9 sticks), then see how many 12 ft wires fit into it. Because the wire is longer than the stick, fewer wires are needed than sticks - the smaller unit needs more counts.

Execute

#7 Identify Subproblems 2.MD.A.1

Each stick is 8 ft, and 9 are laid end to end, so multiply.

9 \times 8 = 72 \text{ ft}

Laying equal sticks end to end adds their lengths, giving the total to cover.

#8 Analyze the Units 2.MD.A.1

Each wire is 12 ft, so see how many 12 ft pieces make 72 ft by dividing.

72 \div 12 = 6

Counting how many copies of the unit fit into the total is measuring with that unit.

Answer: 6 times

Review

Check: 6 wires x 12 ft = 72 ft, which equals 9 sticks x 8 ft = 72 ft. The wire (longer unit) needs fewer counts than 9 sticks, as expected.

Look for a Pattern / repeated addition (tool 5): add 12 + 12 + 12 + 12 + 12 + 12 = 72 ft, reaching the target in 6 wire lengths.

Standards · min grade 2

2.MD.A.1 Measure the length of an object by selecting and using appropriate tools — Measuring the same total length with two different unit lengths (stick, wire).

💡 A longer measuring unit needs fewer counts to reach the same length -- Grade 2 measuring sense!

Variant 9 answer: 8 times

A wooden stick is $4\ \text{ft}$ long, and a wire is $5\ \text{ft}$ long. Find how many times you would lay the wire end to end to cover the same length as $10$ lengths of the wooden stick.

Show solution

Understand

A wooden stick is 4 ft long and a wire is 5 ft long. We want to cover the length of 10 sticks laid end to end. How many wire lengths laid end to end cover that same total?

Givens

The wooden stick is 4 ft long.
The wire is 5 ft long.
The target length is 10 sticks laid end to end.

Unknowns

How many wire lengths laid end to end equal the length of 10 sticks.

Constraints

Both units (stick, wire) measure the same total length; the count must come out whole.

Plan

#8 Analyze the Units · also uses: #7 Identify Subproblems

First find the total length to cover (10 sticks), then see how many 5 ft wires fit into it. Because the wire is longer than the stick, fewer wires are needed than sticks - the smaller unit needs more counts.

Execute

#7 Identify Subproblems 2.MD.A.1

Each stick is 4 ft, and 10 are laid end to end, so multiply.

10 \times 4 = 40 \text{ ft}

Laying equal sticks end to end adds their lengths, giving the total to cover.

#8 Analyze the Units 2.MD.A.1

Each wire is 5 ft, so see how many 5 ft pieces make 40 ft by dividing.

40 \div 5 = 8

Counting how many copies of the unit fit into the total is measuring with that unit.

Answer: 8 times

Review

Check: 8 wires x 5 ft = 40 ft, which equals 10 sticks x 4 ft = 40 ft. The wire (longer unit) needs fewer counts than 10 sticks, as expected.

Look for a Pattern / repeated addition (tool 5): add 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 = 40 ft, reaching the target in 8 wire lengths.

Standards · min grade 2

2.MD.A.1 Measure the length of an object by selecting and using appropriate tools — Measuring the same total length with two different unit lengths (stick, wire).

💡 A longer measuring unit needs fewer counts to reach the same length -- Grade 2 measuring sense!

Variant 10 answer: 4 times

A wooden stick is $10\ \text{ft}$ long, and a wire is $15\ \text{ft}$ long. Find how many times you would lay the wire end to end to cover the same length as $6$ lengths of the wooden stick.

Show solution

Understand

A wooden stick is 10 ft long and a wire is 15 ft long. We want to cover the length of 6 sticks laid end to end. How many wire lengths laid end to end cover that same total?

Givens

The wooden stick is 10 ft long.
The wire is 15 ft long.
The target length is 6 sticks laid end to end.

Unknowns

How many wire lengths laid end to end equal the length of 6 sticks.

Constraints

Both units (stick, wire) measure the same total length; the count must come out whole.

Plan

#8 Analyze the Units · also uses: #7 Identify Subproblems

First find the total length to cover (6 sticks), then see how many 15 ft wires fit into it. Because the wire is longer than the stick, fewer wires are needed than sticks - the smaller unit needs more counts.

Execute

#7 Identify Subproblems 2.MD.A.1

Each stick is 10 ft, and 6 are laid end to end, so multiply.

6 \times 10 = 60 \text{ ft}

Laying equal sticks end to end adds their lengths, giving the total to cover.

#8 Analyze the Units 2.MD.A.1

Each wire is 15 ft, so see how many 15 ft pieces make 60 ft by dividing.

60 \div 15 = 4

Counting how many copies of the unit fit into the total is measuring with that unit.

Answer: 4 times

Review

Check: 4 wires x 15 ft = 60 ft, which equals 6 sticks x 10 ft = 60 ft. The wire (longer unit) needs fewer counts than 6 sticks, as expected.

Look for a Pattern / repeated addition (tool 5): add 15 + 15 + 15 + 15 = 60 ft, reaching the target in 4 wire lengths.

Standards · min grade 2

2.MD.A.1 Measure the length of an object by selecting and using appropriate tools — Measuring the same total length with two different unit lengths (stick, wire).

💡 A longer measuring unit needs fewer counts to reach the same length -- Grade 2 measuring sense!

Variant 11 answer: 2 times

A wooden stick is $4\ \text{ft}$ long, and a wire is $6\ \text{ft}$ long. Find how many times you would lay the wire end to end to cover the same length as $3$ lengths of the wooden stick.

Show solution

Understand

A wooden stick is 4 ft long and a wire is 6 ft long. We want to cover the length of 3 sticks laid end to end. How many wire lengths laid end to end cover that same total?

Givens

The wooden stick is 4 ft long.
The wire is 6 ft long.
The target length is 3 sticks laid end to end.

Unknowns

How many wire lengths laid end to end equal the length of 3 sticks.

Constraints

Both units (stick, wire) measure the same total length; the count must come out whole.

Plan

#8 Analyze the Units · also uses: #7 Identify Subproblems

First find the total length to cover (3 sticks), then see how many 6 ft wires fit into it. Because the wire is longer than the stick, fewer wires are needed than sticks - the smaller unit needs more counts.

Execute

#7 Identify Subproblems 2.MD.A.1

Each stick is 4 ft, and 3 are laid end to end, so multiply.

3 \times 4 = 12 \text{ ft}

Laying equal sticks end to end adds their lengths, giving the total to cover.

#8 Analyze the Units 2.MD.A.1

Each wire is 6 ft, so see how many 6 ft pieces make 12 ft by dividing.

12 \div 6 = 2

Counting how many copies of the unit fit into the total is measuring with that unit.

Answer: 2 times

Review

Check: 2 wires x 6 ft = 12 ft, which equals 3 sticks x 4 ft = 12 ft. The wire (longer unit) needs fewer counts than 3 sticks, as expected.

Look for a Pattern / repeated addition (tool 5): add 6 + 6 = 12 ft, reaching the target in 2 wire lengths.

Standards · min grade 2

2.MD.A.1 Measure the length of an object by selecting and using appropriate tools — Measuring the same total length with two different unit lengths (stick, wire).

💡 A longer measuring unit needs fewer counts to reach the same length -- Grade 2 measuring sense!

Variant 12 answer: 3 times

A wooden stick is $7\ \text{ft}$ long, and a wire is $14\ \text{ft}$ long. Find how many times you would lay the wire end to end to cover the same length as $6$ lengths of the wooden stick.

Show solution

Understand

A wooden stick is 7 ft long and a wire is 14 ft long. We want to cover the length of 6 sticks laid end to end. How many wire lengths laid end to end cover that same total?

Givens

The wooden stick is 7 ft long.
The wire is 14 ft long.
The target length is 6 sticks laid end to end.

Unknowns

How many wire lengths laid end to end equal the length of 6 sticks.

Constraints

Both units (stick, wire) measure the same total length; the count must come out whole.

Plan

#8 Analyze the Units · also uses: #7 Identify Subproblems

First find the total length to cover (6 sticks), then see how many 14 ft wires fit into it. Because the wire is longer than the stick, fewer wires are needed than sticks - the smaller unit needs more counts.

Execute

#7 Identify Subproblems 2.MD.A.1

Each stick is 7 ft, and 6 are laid end to end, so multiply.

6 \times 7 = 42 \text{ ft}

Laying equal sticks end to end adds their lengths, giving the total to cover.

#8 Analyze the Units 2.MD.A.1

Each wire is 14 ft, so see how many 14 ft pieces make 42 ft by dividing.

42 \div 14 = 3

Counting how many copies of the unit fit into the total is measuring with that unit.

Answer: 3 times

Review

Check: 3 wires x 14 ft = 42 ft, which equals 6 sticks x 7 ft = 42 ft. The wire (longer unit) needs fewer counts than 6 sticks, as expected.

Look for a Pattern / repeated addition (tool 5): add 14 + 14 + 14 = 42 ft, reaching the target in 3 wire lengths.

Standards · min grade 2

2.MD.A.1 Measure the length of an object by selecting and using appropriate tools — Measuring the same total length with two different unit lengths (stick, wire).

💡 A longer measuring unit needs fewer counts to reach the same length -- Grade 2 measuring sense!