← Consecutive integers around a middle value · Sum of Evenly Spaced Numbers via the Middle

Consecutive integers around a middle value · 12 practice problems

3.OA.D.93.OA.D.8

0 / 12 solved

Show all answers (parent / teacher)

Generated variants — 12

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

Variant 1 medium

Whole numbers listed in a row, such as $1,\ 2,\ 3$ or $99,\ 100,\ 101$ , are called consecutive whole numbers. When the sum of three consecutive whole numbers is $27$ , find the product of these three numbers.

Your answer

Show solution

1 · Understandwhat's really being asked

Three whole numbers in a row (each one bigger than the last) add up to 27. Find the product of those three numbers.

Givens

The three numbers are consecutive whole numbers (like 8, 9, 10).
Their sum is 27.

Unknowns

The three consecutive numbers, and their product.

Constraints

Consecutive whole numbers differ by exactly 1.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #6 Guess and Check

Three consecutive numbers are (middle - 1), (middle), (middle + 1). Their sum is 3 times the middle, so the middle equals the sum divided by 3. Then check by listing the three numbers.

3 · Execute3 carry out the plan

1The sum is 3 times the middle number

#5 Look for a Pattern 3.OA.D.9

Write the three numbers around the middle one: middle - 1, middle, middle + 1. The -1 and +1 cancel, so the sum is exactly 3 times the middle number.

(m-1) + m + (m+1) = 3 \times m = 27

Evenly spaced numbers balance around their center, so their sum is the center times how many there are.

2Find the middle number

#6 Guess and Check 3.OA.C.7

Since 3 times the middle is 27, the middle number is 27 divided by 3.

m = 27 \div 3 = 9

Dividing by 3 undoes the 'times 3', a basic Grade 3 fact.

3List the numbers and multiply

#5 Look for a Pattern 3.OA.C.7

The numbers are 8, 9, 10 (check: 8 + 9 + 10 = 27). Multiply them together.

8 \times 9 \times 10 = 72 \times 10 = 720

Multiply two at a time to reach the final product.

Answer: 720

4 · Reviewdoes it hold up?

8 + 9 + 10 = 27 matches the given sum. The product 720 is close to 9 cubed (729), which makes sense for three numbers near 9.

Another way: Guess and check: try 7,8,9 (sum 24, too small), then 8,9,10 (sum 27). It works, so multiply to get 720.

Standardsmin grade 3

3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Seeing that three consecutive numbers sum to 3 times the middle one.
3.OA.C.7 Fluently multiply and divide within 100 — Dividing 27 by 3 and multiplying the three numbers.

💡Takeaway. Numbers in a row balance around the middle, so the sum is just the middle times how many -- pure Grade 3 sense!

Variant 2 medium

Whole numbers listed in a row, such as $1,\ 2,\ 3$ or $99,\ 100,\ 101$ , are called consecutive whole numbers. When the sum of three consecutive whole numbers is $12$ , find the product of these three numbers.

Your answer

Show solution

1 · Understandwhat's really being asked

Three whole numbers in a row (each one bigger than the last) add up to 12. Find the product of those three numbers.

Givens

The three numbers are consecutive whole numbers (like 3, 4, 5).
Their sum is 12.

Unknowns

The three consecutive numbers, and their product.

Constraints

Consecutive whole numbers differ by exactly 1.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #6 Guess and Check

Three consecutive numbers are (middle - 1), (middle), (middle + 1). Their sum is 3 times the middle, so the middle equals the sum divided by 3. Then check by listing the three numbers.

3 · Execute3 carry out the plan

1The sum is 3 times the middle number

#5 Look for a Pattern 3.OA.D.9

Write the three numbers around the middle one: middle - 1, middle, middle + 1. The -1 and +1 cancel, so the sum is exactly 3 times the middle number.

(m-1) + m + (m+1) = 3 \times m = 12

Evenly spaced numbers balance around their center, so their sum is the center times how many there are.

2Find the middle number

#6 Guess and Check 3.OA.C.7

Since 3 times the middle is 12, the middle number is 12 divided by 3.

m = 12 \div 3 = 4

Dividing by 3 undoes the 'times 3', a basic Grade 3 fact.

3List the numbers and multiply

#5 Look for a Pattern 3.OA.C.7

The numbers are 3, 4, 5 (check: 3 + 4 + 5 = 12). Multiply them together.

3 \times 4 \times 5 = 12 \times 5 = 60

Multiply two at a time to reach the final product.

Answer: 60

4 · Reviewdoes it hold up?

3 + 4 + 5 = 12 matches the given sum. The product 60 is close to 4 cubed (64), which makes sense for three numbers near 4.

Another way: Guess and check: try 2,3,4 (sum 9, too small), then 3,4,5 (sum 12). It works, so multiply to get 60.

Standardsmin grade 3

3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Seeing that three consecutive numbers sum to 3 times the middle one.
3.OA.C.7 Fluently multiply and divide within 100 — Dividing 12 by 3 and multiplying the three numbers.

💡Takeaway. Numbers in a row balance around the middle, so the sum is just the middle times how many -- pure Grade 3 sense!

Variant 3 medium

Whole numbers listed in a row, such as $1,\ 2,\ 3$ or $99,\ 100,\ 101$ , are called consecutive whole numbers. When the sum of three consecutive whole numbers is $18$ , find the product of these three numbers.

Your answer

Show solution

1 · Understandwhat's really being asked

Three whole numbers in a row (each one bigger than the last) add up to 18. Find the product of those three numbers.

Givens

The three numbers are consecutive whole numbers (like 5, 6, 7).
Their sum is 18.

Unknowns

The three consecutive numbers, and their product.

Constraints

Consecutive whole numbers differ by exactly 1.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #6 Guess and Check

Three consecutive numbers are (middle - 1), (middle), (middle + 1). Their sum is 3 times the middle, so the middle equals the sum divided by 3. Then check by listing the three numbers.

3 · Execute3 carry out the plan

1The sum is 3 times the middle number

#5 Look for a Pattern 3.OA.D.9

Write the three numbers around the middle one: middle - 1, middle, middle + 1. The -1 and +1 cancel, so the sum is exactly 3 times the middle number.

(m-1) + m + (m+1) = 3 \times m = 18

Evenly spaced numbers balance around their center, so their sum is the center times how many there are.

2Find the middle number

#6 Guess and Check 3.OA.C.7

Since 3 times the middle is 18, the middle number is 18 divided by 3.

m = 18 \div 3 = 6

Dividing by 3 undoes the 'times 3', a basic Grade 3 fact.

3List the numbers and multiply

#5 Look for a Pattern 3.OA.C.7

The numbers are 5, 6, 7 (check: 5 + 6 + 7 = 18). Multiply them together.

5 \times 6 \times 7 = 30 \times 7 = 210

Multiply two at a time to reach the final product.

Answer: 210

4 · Reviewdoes it hold up?

5 + 6 + 7 = 18 matches the given sum. The product 210 is close to 6 cubed (216), which makes sense for three numbers near 6.

Another way: Guess and check: try 4,5,6 (sum 15, too small), then 5,6,7 (sum 18). It works, so multiply to get 210.

Standardsmin grade 3

3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Seeing that three consecutive numbers sum to 3 times the middle one.
3.OA.C.7 Fluently multiply and divide within 100 — Dividing 18 by 3 and multiplying the three numbers.

💡Takeaway. Numbers in a row balance around the middle, so the sum is just the middle times how many -- pure Grade 3 sense!

Variant 4 medium

Whole numbers listed in a row, such as $1,\ 2,\ 3$ or $99,\ 100,\ 101$ , are called consecutive whole numbers. When the sum of three consecutive whole numbers is $21$ , find the product of these three numbers.

Your answer

Show solution

1 · Understandwhat's really being asked

Three whole numbers in a row (each one bigger than the last) add up to 21. Find the product of those three numbers.

Givens

The three numbers are consecutive whole numbers (like 6, 7, 8).
Their sum is 21.

Unknowns

The three consecutive numbers, and their product.

Constraints

Consecutive whole numbers differ by exactly 1.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #6 Guess and Check

Three consecutive numbers are (middle - 1), (middle), (middle + 1). Their sum is 3 times the middle, so the middle equals the sum divided by 3. Then check by listing the three numbers.

3 · Execute3 carry out the plan

1The sum is 3 times the middle number

#5 Look for a Pattern 3.OA.D.9

Write the three numbers around the middle one: middle - 1, middle, middle + 1. The -1 and +1 cancel, so the sum is exactly 3 times the middle number.

(m-1) + m + (m+1) = 3 \times m = 21

Evenly spaced numbers balance around their center, so their sum is the center times how many there are.

2Find the middle number

#6 Guess and Check 3.OA.C.7

Since 3 times the middle is 21, the middle number is 21 divided by 3.

m = 21 \div 3 = 7

Dividing by 3 undoes the 'times 3', a basic Grade 3 fact.

3List the numbers and multiply

#5 Look for a Pattern 3.OA.C.7

The numbers are 6, 7, 8 (check: 6 + 7 + 8 = 21). Multiply them together.

6 \times 7 \times 8 = 42 \times 8 = 336

Multiply two at a time to reach the final product.

Answer: 336

4 · Reviewdoes it hold up?

6 + 7 + 8 = 21 matches the given sum. The product 336 is close to 7 cubed (343), which makes sense for three numbers near 7.

Another way: Guess and check: try 5,6,7 (sum 18, too small), then 6,7,8 (sum 21). It works, so multiply to get 336.

Standardsmin grade 3

3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Seeing that three consecutive numbers sum to 3 times the middle one.
3.OA.C.7 Fluently multiply and divide within 100 — Dividing 21 by 3 and multiplying the three numbers.

💡Takeaway. Numbers in a row balance around the middle, so the sum is just the middle times how many -- pure Grade 3 sense!

Variant 5 medium

Whole numbers listed in a row, such as $1,\ 2,\ 3$ or $99,\ 100,\ 101$ , are called consecutive whole numbers. When the sum of three consecutive whole numbers is $9$ , find the product of these three numbers.

Your answer

Show solution

1 · Understandwhat's really being asked

Three whole numbers in a row (each one bigger than the last) add up to 9. Find the product of those three numbers.

Givens

The three numbers are consecutive whole numbers (like 2, 3, 4).
Their sum is 9.

Unknowns

The three consecutive numbers, and their product.

Constraints

Consecutive whole numbers differ by exactly 1.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #6 Guess and Check

Three consecutive numbers are (middle - 1), (middle), (middle + 1). Their sum is 3 times the middle, so the middle equals the sum divided by 3. Then check by listing the three numbers.

3 · Execute3 carry out the plan

1The sum is 3 times the middle number

#5 Look for a Pattern 3.OA.D.9

Write the three numbers around the middle one: middle - 1, middle, middle + 1. The -1 and +1 cancel, so the sum is exactly 3 times the middle number.

(m-1) + m + (m+1) = 3 \times m = 9

Evenly spaced numbers balance around their center, so their sum is the center times how many there are.

2Find the middle number

#6 Guess and Check 3.OA.C.7

Since 3 times the middle is 9, the middle number is 9 divided by 3.

m = 9 \div 3 = 3

Dividing by 3 undoes the 'times 3', a basic Grade 3 fact.

3List the numbers and multiply

#5 Look for a Pattern 3.OA.C.7

The numbers are 2, 3, 4 (check: 2 + 3 + 4 = 9). Multiply them together.

2 \times 3 \times 4 = 6 \times 4 = 24

Multiply two at a time to reach the final product.

Answer: 24

4 · Reviewdoes it hold up?

2 + 3 + 4 = 9 matches the given sum. The product 24 is close to 3 cubed (27), which makes sense for three numbers near 3.

Another way: Guess and check: try 1,2,3 (sum 6, too small), then 2,3,4 (sum 9). It works, so multiply to get 24.

Standardsmin grade 3

3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Seeing that three consecutive numbers sum to 3 times the middle one.
3.OA.C.7 Fluently multiply and divide within 100 — Dividing 9 by 3 and multiplying the three numbers.

💡Takeaway. Numbers in a row balance around the middle, so the sum is just the middle times how many -- pure Grade 3 sense!

Variant 6 medium

Whole numbers listed in a row, such as $1,\ 2,\ 3$ or $99,\ 100,\ 101$ , are called consecutive whole numbers. When the sum of three consecutive whole numbers is $30$ , find the product of these three numbers.

Your answer

Show solution

1 · Understandwhat's really being asked

Three whole numbers in a row (each one bigger than the last) add up to 30. Find the product of those three numbers.

Givens

The three numbers are consecutive whole numbers (like 9, 10, 11).
Their sum is 30.

Unknowns

The three consecutive numbers, and their product.

Constraints

Consecutive whole numbers differ by exactly 1.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #6 Guess and Check

Three consecutive numbers are (middle - 1), (middle), (middle + 1). Their sum is 3 times the middle, so the middle equals the sum divided by 3. Then check by listing the three numbers.

3 · Execute3 carry out the plan

1The sum is 3 times the middle number

#5 Look for a Pattern 3.OA.D.9

Write the three numbers around the middle one: middle - 1, middle, middle + 1. The -1 and +1 cancel, so the sum is exactly 3 times the middle number.

(m-1) + m + (m+1) = 3 \times m = 30

Evenly spaced numbers balance around their center, so their sum is the center times how many there are.

2Find the middle number

#6 Guess and Check 3.OA.C.7

Since 3 times the middle is 30, the middle number is 30 divided by 3.

m = 30 \div 3 = 10

Dividing by 3 undoes the 'times 3', a basic Grade 3 fact.

3List the numbers and multiply

#5 Look for a Pattern 3.OA.C.7

The numbers are 9, 10, 11 (check: 9 + 10 + 11 = 30). Multiply them together.

9 \times 10 \times 11 = 90 \times 11 = 990

Multiply two at a time to reach the final product.

Answer: 990

4 · Reviewdoes it hold up?

9 + 10 + 11 = 30 matches the given sum. The product 990 is close to 10 cubed (1000), which makes sense for three numbers near 10.

Another way: Guess and check: try 8,9,10 (sum 27, too small), then 9,10,11 (sum 30). It works, so multiply to get 990.

Standardsmin grade 3

3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Seeing that three consecutive numbers sum to 3 times the middle one.
3.OA.C.7 Fluently multiply and divide within 100 — Dividing 30 by 3 and multiplying the three numbers.

💡Takeaway. Numbers in a row balance around the middle, so the sum is just the middle times how many -- pure Grade 3 sense!

Variant 7 medium

Whole numbers listed in a row, such as $1,\ 2,\ 3$ or $99,\ 100,\ 101$ , are called consecutive whole numbers. When the sum of three consecutive whole numbers is $24$ , find the product of these three numbers.

Your answer

Show solution

1 · Understandwhat's really being asked

Three whole numbers in a row (each one bigger than the last) add up to 24. Find the product of those three numbers.

Givens

The three numbers are consecutive whole numbers (like 7, 8, 9).
Their sum is 24.

Unknowns

The three consecutive numbers, and their product.

Constraints

Consecutive whole numbers differ by exactly 1.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #6 Guess and Check

Three consecutive numbers are (middle - 1), (middle), (middle + 1). Their sum is 3 times the middle, so the middle equals the sum divided by 3. Then check by listing the three numbers.

3 · Execute3 carry out the plan

1The sum is 3 times the middle number

#5 Look for a Pattern 3.OA.D.9

Write the three numbers around the middle one: middle - 1, middle, middle + 1. The -1 and +1 cancel, so the sum is exactly 3 times the middle number.

(m-1) + m + (m+1) = 3 \times m = 24

Evenly spaced numbers balance around their center, so their sum is the center times how many there are.

2Find the middle number

#6 Guess and Check 3.OA.C.7

Since 3 times the middle is 24, the middle number is 24 divided by 3.

m = 24 \div 3 = 8

Dividing by 3 undoes the 'times 3', a basic Grade 3 fact.

3List the numbers and multiply

#5 Look for a Pattern 3.OA.C.7

The numbers are 7, 8, 9 (check: 7 + 8 + 9 = 24). Multiply them together.

7 \times 8 \times 9 = 56 \times 9 = 504

Multiply two at a time to reach the final product.

Answer: 504

4 · Reviewdoes it hold up?

7 + 8 + 9 = 24 matches the given sum. The product 504 is close to 8 cubed (512), which makes sense for three numbers near 8.

Another way: Guess and check: try 6,7,8 (sum 21, too small), then 7,8,9 (sum 24). It works, so multiply to get 504.

Standardsmin grade 3

3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Seeing that three consecutive numbers sum to 3 times the middle one.
3.OA.C.7 Fluently multiply and divide within 100 — Dividing 24 by 3 and multiplying the three numbers.

💡Takeaway. Numbers in a row balance around the middle, so the sum is just the middle times how many -- pure Grade 3 sense!

Variant 8 medium

Whole numbers listed in a row, such as $1,\ 2,\ 3$ or $99,\ 100,\ 101$ , are called consecutive whole numbers. When the sum of three consecutive whole numbers is $36$ , find the product of these three numbers.

Your answer

Show solution

1 · Understandwhat's really being asked

Three whole numbers in a row (each one bigger than the last) add up to 36. Find the product of those three numbers.

Givens

The three numbers are consecutive whole numbers (like 11, 12, 13).
Their sum is 36.

Unknowns

The three consecutive numbers, and their product.

Constraints

Consecutive whole numbers differ by exactly 1.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #6 Guess and Check

Three consecutive numbers are (middle - 1), (middle), (middle + 1). Their sum is 3 times the middle, so the middle equals the sum divided by 3. Then check by listing the three numbers.

3 · Execute3 carry out the plan

1The sum is 3 times the middle number

#5 Look for a Pattern 3.OA.D.9

Write the three numbers around the middle one: middle - 1, middle, middle + 1. The -1 and +1 cancel, so the sum is exactly 3 times the middle number.

(m-1) + m + (m+1) = 3 \times m = 36

Evenly spaced numbers balance around their center, so their sum is the center times how many there are.

2Find the middle number

#6 Guess and Check 3.OA.C.7

Since 3 times the middle is 36, the middle number is 36 divided by 3.

m = 36 \div 3 = 12

Dividing by 3 undoes the 'times 3', a basic Grade 3 fact.

3List the numbers and multiply

#5 Look for a Pattern 3.OA.C.7

The numbers are 11, 12, 13 (check: 11 + 12 + 13 = 36). Multiply them together.

11 \times 12 \times 13 = 132 \times 13 = 1716

Multiply two at a time to reach the final product.

Answer: 1716

4 · Reviewdoes it hold up?

11 + 12 + 13 = 36 matches the given sum. The product 1716 is close to 12 cubed (1728), which makes sense for three numbers near 12.

Another way: Guess and check: try 10,11,12 (sum 33, too small), then 11,12,13 (sum 36). It works, so multiply to get 1716.

Standardsmin grade 3

3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Seeing that three consecutive numbers sum to 3 times the middle one.
3.OA.C.7 Fluently multiply and divide within 100 — Dividing 36 by 3 and multiplying the three numbers.

💡Takeaway. Numbers in a row balance around the middle, so the sum is just the middle times how many -- pure Grade 3 sense!

Variant 9 medium

Whole numbers listed in a row, such as $1,\ 2,\ 3$ or $99,\ 100,\ 101$ , are called consecutive whole numbers. When the sum of three consecutive whole numbers is $15$ , find the product of these three numbers.

Your answer

Show solution

1 · Understandwhat's really being asked

Three whole numbers in a row (each one bigger than the last) add up to 15. Find the product of those three numbers.

Givens

The three numbers are consecutive whole numbers (like 4, 5, 6).
Their sum is 15.

Unknowns

The three consecutive numbers, and their product.

Constraints

Consecutive whole numbers differ by exactly 1.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #6 Guess and Check

Three consecutive numbers are (middle - 1), (middle), (middle + 1). Their sum is 3 times the middle, so the middle equals the sum divided by 3. Then check by listing the three numbers.

3 · Execute3 carry out the plan

1The sum is 3 times the middle number

#5 Look for a Pattern 3.OA.D.9

Write the three numbers around the middle one: middle - 1, middle, middle + 1. The -1 and +1 cancel, so the sum is exactly 3 times the middle number.

(m-1) + m + (m+1) = 3 \times m = 15

Evenly spaced numbers balance around their center, so their sum is the center times how many there are.

2Find the middle number

#6 Guess and Check 3.OA.C.7

Since 3 times the middle is 15, the middle number is 15 divided by 3.

m = 15 \div 3 = 5

Dividing by 3 undoes the 'times 3', a basic Grade 3 fact.

3List the numbers and multiply

#5 Look for a Pattern 3.OA.C.7

The numbers are 4, 5, 6 (check: 4 + 5 + 6 = 15). Multiply them together.

4 \times 5 \times 6 = 20 \times 6 = 120

Multiply two at a time to reach the final product.

Answer: 120

4 · Reviewdoes it hold up?

4 + 5 + 6 = 15 matches the given sum. The product 120 is close to 5 cubed (125), which makes sense for three numbers near 5.

Another way: Guess and check: try 3,4,5 (sum 12, too small), then 4,5,6 (sum 15). It works, so multiply to get 120.

Standardsmin grade 3

3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Seeing that three consecutive numbers sum to 3 times the middle one.
3.OA.C.7 Fluently multiply and divide within 100 — Dividing 15 by 3 and multiplying the three numbers.

💡Takeaway. Numbers in a row balance around the middle, so the sum is just the middle times how many -- pure Grade 3 sense!

Variant 10 medium

Whole numbers listed in a row, such as $1,\ 2,\ 3$ or $99,\ 100,\ 101$ , are called consecutive whole numbers. When the sum of three consecutive whole numbers is $33$ , find the product of these three numbers.

Your answer

Show solution

1 · Understandwhat's really being asked

Three whole numbers in a row (each one bigger than the last) add up to 33. Find the product of those three numbers.

Givens

The three numbers are consecutive whole numbers (like 10, 11, 12).
Their sum is 33.

Unknowns

The three consecutive numbers, and their product.

Constraints

Consecutive whole numbers differ by exactly 1.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #6 Guess and Check

Three consecutive numbers are (middle - 1), (middle), (middle + 1). Their sum is 3 times the middle, so the middle equals the sum divided by 3. Then check by listing the three numbers.

3 · Execute3 carry out the plan

1The sum is 3 times the middle number

#5 Look for a Pattern 3.OA.D.9

Write the three numbers around the middle one: middle - 1, middle, middle + 1. The -1 and +1 cancel, so the sum is exactly 3 times the middle number.

(m-1) + m + (m+1) = 3 \times m = 33

Evenly spaced numbers balance around their center, so their sum is the center times how many there are.

2Find the middle number

#6 Guess and Check 3.OA.C.7

Since 3 times the middle is 33, the middle number is 33 divided by 3.

m = 33 \div 3 = 11

Dividing by 3 undoes the 'times 3', a basic Grade 3 fact.

3List the numbers and multiply

#5 Look for a Pattern 3.OA.C.7

The numbers are 10, 11, 12 (check: 10 + 11 + 12 = 33). Multiply them together.

10 \times 11 \times 12 = 110 \times 12 = 1320

Multiply two at a time to reach the final product.

Answer: 1320

4 · Reviewdoes it hold up?

10 + 11 + 12 = 33 matches the given sum. The product 1320 is close to 11 cubed (1331), which makes sense for three numbers near 11.

Another way: Guess and check: try 9,10,11 (sum 30, too small), then 10,11,12 (sum 33). It works, so multiply to get 1320.

Standardsmin grade 3

3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Seeing that three consecutive numbers sum to 3 times the middle one.
3.OA.C.7 Fluently multiply and divide within 100 — Dividing 33 by 3 and multiplying the three numbers.

💡Takeaway. Numbers in a row balance around the middle, so the sum is just the middle times how many -- pure Grade 3 sense!

Variant 11 medium

Whole numbers listed in a row, such as $1,\ 2,\ 3$ or $99,\ 100,\ 101$ , are called consecutive whole numbers. When the sum of three consecutive whole numbers is $6$ , find the product of these three numbers.

Your answer

Show solution

1 · Understandwhat's really being asked

Three whole numbers in a row (each one bigger than the last) add up to 6. Find the product of those three numbers.

Givens

The three numbers are consecutive whole numbers (like 1, 2, 3).
Their sum is 6.

Unknowns

The three consecutive numbers, and their product.

Constraints

Consecutive whole numbers differ by exactly 1.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #6 Guess and Check

Three consecutive numbers are (middle - 1), (middle), (middle + 1). Their sum is 3 times the middle, so the middle equals the sum divided by 3. Then check by listing the three numbers.

3 · Execute3 carry out the plan

1The sum is 3 times the middle number

#5 Look for a Pattern 3.OA.D.9

Write the three numbers around the middle one: middle - 1, middle, middle + 1. The -1 and +1 cancel, so the sum is exactly 3 times the middle number.

(m-1) + m + (m+1) = 3 \times m = 6

Evenly spaced numbers balance around their center, so their sum is the center times how many there are.

2Find the middle number

#6 Guess and Check 3.OA.C.7

Since 3 times the middle is 6, the middle number is 6 divided by 3.

m = 6 \div 3 = 2

Dividing by 3 undoes the 'times 3', a basic Grade 3 fact.

3List the numbers and multiply

#5 Look for a Pattern 3.OA.C.7

The numbers are 1, 2, 3 (check: 1 + 2 + 3 = 6). Multiply them together.

1 \times 2 \times 3 = 2 \times 3 = 6

Multiply two at a time to reach the final product.

Answer: 6

4 · Reviewdoes it hold up?

1 + 2 + 3 = 6 matches the given sum. The product 6 is close to 2 cubed (8), which makes sense for three numbers near 2.

Another way: Guess and check: try 0,1,2 (sum 3, too small), then 1,2,3 (sum 6). It works, so multiply to get 6.

Standardsmin grade 3

3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Seeing that three consecutive numbers sum to 3 times the middle one.
3.OA.C.7 Fluently multiply and divide within 100 — Dividing 6 by 3 and multiplying the three numbers.

💡Takeaway. Numbers in a row balance around the middle, so the sum is just the middle times how many -- pure Grade 3 sense!

Variant 12 medium

Whole numbers listed in a row, such as $1,\ 2,\ 3$ or $99,\ 100,\ 101$ , are called consecutive whole numbers. When the sum of three consecutive whole numbers is $45$ , find the product of these three numbers.

Your answer

Show solution

1 · Understandwhat's really being asked

Three whole numbers in a row (each one bigger than the last) add up to 45. Find the product of those three numbers.

Givens

The three numbers are consecutive whole numbers (like 14, 15, 16).
Their sum is 45.

Unknowns

The three consecutive numbers, and their product.

Constraints

Consecutive whole numbers differ by exactly 1.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #6 Guess and Check

Three consecutive numbers are (middle - 1), (middle), (middle + 1). Their sum is 3 times the middle, so the middle equals the sum divided by 3. Then check by listing the three numbers.

3 · Execute3 carry out the plan

1The sum is 3 times the middle number

#5 Look for a Pattern 3.OA.D.9

Write the three numbers around the middle one: middle - 1, middle, middle + 1. The -1 and +1 cancel, so the sum is exactly 3 times the middle number.

(m-1) + m + (m+1) = 3 \times m = 45

Evenly spaced numbers balance around their center, so their sum is the center times how many there are.

2Find the middle number

#6 Guess and Check 3.OA.C.7

Since 3 times the middle is 45, the middle number is 45 divided by 3.

m = 45 \div 3 = 15

Dividing by 3 undoes the 'times 3', a basic Grade 3 fact.

3List the numbers and multiply

#5 Look for a Pattern 3.OA.C.7

The numbers are 14, 15, 16 (check: 14 + 15 + 16 = 45). Multiply them together.

14 \times 15 \times 16 = 210 \times 16 = 3360

Multiply two at a time to reach the final product.

Answer: 3360

4 · Reviewdoes it hold up?

14 + 15 + 16 = 45 matches the given sum. The product 3360 is close to 15 cubed (3375), which makes sense for three numbers near 15.

Another way: Guess and check: try 13,14,15 (sum 42, too small), then 14,15,16 (sum 45). It works, so multiply to get 3360.

Standardsmin grade 3

3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Seeing that three consecutive numbers sum to 3 times the middle one.
3.OA.C.7 Fluently multiply and divide within 100 — Dividing 45 by 3 and multiplying the three numbers.

💡Takeaway. Numbers in a row balance around the middle, so the sum is just the middle times how many -- pure Grade 3 sense!

Where next?

← Back to the problem: Consecutive integers around a middle value