Practice · Pair consecutive numbers with constant sum · Sensim Math

Variant 1 easy

Look at the example calculations, then find the number that belongs in each $\square$ . (Each equation is a sum of consecutive whole numbers.)

Example

$9+10+11=30$
$7+8+9+10+11=45$

$\square+\square+\square+\square+\square+\square+\square=63$
$\square+\square+\square+\square+\square=60$

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Show solution

1 · Understandwhat's really being asked

Two examples show sums of consecutive whole numbers (9+10+11=30 and 7+8+9+10+11=45). Fill the blanks so that 7 consecutive numbers add to 63, and 5 consecutive numbers add to 60.

Givens

Examples: 9+10+11 = 30 and 7+8+9+10+11 = 45
Each blank row is a run of consecutive whole numbers
First target sum is 63 with 7 numbers; second is 60 with 5 numbers

Unknowns

The 7 consecutive numbers that sum to 63
The 5 consecutive numbers that sum to 60

Constraints

The numbers in each row are consecutive whole numbers
With an odd count of consecutive numbers, the sum equals the count times the middle number

2 · Planchoose the strategy

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

Consecutive numbers pair up so the outer pairs each have the same sum as twice the middle; for an odd count the sum is the count times the middle number, so dividing the sum by the count finds the middle and then the whole run.

3 · Execute3 carry out the plan

1Spot the middle-number rule

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

In the examples, 9+10+11=30 is 3 x 10 (the middle), and 7+8+9+10+11=45 is 5 x 9 (the middle). For an odd run of consecutive numbers, the sum equals the count times the middle number.

3 \times 10 = 30,\quad 5 \times 9 = 45

Pairing the ends inward, each pair averages the middle, so the whole sum centers on the middle.

2Solve the 7-number row

#5 Look for a Pattern 4.OA.C.5

7 consecutive numbers sum to 63, so the middle is 63 divided by 7 = 9. The run is the 3 before and 3 after 9.

63 \div 7 = 9 \;\Rightarrow\; 6+7+8+9+10+11+12 = 63

Knowing the middle, just step out 3 on each side.

3Solve the 5-number row

#5 Look for a Pattern 4.OA.C.5

5 consecutive numbers sum to 60, so the middle is 60 divided by 5 = 12. The run is the 2 before and 2 after 12.

60 \div 5 = 12 \;\Rightarrow\; 10+11+12+13+14 = 60

The middle is the balance point, so spread 2 numbers to each side.

Answer: 6+7+8+9+10+11+12 = 63 and 10+11+12+13+14 = 60

4 · Reviewdoes it hold up?

Checking sums: 6+7+8+9+10+11+12 = 63 and 10+11+12+13+14 = 60, both correct, and every row stays a run of consecutive whole numbers.

Another way: Guess and check (tool 6): start a 7-number run near the average 9 and adjust; the constant-sum-pair idea (6+12, 7+11 each equal 18, plus the middle 9) confirms 63.

Standardsmin grade 4

4.OA.C.5 Generate a number or shape pattern following a given rule — Building the consecutive runs outward from the middle number
3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Recognizing that an odd run of consecutive numbers sums to count times the middle

💡Takeaway. This only needs Grade 4 pattern sense: divide the sum by how many numbers to find the middle, then count outward!

Variant 2 easy

Look at the example calculations, then find the number that belongs in each $\square$ . (Each equation is a sum of consecutive whole numbers.)

Example

$12+13+14=39$
$9+10+11+12+13=55$

$\square+\square+\square=30$
$\square+\square+\square+\square+\square+\square+\square=70$

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Two examples show sums of consecutive whole numbers (12+13+14=39 and 9+10+11+12+13=55). Fill the blanks so that 3 consecutive numbers add to 30, and 7 consecutive numbers add to 70.

Givens

Examples: 12+13+14 = 39 and 9+10+11+12+13 = 55
Each blank row is a run of consecutive whole numbers
First target sum is 30 with 3 numbers; second is 70 with 7 numbers

Unknowns

The 3 consecutive numbers that sum to 30
The 7 consecutive numbers that sum to 70

Constraints

The numbers in each row are consecutive whole numbers
With an odd count of consecutive numbers, the sum equals the count times the middle number

2 · Planchoose the strategy

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

Consecutive numbers pair up so the outer pairs each have the same sum as twice the middle; for an odd count the sum is the count times the middle number, so dividing the sum by the count finds the middle and then the whole run.

3 · Execute3 carry out the plan

1Spot the middle-number rule

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

In the examples, 12+13+14=39 is 3 x 13 (the middle), and 9+10+11+12+13=55 is 5 x 11 (the middle). For an odd run of consecutive numbers, the sum equals the count times the middle number.

3 \times 13 = 39,\quad 5 \times 11 = 55

Pairing the ends inward, each pair averages the middle, so the whole sum centers on the middle.

2Solve the 3-number row

#5 Look for a Pattern 4.OA.C.5

3 consecutive numbers sum to 30, so the middle is 30 divided by 3 = 10. The run is the 1 before and 1 after 10.

30 \div 3 = 10 \;\Rightarrow\; 9+10+11 = 30

Knowing the middle, just step out 1 on each side.

3Solve the 7-number row

#5 Look for a Pattern 4.OA.C.5

7 consecutive numbers sum to 70, so the middle is 70 divided by 7 = 10. The run is the 3 before and 3 after 10.

70 \div 7 = 10 \;\Rightarrow\; 7+8+9+10+11+12+13 = 70

The middle is the balance point, so spread 3 numbers to each side.

Answer: 9+10+11 = 30 and 7+8+9+10+11+12+13 = 70

4 · Reviewdoes it hold up?

Checking sums: 9+10+11 = 30 and 7+8+9+10+11+12+13 = 70, both correct, and every row stays a run of consecutive whole numbers.

Another way: Guess and check (tool 6): start a 3-number run near the average 10 and adjust; the constant-sum-pair idea (9+11, 10+10 each equal 20, plus the middle 10) confirms 30.

Standardsmin grade 4

4.OA.C.5 Generate a number or shape pattern following a given rule — Building the consecutive runs outward from the middle number
3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Recognizing that an odd run of consecutive numbers sums to count times the middle

💡Takeaway. This only needs Grade 4 pattern sense: divide the sum by how many numbers to find the middle, then count outward!

Variant 3 easy

Look at the example calculations, then find the number that belongs in each $\square$ . (Each equation is a sum of consecutive whole numbers.)

Example

$6+7+8=21$
$13+14+15+16+17=75$

$\square+\square+\square+\square+\square=65$
$\square+\square+\square=36$

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Show solution

1 · Understandwhat's really being asked

Two examples show sums of consecutive whole numbers (6+7+8=21 and 13+14+15+16+17=75). Fill the blanks so that 5 consecutive numbers add to 65, and 3 consecutive numbers add to 36.

Givens

Examples: 6+7+8 = 21 and 13+14+15+16+17 = 75
Each blank row is a run of consecutive whole numbers
First target sum is 65 with 5 numbers; second is 36 with 3 numbers

Unknowns

The 5 consecutive numbers that sum to 65
The 3 consecutive numbers that sum to 36

Constraints

The numbers in each row are consecutive whole numbers
With an odd count of consecutive numbers, the sum equals the count times the middle number

2 · Planchoose the strategy

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

Consecutive numbers pair up so the outer pairs each have the same sum as twice the middle; for an odd count the sum is the count times the middle number, so dividing the sum by the count finds the middle and then the whole run.

3 · Execute3 carry out the plan

1Spot the middle-number rule

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

In the examples, 6+7+8=21 is 3 x 7 (the middle), and 13+14+15+16+17=75 is 5 x 15 (the middle). For an odd run of consecutive numbers, the sum equals the count times the middle number.

3 \times 7 = 21,\quad 5 \times 15 = 75

Pairing the ends inward, each pair averages the middle, so the whole sum centers on the middle.

2Solve the 5-number row

#5 Look for a Pattern 4.OA.C.5

5 consecutive numbers sum to 65, so the middle is 65 divided by 5 = 13. The run is the 2 before and 2 after 13.

65 \div 5 = 13 \;\Rightarrow\; 11+12+13+14+15 = 65

Knowing the middle, just step out 2 on each side.

3Solve the 3-number row

#5 Look for a Pattern 4.OA.C.5

3 consecutive numbers sum to 36, so the middle is 36 divided by 3 = 12. The run is the 1 before and 1 after 12.

36 \div 3 = 12 \;\Rightarrow\; 11+12+13 = 36

The middle is the balance point, so spread 1 numbers to each side.

Answer: 11+12+13+14+15 = 65 and 11+12+13 = 36

4 · Reviewdoes it hold up?

Checking sums: 11+12+13+14+15 = 65 and 11+12+13 = 36, both correct, and every row stays a run of consecutive whole numbers.

Another way: Guess and check (tool 6): start a 5-number run near the average 13 and adjust; the constant-sum-pair idea (11+15, 12+14 each equal 26, plus the middle 13) confirms 65.

Standardsmin grade 4

4.OA.C.5 Generate a number or shape pattern following a given rule — Building the consecutive runs outward from the middle number
3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Recognizing that an odd run of consecutive numbers sums to count times the middle

💡Takeaway. This only needs Grade 4 pattern sense: divide the sum by how many numbers to find the middle, then count outward!

Variant 4 medium

Look at the example calculations, then find the number that belongs in each $\square$ . (Each equation is a sum of consecutive whole numbers.)

Example

$3+4+5=12$
$8+9+10+11+12=50$

$\square+\square+\square+\square+\square+\square+\square+\square+\square=90$
$\square+\square+\square+\square+\square=55$

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Two examples show sums of consecutive whole numbers (3+4+5=12 and 8+9+10+11+12=50). Fill the blanks so that 9 consecutive numbers add to 90, and 5 consecutive numbers add to 55.

Givens

Examples: 3+4+5 = 12 and 8+9+10+11+12 = 50
Each blank row is a run of consecutive whole numbers
First target sum is 90 with 9 numbers; second is 55 with 5 numbers

Unknowns

The 9 consecutive numbers that sum to 90
The 5 consecutive numbers that sum to 55

Constraints

The numbers in each row are consecutive whole numbers
With an odd count of consecutive numbers, the sum equals the count times the middle number

2 · Planchoose the strategy

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

Consecutive numbers pair up so the outer pairs each have the same sum as twice the middle; for an odd count the sum is the count times the middle number, so dividing the sum by the count finds the middle and then the whole run.

3 · Execute3 carry out the plan

1Spot the middle-number rule

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

In the examples, 3+4+5=12 is 3 x 4 (the middle), and 8+9+10+11+12=50 is 5 x 10 (the middle). For an odd run of consecutive numbers, the sum equals the count times the middle number.

3 \times 4 = 12,\quad 5 \times 10 = 50

Pairing the ends inward, each pair averages the middle, so the whole sum centers on the middle.

2Solve the 9-number row

#5 Look for a Pattern 4.OA.C.5

9 consecutive numbers sum to 90, so the middle is 90 divided by 9 = 10. The run is the 4 before and 4 after 10.

90 \div 9 = 10 \;\Rightarrow\; 6+7+8+9+10+11+12+13+14 = 90

Knowing the middle, just step out 4 on each side.

3Solve the 5-number row

#5 Look for a Pattern 4.OA.C.5

5 consecutive numbers sum to 55, so the middle is 55 divided by 5 = 11. The run is the 2 before and 2 after 11.

55 \div 5 = 11 \;\Rightarrow\; 9+10+11+12+13 = 55

The middle is the balance point, so spread 2 numbers to each side.

Answer: 6+7+8+9+10+11+12+13+14 = 90 and 9+10+11+12+13 = 55

4 · Reviewdoes it hold up?

Checking sums: 6+7+8+9+10+11+12+13+14 = 90 and 9+10+11+12+13 = 55, both correct, and every row stays a run of consecutive whole numbers.

Another way: Guess and check (tool 6): start a 9-number run near the average 10 and adjust; the constant-sum-pair idea (6+14, 7+13 each equal 20, plus the middle 10) confirms 90.

Standardsmin grade 4

4.OA.C.5 Generate a number or shape pattern following a given rule — Building the consecutive runs outward from the middle number
3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Recognizing that an odd run of consecutive numbers sums to count times the middle

💡Takeaway. This only needs Grade 4 pattern sense: divide the sum by how many numbers to find the middle, then count outward!

Variant 5 medium

Look at the example calculations, then find the number that belongs in each $\square$ . (Each equation is a sum of consecutive whole numbers.)

Example

$7+8+9=24$
$5+6+7+8+9=35$

$\square+\square+\square+\square+\square=100$
$\square+\square+\square+\square+\square+\square+\square=84$

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Two examples show sums of consecutive whole numbers (7+8+9=24 and 5+6+7+8+9=35). Fill the blanks so that 5 consecutive numbers add to 100, and 7 consecutive numbers add to 84.

Givens

Examples: 7+8+9 = 24 and 5+6+7+8+9 = 35
Each blank row is a run of consecutive whole numbers
First target sum is 100 with 5 numbers; second is 84 with 7 numbers

Unknowns

The 5 consecutive numbers that sum to 100
The 7 consecutive numbers that sum to 84

Constraints

The numbers in each row are consecutive whole numbers
With an odd count of consecutive numbers, the sum equals the count times the middle number

2 · Planchoose the strategy

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

Consecutive numbers pair up so the outer pairs each have the same sum as twice the middle; for an odd count the sum is the count times the middle number, so dividing the sum by the count finds the middle and then the whole run.

3 · Execute3 carry out the plan

1Spot the middle-number rule

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

In the examples, 7+8+9=24 is 3 x 8 (the middle), and 5+6+7+8+9=35 is 5 x 7 (the middle). For an odd run of consecutive numbers, the sum equals the count times the middle number.

3 \times 8 = 24,\quad 5 \times 7 = 35

Pairing the ends inward, each pair averages the middle, so the whole sum centers on the middle.

2Solve the 5-number row

#5 Look for a Pattern 4.OA.C.5

5 consecutive numbers sum to 100, so the middle is 100 divided by 5 = 20. The run is the 2 before and 2 after 20.

100 \div 5 = 20 \;\Rightarrow\; 18+19+20+21+22 = 100

Knowing the middle, just step out 2 on each side.

3Solve the 7-number row

#5 Look for a Pattern 4.OA.C.5

7 consecutive numbers sum to 84, so the middle is 84 divided by 7 = 12. The run is the 3 before and 3 after 12.

84 \div 7 = 12 \;\Rightarrow\; 9+10+11+12+13+14+15 = 84

The middle is the balance point, so spread 3 numbers to each side.

Answer: 18+19+20+21+22 = 100 and 9+10+11+12+13+14+15 = 84

4 · Reviewdoes it hold up?

Checking sums: 18+19+20+21+22 = 100 and 9+10+11+12+13+14+15 = 84, both correct, and every row stays a run of consecutive whole numbers.

Another way: Guess and check (tool 6): start a 5-number run near the average 20 and adjust; the constant-sum-pair idea (18+22, 19+21 each equal 40, plus the middle 20) confirms 100.

Standardsmin grade 4

4.OA.C.5 Generate a number or shape pattern following a given rule — Building the consecutive runs outward from the middle number
3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Recognizing that an odd run of consecutive numbers sums to count times the middle

💡Takeaway. This only needs Grade 4 pattern sense: divide the sum by how many numbers to find the middle, then count outward!

Variant 6 medium

Look at the example calculations, then find the number that belongs in each $\square$ . (Each equation is a sum of consecutive whole numbers.)

Example

$5+6+7=18$
$10+11+12+13+14=60$

$\square+\square+\square+\square+\square=45$
$\square+\square+\square+\square+\square+\square+\square+\square+\square=117$

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Two examples show sums of consecutive whole numbers (5+6+7=18 and 10+11+12+13+14=60). Fill the blanks so that 5 consecutive numbers add to 45, and 9 consecutive numbers add to 117.

Givens

Examples: 5+6+7 = 18 and 10+11+12+13+14 = 60
Each blank row is a run of consecutive whole numbers
First target sum is 45 with 5 numbers; second is 117 with 9 numbers

Unknowns

The 5 consecutive numbers that sum to 45
The 9 consecutive numbers that sum to 117

Constraints

The numbers in each row are consecutive whole numbers
With an odd count of consecutive numbers, the sum equals the count times the middle number

2 · Planchoose the strategy

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

Consecutive numbers pair up so the outer pairs each have the same sum as twice the middle; for an odd count the sum is the count times the middle number, so dividing the sum by the count finds the middle and then the whole run.

3 · Execute3 carry out the plan

1Spot the middle-number rule

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

In the examples, 5+6+7=18 is 3 x 6 (the middle), and 10+11+12+13+14=60 is 5 x 12 (the middle). For an odd run of consecutive numbers, the sum equals the count times the middle number.

3 \times 6 = 18,\quad 5 \times 12 = 60

Pairing the ends inward, each pair averages the middle, so the whole sum centers on the middle.

2Solve the 5-number row

#5 Look for a Pattern 4.OA.C.5

5 consecutive numbers sum to 45, so the middle is 45 divided by 5 = 9. The run is the 2 before and 2 after 9.

45 \div 5 = 9 \;\Rightarrow\; 7+8+9+10+11 = 45

Knowing the middle, just step out 2 on each side.

3Solve the 9-number row

#5 Look for a Pattern 4.OA.C.5

9 consecutive numbers sum to 117, so the middle is 117 divided by 9 = 13. The run is the 4 before and 4 after 13.

117 \div 9 = 13 \;\Rightarrow\; 9+10+11+12+13+14+15+16+17 = 117

The middle is the balance point, so spread 4 numbers to each side.

Answer: 7+8+9+10+11 = 45 and 9+10+11+12+13+14+15+16+17 = 117

4 · Reviewdoes it hold up?

Checking sums: 7+8+9+10+11 = 45 and 9+10+11+12+13+14+15+16+17 = 117, both correct, and every row stays a run of consecutive whole numbers.

Another way: Guess and check (tool 6): start a 5-number run near the average 9 and adjust; the constant-sum-pair idea (7+11, 8+10 each equal 18, plus the middle 9) confirms 45.

Standardsmin grade 4

4.OA.C.5 Generate a number or shape pattern following a given rule — Building the consecutive runs outward from the middle number
3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Recognizing that an odd run of consecutive numbers sums to count times the middle

💡Takeaway. This only needs Grade 4 pattern sense: divide the sum by how many numbers to find the middle, then count outward!

Variant 7 medium

Look at the example calculations, then find the number that belongs in each $\square$ . (Each equation is a sum of consecutive whole numbers.)

Example

$10+11+12=33$
$8+9+10+11+12=50$

$\square+\square+\square+\square+\square=35$
$\square+\square+\square+\square+\square+\square+\square=140$

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Two examples show sums of consecutive whole numbers (10+11+12=33 and 8+9+10+11+12=50). Fill the blanks so that 5 consecutive numbers add to 35, and 7 consecutive numbers add to 140.

Givens

Examples: 10+11+12 = 33 and 8+9+10+11+12 = 50
Each blank row is a run of consecutive whole numbers
First target sum is 35 with 5 numbers; second is 140 with 7 numbers

Unknowns

The 5 consecutive numbers that sum to 35
The 7 consecutive numbers that sum to 140

Constraints

The numbers in each row are consecutive whole numbers
With an odd count of consecutive numbers, the sum equals the count times the middle number

2 · Planchoose the strategy

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

Consecutive numbers pair up so the outer pairs each have the same sum as twice the middle; for an odd count the sum is the count times the middle number, so dividing the sum by the count finds the middle and then the whole run.

3 · Execute3 carry out the plan

1Spot the middle-number rule

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

In the examples, 10+11+12=33 is 3 x 11 (the middle), and 8+9+10+11+12=50 is 5 x 10 (the middle). For an odd run of consecutive numbers, the sum equals the count times the middle number.

3 \times 11 = 33,\quad 5 \times 10 = 50

Pairing the ends inward, each pair averages the middle, so the whole sum centers on the middle.

2Solve the 5-number row

#5 Look for a Pattern 4.OA.C.5

5 consecutive numbers sum to 35, so the middle is 35 divided by 5 = 7. The run is the 2 before and 2 after 7.

35 \div 5 = 7 \;\Rightarrow\; 5+6+7+8+9 = 35

Knowing the middle, just step out 2 on each side.

3Solve the 7-number row

#5 Look for a Pattern 4.OA.C.5

7 consecutive numbers sum to 140, so the middle is 140 divided by 7 = 20. The run is the 3 before and 3 after 20.

140 \div 7 = 20 \;\Rightarrow\; 17+18+19+20+21+22+23 = 140

The middle is the balance point, so spread 3 numbers to each side.

Answer: 5+6+7+8+9 = 35 and 17+18+19+20+21+22+23 = 140

4 · Reviewdoes it hold up?

Checking sums: 5+6+7+8+9 = 35 and 17+18+19+20+21+22+23 = 140, both correct, and every row stays a run of consecutive whole numbers.

Another way: Guess and check (tool 6): start a 5-number run near the average 7 and adjust; the constant-sum-pair idea (5+9, 6+8 each equal 14, plus the middle 7) confirms 35.

Standardsmin grade 4

4.OA.C.5 Generate a number or shape pattern following a given rule — Building the consecutive runs outward from the middle number
3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Recognizing that an odd run of consecutive numbers sums to count times the middle

💡Takeaway. This only needs Grade 4 pattern sense: divide the sum by how many numbers to find the middle, then count outward!

Variant 8 hard

Look at the example calculations, then find the number that belongs in each $\square$ . (Each equation is a sum of consecutive whole numbers.)

Example

$11+12+13=36$
$14+15+16+17+18=80$

$\square+\square+\square+\square+\square+\square+\square=105$
$\square+\square+\square+\square+\square+\square+\square+\square+\square=153$

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Two examples show sums of consecutive whole numbers (11+12+13=36 and 14+15+16+17+18=80). Fill the blanks so that 7 consecutive numbers add to 105, and 9 consecutive numbers add to 153.

Givens

Examples: 11+12+13 = 36 and 14+15+16+17+18 = 80
Each blank row is a run of consecutive whole numbers
First target sum is 105 with 7 numbers; second is 153 with 9 numbers

Unknowns

The 7 consecutive numbers that sum to 105
The 9 consecutive numbers that sum to 153

Constraints

The numbers in each row are consecutive whole numbers
With an odd count of consecutive numbers, the sum equals the count times the middle number

2 · Planchoose the strategy

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

Consecutive numbers pair up so the outer pairs each have the same sum as twice the middle; for an odd count the sum is the count times the middle number, so dividing the sum by the count finds the middle and then the whole run.

3 · Execute3 carry out the plan

1Spot the middle-number rule

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

In the examples, 11+12+13=36 is 3 x 12 (the middle), and 14+15+16+17+18=80 is 5 x 16 (the middle). For an odd run of consecutive numbers, the sum equals the count times the middle number.

3 \times 12 = 36,\quad 5 \times 16 = 80

Pairing the ends inward, each pair averages the middle, so the whole sum centers on the middle.

2Solve the 7-number row

#5 Look for a Pattern 4.OA.C.5

7 consecutive numbers sum to 105, so the middle is 105 divided by 7 = 15. The run is the 3 before and 3 after 15.

105 \div 7 = 15 \;\Rightarrow\; 12+13+14+15+16+17+18 = 105

Knowing the middle, just step out 3 on each side.

3Solve the 9-number row

#5 Look for a Pattern 4.OA.C.5

9 consecutive numbers sum to 153, so the middle is 153 divided by 9 = 17. The run is the 4 before and 4 after 17.

153 \div 9 = 17 \;\Rightarrow\; 13+14+15+16+17+18+19+20+21 = 153

The middle is the balance point, so spread 4 numbers to each side.

Answer: 12+13+14+15+16+17+18 = 105 and 13+14+15+16+17+18+19+20+21 = 153

4 · Reviewdoes it hold up?

Checking sums: 12+13+14+15+16+17+18 = 105 and 13+14+15+16+17+18+19+20+21 = 153, both correct, and every row stays a run of consecutive whole numbers.

Another way: Guess and check (tool 6): start a 7-number run near the average 15 and adjust; the constant-sum-pair idea (12+18, 13+17 each equal 30, plus the middle 15) confirms 105.

Standardsmin grade 4

4.OA.C.5 Generate a number or shape pattern following a given rule — Building the consecutive runs outward from the middle number
3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Recognizing that an odd run of consecutive numbers sums to count times the middle

💡Takeaway. This only needs Grade 4 pattern sense: divide the sum by how many numbers to find the middle, then count outward!

Variant 9 hard

Look at the example calculations, then find the number that belongs in each $\square$ . (Each equation is a sum of consecutive whole numbers.)

Example

$20+21+22=63$
$16+17+18+19+20=90$

$\square+\square+\square+\square+\square+\square+\square=154$
$\square+\square+\square+\square+\square=90$

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Two examples show sums of consecutive whole numbers (20+21+22=63 and 16+17+18+19+20=90). Fill the blanks so that 7 consecutive numbers add to 154, and 5 consecutive numbers add to 90.

Givens

Examples: 20+21+22 = 63 and 16+17+18+19+20 = 90
Each blank row is a run of consecutive whole numbers
First target sum is 154 with 7 numbers; second is 90 with 5 numbers

Unknowns

The 7 consecutive numbers that sum to 154
The 5 consecutive numbers that sum to 90

Constraints

The numbers in each row are consecutive whole numbers
With an odd count of consecutive numbers, the sum equals the count times the middle number

2 · Planchoose the strategy

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

Consecutive numbers pair up so the outer pairs each have the same sum as twice the middle; for an odd count the sum is the count times the middle number, so dividing the sum by the count finds the middle and then the whole run.

3 · Execute3 carry out the plan

1Spot the middle-number rule

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

In the examples, 20+21+22=63 is 3 x 21 (the middle), and 16+17+18+19+20=90 is 5 x 18 (the middle). For an odd run of consecutive numbers, the sum equals the count times the middle number.

3 \times 21 = 63,\quad 5 \times 18 = 90

Pairing the ends inward, each pair averages the middle, so the whole sum centers on the middle.

2Solve the 7-number row

#5 Look for a Pattern 4.OA.C.5

7 consecutive numbers sum to 154, so the middle is 154 divided by 7 = 22. The run is the 3 before and 3 after 22.

154 \div 7 = 22 \;\Rightarrow\; 19+20+21+22+23+24+25 = 154

Knowing the middle, just step out 3 on each side.

3Solve the 5-number row

#5 Look for a Pattern 4.OA.C.5

5 consecutive numbers sum to 90, so the middle is 90 divided by 5 = 18. The run is the 2 before and 2 after 18.

90 \div 5 = 18 \;\Rightarrow\; 16+17+18+19+20 = 90

The middle is the balance point, so spread 2 numbers to each side.

Answer: 19+20+21+22+23+24+25 = 154 and 16+17+18+19+20 = 90

4 · Reviewdoes it hold up?

Checking sums: 19+20+21+22+23+24+25 = 154 and 16+17+18+19+20 = 90, both correct, and every row stays a run of consecutive whole numbers.

Another way: Guess and check (tool 6): start a 7-number run near the average 22 and adjust; the constant-sum-pair idea (19+25, 20+24 each equal 44, plus the middle 22) confirms 154.

Standardsmin grade 4

4.OA.C.5 Generate a number or shape pattern following a given rule — Building the consecutive runs outward from the middle number
3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Recognizing that an odd run of consecutive numbers sums to count times the middle

💡Takeaway. This only needs Grade 4 pattern sense: divide the sum by how many numbers to find the middle, then count outward!

Variant 10 hard

Look at the example calculations, then find the number that belongs in each $\square$ . (Each equation is a sum of consecutive whole numbers.)

Example

$4+5+6=15$
$6+7+8+9+10=40$

$\square+\square+\square+\square+\square=50$
$\square+\square+\square+\square+\square+\square+\square=161$

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Two examples show sums of consecutive whole numbers (4+5+6=15 and 6+7+8+9+10=40). Fill the blanks so that 5 consecutive numbers add to 50, and 7 consecutive numbers add to 161.

Givens

Examples: 4+5+6 = 15 and 6+7+8+9+10 = 40
Each blank row is a run of consecutive whole numbers
First target sum is 50 with 5 numbers; second is 161 with 7 numbers

Unknowns

The 5 consecutive numbers that sum to 50
The 7 consecutive numbers that sum to 161

Constraints

The numbers in each row are consecutive whole numbers
With an odd count of consecutive numbers, the sum equals the count times the middle number

2 · Planchoose the strategy

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

Consecutive numbers pair up so the outer pairs each have the same sum as twice the middle; for an odd count the sum is the count times the middle number, so dividing the sum by the count finds the middle and then the whole run.

3 · Execute3 carry out the plan

1Spot the middle-number rule

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

In the examples, 4+5+6=15 is 3 x 5 (the middle), and 6+7+8+9+10=40 is 5 x 8 (the middle). For an odd run of consecutive numbers, the sum equals the count times the middle number.

3 \times 5 = 15,\quad 5 \times 8 = 40

Pairing the ends inward, each pair averages the middle, so the whole sum centers on the middle.

2Solve the 5-number row

#5 Look for a Pattern 4.OA.C.5

5 consecutive numbers sum to 50, so the middle is 50 divided by 5 = 10. The run is the 2 before and 2 after 10.

50 \div 5 = 10 \;\Rightarrow\; 8+9+10+11+12 = 50

Knowing the middle, just step out 2 on each side.

3Solve the 7-number row

#5 Look for a Pattern 4.OA.C.5

7 consecutive numbers sum to 161, so the middle is 161 divided by 7 = 23. The run is the 3 before and 3 after 23.

161 \div 7 = 23 \;\Rightarrow\; 20+21+22+23+24+25+26 = 161

The middle is the balance point, so spread 3 numbers to each side.

Answer: 8+9+10+11+12 = 50 and 20+21+22+23+24+25+26 = 161

4 · Reviewdoes it hold up?

Checking sums: 8+9+10+11+12 = 50 and 20+21+22+23+24+25+26 = 161, both correct, and every row stays a run of consecutive whole numbers.

Another way: Guess and check (tool 6): start a 5-number run near the average 10 and adjust; the constant-sum-pair idea (8+12, 9+11 each equal 20, plus the middle 10) confirms 50.

Standardsmin grade 4

4.OA.C.5 Generate a number or shape pattern following a given rule — Building the consecutive runs outward from the middle number
3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Recognizing that an odd run of consecutive numbers sums to count times the middle

💡Takeaway. This only needs Grade 4 pattern sense: divide the sum by how many numbers to find the middle, then count outward!

Pair consecutive numbers with constant sum · 10 practice problems

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