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← 4-1 · Pair consecutive numbers with constant sum · Sum of Evenly Spaced Numbers via the Middle

Pair consecutive numbers with constant sum · 10 practice problems

4.OA.C.53.OA.D.9

Generated variants — 10

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

Variant 1 answer: 7+8+9+10+11 = 45 and 9+10+11+12+13+14+15+16+17 = 117

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=185+6+7=18
10+11+12+13+14=6010+11+12+13+14=60

□+□+□+□+□=45\square+\square+\square+\square+\square=45
□+□+□+□+□+□+□+□+□=117\square+\square+\square+\square+\square+\square+\square+\square+\square=117

Show solution

Understand

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

Plan

#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.

Execute

#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.
3×6=18,5×12=603 \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.
#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÷5=9  ⇒  7+8+9+10+11=4545 \div 5 = 9 \;\Rightarrow\; 7+8+9+10+11 = 45
Knowing the middle, just step out 2 on each side.
#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÷9=13  ⇒  9+10+11+12+13+14+15+16+17=117117 \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

Review

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.

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.

Standards · min 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
💡 This only needs Grade 4 pattern sense: divide the sum by how many numbers to find the middle, then count outward!
Variant 2 answer: 5+6+7+8+9 = 35 and 17+18+19+20+21+22+23 = 140

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=3310+11+12=33
8+9+10+11+12=508+9+10+11+12=50

□+□+□+□+□=35\square+\square+\square+\square+\square=35
□+□+□+□+□+□+□=140\square+\square+\square+\square+\square+\square+\square=140

Show solution

Understand

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

Plan

#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.

Execute

#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.
3×11=33,5×10=503 \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.
#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÷5=7  ⇒  5+6+7+8+9=3535 \div 5 = 7 \;\Rightarrow\; 5+6+7+8+9 = 35
Knowing the middle, just step out 2 on each side.
#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÷7=20  ⇒  17+18+19+20+21+22+23=140140 \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

Review

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.

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.

Standards · min 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
💡 This only needs Grade 4 pattern sense: divide the sum by how many numbers to find the middle, then count outward!
Variant 3 answer: 9+10+11 = 30 and 7+8+9+10+11+12+13 = 70

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=3912+13+14=39
9+10+11+12+13=559+10+11+12+13=55

□+□+□=30\square+\square+\square=30
□+□+□+□+□+□+□=70\square+\square+\square+\square+\square+\square+\square=70

Show solution

Understand

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

Plan

#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.

Execute

#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.
3×13=39,5×11=553 \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.
#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÷3=10  ⇒  9+10+11=3030 \div 3 = 10 \;\Rightarrow\; 9+10+11 = 30
Knowing the middle, just step out 1 on each side.
#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÷7=10  ⇒  7+8+9+10+11+12+13=7070 \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

Review

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.

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.

Standards · min 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
💡 This only needs Grade 4 pattern sense: divide the sum by how many numbers to find the middle, then count outward!
Variant 4 answer: 8+9+10+11+12 = 50 and 20+21+22+23+24+25+26 = 161

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=154+5+6=15
6+7+8+9+10=406+7+8+9+10=40

□+□+□+□+□=50\square+\square+\square+\square+\square=50
□+□+□+□+□+□+□=161\square+\square+\square+\square+\square+\square+\square=161

Show solution

Understand

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

Plan

#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.

Execute

#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.
3×5=15,5×8=403 \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.
#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÷5=10  ⇒  8+9+10+11+12=5050 \div 5 = 10 \;\Rightarrow\; 8+9+10+11+12 = 50
Knowing the middle, just step out 2 on each side.
#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÷7=23  ⇒  20+21+22+23+24+25+26=161161 \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

Review

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.

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.

Standards · min 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
💡 This only needs Grade 4 pattern sense: divide the sum by how many numbers to find the middle, then count outward!
Variant 5 answer: 6+7+8+9+10+11+12+13+14 = 90 and 9+10+11+12+13 = 55

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=123+4+5=12
8+9+10+11+12=508+9+10+11+12=50

□+□+□+□+□+□+□+□+□=90\square+\square+\square+\square+\square+\square+\square+\square+\square=90
□+□+□+□+□=55\square+\square+\square+\square+\square=55

Show solution

Understand

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

Plan

#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.

Execute

#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.
3×4=12,5×10=503 \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.
#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÷9=10  ⇒  6+7+8+9+10+11+12+13+14=9090 \div 9 = 10 \;\Rightarrow\; 6+7+8+9+10+11+12+13+14 = 90
Knowing the middle, just step out 4 on each side.
#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÷5=11  ⇒  9+10+11+12+13=5555 \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

Review

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.

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.

Standards · min 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
💡 This only needs Grade 4 pattern sense: divide the sum by how many numbers to find the middle, then count outward!
Variant 6 answer: 19+20+21+22+23+24+25 = 154 and 16+17+18+19+20 = 90

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=6320+21+22=63
16+17+18+19+20=9016+17+18+19+20=90

□+□+□+□+□+□+□=154\square+\square+\square+\square+\square+\square+\square=154
□+□+□+□+□=90\square+\square+\square+\square+\square=90

Show solution

Understand

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

Plan

#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.

Execute

#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.
3×21=63,5×18=903 \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.
#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÷7=22  ⇒  19+20+21+22+23+24+25=154154 \div 7 = 22 \;\Rightarrow\; 19+20+21+22+23+24+25 = 154
Knowing the middle, just step out 3 on each side.
#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÷5=18  ⇒  16+17+18+19+20=9090 \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

Review

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.

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.

Standards · min 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
💡 This only needs Grade 4 pattern sense: divide the sum by how many numbers to find the middle, then count outward!
Variant 7 answer: 12+13+14+15+16+17+18 = 105 and 13+14+15+16+17+18+19+20+21 = 153

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=3611+12+13=36
14+15+16+17+18=8014+15+16+17+18=80

□+□+□+□+□+□+□=105\square+\square+\square+\square+\square+\square+\square=105
□+□+□+□+□+□+□+□+□=153\square+\square+\square+\square+\square+\square+\square+\square+\square=153

Show solution

Understand

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

Plan

#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.

Execute

#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.
3×12=36,5×16=803 \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.
#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÷7=15  ⇒  12+13+14+15+16+17+18=105105 \div 7 = 15 \;\Rightarrow\; 12+13+14+15+16+17+18 = 105
Knowing the middle, just step out 3 on each side.
#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÷9=17  ⇒  13+14+15+16+17+18+19+20+21=153153 \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

Review

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.

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.

Standards · min 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
💡 This only needs Grade 4 pattern sense: divide the sum by how many numbers to find the middle, then count outward!
Variant 8 answer: 18+19+20+21+22 = 100 and 9+10+11+12+13+14+15 = 84

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=247+8+9=24
5+6+7+8+9=355+6+7+8+9=35

□+□+□+□+□=100\square+\square+\square+\square+\square=100
□+□+□+□+□+□+□=84\square+\square+\square+\square+\square+\square+\square=84

Show solution

Understand

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

Plan

#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.

Execute

#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.
3×8=24,5×7=353 \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.
#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÷5=20  ⇒  18+19+20+21+22=100100 \div 5 = 20 \;\Rightarrow\; 18+19+20+21+22 = 100
Knowing the middle, just step out 2 on each side.
#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÷7=12  ⇒  9+10+11+12+13+14+15=8484 \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

Review

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.

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.

Standards · min 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
💡 This only needs Grade 4 pattern sense: divide the sum by how many numbers to find the middle, then count outward!
Variant 9 answer: 6+7+8+9+10+11+12 = 63 and 10+11+12+13+14 = 60

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=309+10+11=30
7+8+9+10+11=457+8+9+10+11=45

□+□+□+□+□+□+□=63\square+\square+\square+\square+\square+\square+\square=63
□+□+□+□+□=60\square+\square+\square+\square+\square=60

Show solution

Understand

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

Plan

#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.

Execute

#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.
3×10=30,5×9=453 \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.
#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÷7=9  ⇒  6+7+8+9+10+11+12=6363 \div 7 = 9 \;\Rightarrow\; 6+7+8+9+10+11+12 = 63
Knowing the middle, just step out 3 on each side.
#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÷5=12  ⇒  10+11+12+13+14=6060 \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

Review

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.

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.

Standards · min 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
💡 This only needs Grade 4 pattern sense: divide the sum by how many numbers to find the middle, then count outward!
Variant 10 answer: 11+12+13+14+15 = 65 and 11+12+13 = 36

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=216+7+8=21
13+14+15+16+17=7513+14+15+16+17=75

□+□+□+□+□=65\square+\square+\square+\square+\square=65
□+□+□=36\square+\square+\square=36

Show solution

Understand

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

Plan

#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.

Execute

#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.
3×7=21,5×15=753 \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.
#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÷5=13  ⇒  11+12+13+14+15=6565 \div 5 = 13 \;\Rightarrow\; 11+12+13+14+15 = 65
Knowing the middle, just step out 2 on each side.
#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÷3=12  ⇒  11+12+13=3636 \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

Review

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.

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.

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