← Sum consecutive numbers via the middle · Sum of Evenly Spaced Numbers via the Middle

Sum consecutive numbers via the middle · 11 practice problems

3.OA.D.93.NBT.A.22.NBT.A.2

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

Variant 1 easy

Find the sum of the two consecutive whole numbers below.

$12 + 13$

Your answer

Show solution

1 · Understandwhat's really being asked

We must add two numbers that increase by 1 each time: 12, 13, and report their total.

Givens

The two consecutive whole numbers are 12, 13
Each number is exactly 1 more than the one before it

Unknowns

The sum of all two numbers

Constraints

The numbers are consecutive (step of 1)
There are exactly two of them

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #7 Identify Subproblems

Consecutive numbers have a balancing pattern: pairing the smallest with the largest gives equal partial sums. Spotting this turns many additions into a simple multiplication.

3 · Execute3 carry out the plan

1Pair from the outside in

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

Pair the first with the last, the second with the second-to-last, and so on. Each pair adds to the same total.

12+13 = 25

When numbers go up by 1, what you add to one end you take from the other, so the outside-in pairs stay equal.

2Count the equal pairs

#7 Identify Subproblems 3.OA.D.9

The 2 numbers split into 1 pairs, and every pair equals 25. So the total is 1 copies of 25.

25 \times 1

Turning repeated equal sums into a multiplication is faster and less error-prone than adding one by one.

3Add it all up

#7 Identify Subproblems 3.NBT.A.2

Finish the arithmetic to get the final total.

25 \times 1 = 25

A small multiplication finishes the job that many separate additions would have done.

Answer: 25

4 · Reviewdoes it hold up?

The numbers cluster around their middle value 12.5, so the sum should be near 2 x 12.5 = 25. Our answer 25 matches exactly, so the magnitude is sensible.

Another way: Use the middle-value idea (tool 9, easier related problem): the average of the two numbers is 12.5, and multiplying it by 2 gives 25, confirming the pairing result.

Standardsmin grade 3

3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Recognizing that outside-in pairs of consecutive numbers are equal
3.NBT.A.2 Fluently add and subtract within 1000 — Adding each pair and the final total
2.NBT.A.2 Count within 1000, skip-count by 5s, 10s, and 100s — Recognizing the consecutive counting pattern of the numbers

💡Takeaway. Pair the ends of consecutive numbers - they all match, so you just multiply one pair by how many pairs there are!

Variant 2 easy

Find the sum of the five consecutive whole numbers below.

$1 + 2 + 3 + 4 + 5$

Your answer

Show solution

1 · Understandwhat's really being asked

We must add five numbers that increase by 1 each time: 1, 2, 3, 4, 5, and report their total.

Givens

The five consecutive whole numbers are 1, 2, 3, 4, 5
Each number is exactly 1 more than the one before it

Unknowns

The sum of all five numbers

Constraints

The numbers are consecutive (step of 1)
There are exactly five of them

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #7 Identify Subproblems

Consecutive numbers have a balancing pattern: pairing the smallest with the largest gives equal partial sums. Spotting this turns many additions into a simple multiplication.

3 · Execute3 carry out the plan

1Pair from the outside in

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

Pair the smallest with the largest, working inward. Each pair adds to the same total, and the middle number 3 is left over.

1+5 = 6,\ 2+4 = 6

When numbers go up by 1, what you add to one end you take from the other, so the outside-in pairs stay equal.

2Count the equal pairs

#7 Identify Subproblems 3.OA.D.9

The 5 numbers split into 2 equal pairs of 6, plus the leftover middle 3.

6 \times 2 + 3

Turning repeated equal sums into a multiplication is faster and less error-prone than adding one by one.

3Add it all up

#7 Identify Subproblems 3.NBT.A.2

Finish the arithmetic to get the final total.

6 \times 2 + 3 = 12 + 3 = 15

A small multiplication finishes the job that many separate additions would have done.

Answer: 15

4 · Reviewdoes it hold up?

The numbers cluster around their middle value 3, so the sum should be near 5 x 3 = 15. Our answer 15 matches exactly, so the magnitude is sensible.

Another way: Use the middle-value idea (tool 9, easier related problem): the average of the five numbers is 3, and multiplying it by 5 gives 15, confirming the pairing result.

Standardsmin grade 3

3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Recognizing that outside-in pairs of consecutive numbers are equal
3.NBT.A.2 Fluently add and subtract within 1000 — Adding each pair and the final total
2.NBT.A.2 Count within 1000, skip-count by 5s, 10s, and 100s — Recognizing the consecutive counting pattern of the numbers

💡Takeaway. Pair the ends of consecutive numbers - they all match, so you just multiply one pair by how many pairs there are!

Variant 3 easy

Find the sum of the four consecutive whole numbers below.

$10 + 11 + 12 + 13$

Your answer

Show solution

1 · Understandwhat's really being asked

We must add four numbers that increase by 1 each time: 10, 11, 12, 13, and report their total.

Givens

The four consecutive whole numbers are 10, 11, 12, 13
Each number is exactly 1 more than the one before it

Unknowns

The sum of all four numbers

Constraints

The numbers are consecutive (step of 1)
There are exactly four of them

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #7 Identify Subproblems

Consecutive numbers have a balancing pattern: pairing the smallest with the largest gives equal partial sums. Spotting this turns many additions into a simple multiplication.

3 · Execute3 carry out the plan

1Pair from the outside in

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

Pair the first with the last, the second with the second-to-last, and so on. Each pair adds to the same total.

10+13 = 23,\ 11+12 = 23

When numbers go up by 1, what you add to one end you take from the other, so the outside-in pairs stay equal.

2Count the equal pairs

#7 Identify Subproblems 3.OA.D.9

The 4 numbers split into 2 pairs, and every pair equals 23. So the total is 2 copies of 23.

23 \times 2

Turning repeated equal sums into a multiplication is faster and less error-prone than adding one by one.

3Add it all up

#7 Identify Subproblems 3.NBT.A.2

Finish the arithmetic to get the final total.

23 \times 2 = 46

A small multiplication finishes the job that many separate additions would have done.

Answer: 46

4 · Reviewdoes it hold up?

The numbers cluster around their middle value 11.5, so the sum should be near 4 x 11.5 = 46. Our answer 46 matches exactly, so the magnitude is sensible.

Another way: Use the middle-value idea (tool 9, easier related problem): the average of the four numbers is 11.5, and multiplying it by 4 gives 46, confirming the pairing result.

Standardsmin grade 3

3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Recognizing that outside-in pairs of consecutive numbers are equal
3.NBT.A.2 Fluently add and subtract within 1000 — Adding each pair and the final total
2.NBT.A.2 Count within 1000, skip-count by 5s, 10s, and 100s — Recognizing the consecutive counting pattern of the numbers

💡Takeaway. Pair the ends of consecutive numbers - they all match, so you just multiply one pair by how many pairs there are!

Variant 4 medium

Find the sum of the three consecutive whole numbers below.

$33 + 34 + 35$

Your answer

Show solution

1 · Understandwhat's really being asked

We must add three numbers that increase by 1 each time: 33, 34, 35, and report their total.

Givens

The three consecutive whole numbers are 33, 34, 35
Each number is exactly 1 more than the one before it

Unknowns

The sum of all three numbers

Constraints

The numbers are consecutive (step of 1)
There are exactly three of them

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #7 Identify Subproblems

Consecutive numbers have a balancing pattern: pairing the smallest with the largest gives equal partial sums. Spotting this turns many additions into a simple multiplication.

3 · Execute3 carry out the plan

1Pair from the outside in

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

Pair the smallest with the largest, working inward. Each pair adds to the same total, and the middle number 34 is left over.

33+35 = 68

When numbers go up by 1, what you add to one end you take from the other, so the outside-in pairs stay equal.

2Count the equal pairs

#7 Identify Subproblems 3.OA.D.9

The 3 numbers split into 1 equal pairs of 68, plus the leftover middle 34.

68 \times 1 + 34

Turning repeated equal sums into a multiplication is faster and less error-prone than adding one by one.

3Add it all up

#7 Identify Subproblems 3.NBT.A.2

Finish the arithmetic to get the final total.

68 \times 1 + 34 = 68 + 34 = 102

A small multiplication finishes the job that many separate additions would have done.

Answer: 102

4 · Reviewdoes it hold up?

The numbers cluster around their middle value 34, so the sum should be near 3 x 34 = 102. Our answer 102 matches exactly, so the magnitude is sensible.

Another way: Use the middle-value idea (tool 9, easier related problem): the average of the three numbers is 34, and multiplying it by 3 gives 102, confirming the pairing result.

Standardsmin grade 3

3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Recognizing that outside-in pairs of consecutive numbers are equal
3.NBT.A.2 Fluently add and subtract within 1000 — Adding each pair and the final total
2.NBT.A.2 Count within 1000, skip-count by 5s, 10s, and 100s — Recognizing the consecutive counting pattern of the numbers

💡Takeaway. Pair the ends of consecutive numbers - they all match, so you just multiply one pair by how many pairs there are!

Variant 5 medium

Find the sum of the five consecutive whole numbers below.

$75 + 76 + 77 + 78 + 79$

Your answer

Show solution

1 · Understandwhat's really being asked

We must add five numbers that increase by 1 each time: 75, 76, 77, 78, 79, and report their total.

Givens

The five consecutive whole numbers are 75, 76, 77, 78, 79
Each number is exactly 1 more than the one before it

Unknowns

The sum of all five numbers

Constraints

The numbers are consecutive (step of 1)
There are exactly five of them

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #7 Identify Subproblems

Consecutive numbers have a balancing pattern: pairing the smallest with the largest gives equal partial sums. Spotting this turns many additions into a simple multiplication.

3 · Execute3 carry out the plan

1Pair from the outside in

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

Pair the smallest with the largest, working inward. Each pair adds to the same total, and the middle number 77 is left over.

75+79 = 154,\ 76+78 = 154

When numbers go up by 1, what you add to one end you take from the other, so the outside-in pairs stay equal.

2Count the equal pairs

#7 Identify Subproblems 3.OA.D.9

The 5 numbers split into 2 equal pairs of 154, plus the leftover middle 77.

154 \times 2 + 77

Turning repeated equal sums into a multiplication is faster and less error-prone than adding one by one.

3Add it all up

#7 Identify Subproblems 3.NBT.A.2

Finish the arithmetic to get the final total.

154 \times 2 + 77 = 308 + 77 = 385

A small multiplication finishes the job that many separate additions would have done.

Answer: 385

4 · Reviewdoes it hold up?

The numbers cluster around their middle value 77, so the sum should be near 5 x 77 = 385. Our answer 385 matches exactly, so the magnitude is sensible.

Another way: Use the middle-value idea (tool 9, easier related problem): the average of the five numbers is 77, and multiplying it by 5 gives 385, confirming the pairing result.

Standardsmin grade 3

3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Recognizing that outside-in pairs of consecutive numbers are equal
3.NBT.A.2 Fluently add and subtract within 1000 — Adding each pair and the final total
2.NBT.A.2 Count within 1000, skip-count by 5s, 10s, and 100s — Recognizing the consecutive counting pattern of the numbers

💡Takeaway. Pair the ends of consecutive numbers - they all match, so you just multiply one pair by how many pairs there are!

Variant 6 medium

Find the sum of the six consecutive whole numbers below.

$47 + 48 + 49 + 50 + 51 + 52$

Your answer

Show solution

1 · Understandwhat's really being asked

We must add six numbers that increase by 1 each time: 47, 48, 49, 50, 51, 52, and report their total.

Givens

The six consecutive whole numbers are 47, 48, 49, 50, 51, 52
Each number is exactly 1 more than the one before it

Unknowns

The sum of all six numbers

Constraints

The numbers are consecutive (step of 1)
There are exactly six of them

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #7 Identify Subproblems

Consecutive numbers have a balancing pattern: pairing the smallest with the largest gives equal partial sums. Spotting this turns many additions into a simple multiplication.

3 · Execute3 carry out the plan

1Pair from the outside in

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

Pair the first with the last, the second with the second-to-last, and so on. Each pair adds to the same total.

47+52 = 99,\ 48+51 = 99,\ 49+50 = 99

When numbers go up by 1, what you add to one end you take from the other, so the outside-in pairs stay equal.

2Count the equal pairs

#7 Identify Subproblems 3.OA.D.9

The 6 numbers split into 3 pairs, and every pair equals 99. So the total is 3 copies of 99.

99 \times 3

Turning repeated equal sums into a multiplication is faster and less error-prone than adding one by one.

3Add it all up

#7 Identify Subproblems 3.NBT.A.2

Finish the arithmetic to get the final total.

99 \times 3 = 297

A small multiplication finishes the job that many separate additions would have done.

Answer: 297

4 · Reviewdoes it hold up?

The numbers cluster around their middle value 49.5, so the sum should be near 6 x 49.5 = 297. Our answer 297 matches exactly, so the magnitude is sensible.

Another way: Use the middle-value idea (tool 9, easier related problem): the average of the six numbers is 49.5, and multiplying it by 6 gives 297, confirming the pairing result.

Standardsmin grade 3

3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Recognizing that outside-in pairs of consecutive numbers are equal
3.NBT.A.2 Fluently add and subtract within 1000 — Adding each pair and the final total
2.NBT.A.2 Count within 1000, skip-count by 5s, 10s, and 100s — Recognizing the consecutive counting pattern of the numbers

💡Takeaway. Pair the ends of consecutive numbers - they all match, so you just multiply one pair by how many pairs there are!

Variant 7 medium

Find the sum of the seven consecutive whole numbers below.

$100 + 101 + 102 + 103 + 104 + 105 + 106$

Your answer

Show solution

1 · Understandwhat's really being asked

We must add seven numbers that increase by 1 each time: 100, 101, 102, 103, 104, 105, 106, and report their total.

Givens

The seven consecutive whole numbers are 100, 101, 102, 103, 104, 105, 106
Each number is exactly 1 more than the one before it

Unknowns

The sum of all seven numbers

Constraints

The numbers are consecutive (step of 1)
There are exactly seven of them

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #7 Identify Subproblems

Consecutive numbers have a balancing pattern: pairing the smallest with the largest gives equal partial sums. Spotting this turns many additions into a simple multiplication.

3 · Execute3 carry out the plan

1Pair from the outside in

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

Pair the smallest with the largest, working inward. Each pair adds to the same total, and the middle number 103 is left over.

100+106 = 206,\ 101+105 = 206,\ 102+104 = 206

When numbers go up by 1, what you add to one end you take from the other, so the outside-in pairs stay equal.

2Count the equal pairs

#7 Identify Subproblems 3.OA.D.9

The 7 numbers split into 3 equal pairs of 206, plus the leftover middle 103.

206 \times 3 + 103

Turning repeated equal sums into a multiplication is faster and less error-prone than adding one by one.

3Add it all up

#7 Identify Subproblems 3.NBT.A.2

Finish the arithmetic to get the final total.

206 \times 3 + 103 = 618 + 103 = 721

A small multiplication finishes the job that many separate additions would have done.

Answer: 721

4 · Reviewdoes it hold up?

The numbers cluster around their middle value 103, so the sum should be near 7 x 103 = 721. Our answer 721 matches exactly, so the magnitude is sensible.

Another way: Use the middle-value idea (tool 9, easier related problem): the average of the seven numbers is 103, and multiplying it by 7 gives 721, confirming the pairing result.

Standardsmin grade 3

3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Recognizing that outside-in pairs of consecutive numbers are equal
3.NBT.A.2 Fluently add and subtract within 1000 — Adding each pair and the final total
2.NBT.A.2 Count within 1000, skip-count by 5s, 10s, and 100s — Recognizing the consecutive counting pattern of the numbers

💡Takeaway. Pair the ends of consecutive numbers - they all match, so you just multiply one pair by how many pairs there are!

Variant 8 medium

Find the sum of the six consecutive whole numbers below.

$251 + 252 + 253 + 254 + 255 + 256$

Your answer

Show solution

1 · Understandwhat's really being asked

We must add six numbers that increase by 1 each time: 251, 252, 253, 254, 255, 256, and report their total.

Givens

The six consecutive whole numbers are 251, 252, 253, 254, 255, 256
Each number is exactly 1 more than the one before it

Unknowns

The sum of all six numbers

Constraints

The numbers are consecutive (step of 1)
There are exactly six of them

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #7 Identify Subproblems

Consecutive numbers have a balancing pattern: pairing the smallest with the largest gives equal partial sums. Spotting this turns many additions into a simple multiplication.

3 · Execute3 carry out the plan

1Pair from the outside in

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

Pair the first with the last, the second with the second-to-last, and so on. Each pair adds to the same total.

251+256 = 507,\ 252+255 = 507,\ 253+254 = 507

When numbers go up by 1, what you add to one end you take from the other, so the outside-in pairs stay equal.

2Count the equal pairs

#7 Identify Subproblems 3.OA.D.9

The 6 numbers split into 3 pairs, and every pair equals 507. So the total is 3 copies of 507.

507 \times 3

Turning repeated equal sums into a multiplication is faster and less error-prone than adding one by one.

3Add it all up

#7 Identify Subproblems 3.NBT.A.2

Finish the arithmetic to get the final total.

507 \times 3 = 1521

A small multiplication finishes the job that many separate additions would have done.

Answer: 1521

4 · Reviewdoes it hold up?

The numbers cluster around their middle value 253.5, so the sum should be near 6 x 253.5 = 1521. Our answer 1521 matches exactly, so the magnitude is sensible.

Another way: Use the middle-value idea (tool 9, easier related problem): the average of the six numbers is 253.5, and multiplying it by 6 gives 1521, confirming the pairing result.

Standardsmin grade 3

3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Recognizing that outside-in pairs of consecutive numbers are equal
3.NBT.A.2 Fluently add and subtract within 1000 — Adding each pair and the final total
2.NBT.A.2 Count within 1000, skip-count by 5s, 10s, and 100s — Recognizing the consecutive counting pattern of the numbers

💡Takeaway. Pair the ends of consecutive numbers - they all match, so you just multiply one pair by how many pairs there are!

Variant 9 hard

Find the sum of the eight consecutive whole numbers below.

$200 + 201 + 202 + 203 + 204 + 205 + 206 + 207$

Your answer

Show solution

1 · Understandwhat's really being asked

We must add eight numbers that increase by 1 each time: 200, 201, 202, 203, 204, 205, 206, 207, and report their total.

Givens

The eight consecutive whole numbers are 200, 201, 202, 203, 204, 205, 206, 207
Each number is exactly 1 more than the one before it

Unknowns

The sum of all eight numbers

Constraints

The numbers are consecutive (step of 1)
There are exactly eight of them

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #7 Identify Subproblems

Consecutive numbers have a balancing pattern: pairing the smallest with the largest gives equal partial sums. Spotting this turns many additions into a simple multiplication.

3 · Execute3 carry out the plan

1Pair from the outside in

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

Pair the first with the last, the second with the second-to-last, and so on. Each pair adds to the same total.

200+207 = 407,\ 201+206 = 407,\ 202+205 = 407,\ 203+204 = 407

When numbers go up by 1, what you add to one end you take from the other, so the outside-in pairs stay equal.

2Count the equal pairs

#7 Identify Subproblems 3.OA.D.9

The 8 numbers split into 4 pairs, and every pair equals 407. So the total is 4 copies of 407.

407 \times 4

Turning repeated equal sums into a multiplication is faster and less error-prone than adding one by one.

3Add it all up

#7 Identify Subproblems 3.NBT.A.2

Finish the arithmetic to get the final total.

407 \times 4 = 1628

A small multiplication finishes the job that many separate additions would have done.

Answer: 1628

4 · Reviewdoes it hold up?

The numbers cluster around their middle value 203.5, so the sum should be near 8 x 203.5 = 1628. Our answer 1628 matches exactly, so the magnitude is sensible.

Another way: Use the middle-value idea (tool 9, easier related problem): the average of the eight numbers is 203.5, and multiplying it by 8 gives 1628, confirming the pairing result.

Standardsmin grade 3

3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Recognizing that outside-in pairs of consecutive numbers are equal
3.NBT.A.2 Fluently add and subtract within 1000 — Adding each pair and the final total
2.NBT.A.2 Count within 1000, skip-count by 5s, 10s, and 100s — Recognizing the consecutive counting pattern of the numbers

💡Takeaway. Pair the ends of consecutive numbers - they all match, so you just multiply one pair by how many pairs there are!

Variant 10 hard

Find the sum of the 10 consecutive whole numbers below.

$101 + 102 + 103 + 104 + 105 + 106 + 107 + 108 + 109 + 110$

Your answer

Show solution

1 · Understandwhat's really being asked

We must add 10 numbers that increase by 1 each time: 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, and report their total.

Givens

The 10 consecutive whole numbers are 101, 102, 103, 104, 105, 106, 107, 108, 109, 110
Each number is exactly 1 more than the one before it

Unknowns

The sum of all 10 numbers

Constraints

The numbers are consecutive (step of 1)
There are exactly 10 of them

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #7 Identify Subproblems

Consecutive numbers have a balancing pattern: pairing the smallest with the largest gives equal partial sums. Spotting this turns many additions into a simple multiplication.

3 · Execute3 carry out the plan

1Pair from the outside in

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

Pair the first with the last, the second with the second-to-last, and so on. Each pair adds to the same total.

101+110 = 211,\ 102+109 = 211,\ 103+108 = 211,\ 104+107 = 211,\ 105+106 = 211

When numbers go up by 1, what you add to one end you take from the other, so the outside-in pairs stay equal.

2Count the equal pairs

#7 Identify Subproblems 3.OA.D.9

The 10 numbers split into 5 pairs, and every pair equals 211. So the total is 5 copies of 211.

211 \times 5

Turning repeated equal sums into a multiplication is faster and less error-prone than adding one by one.

3Add it all up

#7 Identify Subproblems 3.NBT.A.2

Finish the arithmetic to get the final total.

211 \times 5 = 1055

A small multiplication finishes the job that many separate additions would have done.

Answer: 1055

4 · Reviewdoes it hold up?

The numbers cluster around their middle value 105.5, so the sum should be near 10 x 105.5 = 1055. Our answer 1055 matches exactly, so the magnitude is sensible.

Another way: Use the middle-value idea (tool 9, easier related problem): the average of the 10 numbers is 105.5, and multiplying it by 10 gives 1055, confirming the pairing result.

Standardsmin grade 3

3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Recognizing that outside-in pairs of consecutive numbers are equal
3.NBT.A.2 Fluently add and subtract within 1000 — Adding each pair and the final total
2.NBT.A.2 Count within 1000, skip-count by 5s, 10s, and 100s — Recognizing the consecutive counting pattern of the numbers

💡Takeaway. Pair the ends of consecutive numbers - they all match, so you just multiply one pair by how many pairs there are!

Variant 11 hard

Find the sum of the nine consecutive whole numbers below.

$500 + 501 + 502 + 503 + 504 + 505 + 506 + 507 + 508$

Your answer

Show solution

1 · Understandwhat's really being asked

We must add nine numbers that increase by 1 each time: 500, 501, 502, 503, 504, 505, 506, 507, 508, and report their total.

Givens

The nine consecutive whole numbers are 500, 501, 502, 503, 504, 505, 506, 507, 508
Each number is exactly 1 more than the one before it

Unknowns

The sum of all nine numbers

Constraints

The numbers are consecutive (step of 1)
There are exactly nine of them

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #7 Identify Subproblems

Consecutive numbers have a balancing pattern: pairing the smallest with the largest gives equal partial sums. Spotting this turns many additions into a simple multiplication.

3 · Execute3 carry out the plan

1Pair from the outside in

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

Pair the smallest with the largest, working inward. Each pair adds to the same total, and the middle number 504 is left over.

500+508 = 1008,\ 501+507 = 1008,\ 502+506 = 1008,\ 503+505 = 1008

When numbers go up by 1, what you add to one end you take from the other, so the outside-in pairs stay equal.

2Count the equal pairs

#7 Identify Subproblems 3.OA.D.9

The 9 numbers split into 4 equal pairs of 1008, plus the leftover middle 504.

1008 \times 4 + 504

Turning repeated equal sums into a multiplication is faster and less error-prone than adding one by one.

3Add it all up

#7 Identify Subproblems 3.NBT.A.2

Finish the arithmetic to get the final total.

1008 \times 4 + 504 = 4032 + 504 = 4536

A small multiplication finishes the job that many separate additions would have done.

Answer: 4536

4 · Reviewdoes it hold up?

The numbers cluster around their middle value 504, so the sum should be near 9 x 504 = 4536. Our answer 4536 matches exactly, so the magnitude is sensible.

Another way: Use the middle-value idea (tool 9, easier related problem): the average of the nine numbers is 504, and multiplying it by 9 gives 4536, confirming the pairing result.

Standardsmin grade 3

3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Recognizing that outside-in pairs of consecutive numbers are equal
3.NBT.A.2 Fluently add and subtract within 1000 — Adding each pair and the final total
2.NBT.A.2 Count within 1000, skip-count by 5s, 10s, and 100s — Recognizing the consecutive counting pattern of the numbers

💡Takeaway. Pair the ends of consecutive numbers - they all match, so you just multiply one pair by how many pairs there are!

Where next?

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