Sum consecutive numbers via the middle · 11 practice problems

Understand

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

Plan

#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.

Execute

Review

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.

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.

Standards · min 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

Understand

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

Plan

#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.

Execute

Review

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.

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.

Standards · min 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

Understand

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

Plan

#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.

Execute

Review

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.

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.

Standards · min 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

Understand

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.

Plan

#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.

Execute

Review

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.

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.

Standards · min 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

Understand

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

Plan

#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.

Execute

Review

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.

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.

Standards · min 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

Understand

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

Plan

#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.

Execute

Review

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.

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.

Standards · min 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

Understand

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

Plan

#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.

Execute

Review

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.

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.

Standards · min 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

Understand

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

Plan

#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.

Execute

Review

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.

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.

Standards · min 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

Understand

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

Plan

#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.

Execute

Review

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.

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.

Standards · min 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

Understand

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

Plan

#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.

Execute

Review

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.

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.

Standards · min 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

Understand

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

Plan

#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.

Execute

Review

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.

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.

Standards · min 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