← 3-2 · Sum of evenly spaced numbers equals middle times count · Sum of Evenly Spaced Numbers via the Middle

Sum of evenly spaced numbers equals middle times count · 11 practice problems

3.OA.D.93.OA.A.43.OA.C.7

Generated variants — 11

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

Variant 1 answer: square = 100, triangle = 5, dot = 500

In the equation below, find the numbers for $\blacksquare$ , $\blacktriangle$ , and $\bullet$ . (Here $\blacktriangle$ is a single digit that is not $1$ .)

$50 + 75 + 100 + 125 + 150 = \blacksquare \times \blacktriangle = \bullet$

Show solution

Understand

Find the square, the triangle, and the dot where 50 + 75 + 100 + 125 + 150 = (square) x (triangle) = (dot), with the triangle being a single digit that is not 1.

Givens

The sum is 50 + 75 + 100 + 125 + 150 (5 numbers spaced 25 apart).
The sum is written as (square) x (triangle), which equals (dot).
Triangle is a single digit and is not 1.

Unknowns

The values of the square, the triangle, and the dot.

Constraints

There are 5 evenly spaced terms.
Triangle is a one-digit number, not 1.

Plan

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

Evenly spaced numbers balance around their middle term, so their sum is the middle term times the count. Here the count is 5 and the middle is 100, which names the square and the triangle; multiply to get the dot.

Execute

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

The 5 numbers go up by 25 each time, so they balance around the middle one, 100. The sum is the middle value times how many there are: 100 x 5.

50 + 75 + 100 + 125 + 150 = 100 \times 5

Pair the smallest with the largest (50 + 150 = 200 = 2 x 100); every pair averages the middle, so the total is middle times count.

#7 Identify Subproblems 3.OA.A.4

Since the sum equals 100 x 5, the square is 100 and the triangle is 5. The triangle, 5, is a single digit and not 1, which fits the condition.

\blacksquare = 100,\quad \blacktriangle = 5

Writing the sum as middle x count directly reveals both factors.

#7 Identify Subproblems 3.OA.C.7

Multiply 100 by 5 to get the final value (the dot).

100 \times 5 = 500

100 x 5: 500, total 500.

Answer: square = 100, triangle = 5, dot = 500

Review

5 numbers each near 100 should sum to roughly 5 x 100 = 500; adding them directly (50 + 75 + 100 + 125 + 150) also gives 500, confirming the result.

Add the terms straight across to get 500, then factor it as 100 x 5 to read off the square and the triangle.

Standards · min grade 3

3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Seeing that evenly spaced numbers sum to the middle term times the count.
3.OA.A.4 Determine unknown whole number in multiplication or division equation — Identifying the square and triangle from the middle-times-count form.
3.OA.C.7 Fluently multiply and divide within 100 — Computing 100 x 5 to get the final value.

💡 A line of evenly spaced numbers adds up to the middle one times how many -- a slick Grade 3 pattern!

Variant 2 answer: square = 24, triangle = 7, dot = 168

In the equation below, find the numbers for $\blacksquare$ , $\blacktriangle$ , and $\bullet$ . (Here $\blacktriangle$ is a single digit that is not $1$ .)

$15 + 18 + 21 + 24 + 27 + 30 + 33 = \blacksquare \times \blacktriangle = \bullet$

Show solution

Understand

Find the square, the triangle, and the dot where 15 + 18 + 21 + 24 + 27 + 30 + 33 = (square) x (triangle) = (dot), with the triangle being a single digit that is not 1.

Givens

The sum is 15 + 18 + 21 + 24 + 27 + 30 + 33 (7 numbers spaced 3 apart).
The sum is written as (square) x (triangle), which equals (dot).
Triangle is a single digit and is not 1.

Unknowns

The values of the square, the triangle, and the dot.

Constraints

There are 7 evenly spaced terms.
Triangle is a one-digit number, not 1.

Plan

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

Evenly spaced numbers balance around their middle term, so their sum is the middle term times the count. Here the count is 7 and the middle is 24, which names the square and the triangle; multiply to get the dot.

Execute

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

The 7 numbers go up by 3 each time, so they balance around the middle one, 24. The sum is the middle value times how many there are: 24 x 7.

15 + 18 + 21 + 24 + 27 + 30 + 33 = 24 \times 7

Pair the smallest with the largest (15 + 33 = 48 = 2 x 24); every pair averages the middle, so the total is middle times count.

#7 Identify Subproblems 3.OA.A.4

Since the sum equals 24 x 7, the square is 24 and the triangle is 7. The triangle, 7, is a single digit and not 1, which fits the condition.

\blacksquare = 24,\quad \blacktriangle = 7

Writing the sum as middle x count directly reveals both factors.

#7 Identify Subproblems 3.OA.C.7

Multiply 24 by 7 to get the final value (the dot).

24 \times 7 = 168

24 x 7: 140 + 28, total 168.

Answer: square = 24, triangle = 7, dot = 168

Review

7 numbers each near 24 should sum to roughly 7 x 24 = 168; adding them directly (15 + 18 + 21 + 24 + 27 + 30 + 33) also gives 168, confirming the result.

Add the terms straight across to get 168, then factor it as 24 x 7 to read off the square and the triangle.

Standards · min grade 3

3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Seeing that evenly spaced numbers sum to the middle term times the count.
3.OA.A.4 Determine unknown whole number in multiplication or division equation — Identifying the square and triangle from the middle-times-count form.
3.OA.C.7 Fluently multiply and divide within 100 — Computing 24 x 7 to get the final value.

💡 A line of evenly spaced numbers adds up to the middle one times how many -- a slick Grade 3 pattern!

Variant 3 answer: square = 25, triangle = 3, dot = 75

In the equation below, find the numbers for $\blacksquare$ , $\blacktriangle$ , and $\bullet$ . (Here $\blacktriangle$ is a single digit that is not $1$ .)

$20 + 25 + 30 = \blacksquare \times \blacktriangle = \bullet$

Show solution

Understand

Find the square, the triangle, and the dot where 20 + 25 + 30 = (square) x (triangle) = (dot), with the triangle being a single digit that is not 1.

Givens

The sum is 20 + 25 + 30 (3 numbers spaced 5 apart).
The sum is written as (square) x (triangle), which equals (dot).
Triangle is a single digit and is not 1.

Unknowns

The values of the square, the triangle, and the dot.

Constraints

There are 3 evenly spaced terms.
Triangle is a one-digit number, not 1.

Plan

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

Evenly spaced numbers balance around their middle term, so their sum is the middle term times the count. Here the count is 3 and the middle is 25, which names the square and the triangle; multiply to get the dot.

Execute

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

The 3 numbers go up by 5 each time, so they balance around the middle one, 25. The sum is the middle value times how many there are: 25 x 3.

20 + 25 + 30 = 25 \times 3

Pair the smallest with the largest (20 + 30 = 50 = 2 x 25); every pair averages the middle, so the total is middle times count.

#7 Identify Subproblems 3.OA.A.4

Since the sum equals 25 x 3, the square is 25 and the triangle is 3. The triangle, 3, is a single digit and not 1, which fits the condition.

\blacksquare = 25,\quad \blacktriangle = 3

Writing the sum as middle x count directly reveals both factors.

#7 Identify Subproblems 3.OA.C.7

Multiply 25 by 3 to get the final value (the dot).

25 \times 3 = 75

25 x 3: 60 + 15, total 75.

Answer: square = 25, triangle = 3, dot = 75

Review

3 numbers each near 25 should sum to roughly 3 x 25 = 75; adding them directly (20 + 25 + 30) also gives 75, confirming the result.

Add the terms straight across to get 75, then factor it as 25 x 3 to read off the square and the triangle.

Standards · min grade 3

3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Seeing that evenly spaced numbers sum to the middle term times the count.
3.OA.A.4 Determine unknown whole number in multiplication or division equation — Identifying the square and triangle from the middle-times-count form.
3.OA.C.7 Fluently multiply and divide within 100 — Computing 25 x 3 to get the final value.

💡 A line of evenly spaced numbers adds up to the middle one times how many -- a slick Grade 3 pattern!

Variant 4 answer: square = 210, triangle = 3, dot = 630

In the equation below, find the numbers for $\blacksquare$ , $\blacktriangle$ , and $\bullet$ . (Here $\blacktriangle$ is a single digit that is not $1$ .)

$200 + 210 + 220 = \blacksquare \times \blacktriangle = \bullet$

Show solution

Understand

Find the square, the triangle, and the dot where 200 + 210 + 220 = (square) x (triangle) = (dot), with the triangle being a single digit that is not 1.

Givens

The sum is 200 + 210 + 220 (3 numbers spaced 10 apart).
The sum is written as (square) x (triangle), which equals (dot).
Triangle is a single digit and is not 1.

Unknowns

The values of the square, the triangle, and the dot.

Constraints

There are 3 evenly spaced terms.
Triangle is a one-digit number, not 1.

Plan

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

Evenly spaced numbers balance around their middle term, so their sum is the middle term times the count. Here the count is 3 and the middle is 210, which names the square and the triangle; multiply to get the dot.

Execute

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

The 3 numbers go up by 10 each time, so they balance around the middle one, 210. The sum is the middle value times how many there are: 210 x 3.

200 + 210 + 220 = 210 \times 3

Pair the smallest with the largest (200 + 220 = 420 = 2 x 210); every pair averages the middle, so the total is middle times count.

#7 Identify Subproblems 3.OA.A.4

Since the sum equals 210 x 3, the square is 210 and the triangle is 3. The triangle, 3, is a single digit and not 1, which fits the condition.

\blacksquare = 210,\quad \blacktriangle = 3

Writing the sum as middle x count directly reveals both factors.

#7 Identify Subproblems 3.OA.C.7

Multiply 210 by 3 to get the final value (the dot).

210 \times 3 = 630

210 x 3: 600 + 30, total 630.

Answer: square = 210, triangle = 3, dot = 630

Review

3 numbers each near 210 should sum to roughly 3 x 210 = 630; adding them directly (200 + 210 + 220) also gives 630, confirming the result.

Add the terms straight across to get 630, then factor it as 210 x 3 to read off the square and the triangle.

Standards · min grade 3

3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Seeing that evenly spaced numbers sum to the middle term times the count.
3.OA.A.4 Determine unknown whole number in multiplication or division equation — Identifying the square and triangle from the middle-times-count form.
3.OA.C.7 Fluently multiply and divide within 100 — Computing 210 x 3 to get the final value.

💡 A line of evenly spaced numbers adds up to the middle one times how many -- a slick Grade 3 pattern!

Variant 5 answer: square = 864, triangle = 5, dot = 4320

In the equation below, find the numbers for $\blacksquare$ , $\blacktriangle$ , and $\bullet$ . (Here $\blacktriangle$ is a single digit that is not $1$ .)

$860 + 862 + 864 + 866 + 868 = \blacksquare \times \blacktriangle = \bullet$

Show solution

Understand

Find the square, the triangle, and the dot where 860 + 862 + 864 + 866 + 868 = (square) x (triangle) = (dot), with the triangle being a single digit that is not 1.

Givens

The sum is 860 + 862 + 864 + 866 + 868 (5 numbers spaced 2 apart).
The sum is written as (square) x (triangle), which equals (dot).
Triangle is a single digit and is not 1.

Unknowns

The values of the square, the triangle, and the dot.

Constraints

There are 5 evenly spaced terms.
Triangle is a one-digit number, not 1.

Plan

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

Evenly spaced numbers balance around their middle term, so their sum is the middle term times the count. Here the count is 5 and the middle is 864, which names the square and the triangle; multiply to get the dot.

Execute

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

The 5 numbers go up by 2 each time, so they balance around the middle one, 864. The sum is the middle value times how many there are: 864 x 5.

860 + 862 + 864 + 866 + 868 = 864 \times 5

Pair the smallest with the largest (860 + 868 = 1728 = 2 x 864); every pair averages the middle, so the total is middle times count.

#7 Identify Subproblems 3.OA.A.4

Since the sum equals 864 x 5, the square is 864 and the triangle is 5. The triangle, 5, is a single digit and not 1, which fits the condition.

\blacksquare = 864,\quad \blacktriangle = 5

Writing the sum as middle x count directly reveals both factors.

#7 Identify Subproblems 3.OA.C.7

Multiply 864 by 5 to get the final value (the dot).

864 \times 5 = 4320

864 x 5: 4000 + 300 + 20, total 4320.

Answer: square = 864, triangle = 5, dot = 4320

Review

5 numbers each near 864 should sum to roughly 5 x 864 = 4320; adding them directly (860 + 862 + 864 + 866 + 868) also gives 4320, confirming the result.

Add the terms straight across to get 4320, then factor it as 864 x 5 to read off the square and the triangle.

Standards · min grade 3

3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Seeing that evenly spaced numbers sum to the middle term times the count.
3.OA.A.4 Determine unknown whole number in multiplication or division equation — Identifying the square and triangle from the middle-times-count form.
3.OA.C.7 Fluently multiply and divide within 100 — Computing 864 x 5 to get the final value.

💡 A line of evenly spaced numbers adds up to the middle one times how many -- a slick Grade 3 pattern!

Variant 6 answer: square = 301, triangle = 3, dot = 903

In the equation below, find the numbers for $\blacksquare$ , $\blacktriangle$ , and $\bullet$ . (Here $\blacktriangle$ is a single digit that is not $1$ .)

$300 + 301 + 302 = \blacksquare \times \blacktriangle = \bullet$

Show solution

Understand

Find the square, the triangle, and the dot where 300 + 301 + 302 = (square) x (triangle) = (dot), with the triangle being a single digit that is not 1.

Givens

The sum is 300 + 301 + 302 (3 numbers spaced 1 apart).
The sum is written as (square) x (triangle), which equals (dot).
Triangle is a single digit and is not 1.

Unknowns

The values of the square, the triangle, and the dot.

Constraints

There are 3 evenly spaced terms.
Triangle is a one-digit number, not 1.

Plan

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

Evenly spaced numbers balance around their middle term, so their sum is the middle term times the count. Here the count is 3 and the middle is 301, which names the square and the triangle; multiply to get the dot.

Execute

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

The 3 numbers go up by 1 each time, so they balance around the middle one, 301. The sum is the middle value times how many there are: 301 x 3.

300 + 301 + 302 = 301 \times 3

Pair the smallest with the largest (300 + 302 = 602 = 2 x 301); every pair averages the middle, so the total is middle times count.

#7 Identify Subproblems 3.OA.A.4

Since the sum equals 301 x 3, the square is 301 and the triangle is 3. The triangle, 3, is a single digit and not 1, which fits the condition.

\blacksquare = 301,\quad \blacktriangle = 3

Writing the sum as middle x count directly reveals both factors.

#7 Identify Subproblems 3.OA.C.7

Multiply 301 by 3 to get the final value (the dot).

301 \times 3 = 903

301 x 3: 900 + 3, total 903.

Answer: square = 301, triangle = 3, dot = 903

Review

3 numbers each near 301 should sum to roughly 3 x 301 = 903; adding them directly (300 + 301 + 302) also gives 903, confirming the result.

Add the terms straight across to get 903, then factor it as 301 x 3 to read off the square and the triangle.

Standards · min grade 3

3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Seeing that evenly spaced numbers sum to the middle term times the count.
3.OA.A.4 Determine unknown whole number in multiplication or division equation — Identifying the square and triangle from the middle-times-count form.
3.OA.C.7 Fluently multiply and divide within 100 — Computing 301 x 3 to get the final value.

💡 A line of evenly spaced numbers adds up to the middle one times how many -- a slick Grade 3 pattern!

Variant 7 answer: square = 11, triangle = 3, dot = 33

In the equation below, find the numbers for $\blacksquare$ , $\blacktriangle$ , and $\bullet$ . (Here $\blacktriangle$ is a single digit that is not $1$ .)

$10 + 11 + 12 = \blacksquare \times \blacktriangle = \bullet$

Show solution

Understand

Find the square, the triangle, and the dot where 10 + 11 + 12 = (square) x (triangle) = (dot), with the triangle being a single digit that is not 1.

Givens

The sum is 10 + 11 + 12 (3 numbers spaced 1 apart).
The sum is written as (square) x (triangle), which equals (dot).
Triangle is a single digit and is not 1.

Unknowns

The values of the square, the triangle, and the dot.

Constraints

There are 3 evenly spaced terms.
Triangle is a one-digit number, not 1.

Plan

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

Evenly spaced numbers balance around their middle term, so their sum is the middle term times the count. Here the count is 3 and the middle is 11, which names the square and the triangle; multiply to get the dot.

Execute

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

The 3 numbers go up by 1 each time, so they balance around the middle one, 11. The sum is the middle value times how many there are: 11 x 3.

10 + 11 + 12 = 11 \times 3

Pair the smallest with the largest (10 + 12 = 22 = 2 x 11); every pair averages the middle, so the total is middle times count.

#7 Identify Subproblems 3.OA.A.4

Since the sum equals 11 x 3, the square is 11 and the triangle is 3. The triangle, 3, is a single digit and not 1, which fits the condition.

\blacksquare = 11,\quad \blacktriangle = 3

Writing the sum as middle x count directly reveals both factors.

#7 Identify Subproblems 3.OA.C.7

Multiply 11 by 3 to get the final value (the dot).

11 \times 3 = 33

11 x 3: 30 + 3, total 33.

Answer: square = 11, triangle = 3, dot = 33

Review

3 numbers each near 11 should sum to roughly 3 x 11 = 33; adding them directly (10 + 11 + 12) also gives 33, confirming the result.

Add the terms straight across to get 33, then factor it as 11 x 3 to read off the square and the triangle.

Standards · min grade 3

3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Seeing that evenly spaced numbers sum to the middle term times the count.
3.OA.A.4 Determine unknown whole number in multiplication or division equation — Identifying the square and triangle from the middle-times-count form.
3.OA.C.7 Fluently multiply and divide within 100 — Computing 11 x 3 to get the final value.

💡 A line of evenly spaced numbers adds up to the middle one times how many -- a slick Grade 3 pattern!

Variant 8 answer: square = 36, triangle = 9, dot = 324

In the equation below, find the numbers for $\blacksquare$ , $\blacktriangle$ , and $\bullet$ . (Here $\blacktriangle$ is a single digit that is not $1$ .)

$12 + 18 + 24 + 30 + 36 + 42 + 48 + 54 + 60 = \blacksquare \times \blacktriangle = \bullet$

Show solution

Understand

Find the square, the triangle, and the dot where 12 + 18 + 24 + 30 + 36 + 42 + 48 + 54 + 60 = (square) x (triangle) = (dot), with the triangle being a single digit that is not 1.

Givens

The sum is 12 + 18 + 24 + 30 + 36 + 42 + 48 + 54 + 60 (9 numbers spaced 6 apart).
The sum is written as (square) x (triangle), which equals (dot).
Triangle is a single digit and is not 1.

Unknowns

The values of the square, the triangle, and the dot.

Constraints

There are 9 evenly spaced terms.
Triangle is a one-digit number, not 1.

Plan

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

Evenly spaced numbers balance around their middle term, so their sum is the middle term times the count. Here the count is 9 and the middle is 36, which names the square and the triangle; multiply to get the dot.

Execute

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

The 9 numbers go up by 6 each time, so they balance around the middle one, 36. The sum is the middle value times how many there are: 36 x 9.

12 + 18 + 24 + 30 + 36 + 42 + 48 + 54 + 60 = 36 \times 9

Pair the smallest with the largest (12 + 60 = 72 = 2 x 36); every pair averages the middle, so the total is middle times count.

#7 Identify Subproblems 3.OA.A.4

Since the sum equals 36 x 9, the square is 36 and the triangle is 9. The triangle, 9, is a single digit and not 1, which fits the condition.

\blacksquare = 36,\quad \blacktriangle = 9

Writing the sum as middle x count directly reveals both factors.

#7 Identify Subproblems 3.OA.C.7

Multiply 36 by 9 to get the final value (the dot).

36 \times 9 = 324

36 x 9: 270 + 54, total 324.

Answer: square = 36, triangle = 9, dot = 324

Review

9 numbers each near 36 should sum to roughly 9 x 36 = 324; adding them directly (12 + 18 + 24 + 30 + 36 + 42 + 48 + 54 + 60) also gives 324, confirming the result.

Add the terms straight across to get 324, then factor it as 36 x 9 to read off the square and the triangle.

Standards · min grade 3

3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Seeing that evenly spaced numbers sum to the middle term times the count.
3.OA.A.4 Determine unknown whole number in multiplication or division equation — Identifying the square and triangle from the middle-times-count form.
3.OA.C.7 Fluently multiply and divide within 100 — Computing 36 x 9 to get the final value.

💡 A line of evenly spaced numbers adds up to the middle one times how many -- a slick Grade 3 pattern!

Variant 9 answer: square = 108, triangle = 5, dot = 540

In the equation below, find the numbers for $\blacksquare$ , $\blacktriangle$ , and $\bullet$ . (Here $\blacktriangle$ is a single digit that is not $1$ .)

$100 + 104 + 108 + 112 + 116 = \blacksquare \times \blacktriangle = \bullet$

Show solution

Understand

Find the square, the triangle, and the dot where 100 + 104 + 108 + 112 + 116 = (square) x (triangle) = (dot), with the triangle being a single digit that is not 1.

Givens

The sum is 100 + 104 + 108 + 112 + 116 (5 numbers spaced 4 apart).
The sum is written as (square) x (triangle), which equals (dot).
Triangle is a single digit and is not 1.

Unknowns

The values of the square, the triangle, and the dot.

Constraints

There are 5 evenly spaced terms.
Triangle is a one-digit number, not 1.

Plan

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

Evenly spaced numbers balance around their middle term, so their sum is the middle term times the count. Here the count is 5 and the middle is 108, which names the square and the triangle; multiply to get the dot.

Execute

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

The 5 numbers go up by 4 each time, so they balance around the middle one, 108. The sum is the middle value times how many there are: 108 x 5.

100 + 104 + 108 + 112 + 116 = 108 \times 5

Pair the smallest with the largest (100 + 116 = 216 = 2 x 108); every pair averages the middle, so the total is middle times count.

#7 Identify Subproblems 3.OA.A.4

Since the sum equals 108 x 5, the square is 108 and the triangle is 5. The triangle, 5, is a single digit and not 1, which fits the condition.

\blacksquare = 108,\quad \blacktriangle = 5

Writing the sum as middle x count directly reveals both factors.

#7 Identify Subproblems 3.OA.C.7

Multiply 108 by 5 to get the final value (the dot).

108 \times 5 = 540

108 x 5: 500 + 40, total 540.

Answer: square = 108, triangle = 5, dot = 540

Review

5 numbers each near 108 should sum to roughly 5 x 108 = 540; adding them directly (100 + 104 + 108 + 112 + 116) also gives 540, confirming the result.

Add the terms straight across to get 540, then factor it as 108 x 5 to read off the square and the triangle.

Standards · min grade 3

3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Seeing that evenly spaced numbers sum to the middle term times the count.
3.OA.A.4 Determine unknown whole number in multiplication or division equation — Identifying the square and triangle from the middle-times-count form.
3.OA.C.7 Fluently multiply and divide within 100 — Computing 108 x 5 to get the final value.

💡 A line of evenly spaced numbers adds up to the middle one times how many -- a slick Grade 3 pattern!

Variant 10 answer: square = 55, triangle = 5, dot = 275

In the equation below, find the numbers for $\blacksquare$ , $\blacktriangle$ , and $\bullet$ . (Here $\blacktriangle$ is a single digit that is not $1$ .)

$33 + 44 + 55 + 66 + 77 = \blacksquare \times \blacktriangle = \bullet$

Show solution

Understand

Find the square, the triangle, and the dot where 33 + 44 + 55 + 66 + 77 = (square) x (triangle) = (dot), with the triangle being a single digit that is not 1.

Givens

The sum is 33 + 44 + 55 + 66 + 77 (5 numbers spaced 11 apart).
The sum is written as (square) x (triangle), which equals (dot).
Triangle is a single digit and is not 1.

Unknowns

The values of the square, the triangle, and the dot.

Constraints

There are 5 evenly spaced terms.
Triangle is a one-digit number, not 1.

Plan

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

Evenly spaced numbers balance around their middle term, so their sum is the middle term times the count. Here the count is 5 and the middle is 55, which names the square and the triangle; multiply to get the dot.

Execute

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

The 5 numbers go up by 11 each time, so they balance around the middle one, 55. The sum is the middle value times how many there are: 55 x 5.

33 + 44 + 55 + 66 + 77 = 55 \times 5

Pair the smallest with the largest (33 + 77 = 110 = 2 x 55); every pair averages the middle, so the total is middle times count.

#7 Identify Subproblems 3.OA.A.4

Since the sum equals 55 x 5, the square is 55 and the triangle is 5. The triangle, 5, is a single digit and not 1, which fits the condition.

\blacksquare = 55,\quad \blacktriangle = 5

Writing the sum as middle x count directly reveals both factors.

#7 Identify Subproblems 3.OA.C.7

Multiply 55 by 5 to get the final value (the dot).

55 \times 5 = 275

55 x 5: 250 + 25, total 275.

Answer: square = 55, triangle = 5, dot = 275

Review

5 numbers each near 55 should sum to roughly 5 x 55 = 275; adding them directly (33 + 44 + 55 + 66 + 77) also gives 275, confirming the result.

Add the terms straight across to get 275, then factor it as 55 x 5 to read off the square and the triangle.

Standards · min grade 3

3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Seeing that evenly spaced numbers sum to the middle term times the count.
3.OA.A.4 Determine unknown whole number in multiplication or division equation — Identifying the square and triangle from the middle-times-count form.
3.OA.C.7 Fluently multiply and divide within 100 — Computing 55 x 5 to get the final value.

💡 A line of evenly spaced numbers adds up to the middle one times how many -- a slick Grade 3 pattern!

Variant 11 answer: square = 13, triangle = 7, dot = 91

In the equation below, find the numbers for $\blacksquare$ , $\blacktriangle$ , and $\bullet$ . (Here $\blacktriangle$ is a single digit that is not $1$ .)

$7 + 9 + 11 + 13 + 15 + 17 + 19 = \blacksquare \times \blacktriangle = \bullet$

Show solution

Understand

Find the square, the triangle, and the dot where 7 + 9 + 11 + 13 + 15 + 17 + 19 = (square) x (triangle) = (dot), with the triangle being a single digit that is not 1.

Givens

The sum is 7 + 9 + 11 + 13 + 15 + 17 + 19 (7 numbers spaced 2 apart).
The sum is written as (square) x (triangle), which equals (dot).
Triangle is a single digit and is not 1.

Unknowns

The values of the square, the triangle, and the dot.

Constraints

There are 7 evenly spaced terms.
Triangle is a one-digit number, not 1.

Plan

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

Evenly spaced numbers balance around their middle term, so their sum is the middle term times the count. Here the count is 7 and the middle is 13, which names the square and the triangle; multiply to get the dot.

Execute

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

The 7 numbers go up by 2 each time, so they balance around the middle one, 13. The sum is the middle value times how many there are: 13 x 7.

7 + 9 + 11 + 13 + 15 + 17 + 19 = 13 \times 7

Pair the smallest with the largest (7 + 19 = 26 = 2 x 13); every pair averages the middle, so the total is middle times count.

#7 Identify Subproblems 3.OA.A.4

Since the sum equals 13 x 7, the square is 13 and the triangle is 7. The triangle, 7, is a single digit and not 1, which fits the condition.

\blacksquare = 13,\quad \blacktriangle = 7

Writing the sum as middle x count directly reveals both factors.

#7 Identify Subproblems 3.OA.C.7

Multiply 13 by 7 to get the final value (the dot).

13 \times 7 = 91

13 x 7: 70 + 21, total 91.

Answer: square = 13, triangle = 7, dot = 91

Review

7 numbers each near 13 should sum to roughly 7 x 13 = 91; adding them directly (7 + 9 + 11 + 13 + 15 + 17 + 19) also gives 91, confirming the result.

Add the terms straight across to get 91, then factor it as 13 x 7 to read off the square and the triangle.

Standards · min grade 3

3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Seeing that evenly spaced numbers sum to the middle term times the count.
3.OA.A.4 Determine unknown whole number in multiplication or division equation — Identifying the square and triangle from the middle-times-count form.
3.OA.C.7 Fluently multiply and divide within 100 — Computing 13 x 7 to get the final value.

💡 A line of evenly spaced numbers adds up to the middle one times how many -- a slick Grade 3 pattern!