Practice · Sum of evenly spaced numbers equals middle times count · Sensim Math

Variant 1 easy

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

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

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

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.

2 · Planchoose the strategy

#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.

3 · Execute3 carry out the plan

1Sum equals middle times count

#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.

2Match the square and the triangle

#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.

3Compute the dot

#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

4 · Reviewdoes it hold up?

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.

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

Standardsmin 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.

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

Variant 2 easy

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

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

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

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.

2 · Planchoose the strategy

#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.

3 · Execute3 carry out the plan

1Sum equals middle times count

#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.

2Match the square and the triangle

#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.

3Compute the dot

#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

4 · Reviewdoes it hold up?

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.

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

Standardsmin 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.

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

Variant 3 easy

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

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

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

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.

2 · Planchoose the strategy

#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.

3 · Execute3 carry out the plan

1Sum equals middle times count

#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.

2Match the square and the triangle

#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.

3Compute the dot

#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

4 · Reviewdoes it hold up?

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.

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

Standardsmin 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.

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

Variant 4 medium

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

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

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

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.

2 · Planchoose the strategy

#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.

3 · Execute3 carry out the plan

1Sum equals middle times count

#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.

2Match the square and the triangle

#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.

3Compute the dot

#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

4 · Reviewdoes it hold up?

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.

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

Standardsmin 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.

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

Variant 5 medium

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

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

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

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.

2 · Planchoose the strategy

#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.

3 · Execute3 carry out the plan

1Sum equals middle times count

#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.

2Match the square and the triangle

#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.

3Compute the dot

#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

4 · Reviewdoes it hold up?

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.

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

Standardsmin 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.

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

Variant 6 medium

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

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

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

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.

2 · Planchoose the strategy

#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.

3 · Execute3 carry out the plan

1Sum equals middle times count

#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.

2Match the square and the triangle

#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.

3Compute the dot

#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

4 · Reviewdoes it hold up?

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.

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

Standardsmin 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.

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

Variant 7 medium

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

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

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

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.

2 · Planchoose the strategy

#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.

3 · Execute3 carry out the plan

1Sum equals middle times count

#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.

2Match the square and the triangle

#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.

3Compute the dot

#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

4 · Reviewdoes it hold up?

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.

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

Standardsmin 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.

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

Variant 8 medium

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

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

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

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.

2 · Planchoose the strategy

#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.

3 · Execute3 carry out the plan

1Sum equals middle times count

#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.

2Match the square and the triangle

#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.

3Compute the dot

#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

4 · Reviewdoes it hold up?

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.

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

Standardsmin 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.

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

Variant 9 hard

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

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

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

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.

2 · Planchoose the strategy

#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.

3 · Execute3 carry out the plan

1Sum equals middle times count

#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.

2Match the square and the triangle

#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.

3Compute the dot

#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

4 · Reviewdoes it hold up?

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.

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

Standardsmin 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.

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

Variant 10 hard

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

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

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

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.

2 · Planchoose the strategy

#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.

3 · Execute3 carry out the plan

1Sum equals middle times count

#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.

2Match the square and the triangle

#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.

3Compute the dot

#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

4 · Reviewdoes it hold up?

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.

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

Standardsmin 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.

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

Variant 11 hard

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

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

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

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.

2 · Planchoose the strategy

#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.

3 · Execute3 carry out the plan

1Sum equals middle times count

#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.

2Match the square and the triangle

#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.

3Compute the dot

#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

4 · Reviewdoes it hold up?

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.

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

Standardsmin 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.

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

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

Generated variants — 11

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Where next?