Sum of evenly spaced numbers equals middle times count

3.OA.D.93.OA.C.7 · take · grade 3

Archetype: Sum of Evenly Spaced Numbers via the Middle · step in a 5-type progression

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 (five 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 five 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 five 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: 800 x 5 = 4000, 60 x 5 = 300, 4 x 5 = 20, total 4320.

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

Review

Five 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!