Problem
Sum of evenly spaced numbers equals middle times count
Find ■, ▲, and ● where 860 + 862 + 864 + 866 + 868 = ■ x ▲ = ●, with ▲ a single digit that is not 1.
Operations
Your answer
Watch out The laziest split is ■ = 4320, ▲ = 1 (anything times 1 equals itself) — but ▲ can't be 1, so you need the real factoring, 864 × 5.
How to solve
Strategy Look for a Pattern — 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.
1STEP 1
Sum equals middle times count
Rising by 2 each time, the five numbers balance around the middle 864 — so the sum is 864 x 5.
860 + 862 + 864 + 866 + 868 = 864 × 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.
3.OA.D.9Look For A PatternThe five evenly spaced numbers 860, 862, 864, 866, 868 add up to the middle number 864 taken 5 times, which is 864 times 5.
Why?
You may add the five numbers in any order and group them however you like, so pair the outer two, then the next two, and leave the middle 864 by itself.
Why?
Changing the order in which you add the numbers never changes the total.
🧱Commutativity of additionPutting a with b or b with a piles up the same total — order does not change a sum.
Why?
Choosing which numbers to push together first never changes the total.
🧱Associativity of additionAdding three piles, it does not matter which two you push together first.
Why?
Each outer pair adds up to exactly two 864's, because one number sits as far below 864 as its partner sits above: 860 is 4 under and 868 is 4 over, and 862 is 2 under while 866 is 2 over.
Why?
Going down by an amount and back up by the same amount returns you to where you started, so the below-part of one number and the above-part of its partner cancel and leave two clean 864's.
🧱Inverse operationsSubtraction undoes addition and division undoes multiplication — they reverse each other.
Why?
The two outer pairs give four 864's and the lone middle adds a fifth, and adding 864 five times is exactly what 864 times 5 means.
Why?
Adding the same amount a fixed number of times is what multiplying by that count stands for.
🧱Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.
2STEP 2
Match ■ and ▲
Since the sum is 864 x 5, ■ = 864 and ▲ = 5, a single digit that isn't 1.
\blacksquare = 864, \blacktriangle = 5
Writing the sum as middle x count directly reveals both factors.
3.OA.A.4Identify Subproblems3STEP 3
Compute ●
Multiply 864 by 5 to get the final value ●.
864 × 5 = 4320
864 x 5: 800 x 5 = 4000, 60 x 5 = 300, 4 x 5 = 20, total 4320.
3.OA.C.7Identify SubproblemsAnswer
■ = 864, ▲ = 5, ● = 4320
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.
Takeaway
A line of evenly spaced numbers adds up to the middle one times how many -- a slick Grade 3 pattern!
- Sum equals middle times count
- Match ■ and ▲
- Compute ●
Where next?
Another one like thissuggested
Step back and solidifysuggested