Problem
Sum of a rectangle in a number grid
A number grid lists 11 to 80, ten per row. The shaded block is the 5 by 5 rectangle running from 24-28 along its top row down to 64-68 along its bottom row. Find the total of the 25 shaded numbers.
Reasoning · Grade 3-2 Finding Sums Using Products
Your answer
Watch out Adding all 25 is a slog — the block is symmetric, so the centre is the average and one multiplication suffices.
How to solve
Strategy Look for a Pattern — The shaded numbers are evenly spaced and laid out as a symmetric square, so they balance around their middle value 46. I find that center pattern, then break the work into easy row sums to confirm.
1STEP 1
See the square is balanced around its center
Opposite pairs like 24 and 68 all average the centre, 46.
24 + 68 = 92 = 2 × 46
Evenly spaced numbers around a center cancel out their highs and lows, so the average is just the middle number.
3.OA.D.9Look For A PatternThe shaded square is balanced around its centre cell, so the centre number is the average of all 25.
Why?
Everything above the centre gives away exactly as much as everything below it needs, so the centre is the level after fair sharing.
🧱The mean is the total shared out equallyThe average is the total divided by how many there are, so the total is the average times the count — and everything above that level gives away exactly what everything below it needs.
Why?
Each cell paired with the cell directly opposite it through the centre gives the same total, which is twice the centre.
🧱Pairing from both ends gives a constant sumIn an evenly spaced list the smallest with the largest, the next with the next, always make the same total, so a long sum becomes a multiplication.
2STEP 2
Multiply the center by how many cells
With 25 cells the total is 25 × 46 = 1150.
25 × 46 = 1150
Counting equal-size groups is multiplication: 25 groups that average 46 is 25 × 46.
3.OA.C.7Look For A Pattern3STEP 3
Confirm by adding the five rows
Row by row, 130 + 180 + … also gives 1150.
130 + 180 + 230 + 280 + 330 = 1150
Splitting a hard sum into five small row sums lets me add safely and cross-check the pattern answer.
3.NBT.A.2Identify SubproblemsAnswer
1150
25 × 46 = 1150
The smallest shaded number is 24 and the largest is 68, so the average must sit between them; 1150 ÷ 25 = 46, which is exactly the center value and lies right in that range. The magnitude (about a thousand) is sensible for adding 25 two-digit numbers.
Takeaway
When numbers spread out evenly around a middle number, the whole pile equals that middle number times how many there are — Grade 3 multiplication you already know!
- See the square is balanced around its center
- Multiply the center by how many cells
- Confirm by adding the five rows