Problem
Sum shaded numbers in an array
A 10×10 table lists 1 to 100, ten per row, with some cells shaded. Each shaded number is added exactly once. Find the total of the shaded numbers.
Reasoning · Grade 3-1 Applications of Multiplication and Division
Your answer
Watch out If your groups overlap, a number gets added twice — check the groups are disjoint.
How to solve
Strategy Identify Subproblems — Rather than add 33 numbers one at a time, I will split the shaded cells into tidy groups: the evenly spaced 9-column band, the run 11 to 17 in row 2, and the leftover cells gathered into easy chunks. Each group has a quick total, and adding the group totals gives the answer.
1STEP 1
Add the slanted 9-column band
The stripe ending in 9 centres on 49: 49 × 9 = 441.
9+19+29+39+49+59+69+79+89 = 49 × 9 = 441
Evenly spaced numbers balance around their middle, so I can multiply the middle by how many there are instead of adding nine times.
3.OA.D.9Look For A PatternThe slanted band of nine evenly spaced numbers totals its middle number nine times over.
Why?
The numbers sit balanced around the middle one, so everything above it gives away exactly what everything below it needs.
🧱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?
Nine numbers all standing at the same level is nine equal groups of that level, which is what multiplying by nine counts.
🧱Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.
2STEP 2
Add the row-2 run 11 to 17
Row 2's 11 to 17 centres on 14: 14 × 7 = 98.
11+12+13+14+15+16+17 = 14 × 7 = 98
A short run of consecutive numbers also balances around its middle, turning seven adds into one easy multiplication.
3.OA.D.9Look For A Pattern3STEP 3
Add the leftover cells in chunks
The four triples and the singles come to 1172.
156 + 228 + 258 + 288 + 242 = 1172
Listing the leftovers in small groups keeps every number counted once and the addition manageable.
3.NBT.A.2Introduce A Variable4STEP 4
Add the three group totals
The three parts together make 1711.
441 + 98 + 1172 = 1711
Three friendly subtotals add up far more easily than 33 separate numbers.
3.NBT.A.2Identify SubproblemsAnswer
1711
There are 33 shaded cells. A rough average value is about 50, and 33 x 50 = 1650, close to 1711, so the size is sensible. Re-adding the subtotals a second way, 441 + 1172 = 1613 and 1613 + 98 = 1711, confirms the result.
Takeaway
Group the shaded numbers into evenly spaced bands and neighbor chunks, then multiply and add the small totals instead of adding all 33 one by one!
- Add the slanted 9-column band
- Add the row-2 run 11 to 17
- Add the leftover cells in chunks
- Add the three group totals