Problem
Express a number as consecutive sums
Write 35 as a sum of two or more consecutive numbers. Each run climbs by 1 with no gaps. Reordering the same numbers is not a new way. Find three different ways.
Reasoning · Grade 3-1 Applications of Multiplication and Division
Your answer
Watch out Sum ÷ count is not always a term: with an even count the middle falls between two numbers.
How to solve
Strategy Make a Systematic List — A run of consecutive numbers is set once I choose how many numbers it has, so I will list the possible lengths (2, 3, 4, 5, ...) in order and test each one. For each length there is a quick pattern: the middle number times the count equals the sum, so I can check a length almost instantly instead of adding blindly.
1STEP 1
Use the middle-number shortcut
A consecutive run sums to middle × count.
sum = (middle number) × (count)
Pairing the smallest with the largest, the next-smallest with the next-largest, and so on always gives the same pair-total, which is twice the middle — this is a Grade 3 pattern about balanced sums.
3.OA.D.9Look For A PatternA run of consecutive numbers totals its middle number multiplied by how many numbers there are.
Why?
Pairing the smallest with the largest, then the next pair inward, always gives the same total, because one rises exactly as much as the other falls.
🧱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.
Why?
The middle number is the level everything would sit at after fair sharing, so the whole run is that level repeated once per number.
🧱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.
2STEP 2
Try a run of 2 numbers
Two terms sit either side of half of 35: 17 + 18.
17 + 18 = 35
17 and 18 sit right around half of 35, so their sum should land on 35 — and it does.
3.NBT.A.2Guess And Check3STEP 3
Try a run of 5 numbers
Five terms centre on 7: 5 + 6 + 7 + 8 + 9.
5 + 6 + 7 + 8 + 9 = 35
Since 5 × 7 = 35, a run of five whose middle is 7 must total 35.
3.NBT.A.2Make A Systematic List4STEP 4
Try a run of 7 numbers
Seven terms centre on 5: 2 through 8.
2 + 3 + 4 + 5 + 6 + 7 + 8 = 35
Since 7 × 5 = 35, a run of seven whose middle is 5 must total 35.
3.NBT.A.2Make A Systematic List5STEP 5
Collect three different ways
That is all three different ways.
17+18 = 5+6+7+8+9 = 2+3+4+5+6+7+8 = 35
Lengths 2, 5, and 7 each gave a whole-number middle, so each one works; that is three distinct answers.
3.OA.D.9Make A Systematic ListAnswer
17+18, 5+6+7+8+9, 2+3+4+5+6+7+8 (three ways)
Each run re-adds to 35: 17+18=35; 5+6+7+8+9=35; 2+3+4+5+6+7+8=35. All addends are whole numbers in order with no gaps, and the three runs use different numbers, so they are three different ways.
Takeaway
A run of numbers in a row balances around its middle, so middle x how-many gives the sum — that is all Grade 3 pattern sense!
- Use the middle-number shortcut
- Try a run of 2 numbers
- Try a run of 5 numbers
- Try a run of 7 numbers
- Collect three different ways