Problem
Sum evenly spaced runs via the middle
Two long addition strings have to be worked out. The first climbs from 12 to 30 in steps of 2. The second is five blocks of three consecutive numbers. Find the value of each.
Reasoning · Grade 5-2 Averages
Your answer
Watch out The second looks like one run, but it is five separate blocks, so pairing fails.
How to solve
Strategy Look for a Pattern — Both strings are too long to add one number at a time without risking a slip, and both hide a pattern that turns the adding into multiplying. In (1) the even spacing means the outermost pair, the next pair in, and so on all give the same total, so the pattern does the work in one step. String (2) has no such pattern running through the whole line, so I first cut it into its five natural blocks — that is the subproblem move — and then look at a single tiny block of three consecutive numbers, where I can check by plain addition that the low number's shortfall exactly cancels the high number's excess. Once that easier case is settled, the same middle-times-three rule fires five times and the five block totals are all that is left to add.
1STEP 1
(1) Check the spacing and count the numbers
The first string holds 10 numbers.
30-12 = 18, 18 ÷ 2 = 9, 9+1 = 10
It is the fence-post idea — 9 gaps need 10 posts — and getting the count right matters more than anything else here, because every shortcut multiplies by that count.
4.OA.C.5Look For A Pattern2STEP 2
(1) Pair the ends inward: every pair makes 42
Pairing the ends, each pair makes 42.
12+30 = 14+28 = 16+26 = 18+24 = 20+22 = 42
The 2 that one number gains is exactly the 2 the other loses, so the trade is fair — that is the whole reason an evenly spaced list pairs up so neatly.
3.OA.D.9Look For A PatternPairing the run from both ends inward gives 42 every time.
Why?
Moving inward raises one partner by the step and lowers the other by the same step, so every pair lands on one fixed total.
🧱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?
That fixed pair total is twice the middle value, because the middle is the level everything balances around.
🧱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.
3STEP 3
(1) Turn the five pairs into one multiplication
Five pairs give 210.
42 × 5 = 210
Nine additions have shrunk to one multiplication of a two-digit number by 5, which a fourth grader can do in a single line.
4.NBT.B.5Look For A Pattern4STEP 4
(1) Cross-check with the middle value
The middle-value method also gives 210.
(20+22) ÷ 2 = 21, 21 × 10 = 210
Evenly spaced numbers balance around their middle, so replacing all ten by the middle value changes nothing about the total — and a second route to 210 catches any arithmetic slip.
3.OA.D.9Look For A Pattern5STEP 5
(2) Cut the long string into its five blocks
Cut the second into five blocks.
(17+18+19)+(24+25+26)+(32+33+34)+(40+41+42)+(52+53+54)
Regrouping the numbers before adding is allowed because addition can be done in any order and grouping, and here the grouping that matches the pattern is the one that makes the work easy.
3.OA.D.9Identify Subproblems6STEP 6
(2) Try one small block first: three in a row equals middle times 3
Three in a row make three times the middle.
17+18+19 = 54, 18 × 3 = 54
Testing the rule on one block small enough to add by hand is what makes it safe to trust on the other four blocks without checking them all.
3.OA.D.9Solve An Easier Related Problem7STEP 7
(2) Apply the rule to all five blocks
The five blocks give 54, 75, 99, 123 and 159.
18 × 3 = 54, 25 × 3 = 75, 33 × 3 = 99, 41 × 3 = 123, 53 × 3 = 159
Each block is now one small multiplication instead of two additions, and the same rule fires five times, so there is only one idea to remember.
4.NBT.B.5Look For A Pattern8STEP 8
(2) Add the five block totals
Adding them gives 510.
54+75+99+123+159 = 510
Only four additions are left, and each is a friendly three-digit sum, so the chance of a slip is far smaller than adding fifteen numbers one by one.
3.NBT.A.2Identify Subproblems9STEP 9
(2) Cross-check by adding the middles first
Adding the middles first also gives 510.
18+25+33+41+53 = 170, 170 × 3 = 510
Tripling five numbers and adding, or adding them and tripling once, must agree — that is the distributive property, and the agreement confirms both the rule and the arithmetic.
3.OA.D.9Look For A PatternAnswer
210, 510
42 × 5 = 210, 170 × 3 = 510
Both answers are plain sums of whole numbers, so they carry no units and must be whole numbers, and both are. For (1): ten numbers, the smallest 12 and the largest 30, so the total has to sit between 10 x 12 = 120 and 10 x 30 = 300; a middle value of about 21 predicts about 10 x 21 = 210, and 210 lands exactly there. For (2): fifteen numbers between 17 and 54, so the total has to sit between 15 x 17 = 255 and 15 x 54 = 810; the numbers average roughly 34, predicting about 15 x 34 = 510, and 510 lands exactly there. Both totals are also even-looking sanity checks: (1) adds ten even numbers, so it must be even, and 210 is.
Takeaway
When numbers are evenly spaced, pair the ends or use the middle number — a long addition line turns into one quick multiplication.
- (1) Check the spacing and count the numbers
- (1) Pair the ends inward: every pair makes 42
- (1) Turn the five pairs into one multiplication
- (1) Cross-check with the middle value
- (2) Cut the long string into its five blocks
- (2) Try one small block first: three in a row equals middle times 3
- (2) Apply the rule to all five blocks
- (2) Add the five block totals
- (2) Cross-check by adding the middles first