Pair consecutive numbers with constant sum

4.OA.C.53.OA.D.9 · take · grade 4

Archetype: Sum of Evenly Spaced Numbers via the Middle · step in a 5-type progression

Look at the example calculations, then find the number that belongs in each $\square$ . (Each equation is a sum of consecutive whole numbers.)

Example

$10+11+12=33$
$8+9+10+11+12=50$

$\square+\square+\square+\square+\square=35$
$\square+\square+\square+\square+\square+\square+\square=140$

Show solution

Understand

Two examples show sums of consecutive whole numbers (10+11+12=33 and 8+9+10+11+12=50). Fill the blanks so that five consecutive numbers add to 35, and seven consecutive numbers add to 140.

Givens

Examples: 10+11+12 = 33 and 8+9+10+11+12 = 50
Each blank row is a run of consecutive whole numbers
First target sum is 35 with 5 numbers; second is 140 with 7 numbers

Unknowns

The five consecutive numbers that sum to 35
The seven consecutive numbers that sum to 140

Constraints

The numbers in each row are consecutive whole numbers
With an odd count of consecutive numbers, the sum equals the count times the middle number

Plan

#5 Look for a Pattern · also uses: #6 Guess and Check

Consecutive numbers pair up so the outer pairs each have the same sum as twice the middle; for an odd count the sum is the count times the middle number, so dividing the sum by the count finds the middle and then the whole run.

Execute

#5 Look for a Pattern 3.OA.D.9

In the examples, 10+11+12=33 is 3 x 11 (the middle), and 8+9+10+11+12=50 is 5 x 10 (the middle). For an odd run of consecutive numbers, the sum equals the count times the middle number.

3 \times 11 = 33,\quad 5 \times 10 = 50

Pairing the ends inward, each pair averages the middle, so the whole sum centers on the middle.

#5 Look for a Pattern 4.OA.C.5

Five consecutive numbers sum to 35, so the middle is 35 divided by 5 = 7. The run is the two before and two after 7.

35 \div 5 = 7 \;\Rightarrow\; 5+6+7+8+9 = 35

Knowing the middle, just step out two on each side.

#5 Look for a Pattern 4.OA.C.5

Seven consecutive numbers sum to 140, so the middle is 140 divided by 7 = 20. The run is the three before and three after 20.

140 \div 7 = 20 \;\Rightarrow\; 17+18+19+20+21+22+23 = 140

The middle is the balance point, so spread three numbers to each side.

Answer: 5+6+7+8+9 = 35 and 17+18+19+20+21+22+23 = 140

Review

Checking sums: 5+6+7+8+9 = 35 and 17+18+19+20+21+22+23 = 140, both correct, and every row stays a run of consecutive whole numbers.

Guess and check (tool 6): start a five-number run near the average 7 (for example 5 to 9) and adjust; the constant-sum-pair idea (5+9, 6+8 each equal 14, plus the middle 7) confirms 35.

Standards · min grade 4

4.OA.C.5 Generate a number or shape pattern following a given rule — Building the consecutive runs outward from the middle number
3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Recognizing that an odd run of consecutive numbers sums to count times the middle

💡 This only needs Grade 4 pattern sense: divide the sum by how many numbers to find the middle, then count outward!