Find patterns in mixed-number sequences and sum

4.NF.B.34.OA.C.5 · take · grade 4

Archetype: Generalize a Growing Pattern into a Rule · step in a 12-type progression

The numbers below are listed according to a rule. Find the sum of all the numbers in the list.

$1\frac{2}{13},\ 3\frac{4}{13},\ 5\frac{6}{13},\ \ldots,\ 11\frac{12}{13}$

Show solution

Understand

A list of mixed numbers follows a rule: 1 2/13, 3 4/13, 5 6/13, ..., up to 11 12/13. I must figure out the rule, list every term, and add them all up.

Givens

The list is 1 2/13, 3 4/13, 5 6/13, ..., 11 12/13.
All fraction parts share the denominator 13.
The last listed term is 11 12/13.

Unknowns

The sum of all the numbers in the list.

Constraints

Only the terms that actually fit the rule up to 11 12/13 are included.

Plan

#5 Look for a Pattern · also uses: #2 Make a Systematic List#7 Identify Subproblems

Find the rule (whole parts are odd numbers, numerators are even numbers), list all six terms, then split the sum into a whole-number part and a fraction part — two easy subproblems.

Execute

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

The whole parts go 1, 3, 5, 7, 9, 11 (odd numbers) and the numerators go 2, 4, 6, 8, 10, 12 (the next even number), all over 13. So the full list is 1 2/13, 3 4/13, 5 6/13, 7 8/13, 9 10/13, 11 12/13 — six terms.

1\tfrac{2}{13},\,3\tfrac{4}{13},\,5\tfrac{6}{13},\,7\tfrac{8}{13},\,9\tfrac{10}{13},\,11\tfrac{12}{13}

Spotting that each whole part jumps by 2 and each numerator jumps by 2 makes the pattern obvious.

#7 Identify Subproblems 4.OA.A.3

Add the whole parts: 1 + 3 + 5 + 7 + 9 + 11 = 36.

1+3+5+7+9+11=36

Pairing 1+11, 3+9, 5+7 gives three 12s = 36 — a quick whole-number sum.

#7 Identify Subproblems 4.NF.B.3

Add the fraction parts (same denominator 13): (2+4+6+8+10+12)/13 = 42/13. Since 42/13 = 3 with remainder 3, this is 3 3/13.

\dfrac{2+4+6+8+10+12}{13}=\dfrac{42}{13}=3\tfrac{3}{13}

Like denominators let you add numerators, then regroup the improper fraction into a mixed number.

#7 Identify Subproblems 4.NF.B.3

Add the whole-part total to the fraction-part total: 36 + 3 3/13 = 39 3/13.

36+3\tfrac{3}{13}=39\tfrac{3}{13}

Putting the whole and fractional totals back together gives the full mixed-number sum.

Answer: 39 3/13

Review

Six terms averaging about 6.5 each give roughly 6 × 6.5 ≈ 39, which matches 39 3/13. Units are consistent (all thirteenths), and the fraction part 3/13 is a proper fraction, so the answer is sensible.

Pair the smallest and largest terms (tool 15): 1 2/13 + 11 12/13 = 13 1/13, 3 4/13 + 9 10/13 = 13 1/13, 5 6/13 + 7 8/13 = 13 1/13. Three pairs of 13 1/13 give 39 3/13.

Standards · min grade 4

4.OA.C.5 Generate a number or shape pattern following a given rule — Discovering the odd-whole / even-numerator rule and listing all six terms.
4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Summing the whole-number parts 1+3+5+7+9+11.
4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Adding the like-denominator fraction parts and regrouping 42/13 into a mixed number.

💡 This only needs Grade 4 pattern-spotting plus same-denominator fraction adding — split the wholes from the thirteenths!