Reasoning · Grade 6-2 Mixed Fraction and Decimal Calculations
Problem
Read the examples' rule backwards.
This is nothing more than the usual common-denominator recipe for adding or subtracting fractions, written the other way round: to combine 1/a and 1/b you use a × b underneath and b - a or b + a on top, so seeing that shape means you can uncombine it.
5.NF.A.1Look For A PatternSplit, (1) telescopes to 5/6.
Adding a number and then subtracting the same number leaves you where you started, so once the terms are written as differences the middle of the chain simply disappears — no common denominator of 60 is ever needed.
5.NF.A.1Organize Information In More WaysSplitting each term into a difference of two fractions makes the middle pieces cancel and leaves only the ends.
Each term's second piece is exactly the next term's first piece, so the pair adds to nothing and drops out.
Two fractions with the same denominator subtract by counting equal pieces, which makes the cancelling exact rather than approximate.
The top matches the gap, so (2) is 10/11.
The rule never said the two denominators had to be next-door neighbours — it only said the numerator must be their difference. Rewriting 2 as 3 - 1 is the check that the condition really holds.
5.NF.A.1Look For A PatternSplit (3) into whole parts and fractions.
Addition can be done in any order and in any grouping, so all the whole parts may be collected together and all the fraction parts together — and the odd numbers pair up from the outside in to give equal sums.
1.OA.B.3Identify SubproblemsFactoring the bottoms gives 64 7/18.
Finding the factor pairs of 6, 12, 20, 30, 42, 56 and 72 is ordinary times-table work, and it is what reveals that these seven scattered-looking fractions are really one tidy chain.
4.OA.B.4Look For A Pattern(4) must be split in the sum form.
The two rules look almost the same, and the only thing that separates them is whether the numerator is the sum or the difference of the two factors underneath — so that check has to come first, every time.
5.NF.A.1Look For A PatternAlternating signs cancel, leaving 3/5.
Once the brackets are gone the whole thing is a list of unit fractions with plus and minus signs, and matching each one with its opposite is bookkeeping a fifth grader can do on a single line.
5.NF.A.1Organize Information In More WaysCheck the numerator against the two factors underneath: if it is their difference the fraction splits into a subtraction, if it is their sum it splits into an addition — and then whole rows of fractions wipe each other out.