Reasoning · Grade 6-2 Mixed Operations with Fractions
Problem
In (1) neighbours cancel, leaving 12/25.
Working the brackets out first is the trap: it would leave 24 unlike fractions to add. Erasing them instead costs nothing, because adding a bracket never changes a sign, and then the cancelling is visible to anyone who can see that -1/4 + 1/4 = 0.
5.OA.A.1Organize Information In More WaysFlipping the signs, (2) is 11/16.
The one thing to be careful about is the flipped sign — the second fraction in each bracket comes out positive. After that it is just three fractions over a common denominator of 48, which is Grade 5 work.
5.NF.A.1Organize Information In More WaysSubtracting a bracket flips the sign of everything inside it, which is why the second fraction comes out positive.
Taking away a whole group means taking away each of its members, so a shared minus reaches every term inside.
Subtracting reverses adding, so taking away something that was itself being subtracted puts it back as an addition.
Dividing factor by factor, (3) is 8.
Matching denominators is a signal from the problem-setter that these fractions are meant to meet. When two fractions share a denominator, dividing them is as easy as dividing whole numbers: 4 sevenths shared into groups of 2 sevenths makes 2 groups.
6.NS.A.1Organize Information In More WaysPulling the repeat out, (4) is 10 5/9.
This is the distributive property read backwards: a x 1 + a x 2 + ... + a x 10 = a x (1 + 2 + ... + 10). Seeing the invisible x 1 on the first term is the only subtle part, and once the bracket is formed the sum inside is a Grade 4 pairing trick.
6.EE.A.3Look For A PatternFactoring top and bottom alike, (5) is 1/3.
The huge sum of fifty squares never gets added up, because the same sum appears on both sides of the bar and a fraction with equal top and bottom parts cancels. All that matters is the constant riding on each one: 2 above, 6 below.
6.EE.A.3Identify SubproblemsShrunken versions confirm every answer.
A shrunken copy of the same expression can be computed the slow, honest way in a few seconds, and if the shortcut gives the same number there, the shortcut is being applied correctly.
5.OA.A.1Solve An Easier Related ProblemBefore you calculate, rearrange: drop the brackets, put matching partners next to each other, and pull out whatever every term shares.