Depth · one concept from first principles · step 2 of 4
Write 1 4/7 as 11/7 and subtract the left end. Subtracting a negative adds its size.
The distance from the first point to the last is 12/7.
7.NS.A.1Draw A DiagramFour points have three equal gaps between them.
Each gap is 4/7.
7.NS.A.1Identify SubproblemsThe stretch from to is made of three equal gaps, so one gap is a third of that stretch.
Each gap can be matched with the point at its right end, and every point except the first is the right end of exactly one gap, so four points give three gaps.
The three gaps lie end to end with no overlap and fill the whole stretch, so together they make the full length.
Three equal gaps making the full length means the length is three groups of one gap, so one gap is the length divided by .
Start at -1/7 and add one gap at a time.
One more step gives 1 + 4/7 = 1 4/7, the right end, as it should.
6.NS.C.6Draw A DiagramNow add x and y.
So x+y=10/7.
7.NS.A.1Identify SubproblemsFor evenly spaced numbers, the two middle ones add up to the same as the two ends.
A parent dashboard for the family lives at sensimlab.com.
The same shape shows up later on the AMC, as Arithmetic Expression Decomposition, Lattice / Grid Pattern from Small Cases, Pair-Sum Invariant, Small-Domain Constrained Search. Why the shapes repeat →