Signed Position and Distance
Every point on the number line (and each coordinate in the plane) is a signed position. On the line the distance between two points is the absolute value of their difference; in the plane the horizontal and vertical gaps are each the absolute difference of one coordinate. Directions become signs: reflecting in an axis flips the sign of one coordinate, reflecting in the origin flips both. |x - a| <= b is just 'within b of a'. Each absolute value hides a two-case split -- this side or that side -- and the extremes come from choosing the cases.
grade 7–8 Number systemGeometryExpressions Draw a DiagramMake a Systematic List
Progression (6)
g7 1. Turn directions and distances into signed coordinates Number system
g7 2. Opposite signs turn |b|=2|a| into b=-2a Number system
g7 3. Combine sign cases to get extreme distances Number system
g7 4. Three reflections make a rectangle sized by absolute values Number system
g7 5. A square on a given side has two sides to grow GeometryNumber system
g8 6. Match interval endpoints, splitting on the sign Number systemExpressions