Data & Graphs

Pictographs, bar graphs, and line graphs encode values through a scale — symbol value or grid-square value — that must be read before any total, difference, or comparison is computed. Learners translate between tables and graphs, choose scales, read double bars and two line graphs, and estimate intermediate values. Decoding the scale is the load-bearing step at every grade.

Use the relationship sum = average × count to recover a total, a missing data value, or a combined/weighted average, including changes when data is added or removed.

Express the probability of an event as the fraction of equally likely favorable outcomes out of all outcomes, and compare such probabilities.

A function is a correspondence that gives every input one output and only one. That single condition decides whether a rule is a function at all, which constant makes it one, and how many functions exist between two finite sets (multiply the allowed choices for each input). Running the rule backwards recovers every input that lands on a given output.

A linear function changes by the same amount for every unit step, so any two points on it give the same difference quotient -- the slope -- and a graph through the origin is a direct proportion whose slope is the unit rate. Shifting a line or reading a runner's speed off a graph uses that constant rate; a graph made of straight pieces, like an area that grows and then shrinks as a point moves, has its own constant slope on each piece and changes rate only where the pieces join.

When y = a/x, the product xy is the same at every point, so every rectangle drawn from a point on the curve to the axes has the same area, the curve is symmetric about the origin, and its lattice points are the factor pairs of a -- while points between the curve and the axes satisfy a product inequality instead. Meshed gears are the same law: teeth times turns is shared.