Algebraic Thinking and the Coordinate Plane
Core content
Algebraic thinking includes translating verbal relationships into expressions and equations, simplifying with exponent rules, solving linear equations and inequalities, graphing lines, and interpreting functions. Know slope as rate of change, slope-intercept form y = mx + b, standard form, and how to find intercepts. Systems of equations can be solved by substitution, elimination, or graphing; the solution is the intersection point. For quadratics, connect factoring, zeros, and the graph. On the coordinate plane, be fluent with quadrants, ordered pairs, distance, midpoint, and transformations.
Traps
Frequent errors include distributing a negative incorrectly, such as -(x - 3) = -x + 3 rather than -x - 3, and forgetting to reverse an inequality when multiplying or dividing by a negative. Students also confuse slope with y-intercept. A line with slope 0 is horizontal, while an undefined slope is vertical. Parallel lines have equal slopes; perpendicular slopes are negative reciprocals. When evaluating a function, substitute the input for every x, not just the first occurrence. Check whether a question asks for an x-intercept or a y-intercept.
Worked example
Find the equation of the line through (2, 5) and (6, 13). Slope = (13 - 5) / (6 - 2) = 8 / 4 = 2. Using y = mx + b: 5 = 2(2) + b, so b = 1. The line is y = 2x + 1. Verify with the second point: 2(6) + 1 = 13.
Rule of thumb
For a linear relationship, start with slope = change in y divided by change in x; then substitute one point to find b.