Linear Functions

What this covers

• Calculating slope from two points, a table, a graph, or a description, and reading it as a rate with units (dollars per hour, feet per second)
• Moving fluently between slope-intercept, point-slope, and standard form, including rearranging Ax + By = C to find slope and both intercepts
• Writing the equation of a line from two points, from a point and a slope, or from a parallel/perpendicular condition
• Function notation for linear rules: evaluating f(3), solving f(x) = 12, and interpreting f(0) and the x-intercept in context
• Recognising whether a table, graph, or situation is actually linear by testing for constant first differences
• Graphing linear inequalities in two variables and describing the shaded half-plane, including strict vs. inclusive boundaries
• Interpreting linear models: what the slope and intercept mean in the real situation, and where the model stops making sense

Where learners get stuck

Reading 'm' and 'b' straight off an equation that isn't in y = mx + b form

Students learn slope-intercept form first and pattern-match on position rather than structure, so in 2x + 3y = 12 or y = 7 - 4x they report the slope as 2 or 7. Practising the rearrangement out loud, every time, breaks the habit.

Assuming any table where y keeps increasing is linear

'Going up steadily' feels like a straight line. Without checking that the x-values are equally spaced and the first differences are equal, students label quadratic or exponential tables as linear. Evelyn insists on the two-part check.

Flipping the fraction for perpendicular slopes but forgetting the sign (or vice versa)

'Negative reciprocal' is two operations packed into one phrase, so one gets dropped — especially when the original slope is negative, an integer, or a unit fraction. Sketching the two lines as a sanity check catches it.

Computing slope as (x₂ − x₁)/(y₂ − y₁) or mixing points between numerator and denominator

The formula is memorised as a shape rather than as 'change in output over change in input', so under time pressure the ordering slips. Naming the quantities before substituting fixes it more reliably than re-memorising.

What a session looks like

A typical 25–40 minute session starts with a quick diagnostic — Evelyn gives a line in an unfamiliar form and asks for the slope, intercepts, and a rough sketch, listening to how you get there. From there the work is problem-driven: you narrate your steps, Evelyn probes the ones that came from habit rather than reasoning, and harder variations follow when a skill is solid. Word problems get a full unit check ('slope of 15 what, per what?'). You can share a homework question or a photo of a graph and work from that instead.

Helpful to know first

• Solving multi-step linear equations, including ones with variables on both sides
• Plotting points on the coordinate plane and reading coordinates off a graph
• Working with negative numbers and fractions, especially fraction arithmetic when slopes aren't whole numbers
• Basic familiarity with the idea of a function as an input-output rule

Questions

What's the difference between a linear equation and a linear function?

A linear equation is any statement like 3x + 2y = 8. It becomes a linear function when you treat one variable as the input and the other as the output, so each x gives exactly one y — which is why x = 4 is a linear equation but not a function. Evelyn works through the vertical line test and where it matters.

My child can graph lines but freezes on word problems. Can this help?

That gap is usually about translation, not graphing. Sessions spend time on identifying which quantity is the input, what the slope means with its units, and what the y-intercept represents at time zero — then the graphing is the easy part.

Which form should I use — slope-intercept, point-slope, or standard?

It depends on what you're given. Point-slope is fastest from a point and a slope or from two points; slope-intercept is best for graphing and comparing; standard form appears in intercept problems and systems. Evelyn drills the choice, not just the mechanics.

How does this fit with systems of equations and quadratics?

Solid work on slope, intercepts, and equation forms is what makes systems solvable by graphing and substitution, and it's the baseline you compare against when curves appear in quadratics. Those topics have their own briefs; this one stays on the single line.