040i - Graphing Linear Relations with Digital Tools

Lesson 40 — Graphing Linear Relations Efficiently Using Digital Tools

Strand: Algebra | Descriptor: AC9M8A02 | Duration: 45 minutes

Learning Intentions

• To use a digital graphing tool to graph linear relations quickly and accurately.
• To compare families of linear relations on the same axes using a digital tool.
• To verify a hand-drawn graph using digital technology, and identify the source of any error.

Success Criteria

I can:

1. Enter a linear relation into a digital graphing tool and read its gradient and intercept from the display.
2. Graph several linear relations on the same axes to compare their gradients and intercepts.
3. Use a slider to investigate how changing or affects an entire family of lines.
4. Use a digital tool to check a hand-drawn graph or table of values, and explain the cause of any discrepancy.

Warmup

(5 minutes — retrieval, mini whiteboards)

1. State the gradient and -intercept of .
2. Without graphing, predict which of these lines is steepest: , , .
3. Name a digital tool you could use to check your prediction.
4. Why might a digital tool be more efficient than hand-plotting when comparing several lines at once?

Answers: 1. ; 2. is steepest, since is the largest gradient size; 3. e.g. Desmos or GeoGebra; 4. Hand-plotting every line needs its own table of values and careful ruling, which is slow and leaves room for error; a digital tool draws each line instantly and exactly, freeing you to focus on comparing features rather than the mechanics of plotting.

Activities

Activity 1 — Explicit Instruction: Entering and Reading Relations in a Digital Tool (10 min)

The routine. In a digital graphing tool (Desmos or GeoGebra), typing an equation such as directly into the input bar instantly plots the full line — correctly extended in both directions — with no table of values or ruler required.

I do. Type . Click on the line to display the coordinates of any point on it. Note where it crosses the -axis: , matching read directly from the equation.

I do — adding a second relation. Type on the same axes. Compare the two lines directly on screen: one climbs steeply left to right, the other falls gently.

Non-negotiable habits, modelled explicitly:

• Always predict the gradient, direction and intercept before graphing, using the equation alone — the tool should confirm your thinking, not replace it.
• Adjust the viewing window if a key feature (such as an intercept) is not visible.
• Click or hover to read exact coordinates rather than estimating by eye.

We do: Enter and on the same axes. Predict each line’s direction and intercept first, then check.

You do: Using a digital tool, graph the following on the same set of axes. For each, record the gradient, the -intercept, and whether it slopes up or down, left to right.

(Answers: 1. , up; 2. , down; 3. , up — the gentlest slope of the three.)

Activity 2 — Guided Practice: Exploring Families of Lines with Sliders (10 min)

Pairs, using a digital graphing tool with slider functionality.

Investigation A — a slider for . Type , and let the tool create a slider for . Drag the slider from to and observe.

1. What point does every version of the line pass through, no matter the value of ?
2. Describe, in words, what happens to the line as changes sign, and as increases.

Investigation B — a slider for . Type , with a slider for . Drag it from to .

1. What stays exactly the same as changes?
2. What moves, and in which direction?

Connecting back: Compare your observations to Lesson 39’s Investigations A and B, where you typed separate lines one at a time. Does the slider confirm the same pattern, now shown continuously rather than as separate snapshots?

Extension — matching a family to a point. Using , find the value of needed so the line passes through each of these points, then check each one on the digital tool: ; ; .

Answers: Investigation A — every line passes through the fixed point , since substituting always gives whatever is; negative flips the line’s direction, and larger makes it steeper. Investigation B — the gradient () stays exactly the same, so every line is parallel; the line slides vertically up or down as increases or decreases. Extension — for : ; for : ; for : .

Activity 3 — Inquiry: Spot the Plotting Error, Using a Digital Tool to Verify (15 min)

Pairs, then whole-class share.

A student hand-plots a table of values for the rule , using to :

1. Use a digital tool to graph the true rule .
2. Plot the student’s five points on the same axes (or compare each one against the graphed line). Which point does not lie on the line?
3. Find the correct value, and suggest what error the student most likely made.

Socratic scaffolding:

Prompt	Purpose
Understand: what digital method checks every point at once, rather than substituting one at a time?	Graph the rule directly — every genuinely correct point from the table should sit exactly on the drawn line.
Devise a plan	Plot all five of the student’s points on the same digital graph as the true line, and see which one falls off it.
Carry it out	, , and all sit exactly on the line; sits one unit above it.
Looking back — how could you pinpoint the correct value without re-plotting?	Substitute directly into the rule: , confirming the point should be , not .
Looking back — what does the size and direction of the error suggest about the mistake?	The point is exactly unit too high — consistent with a small final-step slip (e.g. computing as instead of ), not a wrong gradient or a completely different rule, which would have thrown off every point, not just one.

Answers: 2. does not lie on the line. 3. The correct value is , so the point should be ; the error is an isolated arithmetic slip in the final subtraction step, not a misunderstanding of the rule itself.

Checks for Understanding

(5 minutes — exit ticket, collected)

1. Name two features of a digital graphing tool that make it more efficient than hand-plotting when comparing several lines at once.
2. Without using a tool, state the gradient and -intercept you would expect for , and describe what the graph should look like.
3. A digital graph shows . A student claims the point lies on this line. Explain how you would check this claim, and state whether it is correct.
4. Reasoning. Explain how you would use a digital tool with a slider to demonstrate that changing in always produces a family of parallel lines.

Answers: 1. E.g. instantly and accurately plots without a table or ruler; allows several relations to be compared directly on the same axes; provides exact coordinate readouts on click or hover; 2. — a line that falls left to right, crossing the -axis at ; 3. Substitute into the rule: , not , so the claim is false — checking on the graph, the point would appear visibly above the line; 4. Set up a slider for with fixed; as varies, every version of the line keeps the same gradient () but slides vertically, so every line in the family remains parallel to every other, since parallel lines are defined by sharing an identical gradient.

Common Misconceptions

Misconception	How to pre-empt it
Believing a digital tool removes the need to understand gradient and intercept.	Insist on a written prediction of , and direction before every graph is drawn, as modelled in Activity 1.
Mistyping an equation (e.g. entering instead of ) and not noticing the resulting graph looks different from expected.	Always compare the graphed result against the predicted intercept and gradient — a mismatch is a sign of a typing error, not a maths error.
Assuming two lines that “look parallel” on screen are genuinely parallel, without checking their gradients.	Warn that an unadjusted or stretched viewing window can visually distort how steep a line appears; always confirm using the equation’s value.
Not adjusting the viewing window, so a key feature such as an intercept is off-screen and missed entirely.	Model zooming and panning explicitly, and make “check the whole graph is visible” a standard first step.
Treating a slider demonstration as a stand-alone “proof” without connecting it back to the algebraic reasoning about and .	Always follow a slider investigation with a written explanation in terms of substituting into , as required in Checks for Understanding Q4.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). A family of lines all pass through with different gradients. One of these lines also passes through . Find its gradient.

Answer

E2 (AMC Junior style). Two lines, and , are graphed on the same axes and are units apart vertically. If , find the two possible values of .

Answer

or .

E3 (Challenge). Three lines are graphed, all passing through the point , with gradients , and respectively. Find each line’s equation in the form .

Answer

Using for each gradient:

E4 (Investigation). Using a digital tool’s slider for in (note there is no separate term here), explain why the line always passes through the origin, whatever value the slider takes.

Answer

Substituting into gives for every value of , so the point satisfies the rule regardless of the slider’s position — the whole family of lines pivots about the origin.

Homework

1. Using a digital graphing tool, graph , , and on the same axes. What do you notice about all three lines?
2. Using a digital graphing tool, graph , , and on the same axes. What do you notice?
3. State the gradient and -intercept you would predict for , then check using a digital tool.
4. A hand-drawn table of values for contains the point . Use substitution to check whether this point is correct. If it is wrong, state the correct point, and explain how a digital graph would reveal the error visually.
5. Reasoning. Explain one advantage and one limitation of using a digital tool, rather than plotting by hand, to graph and compare linear relations.
6. Challenge. Three lines all pass through the point . Their gradients are , , and . Find each line’s equation in the form .

Answers: Q1 — all three lines are parallel, sharing gradient , differing only in their -intercept. Q2 — all three lines pass through the same point, , since they share the same -intercept; only their steepness and direction differ. Q3 — . Q4 — , not , so the point is incorrect; the correct point is ; on a digital graph, the plotted point would appear visibly below the line, one unit off it. Q5 — advantage: e.g. fast, exact comparison of many relations without the risk of manual plotting error; limitation: e.g. relying on the tool without understanding gradient and intercept means a mistyped equation can go unnoticed, and manual plotting builds a deeper foundational sense of what the graph represents. Q6 — using with point , so : ; (a horizontal line); .