047i - Exploring Families of Linear Functions

Lesson 47 — Exploring Families of Linear Functions Using Digital Tools

Strand: Algebra | Descriptor: AC9M8A04 | Duration: 45 minutes

Learning Intentions

• To use a digital graphing tool to explore how varying the gradient and -intercept affects the graph of a linear function.
• To describe, using precise mathematical language, the visual effect of changing parameters in .

Success Criteria

I can:

1. Use a graphing tool (e.g. Desmos) to plot and vary linear functions.
2. Describe what happens to a line as or changes, using vocabulary such as steeper, flatter, shifts up/down, reflects.
3. Record my noticings as informal conjectures for later testing.
4. Compare a family of lines that share a common feature — the same , or the same .

Warmup

(5 minutes — quick recall, mini whiteboards)

Recall lessons 38–40.

1. In , what does represent?
2. In , what does represent?
3. Sketch (or describe) without a table.
4. If two lines have the same gradient but different intercepts, what do you predict about their graphs?

Answers: 1. Gradient — the rate of change, how much increases per unit increase in ; 2. -intercept — where the line crosses the -axis, the value of when ; 3. Passes through , rising units for every across; 4. Prediction — students may say “parallel”; hold this as today’s first conjecture to test.

Activities

Activity 1 — Teacher Demonstration: Meet the Sliders (10 min)

Whole class, teacher at the display, using a graphing tool such as Desmos.

Teacher demonstrates: type into the graphing tool and let it generate sliders for and . Set , to start, matching the warmup sketch.

Notice and wonder — first round. Drag the slider slowly from to while stays fixed at .

Prompt	Record on the board
What do you notice as changes?	e.g. “the line moves up and down”; “it never changes its steepness”
What do you wonder?	e.g. “does it always move the same way?”; “what if is negative?”

Teacher demonstrates: now reset and drag the slider slowly from to .

Prompt	Record on the board
What do you notice?	e.g. “the line gets steeper”; “at it goes flat”; “negative flips the direction”
What do you wonder?	e.g. “is there a fixed point every line passes through?”; “what does actually mean?”

Class discussion: circle any noticing that could become a testable claim — these will become today’s working conjectures, tested more fully in Lesson 48.

Activity 2 — Guided Pairs Investigation: Two Families (14 min)

Pairs, at a device with a graphing tool. Structured recording sheet.

Family 1 — same gradient, varying intercept. Graph for on the same axes (use a slider or type each separately).

Record:

1. What stays the same across all four lines?
2. What changes?
3. Describe the relationship between the four lines using a geometric word.

Family 2 — same intercept, varying gradient. Graph for on the same axes.

Record:

1. What point do all five lines share? Why does this make sense from the equation?
2. Order the lines from flattest to steepest. Where does sit in this order?
3. What is different about the lines with negative compared to positive ?

Socratic scaffolding for question 1 of Family 2 (the shared point):

Prompt	Purpose
Understand: what do all five equations have in common?	Every one ends in ; only the coefficient of differs.
What happens to every equation when ?	, regardless of .
So what point must every line pass through?	— the shared -intercept.
Devise a test	Check the graph: do all five lines cross the -axis at the same point?
Look back	This explains why varying alone, with fixed, produces a “fan” of lines through one point.

Answers: Family 1 — gradient () stays the same; intercept changes; the lines are parallel. Family 2 — all pass through , the shared -intercept; ordering from flattest to steepest by : (horizontal), , , , ; negative- lines fall from left to right, positive- lines rise.

Activity 3 — Gallery Share and Predict-then-check (10 min)

Whole class.

Each pair shares one noticing from Activity 2 with the class; the teacher compiles a running list of class conjectures on the board (to be formally tested in Lesson 48).

Predict-then-check task: without using the tool, predict what will look like compared to . Write your prediction, then verify with the graphing tool.

Prompts to guide prediction:

• Same intercept — what does that mean for where they start?
• Opposite-sign gradients — what does that mean for direction?

Answer: Both cross the -axis at ; rises to the right, falls to the right — they are mirror images of each other in a vertical sense through their shared point.

Checks for Understanding

(6 minutes — exit ticket, collected)

1. In the family , what feature of every graph stays fixed as varies?
2. In the family , what point do all the graphs share?
3. Describe what happens to a line’s steepness as increases from to .
4. A line has . Describe its graph in one sentence.
5. Reasoning. Two lines are and . Without graphing, explain what you can already say about how these two lines relate to each other.

Answers: 1. The gradient (steepness/direction) stays fixed; 2. ; 3. The line becomes progressively steeper; 4. It is a horizontal line, since does not change as changes; 5. They share the same gradient (), so they must be parallel, but have different intercepts ( and ), so they never touch.

Common Misconceptions

Misconception	How to pre-empt it
Believing affects the steepness of the line.	Contrast Family 1 (same ) directly against Family 2 (same ) side by side.
Believing means “no line” or “nothing to graph.”	Demonstrate explicitly that simplifies to , a valid horizontal line.
Assuming a steeper-looking line always has a larger gradient, without checking sign.	Compare (steep, falling) with (less steep, rising); “steep” refers to $
Thinking every family of lines through one point must share the same intercept, rather than any common point.	Show a counter-example later in Lesson 48: lines can share a different common point if built differently.
Confusing “the lines get closer together” with “the lines will eventually meet,” when they are actually parallel.	Zoom out on the graphing tool to show parallel lines never converge, however close they appear locally.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). The lines and are graphed and appear identical. What must be true of ?

Answer

— identical equations produce identical (coincident) lines, not just parallel ones.

E2 (AMC Junior style). A family of lines is given by . For which value of does the line pass through the point ?

Answer

.

E3 (Challenge). Two lines from the family pass through and respectively. Find both gradients and explain why they cannot be the same line.

Answer

First: . Second: . Different gradients mean these are two different lines from the family, both sharing the intercept but crossing at different -values.

E4 (Challenge). A line in the family passes through both and . Find and .

Answer

From : . From : . So .

Homework

1. For the family , describe what stays the same and what changes as varies.
2. For the family , state the one point every graph shares.
3. Predict, without graphing, how and compare. Check your prediction using a graphing tool if available.
4. A line from the family passes through . Find .
5. Reasoning. Explain why two lines with the same gradient but different intercepts never intersect, using the idea of constant difference between their -values.
6. Challenge. Two members of the family pass through and . Find both gradients, and state whether either line could also belong to the family .

Answers: Q1 — gradient () stays fixed; intercept changes; lines are parallel. Q2 — . Q3 — both cross at ; rises to the right, falls to the right — mirror images through the shared point. Q4 — . Q5 — since both lines have the same gradient, the difference in their -values for any given is always equal to the difference in their intercepts — a constant, non-zero gap that never closes to zero, so they never meet. Q6 — first: ; second: ; different gradients, so they are different lines; neither could belong to unless its intercept were , but both have intercept , so no.