Linear functions: a week of ready-to-teach maths

Five lessons for Year 8 Algebra. Print this pack and the week is prepared: each day has a one-page plan and a student worksheet with practice and a stretch section, plus printed grids, conjecture cards, graph match cards, a mini-check and every answer with its working.

Start here: five minutes to Monday

1. Skim the week at a glance on the next page.
2. Print the five days. Each day is two A4 sheets: a plan and a worksheet. The grids are printed on the worksheets.
3. Print the two cut-out sheets, one copy per pair, and cut them before Day 4: the conjecture cards are for Day 4, the graph match for Day 5.
4. Open the free interactive unit on the board: it is the digital tool for the week. Every plan says which buttons to press.
5. Teach straight from the plan. Timings, talk prompts, misconceptions and worked answers are all on the one page.

The idea in plain words

Every linear function can be written y = mx + c, and the two numbers do two different jobs. The slope m sets the steepness and the direction: a bigger m is steeper, a negative m falls from left to right, and m = 0 is flat. The intercept c sets where the line crosses the y-axis, at (0, c). Changing m with c fixed swings the line round the point (0, c); changing c with m fixed slides the line up or down without turning it, so lines with the same m are parallel. A digital graphing tool lets students turn these dials quickly, but the mathematics is in what they notice: they make a conjecture, test it on new cases, look for a counterexample, and only then state it as a general pattern.

One day, one lesson

The five lessons fill a week of maths, one lesson of about 60 minutes a day. Run them in order: each day stands on the one before. Every plan marks a split point for two shorter sessions, and every worksheet ends with a stretch section for students who finish early.

Digital tools in this pack

The interactive unit on the board is the shared digital tool: each plan names the buttons to press. If students have graphing software or graphing calculators, let them test their own cases on Days 2 to 4 after predicting on paper. Every grid students need is printed, so the week works without devices. We call m the slope and c the y-intercept.

The week at a glance

One lesson a day for a week. Each day stands on the day before, so run them in order.

How the week builds

Day 1 turns both dials and predicts; Days 2 and 3 turn one dial at a time so the effect of each number is clear; Day 4 makes the noticing into claims and tests them honestly; and Day 5 states the general pattern and uses it. It builds on graphing linear relations earlier in Year 8 and prepares gradient, intercepts and the gradient formula in Year 9, and simultaneous equations in Year 10.

Materials for the week (one trip)

• From the classroom: rulers, sharp pencils, three coloured pencils each, scissors, this pack printed.
• Cut-out sheet 1, one per pair: 12 conjecture cards (Day 4).
• Cut-out sheet 2, one per pair: 8 rule cards and 8 graph cards (Day 5).
• Optional: graphing software or calculators for testing cases. No other equipment to buy.

Words for the word wall

linear function, slope, steepness, direction, rises, falls, y-intercept, parallel, family of lines, pivot, shift, conjecture, test case, counterexample, always true, generalise, pattern, digital tool

Dear families

This week in maths, Year 8 is experimenting with straight-line graphs, y = mx + c, using a graphing tool. Students change one number at a time and watch what happens: m makes the line steeper or flatter, and c moves it up or down. Then they make claims (conjectures) about lines and test them, looking for an example that breaks the claim.

Try this at home

• If you have a free graphing app or website, type in y = x, then y = 2x, then y = 3x. What changes?
• Ask: which is steeper, y = 2x + 1 or y = 5x − 3? (y = 5x − 3: the bigger m.)
• Ask: where does y = 3x + 4 cross the y-axis? (At (0, 4).)
• Ask: someone says every line goes through (0, 0). Can you find a line that does not? (y = x + 1.)

My lines this week

Fill one row a day. Write a rule, predict its graph, then check by plotting two points.

Printed from the free seegongsik Linear functions teaching pack · seegongsik.com/au/y8/algebra/AC9M8A04/pack

Turning the dials on y = mx + c

Two numbers control every straight line. Students turn both dials on the board, predict before they plot, draw a family of lines y = mx, and describe lines from their rules alone.

We are learning to

• predict how a line will look from its rule, then check by plotting,
• describe a line by its direction, its steepness and where it crosses the y-axis,
• say which number, m or c, causes each change in a graph.

Success criteria

• I can say that y = −3x + 2 falls steeply and crosses the y-axis at 2, before plotting it.
• I can explain why y = 3x is steeper than y = x.

You need

The worksheet, a ruler and three coloured pencils each.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part A; start the second with Part B.

Watch for these ideas

• Reading c as the steepness and m as the crossing point.
• Thinking a negative slope means the line is below the x-axis.
• Thinking y = 3 is a vertical line: it is flat, crossing the y-axis at 3.

Answers

• Part A: y = x: −2, −1, 0, 1, 2; y = 2x: −4, −2, 0, 2, 4; y = 3x: −6, −3, 0, 3, 6; all pass through (0, 0), each steeper than the last.
• Part B: y = 4x − 1: rises, steep, crosses the y-axis at −1; y = −2x + 3: falls, steep, crosses the y-axis at 3; y = x + 5: rises, medium, crosses the y-axis at 5; y = −x − 2: falls, medium, crosses the y-axis at −2; y = 3: flat, crosses the y-axis at 3; y = ½x + 1: rises, gentle, crosses the y-axis at 1.
• Part C: m changes the steepness and direction; c changes where the line crosses the y-axis.
• Part D: m = 0 gives a flat line y = c; a negative m makes the line fall from left to right.

Turning the dials on y = mx + c

A. A family of lines

Predict first: which will be steepest? Then complete the tables and plot all three lines on the grid in three colours.

B. Describe without plotting

For each line: does it rise, fall or stay flat? Steep, medium or gentle? Where does it cross the y-axis?

C. Which dial?

Finish the sentences. “Changing m changes ______________________.” “Changing c changes ______________________.”

D. Stretch: m = 0 and negative m

Predict, then sketch on your grid: y = 0x + 2 and y = −2x. What does m = 0 do? What does a negative m do?

Changing m with c fixed

Keep c and change m, and every line passes through (0, c): the line pivots. Students plot such a family, measure each slope as rise per step, find slopes from two points and order lines by steepness.

We are learning to

• explain why lines y = mx + c with the same c all pass through (0, c),
• measure a slope as the change in y for each step of 1 in x,
• find the slope between two points and order lines by steepness, ignoring the sign.

Success criteria

• I can find the slope through (0, 2) and (3, 8) and explain what it means.
• I can say which is steeper, y = −3x + 1 or y = 2x + 1, and why.

You need

The worksheet, a ruler and three coloured pencils each.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part A; start the second with Part B.

Watch for these ideas

• Thinking −2 is less steep than 1: steepness is the size of m.
• Reading the rise over the wrong number of steps.
• Mixing up rise over run with run over rise.

Answers

• Part A: y = x + 1: −1, 0, 1, 2, 3; y = 2x + 1: −3, −1, 1, 3, 5; y = −x + 1: 3, 2, 1, 0, −1; all pass through (0, 1).
• Part B: steepest y = 2x + 1 (slope 2); then y = x + 1 and y = −x + 1, equally steep (slope size 1), one rising and one falling.
• Part C: (0, 1) to (2, 7): 6 ÷ 2 = 3; (1, 5) to (3, 1): −4 ÷ 2 = −2; (−2, −3) to (2, 5): 8 ÷ 4 = 2.
• Part D: y = ½x + 1 rises 1 for every 2 across, gentler than y = x + 1 and still through (0, 1).

Changing m with c fixed

A. One point, three slopes

Complete the tables and plot the three lines in three colours. Then draw a step-and-rise triangle on each.

B. Order by steepness

Order your three lines from steepest to least steep. Which two are equally steep, and how do they differ?

C. Slope from two points

Slope = change in y ÷ change in x.

D. Stretch: a gentle slope

Plot y = ½x + 1 on your grid. How does it compare with y = x + 1? Describe a slope of ½ in words.

Changing c with m fixed

Keep m and change c, and the line slides up or down without turning, so the lines are parallel. Students plot a parallel family, measure the shift, and write rules for parallel lines.

We are learning to

• explain why lines with the same m are parallel,
• describe changing c as a vertical shift of the whole line,
• write the rule of a line parallel to a given line through a given y-intercept.

Success criteria

• I can write the rule of the line parallel to y = 3x − 2 that crosses the y-axis at 5.
• I can say how far apart, vertically, y = x + 4 and y = x − 1 are.

You need

The worksheet, a ruler and three coloured pencils each.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part A; start the second with Part B.

Watch for these ideas

• Thinking a bigger c makes a line steeper.
• Measuring the gap between parallel lines across instead of up.
• Changing m when asked for a parallel line.

Answers

• Part A: y = x − 2: −4, −3, −2, −1, 0; y = x: −2, −1, 0, 1, 2; y = x + 3: 1, 2, 3, 4, 5; all have slope 1.
• Part B: each line is the one below it moved up; y = x + 3 is 5 above y = x − 2 at every x.
• Part C: parallel to y = 2x − 1 through (0, 4): y = 2x + 4.
• Part D: lines y = mx + c and y = mx + d stay |c − d| apart vertically, because at every x their y values differ by c − d.

Changing c with m fixed

A. A parallel family

Complete the tables and plot the three lines in three colours.

B. Measure the shift

How far above y = x − 2 is y = x + 3 at x = −2, at x = 0 and at x = 2? What do you notice?

C. Write a parallel rule

Write the rule of the line parallel to y = 2x − 1 that crosses the y-axis at 4. ______________________

Write two more lines parallel to y = −x + 2. ______________________

D. Stretch: how far apart?

Explain why y = mx + c and y = mx + d are always the same vertical distance apart. Use the tables to help.

Testing a conjecture

A conjecture is a claim we think is true. Students test claims about lines with cases they choose, hunt for counterexamples, and sort conjecture cards into always true and not always true.

We are learning to

• test a conjecture with several well-chosen cases, including unusual ones,
• use one counterexample to show a conjecture is not always true,
• explain why passing tests makes a conjecture more believable but does not prove it.

Success criteria

• I can test “Lines with the same c cross the y-axis at the same point.” with three different cases.
• I can give a counterexample to “every line goes through the origin”.

You need

The worksheet; the conjecture cards (cut-out sheet 1), one set per pair; graphing tools if available.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after the sort; start the second with Part A.

Watch for these ideas

• Testing only easy cases (small positive whole numbers).
• Thinking several passes prove a claim; one counterexample disproves it.
• Giving a counterexample that does not actually break the claim.

Answers

• Part A: 1. holds in every case (each pair rises by the same amount per step, so the gap stays the same); 2. fails (y = x + 3 gives 4 at x = 1, a counterexample); 3. holds in every case (swapping x for −x swaps the two rules).
• Part B: for example y = x + 1 (not through the origin); y = 2x + 5 and y = x + 1 (steeper, smaller c); y = −x + 4 (negative m but above the x-axis for x < 4).
• Card sort (printed order): not always are cards 2, 5, 6, 9, 10; the other 7 are always true.
• Part D: tests make a conjecture likely; a reason that works for every m and c (such as comparing the rules) is what makes it certain.

Testing a conjecture

A. Three tests each

Test each conjecture with the cases given. Write what you found and whether it held.

B. Break it

Each claim is false. Give one counterexample and show why it breaks the claim.

1. Every line goes through the origin.   2. A steeper line always has a bigger c.   3. A line with negative m is below the x-axis.

C. Your own conjecture

Write a conjecture about lines y = mx + c. Test it with three cases, including one unusual case. ______________________

D. Stretch: tests or proof?

Conjecture 1 passed every test. Does that prove it? What would convince you it is true for every pair of lines?

Stating the general pattern

After a week of experiments, students state in general what m and c do, read rules off graphs, match eight graphs to their rules, and play the quiz as a class game.

We are learning to

• state the roles of m and c in y = mx + c as general rules,
• write the rule of a line from its graph by reading c and the slope,
• use the general pattern to describe or sketch any line quickly.

Success criteria

• I can write four general statements about m and c without looking.
• I can write the rule of a line from its graph.

You need

The worksheet; the graph match cards (cut-out sheet 2), one set per pair; the quiz on the unit page.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after the match; start the second with Part A and the quiz.

Watch for these ideas

• Reading c from where the line meets the x-axis.
• Writing the slope as positive for a falling line.
• Stating a pattern from one example instead of many.

Answers

• Part A: m sets the steepness and direction; c sets where the line crosses the y-axis; the same m means parallel lines; changing c shifts the line up or down.
• Part B: A: y = 2x − 2; B: y = −x + 2; C: y = ½x.
• Part C: a rises steeply and crosses at 3 (y = 4x + 3 for example); b is parallel to y = 2x through the origin (y = 2x); c is flat at −1 (y = −1).
• Part D: y = −x + c for any c, and y = mx + 4 for any m: one family is parallel, the other pivots through (0, 4).

Stating the general pattern

A. The general pattern

Complete each statement so it is true for every line y = mx + c.

1. The number m sets ______________________.   2. The number c sets ______________________.

3. Lines with the same m are ______________________.   4. Changing c moves the line ______________________.

B. Rules from graphs

C. Write a rule

D. Stretch: two families

Describe every line with slope −1, and every line through (0, 4). How are the two families different?

Conjecture cards

For Day 4, one set per pair: 12 claims about lines

Sort into always true and not always true. On every not-always card, write a counterexample on the back.

Teacher note: always true: 7 cards; not always: A line y = mx + c goes through the origin. / A steeper line has a bigger c. / Two lines with the same c are parallel. / A negative m means the line is below the x-axis. / Doubling c doubles every y value.

Graph match cards

For Day 5, one set per pair: eight rules

Eight graphs

Teacher note: the graphs, in printed order, are y = −2x + 4, y = −2, y = x − 3, y = 2x − 1, y = ½x + 1, y = x + 2, y = 3x, y = −x + 1.

What we know: Linear functions

Work on your own. Show your working.

1. Describe y = 5x − 2 and y = −3x + 4: rising or falling, and where each crosses the y-axis. ______________________
2. Order from steepest to least steep: y = 2x + 1,   y = −4x + 1,   y = ½x + 1. ______________________
3. Write a rule for a line parallel to y = 4x + 1 that crosses the y-axis at −3. ______________________
4. Find the slope of the line through (1, 2) and (4, 11). ______________________
5. Without plotting, where does y = 3x − 1 cross the y-axis, and what happens to y when x goes up by 1? ______________________
6. Is this always true? “Lines with the same c are parallel.” Explain. ______________________
7. Give a counterexample to “every line with a positive slope passes through (1, 1)”. ______________________
8. Complete: in y = mx + c, m sets the ________ and c sets the ________. ______________________

Answers and marking guide

Answers with working

1. y = 5x − 2 rises steeply and crosses at −2; y = −3x + 4 falls steeply and crosses at 4.
2. y = −4x + 1, y = 2x + 1, y = ½x + 1 (size of m: 4, 2, ½).
3. y = 4x − 3.
4. (11 − 2) ÷ (4 − 1) = 3.
5. At (0, −1); y goes up by 3.
6. Not always: y = x + 2 and y = 3x + 2 share c = 2 but cross at (0, 2).
7. For example y = x + 5: at x = 1, y = 6, not 1.
8. m sets the steepness and direction; c sets the y-intercept.

A quick three-level guide

Eight questions, four ideas. A student at standard answers most questions with working and backs a claim with a test case or a counterexample.

Weekly class record

Jot a tick as you move around the room; the mini-check fills any gaps. A tick a day is plenty.

The five columns are the five days: both dials, changing m, changing c, conjectures and generalising.