Linear equations and graphs: 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, step cards, inequality 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 2: the step cards are for Day 2, the match cards for Day 4.
4. Open the free interactive unit on the board. Every plan says which picture to show and 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

A linear relation such as y = 2x + 1 gives one y for every x, and its points all lie on one straight line. Setting y to a number turns the relation into an equation: 2x + 1 = 7 asks which x gives y = 7. There are two ways to answer. Algebraically, undo the operations in reverse order, doing the same to both sides: subtract 1, then divide by 2, so x = 3. Graphically, find where the line meets the horizontal line y = 7. An inequality such as 2x + 1 > 7 asks for every x that works, so its answer is a range, x > 3, shown on a number line. One rule is new: multiplying or dividing both sides by a negative number reverses the inequality sign. Any answer can be checked by putting it back into the original.

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

The interactive unit draws the graphs on the board. If your class has graphing software or a graphing calculator, students can check their Day 1 and Day 3 graphs with it after drawing them by hand; the pack does not depend on it. On number lines an open circle means the end value is not included (> or <) and a filled circle means it is (≥ or ≤).

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 draws linear relations; Day 2 solves equations by undoing; Day 3 shows the same solutions on a graph, so the two methods check each other; Day 4 stretches equations into inequalities; and Day 5 makes checking by substitution a habit. It builds on the linear expressions unit and Year 7 work on the Cartesian plane and one-step equations. It prepares linear modelling later in Year 8, gradient and intercepts in Year 9, and simultaneous equations in Year 10.

Materials for the week (one trip)

• From the classroom: rulers, sharp pencils, coloured pencils, scissors, this pack printed.
• Cut-out sheet 1, one per pair: 12 step cards for three equations (Day 2).
• Cut-out sheet 2, one per pair: 8 inequality cards and 8 number-line cards (Day 4).
• Optional: graphing software or calculators for checking graphs. No other equipment to buy.

Words for the word wall

Cartesian plane, axis, origin, coordinates, table of values, linear relation, straight line, equation, solution, inverse operation, both sides, substitute, verify, inequality, greater than, less than, at least, at most, open circle, closed circle, number line

Dear families

This week in maths, Year 8 is solving equations such as 2x + 3 = 11 in two ways: by undoing the operations step by step (subtract 3, then divide by 2, so x = 4) and by reading where two lines cross on a graph. We also solve inequalities such as 2x + 1 > 7, whose answer is a range of numbers, and we always check answers by putting them back into the start.

Try this at home

• Ask: I think of a number, multiply by 3 and add 5. I get 26. What was my number? (7: take away 5, then divide by 3.)
• Ask: a taxi costs $4 plus $2 a kilometre. How far can you go for $20? (8 km.)
• Ask: which whole numbers make 2x < 9 true? (4 and below.)
• Ask your child to check an answer by substituting it back. Does it make both sides equal?

My equations this week

Fill one row a day. Write an equation from real life, solve it, and check by substituting.

Printed from the free seegongsik Linear equations and graphs teaching pack · seegongsik.com/au/y8/algebra/AC9M8A02/pack

A line on the Cartesian plane

A linear relation gives one y for each x, and its points lie on a straight line. Students make tables of values with negative x, plot two relations on one grid, test points by substituting, and read values off the graph.

We are learning to

• complete a table of values for a linear relation, including negative x,
• plot the points and rule a straight line through them,
• decide by substituting whether a point lies on a line, and read values from a graph.

Success criteria

• I can plot y = 2x + 1 from a table and explain why the points line up.
• I can show by substituting that (2, 4) is on y = 3x − 2.

You need

The worksheet, a ruler and two coloured pencils each.

Lesson flow (about 60 minutes)

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

Watch for these ideas

• Plotting (y, x) instead of (x, y): across first, then up or down.
• Getting negative y wrong: 3 × (−2) − 2 = −8.
• Joining the points dot to dot with wobbles instead of ruling one straight line.

Answers

• Part A: y = 3x − 2: −8, −5, −2, 1, 4, 7; y = −x + 4: 6, 5, 4, 3, 2, 1.
• Part B: (2, 4) on; (0, 2) not on (gives −2); (−1, −5) on; (4, 10) on; (1, 3) not on (gives 1).
• Part C: y = 2.5 when x = 1.5, and x = 3 when y = 7 (3 × 3 − 2 = 7).
• Part D: from x = 1 (1 against 3) to x = 2 (4 against 2) the order swaps; at x = 1.5 both give 2.5, so they cross at (1.5, 2.5).

A line on the Cartesian plane

A. Tables, then graphs

B. On y = 3x − 2 or not?

Substitute each x and compare with the y. Circle on or not.

C. Read the graph

On your graph of y = 3x − 2: what is y when x = 1.5? What is x when y = 7? Check both by substituting.

D. Stretch: where do they cross?

Between which two x values in your tables do the lines cross? Try the x halfway between and check it in both rules.

Solving equations step by step

An equation stays balanced if both sides get the same treatment. Students undo the operations in reverse order, solve equations with brackets and fractions, and turn stories into equations.

We are learning to

• solve a two-step equation by undoing the operations in reverse order on both sides,
• solve equations with a bracket, a fraction or a negative coefficient,
• write an equation for a story and answer the question in words.

Success criteria

• I can solve 3x + 5 = 26 and say what I did to both sides each time.
• I can write and solve an equation for a taxi or shopping story.

You need

The worksheet; the step cards (cut-out sheet 1), one set per pair. No calculators.

Lesson flow (about 60 minutes)

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

Watch for these ideas

• Doing an operation to one side only.
• Undoing in the wrong order: in 4x + 7 = 35, divide by 4 only after taking 7 away.
• Dividing by a negative and losing the sign: −2x = −8 gives x = 4.

Answers

• Part A: 4x + 7 = 35: x = 7; 5x − 3 = 22: x = 5; 3x + 12 = 0: x = −4; −2x + 9 = 1: x = 4; 6x − 5 = −23: x = −3; 7x + 4 = 4: x = 0.
• Part B: 3(x − 2) = 12: x − 2 = 4, x = 6; x/3 + 2 = 5: x/3 = 3, x = 9; 2(x + 5) = 6: x + 5 = 3, x = −2.
• Part C: 2x + 4 = 30, so 13 km; 3x + 2 = 20, so 6 kg; −6x + 50 = 14, so 6 weeks.
• Part D: 5x + 3 = 2x + 15: 3x = 12, x = 4; 7x − 4 = 3x + 8: 4x = 12, x = 3.

Solving equations step by step

A. Two steps

Show what you do to both sides at each step.

B. Brackets and fractions

C. Stories into equations

Write an equation, solve it, and answer in a sentence.

D. Stretch: x on both sides

Solve 5x + 3 = 2x + 15 and 7x − 4 = 3x + 8. First move the x terms to one side.

The same equation solved with a graph

Solving 3x − 1 = r asks where the line y = 3x − 1 reaches the height r. Students rule horizontal lines on a printed graph, read off solutions, estimate ones that are not whole, and check each against the algebra.

We are learning to

• solve an equation by finding where a line meets a horizontal line,
• estimate a solution that is not a whole number and refine it with algebra,
• find where two lines cross and explain what the crossing point solves.

Success criteria

• I can show on a graph that 2x + 3 = 11 has the solution x = 4.
• I can say why a graph gives an estimate and the algebra gives the exact answer.

You need

The worksheet, a ruler and a coloured pencil 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 the y of the crossing instead of the x.
• Ruling a vertical line instead of a horizontal one for y = 5.
• Trusting a graph reading to more accuracy than the grid allows.

Answers

• Parts A and B: 3x − 1 = 5: x = 2; 3x − 1 = −4: x = −1; 3x − 1 = 8: x = 3; 3x − 1 = 0.5: x = 0.5.
• Part C: a graph shows every solution at once and gives a quick estimate; algebra gives exact answers, especially when they are not whole numbers.
• Part D: y = x + 4 and y = −2x + 10 cross at (2, 6), which solves x + 4 = −2x + 10: 3x = 6, x = 2.

The same equation solved with a graph

A. Read the solutions

The line is y = 3x − 1. For each equation, rule the horizontal line, mark where it crosses, and read off x.

B. Check with algebra

C. Graph or algebra?

Give one thing a graph does better and one thing algebra does better. ______________________________________________

D. Stretch: two sloping lines

On the same grid, draw y = x + 4 and y = −2x + 10. Where do they cross? Which equation does that point solve?

Solving a linear inequality

An inequality is solved like an equation, with one new rule: multiplying or dividing by a negative reverses the sign. Students solve inequalities, show the range on a number line and test values.

We are learning to

• solve a linear inequality using inverse operations on both sides,
• reverse the inequality sign when multiplying or dividing by a negative number,
• show a solution on a number line with an open or closed circle.

Success criteria

• I can solve −2x + 1 ≤ 9 and explain why the answer is x ≥ −4.
• I can draw x ≤ 2 and x > 2 on number lines and say how they differ.

You need

The worksheet; the inequality match cards (cut-out sheet 2), one set per pair.

Lesson flow (about 60 minutes)

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

Watch for these ideas

• Forgetting to flip when dividing by a negative.
• Flipping when subtracting a number: only multiplying or dividing by a negative flips.
• Using a closed circle for > or <.

Answers

• Part A: 3x + 2 > 14: x > 4; 2x − 5 ≤ 3: x ≤ 4; −4x < 12: x > −3; 5x + 1 ≥ −9: x ≥ −2; −3x + 6 > 0: x < 2; x + 7 < 4: x < −3.
• Part B: x = 6 yes; x = 4 no; x = 10 yes; x = 0 no; x = −2 no; x = 5 yes (the solution is x > 4).
• Part C: 2.5n ≤ 30 gives n ≤ 12, so at most 12 songs; 4h + 10 ≥ 50 gives h ≥ 10, at least 10 hours.
• Part D: with x = −4, −2x = 8 > 6 is true and x < −3 is true; with x = 0, −2x = 0 > 6 is false. Only the flipped sign keeps the true values.

Solving a linear inequality

A. Solve and draw

Solve, then sketch the solution on a number line with an open or closed circle.

B. Test values

Is each value a solution of 2x − 3 > 5? Substitute, then circle yes or no.

C. At most, at least

1. Songs cost $2.50 each and you have $30. Write and solve an inequality for the number of songs, n.

2. A job pays $4 an hour plus a $10 bonus. How many hours, h, earn at least $50?

D. Stretch: prove the flip

For −2x > 6, test x = −4 and x = 0 in both the inequality and the answer x < −3. What would go wrong without the flip?

Checking a solution by substitution

Every solution can be checked by putting it back into the original equation. Students test candidate answers, find the slip in worked solutions, test values in inequalities, and play the quiz as a class game.

We are learning to

• verify a solution by substituting it into the original equation, left side and right side,
• locate and correct the step where a worked solution went wrong,
• write equations that have a given solution.

Success criteria

• I can show by substitution that x = 3 solves 5x − 4 = 11 and x = 2 does not.
• I can find the wrong step in a worked solution and fix it.

You need

The worksheet, one per student; the self-check quiz on the unit page for the game.

Lesson flow (about 60 minutes)

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

Watch for these ideas

• Checking in a later line, where a slip has already happened.
• Forgetting brackets when substituting a negative.
• Thinking an inequality has one answer: test several values.

Answers

• Part A: 4x − 3 = 17, x = 5: LHS 17, correct; 2x + 9 = 1, x = 4: LHS 17, wrong (x = −4); −3x + 2 = 14, x = −4: LHS 14, correct; 5x + 5 = 0, x = −1: LHS 0, correct; 6x − 1 = 11, x = 3: LHS 17, wrong (x = 2).
• Part B: 3x + 4 = 19: added 4 instead of subtracting: 3x = 15, x = 5; 2x − 6 = 10: subtracted 6 instead of adding: 2x = 16, x = 8; −5x = 20: lost the sign: x = −4.
• Part C: 3 < 7 − x is true for x = 2 and false for x = 4 and x = 5; the solution is x < 4.
• Part D: for example 3x + 1 = −5, x − 4 = −6 and 2(x + 5) = 6, each checked by substituting −2.

Checking a solution by substitution

A. Right or wrong?

Substitute the value into the left side and compare with the right side. If it is wrong, find the true solution.

B. Find the slip

C. Test an inequality

Which of x = 2, 4 and 5 make 3 < 7 − x true? Use your results to write the solution of the inequality.

D. Stretch: work backwards

Write three different equations whose solution is x = −2. Check each one by substitution.

Equation step cards

For Day 2, one set per pair: three equations, four cards each

Cut out each row’s cards. Lay each equation’s four cards in the order you would solve it, then check the answer by substituting.

Teacher note: 4x − 7 = 13: add 7 to both sides, 4x = 20, divide both sides by 4, x = 5; 2(x + 3) = 18: divide both sides by 2, x + 3 = 9, subtract 3 from both sides, x = 6; x/5 − 1 = 3: add 1 to both sides, x/5 = 4, multiply both sides by 5, x = 20.

Inequality match cards

For Day 4, one set per pair: eight inequalities

Eight number lines

Teacher note: the number lines, in printed order, show x ≤ 3, x < 0, x > 2, x ≥ −1, x ≥ 3, x > 0, x < 2, x ≤ −1.

What we know: Linear equations and graphs

Work on your own, without a calculator. Show your working.

1. Complete the table for y = −2x + 5 at x = −1, 0, 2. ______________________
2. Is the point (3, −1) on y = −2x + 5? Show how you know. ______________________
3. Solve 5x − 8 = 27. ______________________
4. Solve −3x + 4 = 19. ______________________
5. Solve 4(x + 1) = 28. ______________________
6. Solve 2x + 5 < 17 and show it on a number line. ______________________
7. Solve −5x ≥ 15. ______________________
8. Someone says x = 5 solves 3x − 7 = 11. Check by substitution, and give the true solution. ______________________

Answers and marking guide

Answers with working

1. x = −1: y = 7; x = 0: y = 5; x = 2: y = 1.
2. At x = 3, y = −1, so (3, −1) is on the line.
3. 5x = 35, x = 7.
4. −3x = 15, x = −5.
5. x + 1 = 7, x = 6.
6. 2x < 12, x < 6: an open circle at 6 with the arrow to the left.
7. Divide by −5 and flip: x ≤ −3.
8. LHS = 3 × 5 − 7 = 8, not 11: wrong. 3x = 18, x = 6.

A quick three-level guide

Eight questions, four ideas. A student at standard answers most questions with working and checks each solution by substitution.

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: graphing, solving step by step, solving with a graph, inequalities and checking.