The shape of the lesson
By the end of the lesson, learners can
- Gradient and intercept: In the lab, set \(m\) and \(c\) with the sliders.
- Rearrange to y =: (Extended) A line may be given as \(ax + by = c\), such as \(5x + 4y = 8\).
- Finding the equation: Whatever you are given, the plan is the same: find the gradient \(m\), substitute the coordinates of one point on the line into \(y = mx + c\) to find \(c\), then write the equation in its simplest form.
- Read: in \(y = mx + c\), \(m\) is the gradient and \(c\) the \(y\)-intercept.
- Rearrange (Extended): make \(y\) the subject, dividing every term.
- Gradient: from the graph, or \(\dfrac{y_2 - y_1}{x_2 - x_1}\) from two points.
- Intercept: substitute one point into \(y = mx + c\) and solve for \(c\).
- Check: the other point must fit; give the equation fully simplified.
One picture every idea returns to
In \(y = mx + c\), \(m\) is the gradient and \(c\) is the \(y\)-intercept (the line crosses the \(y\)-axis at \((0, c)\)). To find an equation: get the gradient, then substitute one point to find \(c\). A vertical line has equation \(x = k\).
Forty-five minutes, phase by phase
| Time | Phase | What happens in the room | Grouping |
|---|---|---|---|
| 0 to 5 min | Hook: predict | Open with the question in the title, Two numbers, one line.. Every non-vertical straight line can be written as \(y = mx + c\). Put the figure on the board: it is the one picture every idea in the lesson returns to. | Whole class, then pairs |
| 5 to 13 min | Teach and check: Gradient and intercept | In the lab, set \(m\) and \(c\) with the sliders. Project tab 3 of the lab (Straight lines lab, equation of a line) and let a student drive it while the class predicts. Model the worked example Reading the gradient and intercept on the board, one line at a time. | Whole class, then pairs |
| 13 to 21 min | Teach and check: Rearrange to y = | (Extended) A line may be given as \(ax + by = c\), such as \(5x + 4y = 8\). Model the worked example From ax + by = c (Extended) on the board, one line at a time. Your turn: (Extended) Find the gradient and \(y\)-intercept of \(3x - 2y = 6\). | Pairs |
| 21 to 30 min | Teach and check: Finding the equation | Whatever you are given, the plan is the same: find the gradient \(m\), substitute the coordinates of one point on the line into \(y = mx + c\) to find \(c\), then write the equation in its simplest form. Model the worked example Gradient and one point on the board, one line at a time. | Jigsaw |
| 30 to 36 min | Check | The four Stage 1 questions from the student lesson, answered on mini whiteboards or by show of fingers. Re-teach the one most get wrong. | Individual, then whole class |
| 36 to 43 min | Plenary | The exam-style questions from Stage 2, one at a time: thirty seconds alone, then a random call. The feedback lines in the key give the examiner's wording. | Individual, random call |
| 43 to 45 min | Exit and homework | Set the worksheet below (6 questions, 17 marks). Learners self-assess against the objectives before they leave. | Individual |
Protect the parts that carry the learning
The sequence carries about fifty minutes of material, so the lab is driven by one student while the class predicts rather than explored freely. Times are a guide.
Protected: the check at 30 minutes. It is the only individual read on whether the key idea landed, and it is not cut.
Cut first if the build overruns: the last check in Finding the equation, never the check or the plenary.
In a 60 minute block: run worksheet questions 1 to 3 in class as a Rally Coach set and model both the hardest worked example and its check live.
Jigsaw
Split the content into expert pieces. Expert groups master one piece each, then regroup so every home team has one expert per piece, and each expert teaches their piece to the team.
The full structure, with its variations and the other twelve, is on the cooperative-learning page.
What to head off, and how
| Trap learners fall into | Teaching move that pre-empts it |
|---|---|
| Taking the first number as the gradient. | In \(y = 5 - 2x\) the gradient is \(-2\), not 5: the gradient is always the number in front of \(x\), with its sign. |
| Dividing only some of the terms. | From \(4y = -5x + 8\), the equation \(y = -5x + 2\) divides only the 8 by 4. Every term on the right must be divided. |
| Putting the coordinates in the wrong places: for the point \((2, 1)\), \(x = 2\) and \(y = 1\). | Writing \(2 = 3 \times 1 + c\) gives the wrong \(c\). Always check your final equation with the point. |
The equation of a straight line: worksheet
6 questions, 17 marks. Written in the style of the papers; no past paper question is reproduced.
Write down the gradient and the \(y\)-intercept of
(a) \(y = 6x + 3\) (b) \(y = 4 - \dfrac{1}{2}x\).
Find the equation of the line with gradient \(-4\) that passes through \((3, -5)\).
(Extended) The equation of a line is \(4x + 3y = 12\).
(a) Find the gradient of the line.
(b) Find the coordinates of the point where the line crosses the \(x\)-axis.
(Extended) Find the equation of the line through \((-3, 9)\) and \((3, -3)\). Give your answer in the form \(y = mx + c\).
A straight line is drawn on a grid. It passes through \((0, -1)\) and \((3, 5)\).
(a) Find the equation of the line.
(b) Does the point \((10, 19)\) lie on the line? Show your working.
A plumber's charge \(C\) dollars for a job lasting \(h\) hours is shown by a straight-line graph. A 1-hour job costs $70 and a 3-hour job costs $130.
(a) Find the equation of the line in the form \(C = mh + c\).
(b) Explain what the values of \(m\) and \(c\) mean.
(c) Find the charge for a job lasting 5.5 hours.
Answer key and marking notes
- (a) gradient 6, \(y\)-intercept 3
- (b) gradient \(-\dfrac{1}{2}\), \(y\)-intercept 4
- \(-5 = -4 \times 3 + c = -12 + c\), so \(c = 7\).
- \(y = -4x + 7\)
- (a) \(3y = -4x + 12\), so \(y = -\dfrac{4}{3}x + 4\). Gradient \(-\dfrac{4}{3}\).
- (b) Put \(y = 0\): \(4x = 12\), so \(x = 3\). The point is \((3, 0)\).
- \(m = \dfrac{-3 - 9}{3 - (-3)} = \dfrac{-12}{6} = -2\).
- \(9 = -2 \times (-3) + c = 6 + c\), so \(c = 3\).
- \(y = -2x + 3\) (check: \(-2 \times 3 + 3 = -3\)).
- (a) \(c = -1\) and \(m = \dfrac{5 - (-1)}{3} = 2\), so \(y = 2x - 1\).
- (b) When \(x = 10\), \(y = 2 \times 10 - 1 = 19\). Yes, \((10, 19)\) lies on the line.
- (a) \(m = \dfrac{130 - 70}{3 - 1} = 30\). \(70 = 30 \times 1 + c\), so \(c = 40\). \(C = 30h + 40\).
- (b) \(m = 30\): the plumber charges $30 for each hour. \(c = 40\): a fixed $40 call-out charge, even for a job of 0 hours.
- (c) \(C = 30 \times 5.5 + 40 = 165 + 40 = 205\), so the charge is $205.
6 multiple-choice questions on this lesson
From the topic question bank. Assign them in the app for a marked score in your gradebook, or run the topic as a class game.
Write down the gradient and the \(y\)-intercept of the line \(y = 7 - 3x\).
- AGradient \(-3\), \(y\)-intercept 7
- BGradient 7, \(y\)-intercept \(-3\)
- CGradient 3, \(y\)-intercept 7
- DGradient \(-3\), \(y\)-intercept \(-7\)
Find the equation of the straight line in the diagram.
- A\(y = 2x - 2\)
- B\(y = \dfrac{1}{2}x - 2\)
- C\(y = 2x + 2\)
- D\(y = 2x + 4\)
Find the equation of the line with gradient \(-2\) that passes through \((3, 5)\).
- A\(y = -2x + 11\)
- B\(y = -2x + 5\)
- C\(y = -2x - 1\)
- D\(y = -2x + 13\)
Find the equation of the line through \((-3, 1)\) and \((-3, 6)\).
- A\(x = -3\)
- B\(y = -3\)
- C\(y = x + 4\)
- D\(x = 1\)
Find the gradient and the \(y\)-intercept of the line \(3x - 2y = 8\).
- AGradient \(\dfrac{3}{2}\), \(y\)-intercept \(-4\)
- BGradient 3, \(y\)-intercept 8
- CGradient \(-\dfrac{3}{2}\), \(y\)-intercept 4
- DGradient \(\dfrac{3}{2}\), \(y\)-intercept 4
Find the equation of the line through \((-2, 7)\) and \((4, -2)\). Give it in the form \(ax + by = c\).
- A\(3x + 2y = 8\)
- B\(2x + 3y = 17\)
- C\(3x - 2y = -20\)
- D\(3x + 2y = -8\)
Quiz key
\(y = 7 - 3x\) is \(y = -3x + 7\), so \(m = -3\) and \(c = 7\). The tempting gradient 7 takes the first number written as the gradient; the gradient is always the coefficient of \(x\). Gradient 3 drops the minus sign that belongs to \(3x\), and intercept \(-7\) gives the minus sign to the wrong term.
The line crosses the \(y\)-axis at \((0, -2)\), so \(c = -2\). From \((0, -2)\) to \((3, 4)\) it rises 6 for a run of 3, so \(m = 6 \div 3 = 2\): \(y = 2x - 2\). The tempting \(y = \dfrac{1}{2}x - 2\) divides run by rise. \(y = 2x + 2\) loses the sign of the intercept, and \(y = 2x + 4\) uses the \(y\)-coordinate of \((3, 4)\) as the intercept, but that point is not on the \(y\)-axis.
\(5 = -2 \times 3 + c\), so \(5 = -6 + c\) and \(c = 11\): \(y = -2x + 11\). Check: \(-6 + 11 = 5\). The tempting \(y = -2x + 5\) uses the point's \(y\)-coordinate as the intercept, but \((3, 5)\) is not on the \(y\)-axis. \(y = -2x - 1\) subtracts 6 instead of adding it, and \(y = -2x + 13\) swaps the coordinates (\(3 = -2 \times 5 + c\)).
Every point on the line has \(x = -3\), so it is the vertical line \(x = -3\) (its gradient is undefined, so it cannot be written as \(y = mx + c\)). The tempting \(y = -3\) is a horizontal line. \(y = x + 4\) passes through \((-3, 1)\) but not \((-3, 6)\), and \(x = 1\) uses a \(y\)-coordinate.
\(-2y = -3x + 8\), so \(y = \dfrac{-3x}{-2} + \dfrac{8}{-2} = \dfrac{3}{2}x - 4\). Gradient \(\dfrac{3}{2}\), intercept \(-4\). The tempting gradient 3, intercept 8 reads the numbers straight from \(3x - 2y = 8\) without rearranging. \(-\dfrac{3}{2}\) and 4 divide by \(+2\) instead of \(-2\), and \(\dfrac{3}{2}\) with 4 divides only the \(x\) term by \(-2\).
\(m = \dfrac{-2 - 7}{4 - (-2)} = \dfrac{-9}{6} = -\dfrac{3}{2}\). Using \((-2, 7)\): \(7 = 3 + c\), so \(c = 4\) and \(y = -\dfrac{3}{2}x + 4\). Multiply by 2: \(2y = -3x + 8\), so \(3x + 2y = 8\). Check \((4, -2)\): \(12 - 4 = 8\). The tempting \(2x + 3y = 17\) uses the gradient upside down (\(-\dfrac{2}{3}\)), so it fits \((-2, 7)\) but not \((4, -2)\). \(3x - 2y = -20\) loses the minus sign of the gradient, and \(3x + 2y = -8\) has the wrong sign on the constant.
What to project
The student lesson embeds these tabs. Open them full screen on the projector; every slider has a drag handle on the stage.
Slides for this lesson
A projectable deck for every pack is on the way. Until then, present the student lesson full screen: its sections, figure and lab tabs follow the same order as the plan above.