The shape of the lesson
By the end of the lesson, learners can
- Every point has an address: In the lab, set the gradient to 2 and the intercept to \(-1\).
- From equation to table: An equation such as \(y = -2x + 5\) is a rule: put in an \(x\) value, get out the matching \(y\) value.
- Plot, rule, check: Plot every point from the table with a small cross, then draw ONE straight line through them all with a ruler, across the full range of \(x\) asked for.
- Coordinates: \((x, y)\), across first, then up or down.
- Table: substitute each \(x\) carefully; brackets round negative numbers.
- Pattern check: the \(y\) values change by the same amount each step.
- Plot and rule: one straight line through every point, across the full range.
- Special lines: \(x = k\) is vertical, \(y = k\) is horizontal.
One picture every idea returns to
A point is written \((x, y)\): across first, then up or down. To draw a linear graph, choose some \(x\) values, work out \(y\) for each, plot the points and join them with ONE ruled straight line. If a point is off the line, one of your calculations is wrong.
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, Plot the points, rule the line.. Every point on a grid has an address \((x, y)\). 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: Every point has an address | In the lab, set the gradient to 2 and the intercept to \(-1\). 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 Finding the fourth corner on the board, one line at a time. | Whole class, then pairs |
| 13 to 21 min | Teach and check: From equation to table | An equation such as \(y = -2x + 5\) is a rule: put in an \(x\) value, get out the matching \(y\) value. Model the worked example Completing a table on the board, one line at a time. Your turn: Complete the table of values for \(y = 3x - 1\) for \(x = -2, -1, 0, 1, 2\). | Pairs |
| 21 to 30 min | Teach and check: Plot, rule, check | Plot every point from the table with a small cross, then draw ONE straight line through them all with a ruler, across the full range of \(x\) asked for. Model the worked example Using the axis crossings (Extended) on the board, one line at a time. | Cooperative Concept Mapping |
| 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 Plot, rule, check, 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.
Cooperative Concept Mapping
Teams receive the key terms on cards and arrange them on a large sheet, drawing and labelling the links. Compare maps between teams before showing the model map.
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 |
|---|---|
| Writing the coordinates the wrong way round. \((x, y)\) is always across first, then up: 'along the corridor, then up the stairs'. \((-1, 3)\) and \((3, -1)\) are in different quadrants. | Return to Every point has an address: ask a pair to state the correct version and show it on the lab before moving on. |
| Sign slips with negative \(x\): \(-2 \times (-1)\) is \(+2\), not \(-2\). | If one value breaks the steady pattern in your table, recheck that column before you plot. |
| Mixing up \(x = 3\) and \(y = 3\). | The line \(x = 3\) is VERTICAL (every point has \(x = 3\)), even though it crosses the \(x\)-axis. Picture a few points on it before you draw. |
Coordinates and drawing linear graphs: worksheet
6 questions, 17 marks. Written in the style of the papers; no past paper question is reproduced.
(a) Write down the coordinates of the point that is 3 units to the left of \((2, -1)\) and then 4 units up.
(b) Write down the coordinates of the point where the line \(x = -3\) crosses the \(x\)-axis.
(a) Complete the table of values for \(y = 2x - 3\).
\(x\): \(-2, -1, 0, 1, 2, 3\)
(b) Describe how you would draw the graph for \(-2 \le x \le 3\).
(Extended) The line \(2x + 5y = 10\) is drawn on a grid.
(a) Find the coordinates of the points where it crosses the \(x\)-axis and the \(y\)-axis.
(b) Find the value of \(y\) on the line when \(x = -5\).
Show whether each point lies on the line \(y = 2 - 3x\).
(a) \((-3, 11)\) (b) \((4, -9)\)
\(A(-2, 1)\), \(B(3, 1)\) and \(C(5, 4)\) are three vertices of a parallelogram \(ABCD\).
Find the coordinates of \(D\).
A taxi charges a fixed $3 plus $2 per kilometre, so the cost of a journey of \(x\) km is \(C = 2x + 3\) dollars.
(a) Complete a table of values for \(x = 0, 2, 4, 6, 8\).
(b) The graph of \(C\) against \(x\) is drawn. Use the equation to find the length of a journey that costs $13.
(c) Explain why the graph is a straight line.
Answer key and marking notes
- (a) \(x = 2 - 3 = -1\) and \(y = -1 + 4 = 3\), so \((-1, 3)\).
- (b) On the \(x\)-axis \(y = 0\), so \((-3, 0)\).
- (a) \(y\): \(-7, -5, -3, -1, 1, 3\) (each step of 1 in \(x\) adds 2).
- (b) Plot \((-2, -7)\), \((-1, -5)\), \((0, -3)\), \((1, -1)\), \((2, 1)\), \((3, 3)\) and draw one ruled straight line from \(x = -2\) to \(x = 3\).
- (a) \(y = 0\): \(2x = 10\), so \((5, 0)\). \(x = 0\): \(5y = 10\), so \((0, 2)\).
- (b) \(2 \times (-5) + 5y = 10\), so \(5y = 20\) and \(y = 4\).
- (a) \(2 - 3 \times (-3) = 2 + 9 = 11\). This matches, so \((-3, 11)\) is on the line.
- (b) \(2 - 3 \times 4 = 2 - 12 = -10\), not \(-9\), so \((4, -9)\) is not on the line.
- In a parallelogram \(AD\) is the same step as \(BC\).
- From \(B\) to \(C\): 2 across and 3 up.
- From \(A(-2, 1)\), go 2 across and 3 up: \(D(0, 4)\).
- (a) \(C\): \(3, 7, 11, 15, 19\).
- (b) \(2x + 3 = 13\), so \(2x = 10\) and \(x = 5\) km (on the graph, go across from 13 on the \(C\)-axis to the line, then down).
- (c) Each extra kilometre always adds the same $2, so the cost goes up in equal steps: the equation is linear.
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.
What are the coordinates of the point \(P\) in the diagram?
- A\((-3, 2)\)
- B\((2, -3)\)
- C\((3, 2)\)
- D\((-3, -2)\)
A table of values is made for \(y = 5 - 3x\). What is the value of \(y\) when \(x = -2\)?
- A\(11\)
- B\(-1\)
- C\(-4\)
- D\(-11\)
Which of these points lies on the line \(y = 2x - 3\)?
- A\((4, 5)\)
- B\((5, 4)\)
- C\((0, 3)\)
- D\((1, 1)\)
Which of these describes the graph of \(y = -2\)?
- AA horizontal line through \((0, -2)\)
- BA vertical line through \((-2, 0)\)
- CA line through the origin with gradient \(-2\)
- DA horizontal line through \((-2, 0)\)
Where does the line \(4x - 3y = 12\) cross the axes?
- A\((3, 0)\) and \((0, -4)\)
- B\((3, 0)\) and \((0, 4)\)
- C\((-3, 0)\) and \((0, 4)\)
- D\((4, 0)\) and \((0, -3)\)
The point \((a, 7)\) lies on the line \(3x + 2y = 5\). Find the value of \(a\).
- A\(-3\)
- B\(3\)
- C\(-\dfrac{2}{3}\)
- D\(\dfrac{19}{3}\)
Quiz key
From the origin, \(P\) is 3 squares to the LEFT, so \(x = -3\), and 2 squares UP, so \(y = 2\): \(P\) is \((-3, 2)\). The tempting \((2, -3)\) writes the numbers the wrong way round (up first, then across); that is actually the point \(Q\). \((3, 2)\) loses the minus sign for going left, and \((-3, -2)\) gives the wrong sign to the upward step.
\(y = 5 - 3 \times (-2) = 5 - (-6) = 5 + 6 = 11\). The tempting \(-1\) works out \(3 \times (-2) = -6\) but then subtracts 6 instead of subtracting \(-6\). \(-4\) works out \(5 - 3 = 2\) first and multiplies by \(-2\), ignoring the order of operations, and \(-11\) has the sign of the whole answer flipped.
For \((4, 5)\): \(2 \times 4 - 3 = 5\), which matches \(y = 5\), so it lies on the line. The tempting \((5, 4)\) has the same numbers swapped: \(2 \times 5 - 3 = 7\), not 4. For \((0, 3)\) the line gives \(y = -3\) (the intercept is \(-3\), not 3), and for \((1, 1)\) it gives \(2 - 3 = -1\), not 1.
Points such as \((0, -2)\), \((1, -2)\) and \((5, -2)\) all have \(y = -2\), so they lie in a horizontal row 2 units below the \(x\)-axis, crossing the \(y\)-axis at \((0, -2)\). The tempting vertical line through \((-2, 0)\) is \(x = -2\). A line through the origin with gradient \(-2\) is \(y = -2x\), and a horizontal line through \((-2, 0)\) is the \(x\)-axis, \(y = 0\).
Put \(y = 0\): \(4x = 12\), so \(x = 3\), giving \((3, 0)\). Put \(x = 0\): \(-3y = 12\), so \(y = -4\), giving \((0, -4)\). The tempting \((0, 4)\) loses the minus sign when dividing 12 by \(-3\). \((-3, 0)\) and \((0, 4)\) has both signs wrong, and \((4, 0)\) and \((0, -3)\) reads the coefficients as the crossing points instead of dividing 12 by them.
\(3a + 2 \times 7 = 5\), so \(3a + 14 = 5\), \(3a = -9\) and \(a = -3\). Check: \(3 \times (-3) + 14 = 5\). The tempting \(3\) subtracts the wrong way round, \(3a = 14 - 5 = 9\). \(-\dfrac{2}{3}\) forgets to multiply the 7 by 2 (\(3a + 7 = 5\)), and \(\dfrac{19}{3}\) moves the 14 across without changing its sign (\(3a = 5 + 14\)).
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.