The shape of the lesson
By the end of the lesson, learners can
- Pythagoras on a grid: In the lab, drag the end points and watch the right-angled triangle under the segment.
- Halfway between: The midpoint sits halfway across and halfway up, so each of its coordinates is the mean of the two end coordinates: add and halve.
- Putting them together: Exam questions often combine both skills: the centre of a circle is the midpoint of a diameter, the radius is half its length, and a known distance can be turned into an equation for a missing coordinate.
- Steps: work out the change in \(x\) and the change in \(y\), with brackets for negatives.
- Length: \(\sqrt{(\text{change in } x)^2 + (\text{change in } y)^2}\).
- Midpoint: \(\left(\dfrac{x_1 + x_2}{2}, \dfrac{y_1 + y_2}{2}\right)\).
- Missing end point: repeat the step from the known end to the midpoint.
- Accuracy: exact surd or 3 significant figures.
One picture every idea returns to
For \(A(x_1, y_1)\) and \(B(x_2, y_2)\): length \(AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\), and the midpoint is \(\left(\dfrac{x_1 + x_2}{2}, \dfrac{y_1 + y_2}{2}\right)\). Length uses differences; midpoint uses sums.
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, How far, and halfway.. Extended. 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: Pythagoras on a grid | In the lab, drag the end points and watch the right-angled triangle under the segment. Project tab 2 of the lab (Straight lines lab, length and midpoint) and let a student drive it while the class predicts. Model the worked example A whole-number length on the board, one line at a time. | Whole class, then pairs |
| 13 to 21 min | Teach and check: Halfway between | The midpoint sits halfway across and halfway up, so each of its coordinates is the mean of the two end coordinates: add and halve. Model the worked example Midpoint from two points on the board, one line at a time. Your turn: Find the midpoint of \((-5, 6)\) and \((1, -4)\). | Pairs |
| 21 to 30 min | Teach and check: Putting them together | Exam questions often combine both skills: the centre of a circle is the midpoint of a diameter, the radius is half its length, and a known distance can be turned into an equation for a missing coordinate. Model the worked example Centre and radius of a circle on the board, one line at a time. | Round Table |
| 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 Putting them together, 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.
Round Table
One sheet travels round the team. Each student adds one line of working or one idea and passes it on, so the solution is built four hands at a time and every contribution is visible.
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 |
|---|---|
| Adding before squaring. \(\sqrt{6^2 + 8^2}\) is 10, but \(6 + 8 = 14\) and \(\sqrt{6 + 8} = 3.74\) are both wrong. | Square each step first, add, then take the square root of the total. |
| Halving the difference instead of the sum. | For \((-3, 8)\) and \((9, -2)\), \(\dfrac{9 - (-3)}{2} = 6\) is half the run, not the midpoint's \(x\)-coordinate. The midpoint is an average: ADD, then halve. |
| Forgetting the negative square root. \((k - 2)^2 = 25\) gives \(k - 2 = \pm 5\): a point 13 units away can be on either side, so there are usually two answers. | Return to Putting them together: ask a pair to state the correct version and show it on the lab before moving on. |
Length and midpoint of a line segment: worksheet
6 questions, 17 marks. Written in the style of the papers; no past paper question is reproduced.
Find the midpoint of the line segment joining \((-4, 3)\) and \((6, -9)\).
Calculate the length of the line segment joining \((2, -3)\) and \((-4, 5)\).
Calculate the length of the line segment joining \((1, 7)\) and \((6, -1)\). Give your answer correct to 3 significant figures.
\(M(4, -1)\) is the midpoint of \(AB\), where \(A\) is \((-2, 5)\).
(a) Find the coordinates of \(B\).
(b) Calculate the length of \(AB\), correct to 3 significant figures.
The vertices of triangle \(ABC\) are \(A(1, 1)\), \(B(7, 3)\) and \(C(3, 7)\).
Show that triangle \(ABC\) is isosceles.
On a map, one unit on the grid represents 1 km. Town \(P\) is at \((-6, 2)\) and town \(Q\) is at \((10, 14)\). A straight road joins them, and a service station is built exactly halfway along it.
(a) Find the coordinates of the service station.
(b) A driver travels from \(P\) to the service station at an average speed of 60 km/h. How many minutes does this take?
Answer key and marking notes
- \(x = \dfrac{-4 + 6}{2} = 1\), \(y = \dfrac{3 + (-9)}{2} = -3\).
- Midpoint \((1, -3)\).
- Change in \(x = -6\), change in \(y = 8\).
- Length \(= \sqrt{36 + 64} = \sqrt{100} = 10\) units.
- Change in \(x = 5\), change in \(y = -8\).
- Length \(= \sqrt{25 + 64} = \sqrt{89} = 9.43\) units (3 s.f.).
- (a) From \(A\) to \(M\): \(+6\) across, \(-6\) up. So \(B = (4 + 6, -1 - 6) = (10, -7)\).
- (b) Change in \(x = 12\), change in \(y = -12\).
- \(AB = \sqrt{144 + 144} = \sqrt{288} = 16.97\ldots = 17.0\) units (3 s.f.).
- \(AB = \sqrt{6^2 + 2^2} = \sqrt{40}\).
- \(AC = \sqrt{2^2 + 6^2} = \sqrt{40}\).
- \(BC = \sqrt{(-4)^2 + 4^2} = \sqrt{32}\).
- \(AB = AC\), so two sides are equal and the triangle is isosceles.
- (a) \(\left(\dfrac{-6 + 10}{2}, \dfrac{2 + 14}{2}\right) = (2, 8)\).
- (b) \(PQ = \sqrt{16^2 + 12^2} = \sqrt{400} = 20\) km, so \(P\) to the station is 10 km.
- Time \(= 10 \div 60 = \dfrac{1}{6}\) hour \(= 10\) minutes.
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.
Find the length of the line segment joining \(A(-1, 3)\) and \(B(5, -5)\).
- A\(10\)
- B\(14\)
- C\(3.74\)
- D\(4.47\)
Find the midpoint of the line segment joining \((-4, 7)\) and \((8, -3)\).
- A\((2, 2)\)
- B\((6, -5)\)
- C\((4, 4)\)
- D\((-6, 5)\)
Find the length of \(PQ\), where \(P(2, -3)\) and \(Q(-4, 1)\). Give your answer correct to 3 significant figures.
- A\(7.21\)
- B\(4.47\)
- C\(10\)
- D\(2.83\)
\(M(3, -2)\) is the midpoint of \(AB\), where \(A\) is \((-1, 5)\). Find the coordinates of \(B\).
- A\((7, -9)\)
- B\((1, 1.5)\)
- C\((4, -7)\)
- D\((5, 1)\)
\(AB\) is a diameter of a circle, where \(A(-2, 1)\) and \(B(6, 7)\). Find the centre and the radius of the circle.
- ACentre \((2, 4)\), radius 5
- BCentre \((2, 4)\), radius 10
- CCentre \((4, 3)\), radius 5
- DCentre \((2, 4)\), radius 7
The point \(C(k, 4)\) is 10 units from \(D(3, -2)\). Find the possible values of \(k\).
- A\(11\) or \(-5\)
- B\(11\) only
- C\(13\) or \(-7\)
- D\(5\) or \(-11\)
Quiz key
Across: \(5 - (-1) = 6\). Up: \(-5 - 3 = -8\). Length \(= \sqrt{6^2 + (-8)^2} = \sqrt{36 + 64} = \sqrt{100} = 10\). The tempting 14 just adds the steps \(6 + 8\), which is the route along two sides of the triangle, not the straight line. 3.74 is \(\sqrt{6 + 8}\) (adding before squaring), and 4.47 adds the coordinates instead of subtracting them: \(\sqrt{4^2 + (-2)^2}\).
\(x = \dfrac{-4 + 8}{2} = 2\) and \(y = \dfrac{7 + (-3)}{2} = 2\), so the midpoint is \((2, 2)\). The tempting \((6, -5)\) halves the DIFFERENCES (\(\dfrac{8 - (-4)}{2}\) and \(\dfrac{-3 - 7}{2}\)), which gives half the step, not the halfway point. \((4, 4)\) adds but forgets to halve, and \((-6, 5)\) halves the differences taken the other way round.
Across: \(-4 - 2 = -6\). Up: \(1 - (-3) = 4\). \(PQ = \sqrt{36 + 16} = \sqrt{52} = 7.21\) (3 s.f.). The tempting 4.47 is \(\sqrt{36 - 16}\): squaring the negative step as \(-36\) or subtracting the squares. 10 adds the steps \(6 + 4\) without Pythagoras, and 2.83 adds the coordinates instead of subtracting: \(\sqrt{(-2)^2 + (-2)^2}\).
From \(A\) to \(M\): across \(3 - (-1) = 4\), up \(-2 - 5 = -7\). Repeat the step from \(M\): \(B = (3 + 4, -2 - 7) = (7, -9)\). Check: \(\left(\dfrac{-1 + 7}{2}, \dfrac{5 - 9}{2}\right) = (3, -2)\). The tempting \((1, 1.5)\) is the midpoint of \(A\) and \(M\), which goes the wrong way. \((4, -7)\) is only the step from \(A\) to \(M\), and \((5, 1)\) is \(2M + A\) instead of \(2M - A\).
Centre \(= \left(\dfrac{-2 + 6}{2}, \dfrac{1 + 7}{2}\right) = (2, 4)\). Diameter \(= \sqrt{8^2 + 6^2} = \sqrt{100} = 10\), so the radius is 5. The tempting radius 10 gives the whole diameter. Centre \((4, 3)\) halves the differences instead of averaging the coordinates, and radius 7 halves \(8 + 6\) without using Pythagoras.
\((k - 3)^2 + 6^2 = 100\), so \((k - 3)^2 = 64\) and \(k - 3 = \pm 8\), giving \(k = 11\) or \(k = -5\). The tempting \(11\) only forgets the negative square root: \(C\) can be on either side of \(D\). \(13\) or \(-7\) leaves out the \(6^2\) (\((k - 3)^2 = 100\)), and \(5\) or \(-11\) uses \(k + 3\) instead of \(k - 3\).
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.