At a glance. One lesson of about 60 minutes. It assumes Pythagoras' theorem and coordinates in four quadrants. Students find the distance between two points, find a midpoint, and use a midpoint to find a missing end point.
Calculator: GDC optional. The numbers are chosen so that most answers come out whole. The optional section shows GDC users how to check a decimal for a surd, and says that exact form comes first.
What students will learn
Know
- AB = √((x₂ − x₁)² + (y₂ − y₁)²) for A(x₁, y₁) and B(x₂, y₂).
- The midpoint is the average of the x values and the average of the y values.
- The order of subtraction does not matter for distance, because the differences are squared.
Understand
- The distance formula is Pythagoras on the triangle between the points.
- The midpoint is a general rule found by averaging, and it can be run backwards to find an end point.
- A general formula covers every pair of points, including those in different quadrants.
Do
- Find the distance between two points.
- Find the midpoint of two points.
- Find a missing end point or a missing coordinate.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: applying the distance and midpoint formulae (questions 1 to 8).
- Criterion B. Investigating patterns: the figure. Students collect pairs of points with whole-number distances and describe how the legs relate.
- Criterion C. Communicating: setting out across, up, squares, sum and root as separate lines (worked examples, faded example).
- Criterion D. Applying mathematics in real-life contexts: the cable car in question 7 and the reserve in the hook.
The one thing to do with the figure
Settle the hook on the grid. The gates of the hook are at (1, 2) and (7, 10), which the sliders cannot reach, so use the same shape: set A(−3, −4) and B(3, 4), 6 across and 8 up, and ask for the distance before anyone reads it. Then ask why going across and then up is longer.
Ask students to set A and B so that the midpoint is (0, 0) and to describe what is true about A and B. They are opposite points: each has the other's coordinates with signs changed.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: shorter, longer or the same as 14 km? Collect predictions. |
| 12 min | Figure, key idea, warning card. |
| 8 min | Worked examples; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose B, return to the midpoint as an average. |
| 20 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Something to think about, then the exit ticket, question 9. |
The answers
| Warm-up | 5 cm; 2; 8. |
|---|---|
| Hook | 10 km, shorter than 14 km: across 6 and up 8 give √(36 + 64) = 10. |
| Worked examples | 41; (1, 3); (7, −3). |
| Faded example | 3√10 (about 9.49). |
| 1. Finding a midpoint | A. B halves the differences; C does not halve; D swaps x and y. |
| 2. Distance from the origin | 15. √225. |
| 3. Midpoint coordinate | −2. (−7 + 3) ÷ 2. |
| 4. Distance between two points | 17. √(8² + 15²). |
| 5. Midpoint with negatives | −3. (−9 + 3) ÷ 2. |
| 6. Finding a missing end point | 7. 2 × 5 − 3. |
| 7. Distance in context | 1000. m, from √(280² + 960²). |
| 8. Working back from a distance | 8. up = 5. |
| 9. Distance and midpoint together | B. A adds the legs; C does not halve; D halves the differences. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| B, C or D in question 1 | Halved the differences, did not halve, or swapped x and y. | Ask: is the midpoint a sum or an average? Check by testing with 0 and 10: halfway is 5. |
| 21 or 225 in question 2 | Added the legs, or stopped before the square root. | Draw the triangle: the straight line is shorter than the two legs. Then take the root. |
| 5 or −4 in question 3 | Halved the difference, or did not halve. | Write the average as (a + b) ÷ 2 and substitute into the brackets. |
| 23, 7 or 289 in question 4 | Added the legs, subtracted them, or left out the square root. | Ask: what is the triangle's longest side? Which operation gives it? |
| −6 or 6 in question 5 | Did not halve, or halved the difference. | Rewrite (−9 + 3) first, then divide by 2. |
| 4 or 2 in question 6 | Averaged A and M, or stopped after one step. | Mark A, M and B on a number line: equal steps. |
| 1240 or 1 000 000 in question 7 | Added the legs, or left the squared length. | Sense-check: the cable must be shorter than 280 + 960 and much shorter than a million metres. |
| 5 or 16 in question 8 | Gave the up difference, or used 13 as the up difference. | Find the across difference first, then 13² = across² + up². |
| A, C or D in question 9 | Added the legs, did not halve, or halved the differences. | Check each piece separately: the distance first, then the midpoint. |
Questions worth asking
- Why does the order of subtraction not matter for distance but matter for a gradient? Distance squares the differences, so signs vanish; a gradient keeps the sign of both together.
- What does the midpoint formula do to the two x values? Listen for: it averages them; it finds the number halfway between them.
- Is the distance formula just Pythagoras? Yes: the across and up differences are the legs, and the distance is the hypotenuse.
- Can two different segments have the same length and the same midpoint? Yes: (−1, 0) to (1, 0) and (0, −1) to (0, 1) both have length 2 and midpoint (0, 0); turning a segment about its midpoint keeps both. What is fixed is the other end: a midpoint and one end point give exactly one other end point.
Differentiation
A way in for anyone stuck
- Give squared paper and have students draw the triangle between the points and count the legs.
- Write the formula as a template: across = ___, up = ___, squares ___ + ___ = ___, root ___.
- Start with pairs that share an x or a y, so one leg is zero.
Stretch
- Find the point on the x-axis that is the same distance from (0, 1) and (6, 5).
- Points A(0, 0), B(6, 0) and C(3, k) form an equilateral triangle. Find k as a surd.
Students learning in English as an additional language
“Midpoint” means the point in the middle of a line segment. “Distance” means how far apart two points are. “Across” and “up” describe the horizontal and vertical differences. “Average” here means “add, then divide by 2”.
| Word | Plain meaning | Example |
|---|---|---|
| distance | the length of the straight line between two points | AB |
| midpoint | the point exactly halfway along a line segment | M |
| coordinates | a pair of numbers that give a point's position | (2, 5) |
| hypotenuse | the longest side of a right-angled triangle | the line AB |
| segment | the part of a line between two points | PQ |
| average | the sum divided by the number of values | (a + b) ÷ 2 |
Sentence starters for explaining aloud or in writing:
- The across difference is … and the up difference is …
- The midpoint is halfway, so I …
- I can check this because the distance must be …
Two things not to say
“Plug it into the formula.” Say instead: draw the triangle and name its legs.
“Subtract and add.” for the midpoint: it is add and then halve.
Questions to set
Extra practice in three tiers, including one question from earlier work for spacing. Every answer has been checked.
1Fluency
The idea on its own, with friendly numbers. Set these first.
- DistanceFind the distance between (0, 0) and (8, 15).
17. - MidpointFind the midpoint of (2, 4) and (8, 10).
(5, 7).
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextTwo radio masts on a flat island in Fiji are at (1, 1) and (13, 6) on a grid in kilometres. How far apart are they?
√(144 + 25) = 13 km. - Earlier work (Year 9)A ladder 5 m long reaches 4 m up a wall. How far is its foot from the wall?
3 m.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningShow that the point (1, 1) is the same distance from (−2, 5) and (4, −3).
Both distances are 5: √(9 + 16) and √(9 + 16). - Working backwardsThe midpoint of AB is (1, −2) and A is (−3, 4). Find B and the length AB.
B = (5, −8); AB = √(64 + 144) = √208 = 4√13.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
The figure works with the keyboard (tab to a slider, then the arrow keys) and by touch. “Show me the steps” gives the same states in words.
No device? Squared paper is enough: students plot the points and draw the triangle.