At a glance. One lesson of about 60 minutes. Finding the steps across and up between two points, using Pythagoras for the distance, and averaging coordinates for the midpoint.
Calculator: scientific calculator. Distances are often not whole numbers, and the order of operations (differences, squares, sum, root) matters.
What students will learn
Know
- The distance formula, d² = (x₂ − x₁)² + (y₂ − y₁)².
- The midpoint is the average of each coordinate.
- Differences across zero add the absolute values.
Understand
- The distance formula is Pythagoras' theorem applied to a triangle built on the grid.
- The midpoint is the point equally far from both ends along the line.
Do
- Find the distance between two points, with negative coordinates.
- Find a midpoint, or an end point given the midpoint.
- Work backwards to 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: using the distance and midpoint formulas (questions 2 to 6).
- Criterion B. Investigating patterns: with the figure, students explain why the straight distance is always shorter than across plus up.
- Criterion C. Communicating: the triangle is sketched, the steps are labelled and the working shows the squares.
- Criterion D. Applying mathematics in real-life contexts: maps and game grids; ask students to invent a grid scenario.
The one thing to do with the figure
Put A at (−3, 1) and B at (2, 4), then ask for the distance before revealing it. Many will say 8 (the sum of the steps). Move B so the steps are 3 and 4 and ask how this triangle is different.
Finally set both points to the same column and ask what the formula does. The flat triangle is the point where students see that the formula contains plain subtraction as a special case.
A possible order
| 5 min | Warm-up, then answers. |
|---|---|
| 8 min | Hook: the treasure; vote on more, equal or less than 14. |
| 12 min | The figure; the steps and the hidden triangle. |
| 12 min | Worked examples; P and Q in pairs. |
| 5 min | Question 1, the check. |
| 13 min | Your turn, Levels 1 to 3. |
| 5 min | Exit ticket, question 9. |
The answers
| Warm-up | 7; 13; 10.5. |
|---|---|
| Worked examples | Treasure: 10 m. Midpoint: (−2, −1). |
| Faded example | √89 = 9.4. |
| 1. Setting up the distance | A. B is the grid path; C forgets to subtract. |
| 2. Distance | 35. |
| 3. Midpoint | 3. |
| 4. Distance across zero | 17. |
| 5. Distance, 1 d.p. | 8.6. |
| 6. Midpoint | 4. |
| 7. Finding an end point | 7. |
| 8. Working backwards | 7. |
| 9. Midpoint | A, (3, 2). B is the sum without halving; C drops the minus sign on −1. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| 49, 23 or 12 in questions 2, 4, 5 | Adds the steps (the grid path). | Ask: is the straight line longer or shorter than the path along the grid? |
| 1225, 289 or 74 | Stops at d². | Ask what 1225 measures: area. We want length. |
| 7 in question 4 | Subtracts the steps (15 − 8). | Ask: is the distance shorter than the steps? Draw the triangle: the distance is its longest side. |
| 5.1 in question 5 | Reads −3 to 4 as a step of 1. | Draw the y-axis and count across zero. |
| 6 or 7 in question 3; 8 or 6 in question 6 | Does not halve, loses a minus sign, or subtracts. | Draw the two numbers on a number line and find the middle. |
| 4 or 2 in question 7 | Uses the distance to the midpoint only once. | Ask how far B is beyond the midpoint compared with A. |
| 1 in question 8 | Subtracts 25 − 24 instead of squaring. | Write k² + 24² = 25² and ask what each square is. |
Questions worth asking
- Why is the straight line never longer than across plus up? Listen for “a hypotenuse is shorter than the other two sides together”.
- Does the order of the points change the distance? No: the steps are squared. Ask for a check with the figure.
- How do you find the point one third of the way along? Open question for later; see stretch.
Differentiation
A way in for anyone stuck
- Give a grid with the triangle drawn for the first three questions.
- Let students count squares for the steps before using subtraction.
- Pre-write the formula with blanks: d² = (___)² + (___)².
Stretch
- Find the point that is 1/3 of the way from (0, 0) to (9, 12). (3, 4.)
- Find the distance between (1, 2, 3) and (4, 6, 15) by applying the same idea twice. (13.)
Students learning in English as an additional language
Use the words “across” and “up” for the steps before introducing “horizontal” and “vertical”.
| Word | Plain meaning | Example |
|---|---|---|
| coordinates | a pair of numbers giving a position: across, then up | (−2, 1) |
| distance | the length of the straight line between two points | 10 m from start to chest |
| midpoint | the point half-way between two points | (−2, −1) |
| horizontal | level, from left to right | the across step |
| vertical | straight up and down | the up step |
Sentence starters for explaining aloud or in writing:
- The steps are … across and … up, so d² = …
- The midpoint is the average of … and …
- The straight line is shorter than the grid path because …
Two things not to say
“Subtract, square, add, root: follow the recipe without the triangle.” Students lose the triangle and then add the steps. Always sketch it.
“Ignore the minus signs.” That produces the 5.1 slip in question 5.
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 from (0, 0) to (24, 32).
40. - MidpointFind the midpoint of (2, 8) and (10, 4).
(6, 6).
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- MapOn a map grid in km, a village is at (−4, 2) and a town at (8, 7). How far apart are they?
13 km. - Meeting pointTwo friends at (1, 5) and (9, −3) meet half-way. Where?
(5, 1).
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningPoints P(0, 0), Q(6, 0) and R(3, h) form an isosceles triangle with PR = 5. Find h.
h = 4. - From Lesson 3Is a triangle with vertices (0, 0), (3, 0) and (0, 4) right-angled? Which side is longest?
Yes; the side from (3, 0) to (0, 4), length 5.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
The figure works with the keyboard and by touch; “Show me the steps” gives four cases in words.
On paper: squared paper with the axes drawn and two pencil dots does the same job.