At a glance. One lesson of about 60 minutes. It assumes distance, midpoint and gradient from lessons 3 and 4. Students show a triangle is isosceles or right-angled, show a quadrilateral is a parallelogram, and find a point equally far from two points.
Calculator: GDC optional. The lesson compares exact squared lengths, so it is done by hand. The optional section shows GDC users that rounded square roots can hide a difference.
What students will learn
Know
- Two sides are equal when their squared lengths are equal.
- A triangle has a right angle when the two smaller squared lengths add to the largest.
- Diagonals that share a midpoint show a parallelogram.
Understand
- A picture suggests a property but only calculations prove it.
- One example proves a property for those points only; a general proof uses letters.
- Each property has a matching test: lengths, midpoints or gradients.
Do
- Compute squared lengths and compare them.
- Use midpoints and gradients to test a quadrilateral.
- Find an equidistant point by setting squared distances equal.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: applying distance, midpoint and gradient (questions 1 to 8).
- Criterion B. Investigating patterns: the figure. Students find which positions of C give an isosceles triangle and which give a right angle, and describe the pattern.
- Criterion C. Communicating: a short proof with the squares written out and a closing sentence (worked examples, faded example, question 9).
- Criterion D. Applying mathematics in real-life contexts: the surveyor in the hook and the radio masts in question 8.
The one thing to do with the figure
Make students predict, then test. Ask for a position of C that gives an isosceles triangle, then check with the squares. Then ask for a right-angled position. Students find (0, 4) is both, and that the isosceles positions form a vertical line at x = 0 while the right-angle positions with the right angle at A or B form vertical lines at x = −4 and x = 4.
Ask: “Is there a position where all three squares are equal?” An equilateral triangle needs AC² = 64, which no whole-number position gives.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: are any two sides equal? 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 warning card. |
| 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 | Yes; (3, 6); at least two. |
|---|---|
| Hook | Yes: AB² = 45, AC² = 45 and BC² = 18, so AB = AC and the triangle is isosceles. |
| Worked examples | Right-angled at A; a parallelogram (both midpoints are (4, 5)); P = (5, 0). |
| Faded example | Isosceles: PR² = QR² = 20. |
| 1. Testing for a right angle | A. B, C and D have two smaller squares that add to something else. |
| 2. Squared distance | 58. 3² + 7². |
| 3. Squared side length | 50. 7² + 1². |
| 4. A missing vertex of a parallelogram | 2. (4 + x) ÷ 2 = 3. |
| 5. A point equidistant from two points | 4. 12x = 48. |
| 6. A right angle at a vertex | 4. k ÷ (−2) = −2. |
| 7. A diagonal of a square | 34. 3² + 5². |
| 8. A relay point in context | 11. 8y = 88. |
| 9. Reasoning with squared lengths | D. A claims a right angle without the squares adding; B adds the legs; C has a sign slip in EF². |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| B, C or D in question 1 | Added the wrong squares or a rounded value. | Add the two smallest and compare with the biggest. |
| 10 or 4 in question 2 | Added or subtracted the legs. | Draw the triangle and write the legs squared. |
| 8 or 48 in question 3 | Added the legs or subtracted the squares. | Write across and up as separate lines, then square. |
| 3 or 6 in question 4 | Gave the midpoint, or doubled without subtracting. | Write (4 + x) ÷ 2 = 3 and solve. |
| 3 in question 5 | Gave the midpoint of AB. | Ask: is the midpoint on the x-axis? Then set up (x, 0). |
| −4 or 1 in question 6 | Sign slip, or only changed sign. | Find the gradient of AB, then write the perpendicular gradient before using k. |
| 17 or 8 in question 7 | Gave a side, or added the legs. | Ask: which two points does the diagonal join? |
| 3 in question 8 | Gave the midpoint of MN. | Check the midpoint: is it on the y-axis? |
| A, B or C in question 9 | Claimed a right angle without the sum, added legs, or a sign slip. | Recompute each side separately and compare. |
Questions worth asking
- Why compare squared lengths rather than lengths? The squares are exact whole numbers; the roots may be surds or decimals.
- Does one worked example prove the property for every parallelogram? No: it proves it for these four points; a general proof uses letters.
- Which test would you use to show two sides are parallel? Equal gradients.
- Which test shows a right angle in a triangle? The converse of Pythagoras on squared lengths, or a product of gradients of −1.
Differentiation
A way in for anyone stuck
- Give a template: AB² = ___, AC² = ___, BC² = ___, so ___.
- Draw the shape on squared paper before calculating.
- Start with shapes whose sides are horizontal or vertical so the arithmetic is light.
Stretch
- Show that the points (−2, 1), (1, 5), (5, 2) and (2, −2) form a square, using lengths and gradients.
- Prove that the midpoints of the sides of any quadrilateral form a parallelogram, using a general quadrilateral with vertices (0, 0), (2a, 0), (2b, 2c) and (2d, 2e).
Students learning in English as an additional language
“Isosceles” means two sides of equal length. “Bisect” means cut into two equal parts. “Equidistant” means the same distance from two things. “Parallelogram” is a four-sided shape with two pairs of parallel sides.
| Word | Plain meaning | Example |
|---|---|---|
| isosceles | a triangle with two equal sides | XY = XZ |
| right-angled | having one angle of 90 degrees | the corner of a page |
| parallelogram | a quadrilateral with two pairs of parallel sides | a rectangle pushed over sideways, so its corners are no longer right angles |
| bisect | cut into two equal parts | the midpoint bisects AB |
| equidistant | the same distance from two points | same distance from A and B |
| converse | the statement turned around | if a² + b² = c² then right-angled |
Sentence starters for explaining aloud or in writing:
- The squared lengths are … and …, so …
- The midpoints are the same, so the diagonals …
- This is not enough to prove … because …
Two things not to say
“It looks right-angled.” Say instead: the squares add to the largest.
“The example proves it for all parallelograms.” Say instead: it proves it for these points.
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.
- Squared lengthFind the squared length of the segment from (0, 0) to (6, 8).
100. - MidpointFind the midpoint of (1, 9) and (7, 3).
(4, 6).
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextThree beacons on a map of an island in the Philippines are at (0, 0), (6, 0) and (3, 5), in kilometres. Is the triangle they make isosceles?
Yes: both other sides have squared length 9 + 25 = 34. - Earlier work (Year 9)Is a triangle with sides 7, 24 and 25 right-angled?
Yes: 49 + 576 = 625.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningShow that (1, 2), (4, 6) and (8, 3) make a right-angled isosceles triangle.
Squared sides 25, 25 and 50: 25 + 25 = 50. - Working backwardsThe points (0, 0), (4, 0) and (2, k) make an isosceles triangle with the two equal sides meeting at (2, k). Find all k that make the triangle right-angled.
4 + k² + 4 + k² = 16, so k² = 4 and k = 2 or −2.
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 compute the squares.