At a glance. One lesson of about 60 minutes. It assumes distance, midpoint, gradient and the parallel and perpendicular conditions from Year 9 and from the Standard course, and algebraic proof from lesson 2. Students prove properties of the midpoint quadrilateral and of rectangles and rhombuses using letters in the coordinates.
Calculator: no calculator. The lesson is by hand: the coordinates are letters, and the arithmetic with numbers is small.
What students will learn
Know
- The midpoint, squared distance and gradient formulas in general form.
- Parallel lines have equal gradients; perpendicular lines have gradients with product −1.
- The midpoint quadrilateral of any quadrilateral is a parallelogram.
Understand
- A choice of axes does not change a shape, so a proof may place one vertex at the origin.
- A property that follows for letters follows for every case, which a drawing cannot show.
- The special names (rhombus, rectangle, square) depend only on the diagonals of the original.
Do
- Place a shape on axes with letters without making it special.
- Use midpoints, squared distances and gradients to prove a property.
- Finish a coordinate proof with a sentence that names the property.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: midpoint, distance and gradient in numbers and in letters (questions 1 to 8).
- Criterion B. Investigating patterns: the figure. Students move C and D, record the shape named for several positions, and conjecture which conditions on the diagonals give which name.
- Criterion C. Communicating: a chain from the placement of the shape, through the calculation, to a sentence (worked examples, faded example).
- Criterion D. Applying mathematics in real-life contexts: the slab and the inner frame in the hook, and the equidistance result (faded example).
The one thing to do with the figure
Ask for a prediction before every move. Before moving C and D, ask: “What would make the inner shape a rhombus?” and let students look for positions. Collect a short list on the board. Then ask what is the same about all of the positions in the list: the diagonals of the outer shape are equal.
Then ask why the inner shape never has just one pair of parallel sides, and is never a kite unless it is a rhombus: both pairs of opposite sides are always parallel. That is the theorem the lesson proves with letters.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: what is the inner shape always? Collect predictions. |
| 12 min | Figure: move C and D; the key idea; settle the hook; the warning card. |
| 10 min | Worked examples 1 to 3; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose B or D, return to the key idea. |
| 18 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Something to think about, then the exit ticket, question 8. |
The answers
| Warm-up | (2, 3); perpendicular, because 2 × (−1/2) = −1; one letter would make the two numbers equal. |
|---|---|
| Hook | A parallelogram, always. |
| Worked examples | Parallelogram (gradients 1 and −2/3); AC² = BD² = w² + h²; product of gradients −1. |
| Faded example | MO² = MP² = MQ² = a² + b². |
| 1. Midpoints and gradient with letters | A. B uses N alone; C is run over rise; D is the gradient of BC. |
| 2. A midpoint segment's gradient | 1. MN is parallel to AC. |
| 3. Half the diagonal | 13. Half of 26. |
| 4. Area of the midpoint rectangle | 48. 6 × 8. |
| 5. Where the inner diagonals meet | 3.5. The point (4, 3.5). |
| 6. Equal diagonals fix a coordinate | 8. k² = 64. |
| 7. Two sides, one direction | −3. The gradient of BD. |
| 8. Naming the midpoint quadrilateral | B. A is true but not specific; C needs perpendicular diagonals; D needs both. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| B, C or D in question 1 | Used one point instead of two, put run over rise, or used the side BC instead of the midpoints. | Write M and N first, then rise = (c + e) − c and run = (b + d) − b. |
| 4 or 0.25 in question 2 | Used the gradient of a side, not of the midpoint segment. | Find M and N first and then use rise over run. |
| 26, 52 or 6.5 in question 3 | Did not halve, doubled, or halved twice. | Draw triangle ABC with M and N marked: the segment is half of the third side. |
| 192, 96 or 14 in question 4 | Used full diagonals, the original area, or added the sides. | Write the two sides of the rectangle first, then multiply. |
| 3, 4 or 7 in question 5 | Used the midpoint of one outer diagonal, the x-coordinate, or divided by 2. | Find the four midpoints on a sketch, then one inner diagonal. |
| 10, 64 or 136 in question 6 | Gave AC, k², or added. | Write AC² = BD², then the squared distances, then solve for k. |
| 3 or 0.8 in question 7 | Lost the sign, or used the other diagonal. | Mark Q and R on a sketch and count left and up. |
| A, C or D in question 8 | Chose a true but weaker name, or needed perpendicular diagonals. | Ask: what do the diagonals of PQRS tell you about the sides? |
Questions worth asking
- Why can we place a vertex at the origin? Listen for: moving or turning the axes does not change the shape's lengths or angles.
- Why is the midpoint quadrilateral always a parallelogram? Listen for: two sides are parallel to AC and the other two to BD, each pair half as long.
- Is a shape drawn with a ruler a proof of its properties? Listen for: no, drawings are accurate to a small error, and only one case.
- Which diagonals give a square? Listen for: equal and perpendicular diagonals.
Differentiation
A way in for anyone stuck
- Give a table of the three tools (midpoint, squared distance, gradient) with one numeric example each and the letter form beside it.
- Provide a proof frame: “Place …”, “Calculate …”, “So …”.
- Use squared paper to draw one midpoint quadrilateral before using letters.
Stretch
- Prove that the diagonals of a parallelogram bisect each other, using O(0, 0), P(a, 0), Q(a + b, c), R(b, c).
- Prove that the midpoint quadrilateral of any quadrilateral has area half of the original.
Students learning in English as an additional language
“Midpoint” is the point halfway along a segment. “Gradient” is how steep a line is. “Diagonal” is a line joining two corners that are not next to each other. “Bisect” means cut exactly in half. Say “perpendicular” and “parallel” with a gesture.
| Word | Plain meaning | Example |
|---|---|---|
| midpoint | the point halfway between two points | the midpoint of (0, 0) and (10, 6) is (5, 3) |
| gradient | how much a line goes up for each step across | a line through (0, 1) and (3, 7) has gradient 2 |
| diagonal | a line joining two corners that are not neighbours | AC in the quadrilateral ABCD |
| parallel | never meeting, with the same gradient | y = 2x + 1 and y = 2x − 4 |
| perpendicular | at a right angle; gradients multiply to −1 | gradients 3 and −1/3 |
| rhombus | a quadrilateral with four equal sides | a diamond shape |
Sentence starters for explaining aloud or in writing:
- Place the shape with … at the origin, so …
- The gradients are equal, so the sides are …
- This holds for every … because the letters are …
Two things not to say
“Put it on the axes so the numbers are nice.” A nice position can make the shape special and the proof covers less than it claims.
“It looks like a parallelogram.” The proof is the equal gradients, not the look.
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.
- MidpointFind the midpoint of (2a, 6) and (4a, 2).
(3a, 4). - GradientFind the gradient of the line through (1, 4) and (5, 12).
2.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- LengthDiagonals of 20 and 30 units, perpendicular: what are the sides of the midpoint rectangle?
10 and 15. - Earlier work (lesson 2)Expand (n + 1)² − n².
2n + 1.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningShow that the midpoint of the hypotenuse of a right triangle with legs 6 and 8 is 5 from each vertex.
The vertices are (0, 0), (6, 0), (0, 8), the midpoint is (3, 4), and each distance is 5. - ExtendA kite has diagonals that meet at right angles. What is the midpoint quadrilateral and why?
A rectangle: the sides are parallel to the perpendicular diagonals.
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 draw a quadrilateral, mark the four midpoints and test gradients. No calculator is needed.