At a glance. One lesson of about 60 minutes, no calculator. Students derive the exact ratios from half an equilateral triangle and half a square, use them in lengths, areas and equations, and prove tan 15° = 2 − √3 from an isosceles construction.
Calculator: no calculator. No calculator is used: exact values and surds are done by hand, and a decimal is shown only as a check.
What students will learn
Know
- sin, cos and tan of 30°, 45° and 60° exactly.
- The sides of a 30, 60, 90 triangle are k, k√3 and 2k; of a 45, 45, 90 triangle k, k and k√2.
- How to rationalize a denominator, including 1 ÷ (2 + √3).
Understand
- The ratios are the same at every size because the triangles are similar.
- A new exact value can be built by using the known ones inside a larger construction.
Do
- Use exact values to find lengths and areas.
- Prove a statement with surds without rounding.
- Construct an exact value, tan 15°.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: exact values and surd arithmetic (questions 1 to 9).
- Criterion B. Investigating patterns: both figures. Students describe what stays the same when the size changes.
- Criterion C. Communicating: a proof with every reason written, in worked example 3 and the faded example.
- Criterion D. Applying mathematics in real-life contexts: the set square and the mast's shadow.
The one thing to do with the figure
Make students predict, then reveal. Before the first figure, ask them to guess whether the longest side of the 10 cm set square is a whole number. Then set the hypotenuse to 20 and the angle to 30°.
In the second figure, let students change k and say what changes and what does not.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: the set square. Collect votes and reasons. |
| 12 min | First figure; key idea derivation; settle the hook; the warning card. |
| 10 min | Second figure; worked examples 1 to 3; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose A, return to which leg is longer. |
| 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 9. |
The answers
| Warm-up | 4√3; 2√5; 7√2 cm. |
|---|---|
| Worked examples | 25√3 cm²; √3/3; tan 15° = 2 − √3. |
| Faded example | 1/2. |
| Something to think about | Not true: √3/2 against 1; and √2 against 1. |
| 1. A side of a 30, 60, 90 triangle | B. A multiplies by √3; C halves; D treats 9 as the hypotenuse. |
| 2. Area of a 30, 60, 90 triangle from the hypotenuse | 24.5. ½ × 7 × 7√3. |
| 3. A product of exact values | 6. √2 × √3 = √6. |
| 4. Perimeter of a 30, 60, 90 triangle | 37.9 cm. 24 + 8√3. |
| 5. A shadow from the exact tan 15 degrees | 112 m. 30(2 + √3). |
| 6. Solving with an exact value | 30°. tan x = 1 ÷ √3. |
| 7. Area of a regular hexagon | 54. 6 × 9√3. |
| 8. Two special triangles sharing a side | 2.93 cm. 4√3 − 4. |
| 9. Hypotenuse of a right isosceles triangle from its area | C. A divides the leg by √2; B gives the leg; D doubles the leg. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, C or D in question 1 | Multiplied by √3, halved, or treated 9 as the hypotenuse. | Ask: is the side opposite 30° the longest or the shortest leg? Then divide. |
| 49 or 12.25 in question 2 | Left out the half, or halved twice. | Draw the triangle, label both legs, and use half base times height. |
| 3 or 9 in question 3 | Used sin 30° or sin 60° in place of sin 45°. | Ask them to write each exact value as a surd before multiplying. |
| 21.9 or 35.3 in question 4 | Left out the hypotenuse, or used 45° values. | Mark the three sides of the triangle on a sketch and add all three. |
| 8 or 104 in question 5 | Multiplied by tan 15°, or halved tan 30°. | Ask: which side is the mast, which the shadow? Then which ratio? |
| 60 or 45 in question 6 | Multiplied, or dropped the √3. | Divide both sides by √3 and then find the angle in the table. |
| 108 or 27 in question 7 | Left out the half, or counted three triangles. | Draw the six triangles from the centre; count them. |
| 1.27 or 10.93 in question 8 | Took BD as a hypotenuse, or added. | Name the two triangles; find BC and BD separately, then subtract. |
| A, B or D in question 9 | Divided the leg by √2, gave the leg, or doubled it as in a 30, 60, 90 triangle. | Ask: which side is the longest, and what is the ratio of the sides of half a square? |
Questions worth asking
- Why are the ratios the same for every size of the triangle? Listen for “the triangles are similar, so the sides are in the same proportion”.
- Why can a proof not use 1.732 for √3? Listen for “it is only approximately equal; a proof needs exact equality”.
- In the tan 15° construction, why is angle D equal to 15°? Listen for “the triangle ACD is isosceles and the exterior angle at C is 30°, which is the sum of the two equal base angles”.
- How would you find the exact value of tan 75°? Listen for “the complement: tan 75° = 1 ÷ tan 15° = 2 + √3”.
Differentiation
A way in for anyone stuck
- Provide the two special triangles on a card with the sides labelled in terms of k.
- Provide a rationalizing template: multiply top and bottom by the conjugate, then expand.
- Restrict to questions 1 to 4 at first, using whole-number multiples.
Stretch
- Use tan 75° = 2 + √3 to show that a 40 m building's shadow is 40(2 − √3) m when the sun is 75° above the horizon.
- Copy the construction of worked example 3 on a 45, 45, 90 triangle with legs 1: extend a leg beyond the 45° corner by the hypotenuse √2. Show that tan 22.5° = √2 − 1.
Students learning in English as an additional language
“Exact” means written with the square root, not a rounded decimal. “Rationalize” means remove the root from the bottom of a fraction. “Isosceles” has two equal sides; use a picture of the set square and the triangle.
| Word | Plain meaning | Example |
|---|---|---|
| exact value | a number written with its roots and fractions, not rounded | √3/2, not 0.866 |
| surd | a root that does not come out as a whole number or fraction | √5 |
| rationalize | rewrite a fraction so the bottom has no root | 1/√2 becomes √2/2 |
| isosceles | a triangle with two equal sides and two equal angles | a triangle with two sides of 5 cm |
| exterior angle | the angle outside a triangle formed by extending a side | equals the sum of the two opposite interior angles |
Sentence starters for explaining aloud or in writing:
- The exact value is … because the sides are in the ratio …
- I cannot use a decimal here because …
- The new angle is half of 30° because the triangle is …
Two things not to say
“Just memorize the table.” Show where each value comes from.
“Use 1.73.” In a proof, keep the root.
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.
- Exact valueWrite the exact value of cos 45° × sin 30°.
√2/4. - Earlier work (Year 10)Rationalize 6 ÷ √2.
3√2.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- LengthA 30, 60, 90 triangle has a short side of 5 cm. Find the perimeter, as an exact value.
15 + 5√3 cm. - EquationSolve 2 cos x = 1 for an acute x.
60°.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningShow that sin²60° + cos²60° = 1 using exact values.
3/4 + 1/4 = 1. - OpenAn equilateral triangle has height 9 cm. Find its side and its area exactly.
Side 6√3 cm, area 27√3 cm².
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
Both figures work 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? Two set squares from a geometry box give the two special triangles directly.