At a glance. One lesson of about 60 minutes. It assumes square numbers, Pythagoras and the index laws from Year 9. Students multiply and divide roots, simplify a surd using its largest square factor, and add and subtract like surds.
Calculator: GDC optional. The lesson is done by hand. The optional section shows GDC users how to check that two surd forms agree, and why a matching decimal is evidence and not proof.
What students will learn
Know
- √a × √b = √(ab) and √a ÷ √b = √(a ÷ b) for positive a and b.
- A surd is simplified by taking out its largest square factor.
- Only like surds (the same root) can be added or subtracted.
Understand
- √8 = 2√2 because the side of the larger tilted square is two copies of the smaller one.
- √a + √b ≠ √(a + b) because two legs of a triangle are longer than its hypotenuse.
- Like surds behave like like terms in algebra.
Do
- Simplify √n to k√m with m having no square factor.
- Multiply surds and square a surd term.
- Add and subtract surds after simplifying them.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: multiplying, simplifying and collecting surds (questions 1 to 9).
- Criterion B. Investigating patterns: the first figure. Students find which tilted squares have sides that simplify and describe the pattern (the side splits into equal surds when p and q share a factor; it can also simplify without that, as across 3, up 4 gives √25 = 5).
- Criterion C. Communicating: showing the square factor as a separate line, √72 = √36 × √2 = 6√2 (worked examples, faded example).
- Criterion D. Applying mathematics in real-life contexts: tile patterns in the hook and the garden bed in question 8.
The one thing to do with the figure
Settle the hook with the grid, not with a rule. Set across 1, up 1, and ask for the side. Then set across 2, up 2, and ask students to look at the rings: the side passes through a grid point halfway, so it is two of the smaller sides. That is √8 = 2√2, seen before it is calculated.
In the second figure, ask students to find any a and b where the two legs together equal the hypotenuse. They cannot, which is the what-if.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: is √8 twice √2, four times, or neither? Collect votes and reasons. |
| 12 min | First figure; key idea; settle the hook; the warning card. |
| 8 min | Second figure; all three 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 | 12; 7x; 1, 4, 9, 16, 25, 36, 49, 64, 81, 100. |
|---|---|
| Hook | Twice: √8 = √4 × √2 = 2√2. |
| Worked examples | 7√2; 3√10; 9√3. |
| Faded example | 4√5. |
| 1. Simplifying a surd | A. B puts the square factor outside instead of its root; C splits 12 wrongly; D takes out a factor that is not square. |
| 2. Taking out a square factor | 3. √45 = 3√5. |
| 3. Multiplying roots | 6. √36. |
| 4. The largest square factor | 6. 72 = 36 × 2. |
| 5. Adding like surds | 5. 3√7 + 2√7. |
| 6. Squaring a surd term | 20. 4 × 5. |
| 7. Subtracting surds | 5. 10√2 − 5√2. |
| 8. Surds in an area | 18. √324, or 2√3 × 3√3. |
| 9. Adding surds | B. A adds under the root; C multiplies; D drops a term. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| B, C or D in question 1 | Put the square factor outside, split 12 wrongly, or took out a factor that is not square. | Ask: which square number divides 12? What is its square root? |
| 9 in question 2 | Wrote the square factor instead of its root. | Write √45 = √9 × √5 as a separate line, then evaluate √9. |
| 36 or 2 in question 3 | Forgot to take the root, or divided. | Write √3 × √12 = √(3 × 12) and stop there; then evaluate. |
| 36 or 3 in question 4 | Wrote the square factor, or did not use the largest one. | List the square numbers and test from the largest down. |
| 3 or 11 in question 5 | Did not simplify √63, or took 9 out instead of 3. | Simplify every term first, then collect. |
| 10 or 100 in question 6 | Squared only the root, or treated √5 as 5. | Write (2√5)² = 2√5 × 2√5 and multiply numbers and roots separately. |
| 15 or 75 in question 7 | Added instead of subtracting, or used the square factors themselves. | Write each simplified surd on its own line before combining. |
| 324, 6 or 39 in question 8 | Forgot the root, forgot √3 × √3 = 3, or added. | Ask: what is √3 × √3? Then what is 2√3 × 3√3? |
| A, C or D in question 9 | Added under the root, multiplied, or dropped a term. | Use the second figure with a = 2 and b = 8: the two legs together (about 4.24) are longer than the hypotenuse √10. |
Questions worth asking
- Why is √8 = 2√2 and not 4√2? Listen for “8 = 4 × 2 and the root of 4 is 2”, or the grid: two copies of the √2 side.
- Why can 3√2 + 5√2 be simplified but √2 + √3 cannot? The first has a common root, like 3x + 5x; the second is like x + y.
- Why use the largest square factor? Otherwise the root left over still has a square factor, as 3√8 does, and the work is not finished.
- Is √a × √b = √(ab) true for negative a and b? Not in the real numbers: √(−4) is not real. The rule is stated for positive numbers.
Differentiation
A way in for anyone stuck
- Give a card of the square numbers up to 100 and ask students to test each one against the number under the root.
- Write every simplification in three lines: the split, the separate roots, the answer.
- Start with the grid: draw tilted squares by hand and find their areas by subtracting four triangles from the box.
Stretch
- Which tilted squares on the grid have a side of √50? Find two different ones.
- Show that (√3 + √2)(√3 − √2) is a whole number.
Students learning in English as an additional language
“Surd” is a short word for a root that is left exact. “Simplify” here means “write in the simplest exact form”, not “work out a decimal”. Say “root two” and “two root two” aloud so the spoken forms are clear.
| Word | Plain meaning | Example |
|---|---|---|
| surd | a square root left exact because it is irrational | √2, 3√5 |
| square factor | a factor that is a square number | 4 is a square factor of 40 |
| simplest form | the surd with no square factor left under the root | √40 = 2√10 |
| like surds | surds with the same number under the root | 3√2 and 5√2 |
| hypotenuse | the longest side of a right-angled triangle, opposite the right angle | √(a + b) in the second figure |
| exact | written without rounding | √2, not 1.41 |
Sentence starters for explaining aloud or in writing:
- The largest square factor of … is …, so …
- These are like surds because …
- √a + √b is not √(a + b) because …
Two things not to say
“Just split the number into two factors.” Only a square factor comes out; 2√3 from √12 is the result.
“Surds are hard.” Treat them as like terms; the algebra students already know does the work.
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.
- SimplifySimplify √50.
5√2. - MultiplySimplify √2 × √18.
√36 = 6.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA square courtyard in Marrakesh has area 48 m². Write its side in simplest surd form.
√48 = 4√3 m. - Earlier work (Year 9)A right-angled triangle has legs 5 cm and 12 cm. Find the hypotenuse.
13 cm.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningShow that √2 + √8 + √18 = √72.
√2 + 2√2 + 3√2 = 6√2 = √72. - Working backwardsk√3 = √108. Find k.
108 = 36 × 3, so k = 6.
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? Squared paper is enough: students draw tilted squares and find their areas. The With a GDC section is optional.