At a glance. One lesson of about 60 minutes. It assumes simplifying surds and expanding brackets. Students expand with surds, use conjugate pairs and the difference of two squares, and rationalize a denominator with two terms.
Calculator: no calculator. No calculator: the lesson is about exact form, and a decimal would hide the cancelling that makes it work.
What students will learn
Know
- (a + √b)(a − √b) = a² − b.
- (√p + √q)² = p + q + 2√(pq).
- To rationalize a binomial denominator, multiply by its conjugate.
Understand
- The conjugate removes the irrational part because the middle terms cancel.
- Expanding brackets with surds is the same as expanding algebra, with (√b)² = b.
- An exact form often shows structure that a decimal hides.
Do
- Expand and simplify brackets containing surds.
- Rationalize denominators of the form a + √b and √a + √b.
- Use a square of a surd expression to solve for an unknown coefficient.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: expanding with surds and rationalizing (questions 1 to 9).
- Criterion B. Investigating patterns: the second figure and the what-if. Students find when a square of a sum of roots is a whole number.
- Criterion C. Communicating: setting out an expansion so that each product is visible, and justifying why the conjugate works (worked examples 1 to 3).
- Criterion D. Applying mathematics in real-life contexts: exact areas of tiles from irrational edge lengths (the hook and question 6).
The one thing to do with the figure
Ask for a prediction before every slide. With the first figure set to a = 4 and b = 3, ask: “what do I get if I make b bigger?” Then ask why the rectangle's area is whole when both of its sides are not.
In the second figure, ask students to find p and q so that the total is a whole number, and to say what pq has to be.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: the tile with two irrational edges. Collect predictions. |
| 12 min | Both figures with predictions; the key idea. |
| 10 min | Warning card and worked examples; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose C or D, return to the first figure. |
| 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 | 2√3; x² − 9; rational (4). |
|---|---|
| Hook | The area is 25 − 7 = 18 cm², rational; (√2 + √3)² = 5 + 2√6, which still has a root. |
| Worked examples | 11; 2√6 + 2; 11 + 6√2. |
| Faded example | 6 + 2√5. |
| 1. A conjugate pair | B. A adds the 3; C squares 3; D squares the first bracket. |
| 2. Collecting like surds | 6. 5 + 3 − 2. |
| 3. Expanding two brackets | 3. 7 + 3√3. |
| 4. Rationalizing with a conjugate | 6. 24 ÷ 4. |
| 5. Squaring a sum of roots | C. A leaves out the cross term; B does not double it; D multiplies 5 by 20. |
| 6. A square tile | 4. 9 + 4√5. |
| 7. Using a root's square | −4. The √5 terms cancel. |
| 8. Matching coefficients | 2. 6p = 12. |
| 9. Rationalizing a two-term denominator | A. B does not divide by 6; C gives the conjugate itself; D adds 16 and 10. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, C or D in question 1 | Added the 3, squared the 3, or squared the first bracket. | Write the four products of the expansion in a grid and ask what (−√3)(√3) is. |
| 10 or 4 in question 2 | Added √8, or took √8 as 4√2. | Ask for the square factor of 8 and its root. |
| 7 or −3 in question 3 | Ignored the minus sign, or reversed it. | Write each of the four products with its sign before collecting. |
| 24 or 4 in question 4 | Did not divide by the new denominator, or gave it instead. | Write the new bottom 7 − 3 and ask what the top must be divided by. |
| A, B or D in question 5 | Left out the cross term, did not double it, or multiplied. | Draw the four regions of the square of a sum. |
| 2 or 0 in question 6 | Only one cross term, or squared each term separately. | Expand with four products and count the cross terms. |
| 14 or 4 in question 7 | Stopped at the rational part, or lost the sign. | Write x² and 6x with their √5 parts side by side. |
| 4 or 8 in question 8 | Used the wrong cross term, or confused p with 2p². | Expand (3 + p√2)² and list the rational part and the √2 part. |
| B, C or D in question 9 | Did not divide, gave the conjugate as the answer, or added 16 and 10. | Ask what the conjugate is for, and what the bottom becomes. |
Questions worth asking
- Why do the middle terms cancel in (a + √b)(a − √b)? Listen for “one is −a√b and the other is +a√b”.
- Is the conjugate of √7 − √3 the same as its negative? No: the negative is −√7 + √3, which has both signs changed; the conjugate changes only the middle sign.
- When is (√p + √q)² a whole number? When pq is a perfect square, because the cross term is then whole.
- Why can't we write √(a + b) = √a + √b? Test with 9 and 16: 5 is not 7.
Differentiation
A way in for anyone stuck
- Use a four-box grid for every expansion before collecting terms.
- Give the conjugate pairs as cards and have students find a match for each bracket.
- Start with numbers where the root is a perfect square so the cancelling is visible.
Stretch
- Show that 1/(√3 − √2) − 1/(√3 + √2) = 2√2.
- Find x if (3 + √x)(3 − √x) = 5.
Students learning in English as an additional language
“Conjugate” means “joined together as a pair”: the same two terms with the sign between them switched. “Rationalize” means “make rational”. “Expand” means “multiply out the brackets”. “Cross term” is the middle term of a squared sum.
| Word | Plain meaning | Example |
|---|---|---|
| surd | a root that cannot be written as a whole number or a fraction | √11 |
| conjugate | the same two terms with the sign between them switched | 3 + √5 and 3 − √5 |
| rationalize | rewrite a fraction so the denominator is a whole number | 1/√3 = √3/3 |
| like surds | surds with the same root, which can be added or subtracted | 4√3 and 9√3 |
| cross term | the middle term when a sum is squared | the 2ab in (a + b)² |
| difference of two squares | a product that gives one square minus another | (x + 4)(x − 4) |
Sentence starters for explaining aloud or in writing:
- The conjugate of … is …, because …
- The middle terms cancel, so I am left with …
- I can check by …
Two things not to say
“Just multiply by the bottom.” Multiplying by the same bracket squares it; say “multiply by the conjugate”.
“Square roots and plus signs don't mix.” √(a + b) is not √a + √b, and the reason is a test with numbers, not a rule.
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.
- Simplify√75 + √12.
7√3. - Expand(√5 + 1)(√5 − 1).
4.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA square mosaic has sides of (2 + √3) cm. Find its perimeter and its area.
Perimeter 8 + 4√3 cm; area 7 + 4√3 cm². - Earlier work (Lesson 1)Write 0.8181… as a fraction.
81/99 = 9/11.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningShow that 1/(√3 − √2) − 1/(√3 + √2) = 2√2.
(√3 + √2) − (√3 − √2) = 2√2, each bottom being 1. - FindingFind x if (3 + √x)(3 − √x) = 5.
9 − x = 5, so x = 4.
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 calculator is needed; every answer is exact.