At a glance. One lesson of about 60 minutes. It assumes simplifying surds (Lesson 4), expanding double brackets and Pythagoras from Year 9. Students rationalize a/√b and a/(k√b), expand brackets with surds, and use (a + √b)(a − √b) = a² − b.
Calculator: no calculator. None is needed, and none is wanted: rationalizing is about exact form, and a decimal hides the structure. Questions like these are set without a calculator.
What students will learn
Know
- To rationalize a/√b, multiply the top and the bottom by √b.
- √b × √b = b.
- (a + √b)(a − √b) = a² − b.
Understand
- Multiplying by √b/√b multiplies by 1, so the value does not change.
- A rational denominator makes a surd quick to estimate and to combine.
- Expanding brackets with surds follows the same rules as algebra.
Do
- Rationalize denominators of the form a/√b and a/(k√b) and simplify.
- Expand single and double brackets with surds.
- Find exact lengths by Pythagoras in simplest surd form.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: rationalizing and expanding with surds (questions 1 to 9).
- Criterion B. Investigating patterns: the second figure. Students find which a and b give a whole number (b a perfect square whose root divides a, as 4/√4 = 2), and explain why a = b gives √a, which is still a surd.
- Criterion C. Communicating: writing the multiplication by √b/√b as its own line and simplifying in a separate step (worked examples, faded example).
- Criterion D. Applying mathematics in real-life contexts: the tile in the hook, the park path in question 6 and the rectangle in question 7.
The one thing to do with the figure
Use the first figure to answer the hook before any rule. Set the diagonal to 6 and ask for the side by Pythagoras on the shaded triangle (3√2). Then ask: the side is also the diagonal divided by √2, so what is 6/√2?
In the second figure, ask students to predict where 3/√6 lands before they move the sliders, and to explain why both labels point to the same place.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: 10/√2 or 5√2? Collect predictions; ask which is easier to estimate. |
| 10 min | First figure; settle the hook (both are about 7.07); key idea. |
| 10 min | Second figure; the warning card; both worked examples; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose C or D, 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 | 2√7; 3x + 12; x² − 9. |
|---|---|
| Hook | Both are the same length: 10/√2 = 10√2/2 = 5√2, about 7.07 cm. |
| Worked examples | 2√7; 5 + 3√3. |
| Faded example | 3√3. |
| 1. Rationalizing 1/√11 | A. B multiplies the top by 11 instead of by √11; C multiplies only the top; D multiplies only the bottom. |
| 2. A single bracket | 3. 3 + 5√3. |
| 3. Rationalizing a/√b | 2. 10√5/5. |
| 4. A product that is rational | 7. 3² − (√2)² = 9 − 2. |
| 5. Rationalizing a/(k√b) | 1.5. 15√5/10 = 3√5/2. |
| 6. An exact length by Pythagoras | 30. √9000 = 30√10 (about 94.9 m). |
| 7. Rationalizing in context | 4. 24/√6 = 4√6. |
| 8. Squaring a bracket with a surd | 9. (2 + √5)² = 9 + 4√5. |
| 9. Rationalizing 12/√3 | C. A loses the surd; B forgets the denominator; D roots the top as well. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| B, C or D in question 1 | Multiplied the top by the wrong number, or only the top or only the bottom. | Ask: what is √11/√11? Multiplying by it must not change the number. |
| 9 in question 2 | Thinks √3 × √3 = 9. | Ask what number squared gives 3; that is √3, so √3 × √3 = 3. |
| 10 or 50 in question 3 | Forgot the new denominator, or multiplied by it. | Write the fraction 10√5/5 in full before simplifying. |
| 5 or 11 in question 4 | Thinks √2 × √2 = 4, or lost the minus sign. | Write all four products in a grid, signs included. |
| 7.5 or 3 in question 5 | Ignored the √5, or ignored the 2. | Work out the new denominator 2√5 × √5 = 10 on its own line. |
| 900 or 120 in question 6 | Wrote the square factor, or added the sides. | Ask which side the path is; then simplify √9000 as √900 × √10. |
| 24 or 144 in question 7 | Forgot to divide by 6, or multiplied. | Check by multiplying the answer by √6: it must give 24. |
| 7 or 29 in question 8 | Added instead of squaring 2, or squared √5 to 25. | Write (2 + √5)(2 + √5) and all four products. |
| A, B or D in question 9 | Lost the surd, forgot the denominator, or took the root of the top too. | Use the second figure's method: multiply by √3/√3 and write the new fraction before simplifying. |
Questions worth asking
- Why does multiplying by √5/√5 not change the number? Listen for “√5/√5 is 1, and multiplying by 1 changes nothing”.
- Why is 5√2 easier to estimate than 10/√2? √2 is about 1.4, so 5√2 is about 7; dividing by 1.4 in your head is harder.
- When does a/√b come out as a whole number? When b is a perfect square whose root divides a, such as 6/√9 = 2. Otherwise a surd stays: 5/√5 = √5 is still irrational.
- How is (3 + √2)(3 − √2) like (x + 2)(x − 2)? Both are a difference of two squares: the middle terms cancel.
Differentiation
A way in for anyone stuck
- Give a frame: a/√b = (a × √b)/(√b × √b) = …/…, with boxes for each part.
- Use a grid for every double bracket, with the four products written in the cells before collecting.
- Start with a = b (5/√5 = √5) so the simplification can be seen at once.
Stretch
- Rationalize 2/(3 + √5) and give the answer in the form a + b√5.
- Show that 1/√2 + 1/√8 = 3√2/4.
Students learning in English as an additional language
“Rationalize” means “make rational”: here, make the denominator a whole number. “Denominator” is the bottom of a fraction and “numerator” the top. Say “multiply top and bottom by root b” as one phrase.
| Word | Plain meaning | Example |
|---|---|---|
| rationalize | rewrite a fraction so that its denominator has no surd | 2/√7 = 2√7/7 |
| denominator | the bottom number of a fraction | √5 in 1/√5 |
| numerator | the top number of a fraction | 10 in 10/√5 |
| expand | multiply out brackets | √2(√2 + 3) = 2 + 3√2 |
| difference of two squares | a² − b², which factorizes as (a + b)(a − b) | (4 + √3)(4 − √3) = 16 − 3 |
| exact value | a value written without rounding | √7 rather than 2.65 |
Sentence starters for explaining aloud or in writing:
- I multiply the top and the bottom by … because …
- The denominator becomes … because √b × √b = …
- The value does not change because …
Two things not to say
“Get rid of the root.” The root is not removed, it moves to the top; say “make the denominator a whole number”.
“You must always rationalize.” It is a convention that helps estimating and combining; an unrationalized answer is still correct in value.
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.
- RationalizeRationalize 6/√3.
2√3. - ExpandExpand √2(4 + √2).
4√2 + 2.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA square window has a diagonal of 8 dm. Find its side in simplest surd form.
8/√2 = 4√2 dm. - Earlier work (Lesson 4)Simplify √48 + √12.
4√3 + 2√3 = 6√3.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningShow that (√7 + √3)(√7 − √3) is a whole number.
7 − 3 = 4. - Working backwardsk/√2 = 3√2. Find k.
k = 3√2 × √2 = 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? Draw a square of side 5 cm, measure its diagonal (about 7.1 cm) and divide by √2 ≈ 1.41 to see the side return.