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Year 10
Higher

Addition with surds

I can appreciate the structure that underpins addition of surds.

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Year 10
Higher

Addition with surds

I can appreciate the structure that underpins addition of surds.

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Share activities with pupils
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Lesson details

Key learning points

  1. An unknown value is represented by a letter.
  2. An unevaluated surd can be thought of as an unknown value.
  3. Each simplified surd has a unique value.
  4. Surds can therefore be grouped in the same way as like terms.

Keywords

  • Surd - A surd is an irrational number expressed as the root of a rational number.

  • Radical - The root sign is the radical symbol.

  • Radicand - The radicand is the value inside the radical symbol.

Common misconception

Pupils may think like surds need both the radicand and the coefficient to be the same.

Remind them of the distributive law: a(b+c) = ab + ac.


To help you plan your year 10 maths lesson on: Addition with surds, download all teaching resources for free and adapt to suit your pupils' needs...

It can help to use the distributive law to illustrate this point. 2(a) + 3(a) = (2+3)(a) = 5(a). The a can stand for any value and therefore can represent a surd.
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This content is © Oak National Academy Limited (2025), licensed on Open Government Licence version 3.0 except where otherwise stated. See Oak's terms & conditions (Collection 2).

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Starter quiz

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6 Questions

Q1.
A surd is in its when the radicand is an integer with no perfect square factors greater than 1.
Correct Answer: simplest form
Q2.
Which of these surds are in their simplest form?
$$3\sqrt 4$$
Correct answer: $$8\sqrt 3$$
$$\sqrt {100}$$
$$21\sqrt 9$$
Q3.
$$\sqrt {18}$$ simplifies to $$a\sqrt 2$$. What is the value of $$a$$?
Correct Answer: 3, three
Q4.
$$3\sqrt {76}$$ simplifies to $$k\sqrt {19}$$. What is the value of $$k$$?
2
3
Correct answer: 6
19
Q5.
$$3\sqrt {125}$$ simplifies to $$h\sqrt 5$$. What is the value of $$h$$?
Correct Answer: 15, Fifteen
Q6.
$$9\sqrt {98}$$ simplifies to $$d\sqrt 2$$. What is the value of $$d$$?
Correct Answer: 63, sixty-three

Exit quiz

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6 Questions

Q1.
Three of the surds are like surds. Which one is not?
$$-5\sqrt 2$$
$${4 \over 5}\sqrt 2$$
$$b\sqrt 2$$
Correct answer: $$5\sqrt {22}$$
Q2.
Three of the surds are like surds. Which one is not?
$$a\sqrt 5$$
$${2 \over 3}\sqrt 5$$
$$a\sqrt {20}$$
Correct answer: $$-5\sqrt {30}$$
Q3.
$$4\sqrt {6} - a\sqrt {6} = -5\sqrt {6}$$. What is the value of $$a$$?
Correct Answer: 9, nine
Q4.
After gathering like terms in the expression $$4 - 3\sqrt b + 7\sqrt {bc} + 3\sqrt b - 20 - \sqrt {bc}$$, how many terms will be in the resulting expression?
Correct Answer: 2, Two
Q5.
Is the statement "It is always best to simplify before adding surds"
Always true
Correct answer: Sometimes true
Never true
Q6.
Calculate the perimeter of a square with side length $$\sqrt {18} + \sqrt {20}$$. Give your answer as simply as possible.
$$4\sqrt {18} + 4\sqrt {20}$$
Correct answer: $$8\sqrt {5} + 12\sqrt {2}$$
$$4(\sqrt {18} + \sqrt {20})$$
$$3\sqrt {2} + 2\sqrt {5}$$