Your Basket 0 items - £0.00

When learning how to simplify surds students need to understand the difference between a rational and irrational number.

Rational numbers include integers and terminating and repeating decimals. They can be written as a fraction with both the numerator and denominator as integers.

An irrational number is a number which, in its decimal form does not terminate or repeat. This means it cannot expressed as a fraction with two integers. π is one example of an irrational number which the students would have come across when learning about circles.

Another type of irrational number is a Surd. A surd is a number written exactly using square or cube roots. For example, √2 and ^{3}√7 are surds. √16 and ^{3}√8 are not surds, because √16 = 4 and ^{3}√8 = 2.

To introduce surds students use their calculators to find the only irrational from a selection of rational numbers. If the class need some help you could pose the question “Which of the following cannot be written as a fraction using two integers?”

While it is often useful to approximate surds, for example, when finding the length of the hypotenuse in a right-angled triangle or the radius of a circle when given its area students need to be able to calculate with surds in their exact form. This is needed when solving a quadratic equation by completing the square or using the formula. Learning how to simplify surds is also needed for rationalising denominators.

A helpful exam tip I give to students is to keep a number in surd form until they get to the final answer. This way they won’t carry through any rounding errors that would affect the final solution.

In the second slide we discuss how to simplify surds using the following examples.

Multiplying with Surds

√9 × √4 = 3 × 2 = 6

√9 × √4 = √(9 × 4) = √36 = 6

√12 = √4 × √3 = 2 × √3 = 2√3

Dividing with Surds

√100 ÷ √25 = 10 ÷ 5 = 2

√100 ÷ √25 =√4 = 2

As we work through some more examples I challenge the students to define these relationships.

Later, I ask the class to attempt the following questions for themselves on their mini-whiteboards.

Simplify:

We work through each question one at a time using the learning from the previous problem to help progress with the next one. When we have completed the third question students are ready to work through the problems on the third slide independently. I encourage the class to check their answers using a calculator with a natural display.

The plenary challenges students to link working with surds to setting up and solving equations involving the area of a rectangle and triangle. This should be attempted without the use of a calculator so the relationships we have learnt are applied.

August 22, 2019

When getting ready for a new school year I have a list of priorities to work through. Knowing my team have all the information and resources they need to teach their students gives me confidence we will start the term in the best possible way. Mathematics Teaching and Learning Folder All teachers receive a folder […]

July 6, 2019

Earlier this week, my school took part in a trial OFSTED inspection as part of getting ready for the new inspection framework in September 2019. This involved three Lead Inspectors visiting our school over the course of two days. The first day involved a ‘deep dive’ by each of the Lead Inspectors into Mathematics, English […]

June 30, 2019

The method of how to solve quadratics by factorising is now part of the foundational knowledge students aiming for higher exam grades are expected to have. Here is an example of such a question. Solve x2 + 7x – 18 = 0 In my experience of teaching and marking exam papers students often struggle with […]