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.

January 29, 2019

When I teach rounding to a significant figure, I ask the class to discuss in pairs or small groups a definition for the word significant. It is a word that all the students have heard before but not all are able to define. After 2 or 3 minutes of conversation I ask the students to […]

January 13, 2019

When calculating instantaneous rates of change students need to visualise the properties of the gradient for a straight line graph. I use the starter activity to see if they can match four graphs with their corresponding equations. The only clue is the direction and steepness of the red lines in relation to the blue line […]

January 4, 2019

Fractions, decimals and percentages are ways of showing a proportion of something. Any fraction can be written as a decimal, and any decimal can be written as a percentage. In this blog I discuss how to use the place value table and equivalent fractions to illustrate how fractions, decimals and percentages are connected. You can […]