Communicating With Aliens via Recurring Decimals - Joyous Maths Pt. 1

Have you watched Arrival? If not, here’s the synopsis.

12 alien vessels appear unannounced and hover over 12 different locations on Earth. Experts from different countries make contact with the extra-terrestrials, trying to decide if they are friend or foe. The only problem is that the aliens communicate using arcane circular symbols, unintelligible to Earthlings. The breakthrough comes with the discovery of the encrypted repeating decimal of 0.08̅3̅ (i.e. 0.08333…) in the symbols. When converted to fraction, 0.08̅3̅ is , which seems somewhat related to number of alien vessels…

So, learn your recurring decimals well and you may hold the key to establishing communication with aliens yet.

Conversion of repeating decimals to fractions happens to be one of the things students at Joyous Learning pick up in their maths classes. Extract A is taken from our Primary 6 maths worksheet. It is an extension chapter under the topic of fractions.

Extract A

Fractions and Repeating Decimals

Switch on your calculator! Enter the following into your calculator: “1÷ 3 =”. What answer did you get?

This is an example of a repeating decimal and is written like this: 0.̅3̅ (the horizontal bar above the digit 3 is called a vinculum). Alternatively, it can be represented as a fraction: 1/3. Sometimes, repeating decimals are also rounded off to a certain number of decimal places; for example: 0.33.

Rounding off can pose a problem because rounded figures lose some degree of accuracy. Does this degree of accuracy matter? Well, sometimes it doesn’t, sometimes it does.

For example, a builder calculated that for every brick laid, 1/3 kg of cement is required. Say, there are 3000 bricks to be laid.

Scenario A

If the builder assumes 0.33 kg of cement per brick,

he will use 0.33 kg x 3000 = 990 kg of cement in total.

Scenario B

If the builder assumes 1/3 kg of cement per brick,

he will use 1/3 kg x 3000 = 1000 kg of cement in total.

Now, if the builder were building your house, would you rather he use 0.33 kg or 1/3 kg in his calculations?

When a high degree of accuracy is required, fractions are preferred to rounded figures. Converting fractions to decimals is generally simple: switch on your calculator, press the numerator, followed by the divide operation, and finally the denominator. But what if you wished to express a recurring decimal as a fraction instead?

Let’s go back to our example of 0.̅3̅ . Normally, 0.3 converted to a fraction is 3/10. In the case of 0.̅3̅ , there is only one repeating digit, which is 3. For every repeated digit, use one 9 as the denominator. So, 0.̅3̅ becomes 3/9; notice that 3 remains the numerator. And 3/9 can be simplified to 1/3. Now, what happens when there is more than one repeating digit? Take as an example, 0.̅1̅8̅ . There are two repeating digits, so two 9s are used as the denominator. So, 0.̅1̅8̅ becomes 18/99, which is also 2/11.

Besides its eight Oscars nomination, now there is another compelling reason to take your kids to the cinema to catch Arrival. Just be sure to have your calculators on hand.

Providing engaging trigger activities in Maths lessons

It is not often that a good movie comes along to stir students’ interest in Maths. This is why trigger activities form an important component of Maths lessons. Extract B (part 1) is a maths trick that you have probably seen before.

Read Part 2

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