EdPlace's Key Stage 3 Home Learning Maths Lesson: Indices
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Now...onto the lesson!
Why Does Your Child Need to Master Indices?
The concept of 'powers' will have been taught in Key Stage 2 and students should know the meaning of squaring and cubing. In Key Stage 3, this concept is extended further to include calculations of powers multiplied, and powers of decimals. This guide will explain how to introduce these concepts.
We're confident that by following this step-by-step guide your child will be able to:
1) Evaluate numbers up to the power of 5
2) Calculate two powers being multiplied
3) Work out the squares and cubes of decimals
Step 1 - What are Powers?
The first thing to do is consolidate that powers refer to a number being multiplied by itself a given number of times. It's often mistaken for multiplication, so ensure your child can tell the difference! For example, 43 is “4 to the power of 3” and means 4 multiplied by itself three times over (4 x 4 x 4 = 64). It isn’t 4 x 3 = 12. The 'big number' at the front is called the base number and the power, or 'little number' is called the index. The plural of an index is indices.
Step 2 - How To Multiply and Divide Powers
Let’s take this idea one step further. Suppose you have two powers being multiplied together. For example, 43 x 46.
43 = (4 x 4 x 4)
46 = (4 x 4 x 4 x 4 x 4 x 4)
If we multiply these together, we get (4 x 4 x 4) x (4 x 4 x 4 x 4 x 4 x 4) - which leaves us with 4 being multiplied by itself nine times over. (The answer is 262,144). In other words: 43 x 46 = 49.
So the rule to remember is: when you multiply powers together, you add the powers together.
Importantly, this will only work if the base numbers are the same.
For example, 74 x 74 = 78 but we can’t take any shortcuts if we have 74 x 44 due to their different bases.
Next, let us discuss what happens when we divide two powers.
For example, 69 ÷ 67 = (6 x 6 x 6 x 6 x 6 x 6 x 6 x 6 x 6) ÷ (6 x 6 x 6 x 6 x 6 x 6 x 6).
When we divide, the seven 6s in the first bracket (denominator) will cancel out seven 6s in the second bracket (the numerator). This just leaves us with (6 x 6) = 62. So the rule to remember is: when you divide powers together, you subtract the powers from each other. Again, this only works when the base numbers are the same.
Step 3 - The Power of Zero and Decimals
Step 2 leads up to the concept of the power of zero. This may seem quite an odd concept – how can you multiply something by itself zero times over? Many students think the answer must be 0, but if we look at the mathematics behind it, we find it actually gives a different answer!
Imagine you divided 69 by 69 – i.e. dividing a number by itself. The rules state that you have to subtract the powers. 69 ÷ 69 = 60. As we know, anything divided by itself is always equal to 1. So what we see here is 60 = 1. This would work with any base number or index, provided it was divided by itself. The powers must be subtracted to give the power of zero, and anything divided by itself must be equal to 1 (e.g. 47 ÷ 47 = 40 = 1). So, anything to the power of zero is actually equal to 1. This is an important rule for your child to remember.
Finally, let's look at the powers of decimals. The easiest way of multiplying with decimals is to forget they are there, and then re-insert them at the end of the calculation. The thing to remember is that the numbers themselves will not be affected. You just have to put the decimal in the correct place. For example, we know that 52 = 5 x 5 = 25. If you were then asked to find 0.52, you would forget about the decimal for a moment and work out what the numbers in the answer will be: 52 = 5 x 5 = 25. All you then need to do is count how many digits in the question were behind a decimal place.
0.5 x 0.5 = Two decimal places.
So, this is how many decimal places we need to move in the answer.
25 becomes 0.25.
The final answer is 0.52 = 0.25.
Another example: 0.023 = 0.02 x 0.02 x 0.02.
Ignore the decimals for a moment and work out 23 = 2 x 2 x 2 = 8.
Now, count the total decimal digits in the question: 0.02 x 0.02 x 0.02. There are six digits behind the decimal place, so that is how many places we need to move it in the answer. 8 becomes 0.000 008. The final answer is 0.23 = 0.000 008.
Step 4 - Practice Questions
1) Work out 3 cubed plus 7 squared.
2) What is 56 x 58?
3) What is 410 ÷ 42?
4) Work out 0.072
5) What is x0 + y0?
Step 5 - Give it a Go...
Now that you’ve both mastered indices, see if your child can apply their knowledge to the following 5 activities. Have them complete them in the order listed below. All activities are created by teachers and automatically marked.
Plus, with an EdPlace subscription, we can automatically progress your child at a level that's right for them. Sending you progress reports along the way so you can track and measure progress, together - brilliant!
Activity 1 - Evaluate Powers (1)
Activity 2 - Evaluate Powers (2)
Activity 3 - Multiplying and Dividing Indices
Activity 4 - Powers of Decimals
Answers
1) 33 = 27. 72 = 49. So, 27 + 49 = 76.
2) Add the powers = 514.
3) Subtract the powers = 48.
4) 72 = 49. There are 4 decimal digits in 0.07 x 0.07, so move the decimal 4 times in the answer. 0.072 = 0.0049.
5) It doesn’t matter what x and y are. Anything to the power of zero = 1. So the question is just asking 1 + 1. 1 + 1 = 2.
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