Teach Secondary 13.7

[ M A T H S P R O B L E M ] Colin Foster (@colinfoster77) is a Reader in Mathematics Education in the Department of Mathematics Education at Loughborough University. He has written many books and articles for mathematics teachers. foster77.co.uk , blog.foster77.co.uk THE DIFFICULTY Which of these numbers is in standard form and which isn’t? How can you tell? 2.1 × 10⁴ and 37 × 10⁴ The first number is in standard form, because it’s of the form × 10 , where 1 ≤ < 10 and is an integer. The second number isn’t in standard form, because 37 > 10. How would you write 37 × 10⁴ in standard form? In standard form, this number would be 3.7 × 10⁵ . The 37 has been divided by 10 and the 10⁴ has been multiplied by 10 , making 10⁵ , so the product is unchanged.We can imagine a factor of 10 leaving the 37 to join the 10⁴ . How would you add up these two numbers? Students may be unsure, or they may suggest wrong answers, such as 5.8 × 10 5 or 5.8 × 10 10 . THE SOLUTION It’s easier to see the answer to this question if we go back to the way the original two numbers were presented: 2.1 × 10⁴ and 37 × 10⁴ . Both numbers are written as multiples of : we have 2.1 lots of 10⁴ plus 37 lots of 10⁴ . So, we can add 2.1 and 37 , because the ‘units’ are the same: 2.1 × 10⁴ + 37 × 10⁴ = 39.1 × 10⁴ .We can think of this as ‘counting in ten thousands’. Giving the answer in standard form, it would be 3.91 × 10⁵ . In this lesson, students explore alternative ways of adding and subtracting numbers written in standard form How could you do this by ‘counting in hundred thousands’ instead of by ‘counting in ten thousands’? We could write both numbers as multiples of 10⁵ : 2.1 × 10⁴ + 3.7 × 10⁵ = 0.21 × 10⁵ + 3.7 × 10⁵ = 3.91 × 10⁵. Get students to explain what’s happened here. Which way do you think is easier? Why? Make up five examples of additions like this. Solve them in both ways (i.e. by making the powers of 10 match in two different ways , like this). Students may see an analogy with finding common denominators when adding fractions. In both cases, we need to find a ‘common unit’ before we can add. How would you work out the difference between the two numbers that we started with? The process is almost identical. To find the difference, we first need to decide which number is larger, and then subtract the smaller number from this: 37 × 10⁴ is greater than 2.1 × 10⁴ , so we calculate 37 × 10⁴ − 2.1 × 10⁴ = 34.9 × 10⁴ = 3.49 × 10⁵ . Alternatively, in 10⁵ s, we have 3.7 × 10⁵ − 0.21 × 10⁵ = 3.49 × 10⁵ . Checking for understanding How would you find the sum and the difference of these two numbers? 2.1 × 10³ and 37 × 10⁵ What is different this time? The method is the same here, but the numbers happen to be further apart in size. This time, we can choose to work in units of 10³ , units of 10⁵ or units of 10⁶ . However we do it, the sum comes to 3.7021 × 10⁶ and the difference comes to 3.6979 × 10⁶ . Students should sense- check their answers by noting in advance that both answers must be ‘around 4 million’. ADDINGAND SUBTRACTING NUMBERS IN STANDARD FORM Numbers written in standard form can be confusing for students to add or subtract, says Colin Foster 21 teachwire.net/secondary