This is a difficult Starter so there is a prize of a watch if you can complete it.
For each pair of numbers multiply the sum by the difference then divide the answer by 5 without using a calculator.
3, 2 
8, 7 
15, 13 
17, 16 
23, 18 
22, 14 
13.5, 13 
10.5, 10.5 
At least one of your answers is wrong.
Congratulations, all of the answers are correct.
You have won a watch. You can watch this sheep for as long as you want!
A Starter which is similar to this one but slightly harder is Multiply, Add, Subtract and Divide.
Topics: Starter  Mental Methods
In any real situation I can think of, if you were to ask someone for the difference between, say, 3 and 5, the answer would be 2 not 2! We tend to think of differences as positive numbers, and thus the proper rendition of that expression algebraically would be 35 (or 53). This is the same idea as the "distance" between two numbers on the number line, which is always positive.
In textbooks, as you've observed, it seems common in 'word problems' to make a different convention, that 'the difference between a and b' means a  b.
The full discussion can be seen here.
Perhaps we'll wait for a few more comments on this page and then adapt this starter to conform with the opinions of the majority. In the meantime you'll see that all the pairs of numbers above have the largest first to take out the ambiguity.
Thanks once again for your feedback 7 Mandelbrot and Laura.How did you use this starter? Can you suggest
how teachers could present or develop this resource? Do you have any comments? It is always useful to receive
feedback and helps make this free resource even more useful for Maths teachers anywhere in the world.
Click here to enter your comments.
Previous Day  This starter is for 27 October  Next Day
1  3  11.2  6.6 
41  57.6  2.65  0 
Note to teacher: Doing this activity once with a class helps students develop strategies. It is only when they do this activity a second time that they will have the opportunity to practise those strategies. That is when the learning is consolidated. Click the button above to regenerate another version of this starter from random numbers.
Dividing by five is a great deal to ask,
But an easier method will help you.
Dividing by ten is a much easier task,
Then multiply your answer by two!
Your access to the majority of the Transum resources continues to be free but you can help support the continued growth of the website by doing your Amazon shopping using the links on this page. Below is an Amazon search box and some items I have chosen and recommend to get you started. As an Amazon Associate I earn a small amount from qualifying purchases which helps pay for the upkeep of this website.
Teacher, do your students have
access to computers? 

Here a concise URL for a version of this page without the comments.
Transum.org/go/?Start=October27
Here is the URL which will take them to a related student activity.
See the National Curriculum page for links to related online activities and resources.