The Complexity and Simplicity of Numbers and Their Operations

1. The complexity and simplicity of numbers and there operations Paul Cliffe

2. How many lollies are there? You have 30 seconds starting now!

3. How many lollies are there? You have 15 seconds starting now! What is the most important thing we can do to help students understand number? CREATE STRUCTURE

4. Consider the three numbers below. • At what age would you expect a student to understand their value? • What are the important concepts that a student needs to understand their value? Neither 162 or 3.4 look like one number. They are made up of many digits. How do we know they are just one number? AND… 60=6x10 BUT… 162 =100+60+2 100 = 10x10 = 102 AND… BUT… BUT… AND… At the very least can be imagined as or as minus or below zero.

5. It does not matter what number representation is used, they all involve at least one or more of the operations +, -, x and ÷ . What is addition? SO… What is division? What is multiplication? What is subtraction?

6. Your definition for WHAT IS … • A definition that describes how to do it • A definition for your students to remember • Something that makes sense

7. Most common Year 13 students response to WHAT IS… - taking a number away from another - minus a number from a number to get a lower than previous number - when a value is reduced by a certain amount - the difference of two values Subtraction - Times one number with another - Groups of numbers are combined to make a number - The amount of times a value is added to itself - Multiplying two or more numbers together to get a product Multiplication - Dividing a number by another number - Seeing how many times one number fits into another - Division is the opposite of multiplication - A thing split into a number of evenly sized groups Division

8. My definition for WHAT IS… Moving DOWN (the number line) Repeatedly adding the same value How many times does the bottom add to make the top

9. These definitions produce the structure below for students to understand numbers and make calculations. SUBTRACTION: is moving down (the number line) 5 DIVISION: how many times the bottom adds to make the top

10. How do we want out students to visualize 162.

11. How do we want out students to visualize 3.4 If the orange rod can be considered as a whole then each white unit cube is of that so is three orange rods and 4 white rods. Not only that the can easily be seen as two red rods, where each red rod is of the orange 1 rod. This clearly shows that .

12. Without using a calculator evaluate the following and explain what you have done B) why you have done it 2) 3 4) If you finish early, have a go at these 5) ( 5 to the power of zero) 6) ( 5 to the power of negative two) NOTE: You can only add or subtract things that are similar. (ie have the same units)

13. The numerical structure of decimal numbers

14. Common Misconceptions

15. 1) Organise these numbers into as many groups as needed. 2) Organise these numbers into two or three similar sized groups.

16. What is the most important thing about a number?

17. The mixed representations of the numbers have been placed in order of size and shown on the number line. Any representation of a number has very little meaning unless it can be placed on a number line to be compared with other numbers. Numbers we understand better and have more connections to. When it comes to numbers; SIZEreally counts.

18. Important number Properties 1) You can only ADD or SUBTRACT similar things (ie with the same units) 1 metre + 1 metre = 2 metres 1 + 1 = 2 etc… 1 apple + 1 apple = 2 apples 1 thousand + 1 thousand= 2 thousand 1 tenth + 1 tenth = 2 tenths 2) When only adding you can do it in any order (ie commutative law for addition) WHY must this be true? 3) When only multiplying you can do it in any order (ie commutative law for multiplication) WHY must this be true? When a rectangle is rotated by 90° its AREA does not change This would also be true for a cuboid with sides its volume would be the same regardless of how it is rotated. We must draw the students attention to these important properties each time they occur. Students must be able to justify what they are learning.

19. SUMMARISING THOUGHTS I do not think it is good enough just to give students some knowledge and teach them a procedure if they do not understand where it comes from and why it works. Confidence is the solution to improving student outcomes. However, this will only happen if students have a meaningful connection to the concepts being taught and they make sense to them. Rote learning is not the solution to improving student understanding. However, there is a place for rote learning the foundational rules such as Adding is… Subtracting is… Multiplying is… Dividing is…When you can add, When only multiplying you can do it in any order. It is also useful to rote learn basic number fact such as multiplication tables, however they should still make sense. EVERYTHING IN MATHEMATICS COMES FROM SOMEWHERE. IT ALL MAKES SENSE AND CAN BE UNDERSTOOD.

20. I have recently written a book called … It ain’t MATHEMATICS DOESN’T MAKE SENSE if it