Visual Models for Multiplying Fractions (and Decimals) Jim Hogan Secondary Mathematics Advisor SSS, Waikato University MAVCONMAVCON MAVCONMAVCON Visual Models for Multiplying Fractions (and Decimals)

Purpose To learn how to use the array to model multiplication - with whole numbers - fractions - decimals To re-learn how important robust mental models are to learning. MAVCONMAVCON MAVCONMAVCON

3 x 4 Draw a picture of 3 x 4 Make a model of 3 x 4 What does 3 x 4 look like? Ask your classes and staff to do this task and see what mental models are established. MAVCONMAVCON MAVCONMAVCON

The “repeated addition” model is common. Why the Array? By Year 6 students are developing multiplicative ideas – or should be… MAVCONMAVCON MAVCONMAVCON

This model obstructs connections to CL Level 4 mathematics. Useful Array This model connects to factors, multiples, primes, fractions, decimals … MAVCONMAVCON MAVCONMAVCON

This is the array model for 3 x 4. - seeing the 3 and the 4 at the same time - more complex thinking than adding - transfers from and to other subjects 3 x 4 What do you notice? MAVCONMAVCON MAVCONMAVCON

Make a model of 1. Revision of One MAVCONMAVCON MAVCONMAVCON

One can be anything I choose it to be! Flexible One MAVCONMAVCON MAVCONMAVCON

Make a model of 1/3 x 1/4 MAVCONMAVCON MAVCONMAVCON

1 1 The sides have been divided into thirds and quarters. There are 12 parts. Each part is 1 twelfth. How would an “adder” see this problem? MAVCONMAVCON MAVCONMAVCON

The array clearly shows that multiplication of the two fractions. The array is intact. The rectangular shape is preserved. The answer is the orange square. Is it important that the “ones” are the same size? MAVCONMAVCON MAVCONMAVCON

MAVCONMAVCON MAVCONMAVCON Notice…NO RULES!

Make a model of this problem MAVCONMAVCON MAVCONMAVCON

Does your model or drawing show every number, every equals and the answer 3? Where is the 4? MAVCONMAVCON MAVCONMAVCON

This model tips out everything. There are 4x9=36 parts. Twelve parts make up the 1. Joining the scattered parts makes another 1. What is the meaning of 1 complete row? MAVCONMAVCON MAVCONMAVCON

…and so to 0.3 x 0.4 Do you need help? Make a model. MAVCONMAVCON MAVCONMAVCON

Essential knowledge The hundreds board is a very useful device. The answer is the 12 orange squares. A little reflection makes this 12 hundredths and now the problem moves to how we write that answer. MAVCONMAVCON MAVCONMAVCON

…and so to 1.3 x 2.4 Draw a picture of the answer of 1.3 x 2.4 MAVCONMAVCON MAVCONMAVCON

(x+2)(x+4) Curiously, many teachers know and use the array model to expand quadratics. The square of x is clearly visible! The 4 groups of x blue squares and the 2 groups of x yellow squares makes 6x. The 2 groups of 4 green squares makes 8. So (x + 2)(x + 4) = x 2 + 4x + 2x + (2x4) and everything is visible. x 2 4x MAVCONMAVCON MAVCONMAVCON

2(x+4) Curiously many teachers do not use the array model here. The x is represented by a clear line of 3 squares. There are 2 groups of an (x and 4) blue squares. So 2(x + 4) = 2x + 2 x 4 = 2x + 8 and everything is visible. 2 4x Provided the -4 is seen as a number, 2 (x – 4) is the same model. MAVCONMAVCON MAVCONMAVCON

(x+1) 2 This is nearly the end of this presentation and the beginning of squares… Notice the coloured squares and the “extra 1” can be transformed to “two the same “and one more, making an odd number. Between any two consecutive squares is an odd number. What are the pair of squares that are different by 25? Can you see an infinite number of Pythagorean Triples here? MAVCONMAVCON MAVCONMAVCON

And so we move to yet another place… All files are located at MAVCONMAVCON MAVCONMAVCON