# Cross Products

Teacher’s Notes

These exercises and activities are for students to use independently of the teacher to practice number properties.

### Number Framework domain and stage:

Multiplication and division, stages 6 – 7. Advanced additive to advanced multiplicative

### Curriculum reference:

Number, levels 3 to 4

Algebra level 3

Algebra level 5 (exercise 11)

### Numeracy Project book reference:

These activities can be used to follow the teaching of Cross Products, Book 6, page 37 and are for those students who are able to use the associated number properties. These activities use an area model for multiplication arising out of a sound understanding of the array model for multiplication.

### Prior knowledge. Students should be able to:

- recall all their multiplication facts to 100 (at least)
- multiply by multiples of 10
- understand the area model of multiplication
- recognise that different geometric shapes can be used to represent different numbers
- use the conventions of order of operations
- solve multiplication problems involving up to two 1decimal place decimals (exercises 4 – 9)
- multiply algebraic terms, including use simple power notation (exercise 11)
- recognise and collect like terms (exercise 11)

### Background

While the drawings involve using an area model to display the multiplications, (so the size of each box is in some form of proportion to the size of the numbers), it should also be noted that students can do the same problems using a table (or tabular form). In this form area is not considered.

For example: 14 x 51 can be done by suing the following table (where all the boxes are the same size).

So 14 x 51 = 500 + 200 + 10 + 4

= 714

Introducing the tabular form at some stage can be useful as it is easier to use for subtractions. For example:

Exercise 1: Place value but not as we know it

This exercise is designed to follow on from a teaching episode in which the strategy is illustrated and used by the students. Once students have convinced themselves that the strategy works with a range of simple numbers they are ready to explore more difficult numbers including decimals.

Exercise 2: Into the unknown

In this exercise the strategy is used to work backwards. Being able to multiply by multiples of 10 is central.

Exercise 3: Mixing up the unknowns

This exercise would work well as a homework exercise. It builds on exercise 2 using the same strategy to work backwards but introduces diagrams oriented vertically.

Exercise 4: How does it work with decimals?

This introduces a decimal number (1 dp) for one of the factors in the product.

Exercise 5: More rectangles

This exercise uses whole numbers between 10 and 100 for both factors.

Exercise 6: Working in the decimal zone

One of the factors is a one place decimal and the other factor is a whole number between 10 and 100.

Exercise 7: Finally… Two decimals

Both factors are one dp decimals, using numbers less than 10 until the last question which is greater than 10.

Exercise 8: The unknown zone

This exercise combines the previous exercises to find unknowns. Problems involve two decimals to 1dp.

Exercise 9: Word problems

This exercise asks students to use the cross product strategy to solve word problems. Students may need to be encouraged to use units appropriately.

Exercise 10: Investigation

This exercise asks students to investigate the use of the cross product method for numbers in the 100s

Exercise 11: Moving to algebra

This exercise introduces *x* as a length added to both sides of the rectangle and requires students to multiply the algebra and add like terms.