These exercises and activities are for students to use independently of the teacher to practice number properties
explore a pattern involving powers
generalise a numberr pattern
explore a generalisation involving decimals
Multiplication and division, AP (Stage 8)
Prior knowledge
Basic multiplication facts to 100 (at least)
know squares of numbers to 12
solve multiplication problems using a range of strategies
solve multiplication problems involving decimals
use a letter to stand for ‘any number’
Background
This activity investigates the difference of two squares using an array model for multiplication. The number properties arising out of the difference of two squares is called a pattern. Students initially investigate numbers to see if the pattern works for a range of whole numbers. The array model is used to explore why the pattern works. Symbols are used to summarise the pattern. The array model is extended into an area model for multiplication in working with decimals. The pattern is then used to simplify a range of multiplication problems. Figure it out Algebra level 4+, Book 4, page 1, Square number differences could be a useful follow on activity.
Comments on the Exercises
Exercise 1: Tessa’s pattern
Asks students to convince themselves that the pattern seems to work with a range of numbers. This exercise is designed to follow on from a teaching episode in which the pattern is illustrated and students’ curiosity is roused.
Exercise 2: Developing the pattern
Asks students to express the pattern as a rule. Note that they have not yet proved it works for all numbers so this exercise only asks them to write the rule for numbers which they have seen the pattern works for.
Exercise 3: Squares and rectangles
Asks students to begin to explore the notion on mathematical proof. Students should be ready to explore the reason for the pattern working using an array model for multiplication. Students benefit from building the arrays themselves to see the ‘missing’ or ‘extra’ one. The exercise uses symbols to express the rule. Students need to be comfortable with using ‘n’ to stand for ‘any number’. Explaining the rule in words is a good check of student understanding. The term ‘difference of two squares’ is introduced and students are asked to explain why it is used for the pattern.
Exercise 4: What about decimals?
Asks students to apply the rule they know works with whole numbers to decimals. The area model for multiplication is abstract. Students may need to start with a scaled area before moving to the abstract unscaled rectangle.
Exercise 5: Using Squares
Asks students to explore squares. The use of the rule to calculate products from squares provides another opportunity to reinforce understanding of how the numbers work. Students should notice the symmetry of the numbers about the number they are squaring. Decimals are included. The last question asks students to write their own ‘sensible’ problems. A discussion of what we mean by ‘sensible’ may be appropriate.
Exercise 6: Calculating Squares
Asks students to use the rule in reverse to calculate squares from products. Decimals are included. Again the last question asks students to write their own ‘sensible’ problems. To do this students need a real understanding of the number properties.
Exercise 7: Always true, sometimes true, never true?
Asks students to investigate a generalisation. The first question focuses on generalising the rule about the difference of two squares but introduces terminology of consecutive odd numbers.
Students need to be encouraged to try particular examples. Once they have discerned that the rule seems to work, they need to focus on looking for relationships eg how are the relevant square numbers related to the sequence of consecutive odd numbers? Students should be encouraged to look for numbers for which the rule may not hold – suggestions of numbers to investigate are numbers close to zero and negatives.
The exercise progresses to investigating sums of consecutive odd numbers. Students should be encouraged to follow the same format of investigating particular numbers and then looking for relationships. The relationship between sums of consecutive odd numbers and the difference of two squares is worth extending into algebra.
Students should be encouraged to formally write up their investigations as this helps bring their tendency to invent their own mathematics into acceptable classroom practice. It also encourages students to be more rigorous in their creative process – rather than accepting that something works because it ‘got the answer to number 1 correct!’ (An investigation write up sheet is available in the Related Resources).
3 + 5 = 8
32 – 12 = 8
5 + 7 = 12
42 – 22 = 12
(2n – 1) + (2n + 1) = 4n
(n + 1) 2 – (n – 1) 2 = 4n
- providing students can understand this level of notation and manipulation where n stands for a natural number (so 2n – 1 is an odd number and 2n + 1 is the next odd number).