Lesson 8 Ladders and slides
|Reasoning||Resources: Lists A-D of numbers (see page 77) on whiteboard or OHT, Worksheet 1, Worksheet 4 (optional)|
|.||Whole class preparation|
|Recognising that some descriptions are more useful than others in generating unknown numbers of the set.
Using the additive relationships to bring out the multiplicative relationships, because what has to be added is not constant but goes up multiplicatively, e.g. A B 2+ 4= 6, 4+ 8=12, 6+12=18
|Reveal ‘ladder’ A and ask for a description of the set. (Alternatively, start writing the numbers in list A and ask pupils to continue the pattern, then ask for a description.) Likely responses are: 2-times table, going up in 2s, add 2 each time, even numbers, multiples of 2. Ask what the missing numbers are and ask for explanations of how they have been found. This should lead to a general understanding that if you're looking for the nth number in the list, you have to double # to find it. So which is the most useful way of describing the set? Why?
Cover ladder A and repeat with ladder B, either revealing it or generating it in front of them: 6-times table, add 6, multiples of 6,6 times whichever number in the list you want to find. Then list C: 3-times table, add 3, multiples of 3,3 times whichever number in the list you want to find. If a pupil says odds and evens, ask about the pattern of odds and evens and ask: Are all the odds and evens here? And list D: 9-times table, add 9, multiples of 9, 9 times whichever number in the list you want to find.
Show only lists A and B and ask What connects the numbers in lists A and B? The initial response might be ‘add 4! Ask: This works for the first pair, does it work for the others? Establish that the relationship can be expressed in different ways: X3, tripling, adding the number to itself twice. Stress that this is a multiplicative relationship: adding a number to itself twice is the same as tripling it. Look also at the reverse relationship from B to A (third, divide by 3). Use the larger numbers in list A to reinforce the process of getting from a number in list A to a (missing) number in list B. Cover list A and look at lists B and C in the same way.
Key question: What is the relationship between the numbers in the different pairs of lists?
|Pair and group work|
|Handling inverse operations||Give out Worksheet 1, one between two. Pupils repeat looking at lists B and C and then go on to look for the relationship between lists A and C,B and D and A and D.
You may want to stop and discuss the relationship between lists A and C when pupils have had an initial go at this. Because the relationship is not a simple multiplicative one, some discussion at this stage may be helpful.
|Whole class sharing/discussion|
|Finding the relationships between the lists which involve fractional multipliers or alternatively two steps multiply and divide or vice versa.
There is the potential for reading the relationships as ratios.
Moving from additive to multiplicative relationship, with the intermediate step of adding parts of what you start with:
2+1=3 4+2=6 64+3=9
Recognising the same relationships in larger numbers.
|Likely responses for the relationship between A and C are:’halve A and add to itself’ ‘one and a half of A, ‘multiply A by 3 and halve it! Write pupils’ responses to bring out their equivalence:so C + 1/2 A+A, 1 1/2A , or 3/2A. Stress that we are multiplying and dividing to find the relationship. Some pupils may have noticed that 4 produces equivalent fractions; this reinforces the notion of the constant relationship between the numbers in the lists. The fraction in its simplest form gives the multiplier from list C to A. (For some classes you may want to write 4 = 3 and show that if you multiply each side by C this becomes A =2C or 2C. This will depend on their facility with algebraic equations.)
If some pupils are still insisting on seeing the relationship as additive, with what is added changing, repeat the exercise for lists A and C and ask: How can we find what is added if we only know the A number? This should lead to the understanding that half the A number is added. Have a similar discussion about the relationship between lists B and D; this is essentially the same relationship but the larger numbers obscure it.
The relationship between lists A and D is more complex and may be described as: multiply A by 4 and add 4A; multiply A by 44; divide A by 2 and multiply by 9; multiply A by 9 and divide by 2; multiply A by 3 produces equivalent fractions to 4.
End with a brief reflection on the difference between additive and multiplicative relationships. Most pupils would accept that the addition in a sequence is easy but laborious for matching across and finding unknowns.