Key stage 2 mathematics tasks for the more able Number slide solutions and what to look for

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1 Key stage 2 mathematics tasks for the more able Number slide solutions and what to look for Solutions Part 1 (a) One possible solution is as follows: Note: The completed tree diagram should include 16 branches. These may appear in any order. (b) One possible solution is as follows: All of the answers are even, so you are certain to lose the game. KS2/mathematics tasks/above level/number slide 1

2 (c) The best starting numbers to choose are the odd numbers: 1, 3, 5, 7, 9. When starting from an odd number you always add three odd numbers and two even numbers to make a final score that is an odd number. Part 2 (a) (b) It does not matter which number you start on, you will always win. You win the game of Two Slide if you start on an even number. It is impossible to win the game of Three Slide. (c) and (d) are complex questions which are explored in more detail in the What to look for section. Some possible generalisations are given below: (c) (d) If the number of moves is odd, winning will be either certain or impossible. If the number of moves is even, whether or not you win the game depends on whether you start on an odd number or an even number. The best starting numbers to choose depend on the number of moves. The pattern repeats. Number of slides How to win Multiple of 4 Start on an odd number 1 more than a multiple of 4 Start anywhere 2 more than a multiple of 4 Start on an even number 3 more than a multiple of 4 You cannot win (e) You win the game of Fifty Slide only if you start on an even number, because 50 is 2 more than a multiple of 4. You cannot win the game of Ninety-nine Slide, because 99 is 3 more (1 less) than a multiple of 4. You win the game of Million Slide only if you start on an odd number, because is a multiple of 4. KS2/mathematics tasks/above level/number slide 2

3 What to look for Part 1 Decide how best to organise and present findings (Ma4 1f) Explain and justify their methods and reasoning (Ma4 1h) Part 1(a) In this part pupils are asked to complete the tree diagram to show all the possible scores after four moves, starting at 5. This requires some care and persistence. While most pupils complete the diagram, careless adding up can spoil the result. Some checking is useful here. In this example, it seems likely that the initial 5 has been missed in the addition for the last four totals. Part 1(b) This part does not explicitly ask for a tree diagram to display outcomes. Most pupils do make one (careful planning of the layout is necessary to avoid crowding in the later branches); a listing of the numbers in each set of four moves, for example, could be used instead, though pupils who try this often make an error, unless they are systematic. The tree diagram helps to overcome this risk. It is an obvious result for most pupils that starting at 2 always results in a final even number, and so the game is always lost. KS2/mathematics tasks/above level/number slide 3

4 Part 1(c) This is a test of analytical skills. Most pupils are willing to accept that on the basis of an odd start, an odd total will result (so the game is won); and on the basis of an even start an even total will appear (so the game is lost). Not all are clear (or accurate) as to why this is the case. In some cases the perceptions are useful starting points, though the immediate conclusions drawn may not be justified. More able pupils give clear reasons to explain why odd numbers always win the game or why even numbers always lose. KS2/mathematics tasks/above level/number slide 4

5 Part 2 Approach problems flexibly, including trying alternative approaches to overcome any difficulties (Ma4 1b) Identify the data necessary to solve a given problem (Ma4 1c) Part 2(a) Five Slide gives pupils an opportunity to make their own exploration of a game similar to Four Slide. The results for Five Slide are sharply different to Four Slide it makes no difference what the starting number is, you are always adding three even numbers and three odd numbers, so the total is always odd: you win every time. Part 2(b) This continues the mapping out of the mathematical properties of related games, Two Slide and Three Slide. With care, able pupils see that the outcome of Two Slide is dependent on the starting number, since you are either adding up two even numbers and one odd (even number start) to get an odd outcome or two odd numbers and one even (odd number start) to get an even outcome. These results neatly contradict some of the ideas that are likely to be expressed in 1(c), such as starting or finishing on an odd number will automatically give an odd total. Three Slide, in contrast to Five Slide, is impossible to win because you are always adding two even and two odd numbers, resulting in an even total. KS2/mathematics tasks/above level/number slide 5

6 Part 2(c) The answer to whether changing the number of moves makes it easier or harder to win the game is complex. Many pupils attempt to answer the question in simple yes or no terms, despite the variation in the results shown in parts 2(a) and 2(b). Able pupils will see that some further exploration is needed in order that a clearer pattern can emerge. Six Slide and Seven Slide may be explored, although solutions may not be complete. Only the most able pupils are able to describe the entire structure of the game and its variants in one go. Other pupils are more likely to make piecemeal discoveries and then fit them into an emerging pattern. This makes the investigation work well suited to group effort, as different members of a group can explore particular facets the even numbered slides, the odd numbered slides, various starting numbers etc. KS2/mathematics tasks/above level/number slide 6

7 Part 2(d) Again this is a complex question needing careful collection and examination of the results. Sufficient exploration is needed before any generalisation that if the number of slides is 1, 3, 5, 7 etc. then the starting number has no effect you are either bound to win or bound to lose. In the even numbered slides you will win or lose depending on whether you start with an odd or an even number. Some able pupils demonstrate partial understanding of the underlying patterns by generalising correctly about odd and even numbers, but are not then able to go further to identify which games will be won or lost. Part 2(e) Reliable explanations on how to win Fifty Slide, Ninety-nine Slide and Million Slide depend on knowing the rules established in the preceding sections. Many pupils make a lucky guess on one of these, based on partial knowledge but they are unlikely to get all three correct this way, and their explanations will be unsound. Relatively few pupils succeed here without a few pointers in the right direction. For example, some pupils realise that odd numbered slide games are certain to be won or lost independently of the starting number; the challenge is to know which will be won and which will be lost. This leads to an investigation of one part of the slide games that able pupils can undertake. They should be able to see a pattern and generalise it for Ninetynine Slide. Similarly the even-numbered slide games are won or lost depending on the starting number. This can be explored and the result generalised in order to distinguish Fifty Slide from Million Slide. KS2/mathematics tasks/above level/number slide 7

8 Reviewing mathematical achievement Level 5 Pupils working at level 5 are able to complete the tree diagram started for them in Part 1(a) with confidence and sustained accuracy. They are generally successful at making their own diagram (or equivalent) in Part 1(b). They are able to suggest the best starting numbers in Part 1(c) and make some statements to justify this in terms of odd + odd = even etc. Level 5 thinking will involve some engagement with the generality required. In Part 2, most pupils working at level 5 are able to explore Five Slide and report the result, though they are unlikely to offer clear or comprehensive explanations tending to generalise from only one result. This may also be the case with One Slide and Two Slide. Level 6 Level 6 thinking allows pupils to see the reasons behind the results on the basis of their knowledge of odd and even numbers (Part 1(c)). They will be able to offer confident general assertions from looking at sufficient individual cases for a pattern to be seen. They will use this pattern to predict and then check further results in parts 2(b), (c) and (d). In Part 2(e) pupils apply this understanding to predict what cannot easily be checked. What next? Asking what if can modify any task. For example, what if: All the numbers in the grid were increased by 1? The grid became a 4 x 4 square? The numbers were multiplied instead of added? KS2/mathematics tasks/above level/number slide 8

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