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Reasoning in the classroom Year 8 Playing games Support materials for teachers Year 8 Reasoning in the classroom – Playing games These Year 8 activities encourage learners to work systematically to solve problems. In Activity 2 and Activity 3 they also find and generalise number patterns, and seek to justify their answers. Playing games Learners solve a problem based on pairs of teams playing games. Includes: ■■ Playing games question ■■ Markscheme More games They continue the context of playing games to generate the sequence of triangular numbers, then look to explain the nth term. Includes: ■■ Explain and question – instructions for teachers ■■ Whiteboard – Four teams ■■ Whiteboard – Triangular numbers Learners will need access to the internet, either in school or at home. Stacking cans They apply their knowledge of triangular numbers to solve a problem involving stacking cans. Includes: ■■ Explain and question – instructions for teachers ■■ Whiteboard – Stacking cans Reasoning skills required Identify Communicate Review Learners choose their own methods and transfer their mathematical skills to a range of problems. They explain their results, using algebraic language where appropriate. They look to justify their solutions and review published justifications, putting them into their own words. Procedural skills Numerical language ■■ Time ■■ Number patterns ■■ Proof ■■ Simple algebra ■■ Triangular numbers ■■ nth term ■■ Formula ■■ Square numbers Year 8 Reasoning in the classroom: Playing games Introduction Playing games Activity 1 – Playing games or Outline This activity is based on four teams that play together in pairs. Learners work systematically to find the time when the last game is completed. You will need Q Playing games question One page for each learner M Markscheme Year 8 Reasoning in the classroom: Playing games Activity 1 – Playing games – Outline Q There are four teams: red team (R) blue team (B) ●● each team plays the other three teams ●● each game lasts 30 minutes ●● there are 10 minutes between games ●● only one game is played at a time. yellow team (Y) purple team (P) The first game starts at 2pm. At what time does the last game finish? Show how you know. pm 4m Playing games Activity 1 – Playing games – Question M Activity 1 – Playing games – Markscheme Marks 4m Or 3m Answer 5:50pm Shows that there are 6 games but then gives the answer 6pm 7 The only error is to have 6 rather than 5 intervals 7 Accept a total that is greater than 12, adding 40 minutes for each additional game 7 Has included one too many intervals Or Shows that the total time is 3 hours 50 minutes (or 230 minutes) Or 2m Shows that there are 6 games Or Clearly shows an incorrect total number of games greater than 2, then gives their correct finish time (as shown below) Or 1m Games Finish time 3 3:50 4 4:30 5 5:10 6 5:50 7 6:30 8 7:10 9 7:50 10 8:30 11 9:10 12 9:50 Clearly shows an incorrect total number of games greater than 2, but their finish time is 10 minutes more than in the table above Year 8 Reasoning in the classroom: Playing games Activity 1 – Playing games – Markscheme M Activity 1 – Playing games – Exemplars RB RY RP BY BP YP 6 × 30 mins + 50 mins between Correct; 4 marks ●● 10 to 6 10 to 6 is an acceptable alternative to 5:50pm. pm RB RY RP BY BP 6 games finishing at 6pm; 3 marks YP 6 matches of 40 minutes is 240 minutes which is 60 × 4 = 4 hours so they finish at 6pm pm R v B at 2pm RY v P at 2.40pm R v Y at 3.20pm B v P at 4pm R v P at 4.40pm B v Y at 5.20pm R +B + Y = 90 mins +P B +R + Y = 90 mins +P = 110 mins 60 30 – 60 – 30 ●● 5.20 pm P +Y + B = 90 mins +R This learner communicates effectively, showing clearly that there are 6 games. However, 5:20 is the start time not the finish time of the last game. This shows the importance of checking work. 360 8pm 12 games with 9:50pm; 2 marks 30 Y +R + B = 90 mins +P The games are listed systematically and the working is clear. However, this time includes an additional 10 minutes since there should be 5 intervals between games, not 6. 6 games; 2 marks + 60 60 Common error 60 60 60 = 360 60 60 60 9.50 Common error This learner has doubled the number of games by working with, for example, both RB and BR. This is a common error. pm 4 games 4 games with 4:40pm; 1 mark 4 x 30 = 120mins= 2 hrs 10 minute break x 4 = 40 2hrs + 40 mins = 2.40 4.40 pm Year 8 Reasoning in the classroom: Playing games Common error 4 games (because there are 4 teams) is another common error. For 2 marks, the time would need to be 4:30. However, the answer of 4:40 is 10 minutes more than 4:30 showing that the learner has used one too many intervals between games. Activity 1 – Playing games – Exemplars More games Activity 2 – More games or Outline This activity continues the context of playing games introduced in Activity 1 – Playing games. Once the sequence of triangular numbers has been found, learners consider the nth term and review explanations that justify the formula, putting the explanations into their own words. You will need WB Whiteboard – Four teams WB Whiteboard – Triangular numbers Learners will need access to the internet, either in school or at home Year 8 Reasoning in the classroom: Playing games Activity 2 – More games – Outline Activity 2 – More games Show Four teams on the whiteboard and ask learners to work in groups or pairs to find the total number of games played by different numbers of teams. Explain (If needed, stop learners after they have been working for a few minutes, and remind them that working systematically helps them to see number patterns.) When appropriate, share results and show them in a table. For example: Number of teams Total number of games 2 1 3 3 4 6 5 10 +2 +3 +4 Ask if they can explain why the number of games increases in this way. (When the fifth team is added, this new fifth team plays the previous four teams so the number of games increases by four, and so on.) Draw attention to the sequence 1, 3, 6, 10, . . . then show Triangular numbers on the whiteboard. Discuss the pattern, and check understanding of the formula by asking learners to make sure that when n = 1, 2, 3 and 4, the totals of 1, 3, 6 and 10 are correct. Their task is to research why this formula works, and to explain the method(s) in their own words. You can leave this open, or direct learners as appropriate, for example: ■■ a visual proof of the formula can be found at www.maths.surrey.ac.uk/hosted-sites/R.Knott/runsums/triNbProof.html ■■ the Gaussian method of summing two sequences of triangular numbers can be found at nrich.maths.org/2478 Or Print the relevant pages for them. ■■ How can you use a spreadsheet to continue the sequence of triangular numbers? ■■ Why is using a formula more efficient than working out one row at a time? ■■ What is an nth term? Why do we use n rather than a number like 20, or . . . ? Question ■■ How important is it to know why the formula works? ■■ How difficult was it to understand the proof and put it into your own words? Year 8 Reasoning in the classroom: Playing games Activity 2 – More games – Explain and question WB red team (R) blue team (B) yellow team (Y) purple team (P) When four teams play one another there are a total of six games. Choose a different number of teams. How many games would they play? Can you find a rule for any number of teams? Playing games Activity 2 – Four teams – Whiteboard WB 1 3 6 10 n(n + 1) The nth triangular number is 2 Playing games Activity 2 – Triangular numbers – Whiteboard Stacking cans Activity 3 – Stacking cans or Outline This activity is designed to continue from Activity 2 – More games. Learners solve a problem involving stacking cans. They use their knowledge of triangular numbers to explain why the nth term is a square number. You will need WB Year 8 Reasoning in the classroom: Playing games Whiteboard – Stacking cans Activity 3 – Stacking cans – Outline Activity 3 – Stacking cans Show Stacking cans on the whiteboard and tell learners that this stack of cans is five cans high. Explain A shopkeeper wants to make a display like this but the stack of cans will be 10 cans high. How many cans does the shopkeeper need in total? Their task is to find the number of cans and convince you that it must be correct. (100 cans will be needed in total. There are different ways of proving that the answer is 10 2, for example by moving the right-hand side to show a 10 by 10 square. The diagram below shows how a stack that is 5 cans high can be rearranged to form a 5 by 5 square.) R ■■ How are you approaching the problem? ■■ Why is drawing the cans not an efficient method? ■■ 100 is a square number. Can you show me how the cans can be rearranged to show 100 Question as a square number? ■■ What is the nth term for the sequence of cans arranged like this? (n 2) Extension ■■ We can think of the stack of cans as two triangular numbers. How? (See diagram above.) For a stack that is 10 cans high, which two triangular numbers do we need? (10th and 9th) So how would you write these using n? n(n+1) + (n−1)n 22 Can you show me how these two nth terms add together to give n2? ( Year 8 Reasoning in the classroom: Playing games ) Activity 3 – Stacking cans – Explain and question WB Playing games Activity 3 – Stacking cans – Whiteboard