Download Playing games - Hawthorn High School

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