Year 7 investigation homework

Year 7 Investigation Homework
Each investigation is designed to take a minimum of 4 hours and should be extended as much as the pupil is able. The
project should be set in the 1st lesson of week A and collected in at the end of week B. It is the expectation that for each
investigation a student completes a poster or report. The work produced should be levelled and the students should
have a target for improvement that they copy onto the homework record sheet (which is to be kept in the APP folder).

Outline for the year:

Date set Investigation Title Minimum Hours Due in
Week beginning Week beginning
5th Sep 2011 Final scores 4 hours 26th Sep 2011
3rd Oct Ice cream 4 hours 4th Nov 2011
2011
7th Nov A piece of string 4 hours 28th Nov 2011
2011
5th Dec Jumping 4 hours 9th Jan 2012
2011
16th Jan How many triangles? 4 hours 10th Feb 2012
2012
20th Feb Polo Patterns 4 hours 12th Mar 2012
2012
19th Mar Opposite Corners 4 hours 23rd April 2012
2012
30th April Adds in Order 4 hours 21st May 2012
2012
28th May Match Sticks 4 hours 25th June 2012
2012
2nd Jul Fruit Machine 4 hours 16th July 2012
2012

Year 7 Homework Record Sheet
Date set Investigation Level Target for improvement
Week Title
beginning
Final scores
5th Sep
2011

Ice cream
rd
3 Oct
2011

A piece of
7th Nov string
2011

Jumping
5th Dec
2011

How many
16th Jan triangles?
2012

Polo
20th Feb Patterns
2012

Opposite
th
19 Mar Corners
2012

Adds in
30th April order
2012

Match Sticks
th
28 May
2012

Fruit
2nd Jul Machines
2012

Tackling investigations

What are investigations?
In an investigation you are given a starting point and you are expected to explore different avenues for yourself.
Usually, having done this, you will be able to make some general statements about the situation.

Stage 1 ~ Getting Started
Look at the information I have been given.
Follow the instructions.
Can I see a connection?
NOW LET’S BE MORE SYSTEMATIC!

Stage 2 ~ Getting some results systematically
Put your results in a table if it makes them easier to understand or clearer to see.

Stage 3 ~ Making some predictions
I wonder if this always works? Find out…

Stage 4 ~ Making some generalisations
Can I justify this?
Check that what you are saying works for all of them.

Stage 5 ~ Can we find a rule?
Let’s look at the results in another way.

Stage 6 ~ Extend the investigation.
What if you change some of the information you started with, ask your teacher if you are not sure how to extend the
investigation.

Remember your teachers at Queensbury are her to help, if
you get stuck at any stage, come and ask one of the Maths
teachers.

Final Score

When Spain played Belgium in the preliminary round of the men's hockey competition in the 2008
Olympics, the final score was 4−2.

What could the half time score have been?
Can you find all the possible half time scores?
How will you make sure you don't miss any out?

In the final of the men's hockey in the 2000 Olympics, the Netherlands played Korea. The final score
was a draw; 3−3 and they had to take penalties.

Can you find all the possible half time scores for this match?

Investigate different final scores. Is there a pattern?

Final Score Mark Scheme
Level Assessment – what evidence is there? Tick What you have done well….

3 Describe the mathematics used

4 Explain ideas and thinking

5 Identify problem solving strategies used

6 Give a solution to the question

7 Explain how the problem was chunked into smaller tasks

8 Relate solution to the original context

2 Create their own problem and follow it through

3 Discuss the problem using mathematical language

4 Organise work and collect mathematical information What you need to do to improve…

5 Check that results are reasonable

6 Justify the solution using symbols, words & diagrams

7 Clearly explain solutions in writing and in spoken
language

8 Explore the effects of varying values and look for
invariance

2 Use some symbols and diagrams

3 Identify and overcome difficulties

4 Try out own ideas

5 Draw own conclusions and explain reasoning

6 Make connections to different problems with similar
Level for this piece of homework…
structures

7 Refine or extend mathematics used giving reasons

8 Reflect on your own line of enquiry examine
generalisations or solutions

Final Score Teachers Notes
Level
2 a) Finds a way of making or recording at least one combination

b) Records combinations of half time scores that work. Spain V Belgium 15
possibilities. Netherlands V Korea 16 possibilities.
3 a) Adopts a method to move forward in the activity: e.g. finds other combinations
of scores and records them.
b) Describes what they are doing/ have done using correct mathematical words.
c) Clearly records using diagrams, symbols, letter, colours
d) Records several correct half time scores in a logical manner.
4 a) Finds all combination of half time scores
b) Organises results into a table or other form which makes them useable.
c) Explains how they know they have found all the combinations.
d) Generalises that e.g. as more outcomes are used then the number of
combinations increases: the number of combinations increases in a pattern.
5 a) Follows process outlined in task for own selection of colours in an organised/
structured way
b) Presents results in more than one of the following: adds a suitable comment to
table of results: graph with comment: clear description of findings.
c) Makes a generalisation about the number pattern found and predicts and tests
with a further number of colours with accuracy, e.g. predicts next case and checks
it.
6 a) Identifies number pattern in table of results and pursues this
b) Redrafts own account of work to make it clearer or suggests.
c) Gives some sensible justification for why the number of combinations goes up in
the way it does:
d) comes up with the formula total number of half time scores = No. of possible scores for team
1 x No. of possible scores for team 2.
7 a) Investigates for different scores and considering the effect on the resulting
combinations.
b) Produces a formula and tests it for any number of full time scores.
8 Looks for an overall rule to work when there is more than 2 teams playing a game.

Ice Cream
∞ I have started an ice cream parlor.

∞ I am selling double scoop ice creams.

∞ At the moment I am selling 2 flavours, Vanilla and
Chocolate.
I can make the following ice creams:

Vanilla Chocolate Chocolate
+ + +
Vanilla Vanilla Chocolate

∞ Now you choose three flavours.

∞ Each ice cream has a double scoop.

∞ How many different ice creams can you make?

Extension

Suppose you choose 4 flavours or 5 or 6…

What if you sell triple scoops.

How many then???????

Ice Cream Mark Scheme












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Ice Cream Teachers Notes
Level Strand (i) Strand (ii) Strand (iii)

Application Communication Reasoning, logic & proof

1 a) Works on part of the activity b) Talks about what they are doing c) Talks about whether they will be able to
make lots of ice creams or just a few

2 a) Finds a way of making or b) Talks about how they decided to c) Responds to the questions, “do you
recording at least one combination make a particular ice cream or how think you will be able to make more ice
they decided to record it. creams using those flavours”

3 a) Adopts a method to move b) Describes what they are doing/ d) Works with a general statement (see
forward in the activity: e.g. finds have done using correct 4d) given by the teacher and investigates
other combinations of three mathematical words. to see if it’s true
colours and records them
c) Clearly records using diagrams,
symbols, letter, colours

4 a) Finds all combinations of 3 b) Organises results for different c) Explains how they know they have
colours and finds combinations for colours into a table or other form found all the combinations of 3 colours
a higher number of colours which makes them useable
d) Generalises that e.g. as more colours
are used then the number of
combinations increases: the number of
combinations increases in a pattern.

5 a) Follows process outlined in task b)Presents results in more than c) Makes a generalisation about the
for own selection of colours in an one of the following: adds a number pattern found and predicts and
organised/ structured way suitable comment to table of tests with a further number of colours
results: graph with comment: clear with accuracy, e.g. predicts next case and
description of findings. checks it.

6 a) Identifies triangular number b) Redrafts own account of work to c) Gives some sensible justification for
pattern in table of results and make it clearer or suggests why the number of combinations goes up
pursues this improvements to the mathematical in the way it does: e.g. 3 colours had 6
merit of results produced by combinations so with 4 colours there are
others. those plus 3 previous colours with the
new colour and a double scoop of new
colour.

7 a) Investigates for triple scoop but b) Produces a formula and tests it for any
considers the spatial arrangement number of scoops.
of colours in line or in a circle and
considering the effect on the
resulting combinations.

A piece of String
You have a piece of string 20cm long.

1) How many different rectangles can you make?

Here is one

9cm

1cm 1cm

9cm

(Check 1 + 9 + 1 + 9 = 20)

Draw each rectangle on squared paper to show your results.

2) I am going to calculate the area of the rectangle I have drawn.
Area = base x height so for the one above it is 1 x 9 = 9cm².
From the rectangle you’ve drawn, which rectangle has the biggest area?
What is the length and width of this rectangle?
Write a sentence to say which rectangle has the biggest area.

3) Now repeat the ‘problem’ but the piece of string is now 32xm long.

4) Now the string is 40cm long.

5) Now the string is 60cm long.

6) Look at all your answers for the biggest area. What do you notice?

7) Investigate circles when using string of 20cm.

8) Look at your answers for the largest area for each string size. What do you
notice?

A piece of String Mark Scheme












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A Piece of String – Teachers Notes

Level
2 • Find some areas with the given perimeter of 20cm.
e.g. 9x1=9cm² or 5x5=25cm²
• Writes down what they have done and shows working carefully.
3 • Finds all areas of 20cm perimeter. 1x9=9cm², 2x8=16cm², 3x7=21cm²,
4x6=24cm², 5x5=25cm². Doesn’t matter if repeated 9x1=9cm².
• Recognises biggest area is 5x5
• Moves on to investigate string of 32cm and finds at least 1 correctly.
• Clearly shows workings out and explains maths used.
4 • Completes all for string of 32cm. 1x15=15cm², 2x14=28cm², 3x13= 39cm²,
4x12=48cm², 5x11=55cm², 6x10=60cm², 7x9=63cm², 8x8= 63cm²
• Recognises biggest area of 8x8
• Looks at areas in an ordered way to avoid repeats/ missed rectangles.
5 • Completes all for 40cm. 1x19=19cm², 2x18=36cm², 3x17=51cm²,
4x16=64cm², 5x15=75cm², 6x14=84cm², 7x13=91cm², 8x12=96cm², 9x11=99cm²,
10x10=100cm²
• Completes all for 60cm. 1x29=29cm², 2x28=56cm², 3x27=81cm²,
4x26=104cm², 5x25=125cm², 6x24=144cm², 7x23=161cm², 8x22=176cm²,
9x21=189cm², 10x20=200cm², 11x19=209cm², 12x18=216cm², 13x17=221cm²,
14x16=224cm², 15x15=225cm²
• Recognises biggest area is a SQUARE (must use word square).
6 • Puts results in a table and starts to make generalisations.
• Recognises the circumference of a circle is the 20cm piece of string
• Moves on to look at C=πd d=3.2cm
7 • Calculates the area of circles:
• 20cm string = 32cm² (ish)
8 • Justifies that a circle has the biggest are of all shapes and clearly has
shown workings at all stages.

Jumping

Ben is hoping to enter the long jump at his school sports day.
One day I saw him manage quite a good jump.
However, after practicing several days a week he finds that he can jump half as far again as he did
before.
This last jump was 3 75 meters long.
So how long was the first jump that I saw?

Now Mia has been practicing for the high jump.
I saw that she managed a fairly good jump, but after training hard, she managed to jump half as
high again as she did before.
This last jump was 1 20 meters.
So how high was the first jump that I saw?
You should try a trial and improvement method and record you results in a table. Use a number
line to help you.

Please tell us how you worked these out.
Can you find any other ways of finding a solution?
Which way do you prefer? Why?

Jumping Mark Scheme












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Teachers Notes

Level 3 For Ben
Attempt to show numbers being halved, example 3.75 ÷ 2 = 1.875
Add the above to the length of the jump.
For Mia - as above but using 1.2m
LEVEL 4 Shows number line and trials which show the method
//Eg
1m 0.5m  1.5m
//Eg
2 m 1.0m  3.5m
//Eg
2.5m 1.25m  3.75m
So previous jump was 2.5m for Ben
Using similar method to show the previous jump for Mia was 0.8m
Explains the method.
May use other diagrams to illustrate the trial and error method
Records the results in a table
LEVEL 5 Extends the task to show different jump lengths/heights, using the trial and
improvement method.
Uses inverse operations to show how function machines may be used to illustrate the
problem.
- x1.5  3.75
/
- x1.25  3.75

Level 6 Extends the task to investigate ¼ or 1/3 as much increase in jump height/length.
Records results in table

Original Jump
Increase
New Jump

1m
½
1½

2m
½
3

3m

4¼

Investig
ate and find the “multiplying/Dividing factor to find the solution
Level 7 Uses algebra to denote the missing number, shows the reverse function using fraction
Eg 3.75 ÷ 1.5 = 2.5m (the original jump for Ben)
And 1.2÷ 1.5 = 0.8 (the original jump for Mia)

How many triangles?
Look at the shape below, how many triangles can you see?

I can see 5. Am I correct or can you see more or less? Highlight all the triangles
you can see.

How many triangles can you see in the shape below?

Can you draw a triangle like the ones above that have over 20 but less than 150
triangles?

Try and draw it to show if it or is not possible.

How many triangles? Mark Scheme












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Teachers Notes

Answers to the questions:

Triangle base of 2 triangles
Size Frequency
1 5
2 1
Total 5

Size Frequency
1 9
2 3
3 1
Total 13

Size Frequency
1 16
2 7
3 3
4 1
Total 27

Size Frequency
1 25
2 13
3 6
4 3
5 1
Total 48

Size Frequency
1 36
2 21
3 11
4 6
5 3
6 1
Total 78

Size Frequency
1 49

2 31
3 18
4 10
5 6
6 3
7 1
Total 118

8 x 8 is 170 so over 150.

Level Criteria 1 - Application
3 In describing the mathematics used pupils need to sum the number of triangles in the shape given.
4 Pupils need to explain what they are doing and why to reach level 4. This can be done by an explanation of how they get to
the answer.
5 Pupils can identify problem solving strategies by breaking the tasks down into different size triangles as well as or devising
a way to keep track of which triangles the have counted (for example by use of tally chart or highlighting).
6 Pupils need to show clearly where answers are identify a 4x4 5x5 6x6 or 7x7 shape as the ones producing the number of
triangles between 20 - 150
7 Not only must pupils have broken down tasks into easier components to deal with but they must then explain why they
have done this
8

Level Criteria 2 - Communication
2 Pupils must attempt to solve the problem given
3 Pupils must refer to mathematical words such as triangle etc when describing what they are doing.
4 Organise work by use of tally chart or sensible lists in which results are shown next to original shape.
5 Sensible results will not have a bigger shape have less triangles than small shapes
Total numbers of triangles are as follows 2x2 = 5, 3x3 = 13, 4x4 = 27, 5x5 = 48, 6x6 = 78, 7x7 = 118 and 8x8 = 170.
6 Pupils must explain why they have met the criteria of the brief by referring there results to the corresponding tasks. This
may also be done by use of diagrams.
7 A conclusion to the tasks must be written to sum up how and why the pupil has met the tasks.
8 Pupils may extend the project by looking for patterns for different sizes of triangles within a shape. This may be done
numerically or algebraically.

Level Criteria 1 - Reasoning, Logic and Proof
2 Mathematical symbols or tables used.
3 Break topics into stages before completing tasks.
4 Pupils need to attempt own way of solving the problem to reach end of tasks given.
5 Pupils need to explain why they are using certain strategies and once they have come to an answer they need to conclude
what their answers tell them.
6 Pupils can hit level 6 if they can produce their own shape and solve the task in the 3rd section of the worksheet.
7 Pupils can gain level 7 if they extend the project by looking for patterns between size of shape and number of triangles for
example the number of 1x1 triangles follows the square number pattern.
8 After attempting an extension pupils need to analysis whether there extension works for multiple shapes and determine if
they can take the project further from the extra results gained.

Polo Patterns

When the black tiles surround white tiles this is known as a polo
pattern.

You are a tile designer and you have been asked to design different polo
patterns (this is be made by surrounding white tiles with black tiles).
The drawing shows one white tile surrounded by 8 black tiles.

What different polo patterns can you make with 12 black tiles (you can
surround as many white tiles as you like)?

Investigate how the number of tiles in a polo pattern depends on the number of
white tiles.

Polo Patterns Mark Scheme












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Polo Patterns Teachers Notes
Level
2 c) Finds the 2 ways of arranging 12 black tiles in a polo pattern

d) Written what they have done
3 a) Adopts a method to move forward in the activity, arranging and recording other
combinations of polo patterns
b) Describes what they are doing/have done using correct mathematical words.
4 a) Records the polo patterns in a table in a logical order
b) Explains their ideas and thinking clearly
c) Starts to generalise that the number of black tiles increases as the number of
white tiles increases – in a pattern.
5 a) Notices that the number of black tiles can be different for same number of
white tiles which are arranged in different ways – tries to explain why this
happens.
b) Makes predictions from the patterns they have found e.g. predicts next case
and checks it.

6 a) Finds the nth term of the pattern when white tiles are arranged in a
1 x w rectangle. B = 2W + 6
b) Finds the nth term for the pattern for when the white tiles are arranged in a
square B = 4S + 4 (S = 1 when 1x1 sq, S=2 when 2x2 sq etc.)
c) Redrafts own account of work to make it clearer or suggests.
d) Gives some sensible justification for why the number of combinations goes up
in the way it does
7 a) Investigates for different rectangles ways the white tiles could be arranged and
investigate the effect on the resulting combinations.
8 Looks for an overall rule to work no matter how the white tiles are arranged.

Opposite Corners.

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

The diagram shows a 100 square.

A rectangle has been shaded on the 100 square.

The numbers in the opposite corners of the shaded rectangle are
54 and 66 and 64 and 56

The products of the numbers in these opposite corners are

54 x 66 = 3564 and

64 x 56 = 3584

The difference between these products is 3584 – 3564 = 20

Task: Investigate the difference between the products of the numbers in the opposite corners
of any rectangles that can be drawn on a 100 square.

1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20 11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30 21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40 31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50 41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60 51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70 61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80 71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90 81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100 91 92 93 94 95 96 97 98 99 100

1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20 11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30 21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40 31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50 41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60 51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70 61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80 71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90 81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100 91 92 93 94 95 96 97 98 99 100

Opposite Corners Mark Scheme












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Opposite Corners Teachers Notes
Teachers should introduce the task by reference to the example on the students’ sheet and
some other examples for different sized rectangles. Encourage students to develop the general
case using symbolism without actually introducing it (required for top marks at level 7/8).
Possibly extend to different sized grids if students complete task.

Level 3 Students correctly work out the differences for at least two rectangles and notice the
difference is the same for same sized rectangles.

Level 4 Students correctly work out the differences for at least five rectangles of two different
dimensions and notice the difference is the same for same sized rectangles.

Level 5 Students correctly work out the differences for at least eight rectangles of four
different dimensions and notice the difference is the same for same sized rectangles. Students produce
a well-tabulated set of results and comment on the differences for each dimension.

Level 6 Students work strategically on a set of various sizes; 2x2, 2x3, 2x4 ….3x3, 3x4 etc up to
and including 5x5 and produce a well tabulated set of results and comments.

Level 7 Students move into symbolism e.g. for a 2x3 grid can produce

x x+1 x+2
x+10 x+11 x+12
And express the difference correctly as
(x + 10)(x + 2) – x(x + 12) Multiply out double brackets correctly.

Level 8 Generalise further by using
x …………. x+n
x+10 ………..
x + 10m …………. x + n + 10m
Hence produce a general result for an n x m rectangle.

Numbers in order like 7, 8, 9 are called CONSECUTIVE numbers.

4+5=9

12 = 3 + 4 +
6=1+
5
17 = 8 + 9 2+3

17, 9, 6 and 12 have all been made by adding CONSECUTIVE numbers.

What other numbers can you make in this way? Why?

Are there any numbers that you cannot make? Why?

Adds in Order Mark Scheme












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“Add On’s” Teacher Notes
Level

2 a) Finds the 2 ways of recording results.eg 1+2=3, 2+3=5 etc
b) Written what they have done
3 a) Adopts a method to move forward in the activity, arranging and recording other combinations
of consecutive numbers, in three or fours. eg 1+2+3=6
b) Describes what they are doing/have done using correct mathematical words.
4 a) Records the patterns found in a table in a logical order 2 consecutive number, 3 consecutive
etc.
b) Explains their ideas and thinking clearly
c) Starts to generalise two consecutive numbers gives you all the odd numbers. Because you are
adding an odd number to an even number every time so the results will always be odd. 1+2,
2+3, 3+4..
5 a) Notices the patterns when extending to 4 and 5 consecutive numbers– tries to explain why
this happens.
b) Makes predictions from the patterns they have found e.g. predicts next case and checks it.

Consecutive
numbers
pattern

2
Odd numbers
2n -1
1+2=3,
2+3=5,3+4=7..

3
Multiples of 3
3n+3
1+2+3=6,2+3+4
=9..

4
Going up in 4’s
4n+6
1+2+3+4=10,1+
2+3+4+5=14…

5
Multiples of 5
5n+10
1+2+3+4+5=15,
2+3+4+5+6=20..

6 predicted &
checked
Going up in 6’s
6n+15
21,27,22

7

7n+21
28,35,42

6 a) Finds the nth term of the pattern see table above
b) Redrafts own account of work to make it clearer or suggests.
7 a )Extends work looking at consecutive even/odd numbers
c) Explains task and how they have broken it down
8 Looks for an overall rule. Can explain why some of the numbers cannot be made.
Smallest number can be made
With

3
2 numbers

6
3 numbers

10
4 numbers

15
5 numbers

21
6 numbers

Numbers that cannot be made 1,2,4,8,16,28,32,44

Match Sticks
Look at the match stick shape below.

How many match sticks do you expect to be in pattern 2?

Pattern 2 Pattern 3
2 triangles 3 triangles

Draw the next 5 patterns.

What do you notice about the number of matchsticks used, is there a pattern?

Extension - Can you write it in algebra?

How many matchsticks do you need to make the 50th pattern?

What’s the biggest number pattern can you make with 100 matchsticks? Are there
any left over?

Think about different shapes you can make using matchsticks, investigate (as
above).

Match Sticks Mark Scheme












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Match Sticks Teacher Notes

Level
2 a) Be able to increase each pattern by 2 matches.
b) Can draw the next 5 patterns.
c) Clearly drawing the patterns correctly.
3 a) Describe the sequence of the patterns using correct mathematical words.
b) Clearly recording the information in a table.
Pattern
1
2
3
4
5
6
7
8

Matches
3
5
7
9
11
13
15
17

4 a) Be able to write the pattern discovered in algebra.
b) Writing the correct nth rule. 2n+1.
c) Calculating and writing down the correct number of matchsticks needed to make the 50th
pattern.
2 x 50 +1 = 101 matchsticks.
5 a) Follows up from the 50th pattern to find the biggest number pattern that can be made with
100 matchsticks. Also indicating how many matchsticks are left over. 49th Pattern uses 99
matchsticks with 1 matchstick left over.
b) Explains the method how they found the biggest pattern made by 100 matchsticks. Pupil has
already investigated the 50th pattern that uses 101 matchsticks, so if the pupil calculates the
49th term that would be the closest pattern that uses most of the 100 matchsticks.
6 a) Investigate creating different shapes using matchsticks. For example, Squares and
pentagons.
b) Create a table of results for each pattern of shapes.
Squares.

Pattern
1
2
3
4
5
6
7
8

Matches
4
7
10
13
16
19
22
25

Pentagons.

Pattern
1
2
3
4
5
6
7
8

Matches
5
9
13
17
21
25
29
33

c) Describe the sequence of the patterns using correct mathematical words.
7 a) Be able to write the nth rule for the new patterns in algebra.
Squares: 3n+1
Pentagons: 4n+1
b) Clearly explain solutions in writing and in spoken language.
8 a) Be able to explore a relationship between the nth rules for all the different shapes created.
Number of sides on a shape subtract by 1 that would be the number that you multiply the
pattern by. Then always add 1.
Square = 4 sides subtract 1 equals 3. 3 x number of pattern add 1. 3n+1.
Pentagon = 5 sides subtract 1 equals 4. 4 x number of pattern add 1. 4n+1.
b) Investigate an nth rule for another shape like hexagon or octagon.
Hexagon =6 sides subtract 1 equals 5. 5 x number of pattern add 1. 5n+1
Octagon = 8 sides subtract 1 equals 7. 7 x number of pattern add 1. 7n+1.

Fruit Machine
In this task you are going to design your own fruit machine.

Start with a simple one so you can see how it works.

Use two strips for the reels – each reel has three fruits.

Lemon

Banana

Apple

The only way to win on this machine is to get two apples. If you win you get 50 pence back. It costs 10
pence to play.

Is it worth playing?

You need to know how many different combinations of fruits you can get.

Use the worksheet. Carefully cut out two strips and the slotted fruit machine. Fit the strips into the
first two reels of the machine. Start with lemons in both windows. Move reel 2 one space up – now you
have a lemon and an apple. Try to work logically, and record all the possible combinations in a table,
starting like this:

Reel 1 Reel 2
How many different ways can the machine stop? Are you likely to win?
Lemon Lemon Is it worth playing?

Lemon Apple

Lemon

.

Maths Fruit Machine

Cut out this window Cut out this window

Only 10 pence per play.

Match two apples to win 50 pence.

Fruit Machine Mark Scheme












language

invariance



4 Try out own ideas


structures



Fruit Machine Teachers Notes
Level
2 a) Find one set of solutions.
b) Complete the table at the bottom of the question sheet and produce 3 combinations.

3 a) Adopt a method to move on and complete all the combinations for 2 reels and 3
fruits. (9 combinations)
b) Represent the results in a table.
c) Describe what they are doing to get their results.

4 a) Find all the combinations.
b) Represent all the results in a table.
c) Explain method used.
d) Extend by adding another fruit (4 fruits), but still using 3 reels.

5 a) Using 4 fruits find all the combinations. (16 combinations)
b) Represent in a table.
c) Explain thinking and look for patterns from 3 fruits to 4 fruits.
d) Move on to 5 fruits and find combinations.
e) Predict the next set of results for 5 fruits(25 combinations)

6 a) Produce all sets of results in a clear table. Incorporate 3, 4 and 5 fruits.
b) Find the pattern and make a statement about this.
c) Comment on the denominator being a square number
d) Making the link between the number of fruits and the combinations. i.e 3 fruits
would be 3(squared) = 9 combinations, x fruits would x(squared) = x squared
combinations.

7 a) Investigate 3 reels, starting with 3 fruits (27 combs), 4 fruits (64 combs) and building
it up.
b) Representing all the information in a table.
c) Making the link between the number of fruits and the combinations. i.e 3 fruits
would be 3(cubed) = 27 combinations, x fruits would x(cubed) = x cubed
combinations.
d) Making a link between 2 reels, 3 reels and predicting 4 reels, drawing out and
representing in a table.
e) Finding that combinations are cube numbers.

8 a) Finding a general formula, that fits any number of reels and any number of fruits.
b) General formula is F(to the power of R) = number of combinations
F = Number of fruits
R = Number of reels

Year 7 investigation homework

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