Year 7 investigation homework

Year 7 Investigation HomeworkEach 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 setWeek beginning

Investigation Title Minimum Hours Due inWeek beginning

5th Sep 2011 Final scores 4 hours 26th Sep 20113rd Oct 2011

Ice cream 4 hours 4th Nov 2011

7th Nov 2011

A piece of string 4 hours 28th Nov 2011

5th Dec 2011

Jumping 4 hours 9th Jan 2012

16th Jan 2012

How many triangles? 4 hours 10th Feb 2012

20th Feb 2012

Polo Patterns 4 hours 12th Mar 2012

19th Mar 2012

Opposite Corners 4 hours 23rd April 2012

30th April 2012

Adds in Order 4 hours 21st May 2012

28th May 2012

Match Sticks 4 hours 25th June 2012

2nd Jul 2012

Fruit Machine 4 hours 16th July 2012

Year 7 Homework Record Sheet

Date setWeek

beginning

Investigation Title

Level Target for improvement

5th Sep 2011

Final scores

3rd Oct 2011

Ice cream

7th Nov 2011

A piece of string

5th Dec 2011

Jumping

16th Jan 2012

How many triangles?

20th Feb 2012

Polo Patterns

19th Mar 2012

Opposite Corners

30th April 2012

Adds in order

28th May 2012

Match Sticks

2nd Jul 2012

Fruit Machines

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 StartedLook 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 systematicallyPut your results in a table if it makes them easier to understand or clearer to see.

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

Stage 4 ~ Making some generalisationsCan 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

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

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 structures

7 Refine or extend mathematics used giving reasons

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

What you have done well….

What you need to do to improve…

Level for this piece of homework…

Final Score Teachers Notes

Level

2 a) Finds a way of making or recording at least one combinationb) 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, coloursd) Records several correct half time scores in a logical manner.

4 a) Finds all combination of half time scoresb) 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???????

Investigate

http://www.google.co.uk/imgres?imgurl=http://www.foodclipart.com/food_clipart_images/double_scoop_of_ice_cream_in_a_waffle_cone_0515-0906-2714-0345_SMU.jpg&imgrefurl=http://www.foodclipart.com/food_clipart_images/double_scoop_of_ice_cream_in_a_waffle_cone_0515-0906-2714-0345.html&usg=__9bNhylO_ksTgKVTw9j-kSmEF1HA=&h=300&w=111&sz=13&hl=en&start=8&zoom=1&tbnid=0nYK5ID77g23ZM:&tbnh=116&tbnw=43&ei=IcT5TeOGJMG2hQfqmvW3Aw&prev=/search?q=ice+cream+double+scoop&hl=en&biw=762&bih=403&gbv=2&tbm=isch&itbs=1%00%E5%A1%B9%EF%92%81%E1%B4%BB%E4%A1%BF%E2%B2%AF%E5%B6%82%E8%97%84%E6%8C%A7%00%00%EA%AE%A5

Ice Cream Mark Scheme

















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Ice Cream Teachers Notes

Level Strand (i)

Application

Strand (ii)

Communication

Strand (iii)

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 recording at least one combination

b) Talks about how they decided to make a particular ice cream or how they decided to record it.

c) Responds to the questions, “do you think you will be able to make more ice creams using those flavours”

3 a) Adopts a method to move forward in the activity: e.g. finds other combinations of three colours and records them

b) Describes what they are doing/ have done using correct mathematical words.

c) Clearly records using diagrams, symbols, letter, colours

d) Works with a general statement (see 4d) given by the teacher and investigates to see if it’s true

4 a) Finds all combinations of 3 colours and finds combinations for a higher number of colours

b) Organises results for different colours into a table or other form which makes them useable

c) Explains how they know they have found all the combinations of 3 colours

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 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 triangular number pattern in table of results and pursues this

b) Redrafts own account of work to make it clearer or suggests improvements to the mathematical merit of results produced by others.

c) Gives some sensible justification for why the number of combinations goes up in the way it does: e.g. 3 colours had 6 combinations so with 4 colours there are those plus 3 previous colours with the new colour and a double scoop of new colour.

7 a) Investigates for triple scoop but considers the spatial arrangement of colours in line or in a circle and considering the effect on the resulting combinations.

b) Produces a formula and tests it for any number of scoops.

A piece of StringYou have a piece of string 20cm long.

1) How many different rectangles can you make?

Here is one

(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?

9cm

9cm

1cm 1cm

A piece of String Mark Scheme

















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A Piece of String – Teachers Notes




Level2 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) 32cm string = 82cm² (ish) 40cm string = 129cm² (ish) 60cm string = 289cm² (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 BenAttempt to show numbers being halved, example 3.75 ÷ 2 = 1.875Add 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 BenUsing similar method to show the previous jump for Mia was 0.8mExplains the method.May use other diagrams to illustrate the trial and error methodRecords 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 Jump1m ½ 1 ½ 2m ½ 33m 4 ¼

Investigate and find the “multiplying/Dividing factor to find the solutionLevel 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 NotesAnswers to the questions:

Triangle base of 2 trianglesSize Frequency1 52 1Total 5

Triangle base of 3 trianglesSize Frequency1 92 33 1Total 13

Triangle base of 4 trianglesSize Frequency1 162 73 34 1Total 27

Triangle base of 5 trianglesSize Frequency1 252 133 64 35 1Total 48

Triangle base of 6 trianglesSize Frequency1 362 213 114 65 36 1Total 78

Triangle base of 7 trianglesSize Frequency1 492 313 184 105 66 37 1Total 118

8 x 8 is 170 so over 150.

Level Criteria 1 - Application3 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 - 1507 Not only must pupils have broken down tasks into easier components to deal with but they must then explain why they

have done this8

Level Criteria 2 - Communication2 Pupils must attempt to solve the problem given3 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 Proof2 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 patternd) 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 clearlyc) 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 + 6b) 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 are54 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+2x+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 usingx …………. x + nx+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.

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?

17 = 8 + 9

4 + 5 = 9

6 = 1 +

2 + 3

12 = 3 + 4 + 5

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=6b) 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 clearlyc) 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 28,35,426 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 down8 Looks for an overall rule. Can explain why some of the numbers cannot be made.

Smallest number can be made With 3 2 numbers6 3 numbers

10 4 numbers15 5 numbers21 6 numbers

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

Match SticksLook at the match stick shape below.

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

Pattern 2 Pattern 32 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 SchemeLevel Assessment – what evidence is there? Tick
















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Match Sticks Teacher Notes




Level2 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 8Matches 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 8Matches 4 7 10 13 16 19 22 25

Pentagons.Pattern 1 2 3 4 5 6 7 8Matches 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+1Pentagons: 4n+1b) 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+1Octagon = 8 sides subtract 1 equals 7. 7 x number of pattern add 1. 7n+1.

Fruit MachineIn 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:

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

Reel 1 Reel 2

Lemon Lemon

Lemon Apple

Lemon

.

Maths Fruit Machine

Only 10 pence per play.

Match two apples to win 50 pence.

Cut out this window Cut out this window

Fruit Machine Mark Scheme

















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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 numberd) 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 fruitsR = Number of reels

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Year 7 investigation homework