Year 7 Investigation HomeworkEach investigation is designed to take a minimum of 4 hours and should be extended as much as...
Year 7 Homework Record Sheet Date set     Investigation   Level                Target for improvement    Week          Tit...
Tackling investigationsWhat are investigations?In an investigation you are given a starting point and you are expected to ...
Final ScoreWhen Spain played Belgium in the preliminary round of the mens hockey competition in the 2008Olympics, the fina...
Final Score Mark SchemeLevel Assessment – what evidence is there?                     Tick   What you have done well….3   ...
Final Score Teachers NotesLevel 2      a) Finds a way of making or recording at least one combination        b)   Records ...
Ice Cream    ∞ I have started an ice cream parlor.    ∞ I am selling double scoop ice creams.    ∞     At the moment I am ...
Investigate
Ice Cream Mark SchemeLevel Assessment – what evidence is there?                     Tick   What you have done well….3     ...
Ice Cream Teachers NotesLevel                 Strand (i)                             Strand (ii)                          ...
A piece of StringYou have a piece of string 20cm long.  1) How many different rectangles can you make?  Here is one       ...
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 ...
A piece of String Mark SchemeLevel Assessment – what evidence is there?                     Tick   What you have done well...
Level  2     •   Find some areas with the given perimeter of 20cm.            e.g. 9x1=9cm² or 5x5=25cm²        •   Writes...
JumpingBen is hoping to enter the long jump at his school sports day.One day I saw him manage quite a good jump.However, a...
Jumping Mark SchemeLevel Assessment – what evidence is there?                     Tick   What you have done well….3     De...
Teachers NotesLevel 3     For Ben            Attempt to show numbers being halved, example 3.75 ÷ 2 = 1.875            Add...
Original Jump           Increase           New Jump           1m           ½           1½           2m           ½        ...
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 o...
How many triangles? Mark SchemeLevel Assessment – what evidence is there?                     Tick   What you have done we...
Answers to the questions:Triangle base of 2 trianglesSize Frequency1      52      1Total 5Triangle base of 3 trianglesSize...
2       313       184       105       66       37       1Total   1188 x 8 is 170 so over 150.Level   Criteria 1 - Applicat...
Polo PatternsWhen the black tiles surround white tiles this is known as a polopattern.You are a tile designer and you have...
Polo Patterns Mark SchemeLevel Assessment – what evidence is there?                     Tick   What you have done well….3 ...
Polo Patterns Teachers NotesLevel 2      c) Finds the 2 ways of arranging 12 black tiles in a polo pattern        d) Writt...
Opposite Corners.  1      2       3      4       5      6         7     8    9      10 11     12       13     14     15   ...
1    2    3    4    5    6    7    8    9    10    1    2    3    4    5    6    7    8    9    1011   12   13   14   15  ...
Opposite Corners Mark SchemeLevel Assessment – what evidence is there?                     Tick   What you have done well…...
Opposite Corners Teachers NotesTeachers should introduce the task by reference to the example on the students’ sheet andso...
Numbers in order like 7, 8, 9 are called CONSECUTIVE numbers.                  4+5=9                                      ...
Adds in Order Mark SchemeLevel Assessment – what evidence is there?                     Tick   What you have done well….3 ...
“Add On’s” Teacher NotesLevel 2      a) Finds the 2 ways of recording results.eg 1+2=3, 2+3=5 etc        b) Written what t...
Consecutivenumberspattern2Odd numbers2n -11+2=3,2+3=5,3+4=7..3Multiples of 33n+31+2+3=6,2+3+4=9..4Going up in 4’s4n+61+2+3...
6     a) Finds the nth term of the pattern see table above       b) Redrafts own account of work to make it clearer or sug...
Match SticksLook at the match stick shape below.How many match sticks do you expect to be in pattern 2?          Pattern 2...
Match Sticks Mark SchemeLevel Assessment – what evidence is there?                     Tick   What you have done well….3  ...
Level2       a) Be able to increase each pattern by 2 matches.        b) Can draw the next 5 patterns.        c) Clearly d...
Pattern12345678Matches47101316192225Pentagons.
Pattern     1     2     3     4     5     6     7     8     Matches     5     9     13     17     21     25     29     33 ...
Fruit MachineIn this task you are going to design your own fruit machine.Start with a simple one so you can see how it wor...
.    Maths Fruit Machine     Cut out this window      Cut out this window             Only 10 pence per play.    Match two...
Fruit Machine Mark SchemeLevel Assessment – what evidence is there?                     Tick   What you have done well….3 ...
Fruit Machine Teachers NotesLevel 2      a) Find one set of solutions.        b) Complete the table at the bottom of the q...
Year 7 investigation homework
Year 7 investigation homework
Year 7 investigation homework
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Year 7 investigation homework

  1. 1. 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. Theproject 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 eachinvestigation a student completes a poster or report. The work produced should be levelled and the students shouldhave 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 inWeek beginning Week beginning5th Sep 2011 Final scores 4 hours 26th Sep 20113rd Oct Ice cream 4 hours 4th Nov 201120117th Nov A piece of string 4 hours 28th Nov 201120115th Dec Jumping 4 hours 9th Jan 2012201116th Jan How many triangles? 4 hours 10th Feb 2012201220th Feb Polo Patterns 4 hours 12th Mar 2012201219th Mar Opposite Corners 4 hours 23rd April 2012201230th April Adds in Order 4 hours 21st May 2012201228th May Match Sticks 4 hours 25th June 201220122nd Jul Fruit Machine 4 hours 16th July 20122012
  2. 2. Year 7 Homework Record Sheet Date set Investigation Level Target for improvement Week Title beginning Final scores5th Sep2011 Ice cream rd3 Oct2011 A piece of7th Nov string2011 Jumping5th Dec2011 How many16th Jan triangles?2012 Polo20th Feb Patterns2012 Opposite th19 Mar Corners2012 Adds in30th April order2012 Match Sticks th28 May2012 Fruit2nd Jul Machines2012
  3. 3. Tackling investigationsWhat 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 theinvestigation. Remember your teachers at Queensbury are her to help, if you get stuck at any stage, come and ask one of the Maths teachers.
  4. 4. Final ScoreWhen Spain played Belgium in the preliminary round of the mens hockey competition in the 2008Olympics, 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 dont miss any out?In the final of the mens 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?
  5. 5. Final Score Mark SchemeLevel Assessment – what evidence is there? Tick What you have done well….3 Describe the mathematics used4 Explain ideas and thinking5 Identify problem solving strategies used6 Give a solution to the question7 Explain how the problem was chunked into smaller tasks8 Relate solution to the original context2 Create their own problem and follow it through3 Discuss the problem using mathematical language4 Organise work and collect mathematical information What you need to do to improve…5 Check that results are reasonable6 Justify the solution using symbols, words & diagrams7 Clearly explain solutions in writing and in spoken language8 Explore the effects of varying values and look for invariance2 Use some symbols and diagrams3 Identify and overcome difficulties4 Try out own ideas5 Draw own conclusions and explain reasoning6 Make connections to different problems with similar Level for this piece of homework… structures7 Refine or extend mathematics used giving reasons8 Reflect on your own line of enquiry examine generalisations or solutions
  6. 6. Final Score Teachers NotesLevel 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.
  7. 7. 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???????
  8. 8. Investigate
  9. 9. Ice Cream Mark SchemeLevel Assessment – what evidence is there? Tick What you have done well….3 Describe the mathematics used4 Explain ideas and thinking5 Identify problem solving strategies used6 Give a solution to the question7 Explain how the problem was chunked into smaller tasks8 Relate solution to the original context2 Create their own problem and follow it through3 Discuss the problem using mathematical language4 Organise work and collect mathematical information What you need to do to improve…5 Check that results are reasonable6 Justify the solution using symbols, words & diagrams7 Clearly explain solutions in writing and in spoken language8 Explore the effects of varying values and look for invariance2 Use some symbols and diagrams3 Identify and overcome difficulties4 Try out own ideas5 Draw own conclusions and explain reasoning6 Make connections to different problems with similar Level for this piece of homework… structures7 Refine or extend mathematics used giving reasons8 Reflect on your own line of enquiry examine generalisations or solutions
  10. 10. Ice Cream Teachers NotesLevel 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.
  11. 11. A piece of StringYou 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.
  12. 12. 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?
  13. 13. A piece of String Mark SchemeLevel Assessment – what evidence is there? Tick What you have done well….3 Describe the mathematics used4 Explain ideas and thinking5 Identify problem solving strategies used6 Give a solution to the question7 Explain how the problem was chunked into smaller tasks8 Relate solution to the original context2 Create their own problem and follow it through3 Discuss the problem using mathematical language4 Organise work and collect mathematical information What you need to do to improve…5 Check that results are reasonable6 Justify the solution using symbols, words & diagrams7 Clearly explain solutions in writing and in spoken language8 Explore the effects of varying values and look for invariance2 Use some symbols and diagrams3 Identify and overcome difficulties4 Try out own ideas5 Draw own conclusions and explain reasoning6 Make connections to different problems with similar Level for this piece of homework… structures7 Refine or extend mathematics used giving reasons8 Reflect on your own line of enquiry examine generalisations or solutions A Piece of String – Teachers Notes
  14. 14. 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) • 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.
  15. 15. JumpingBen 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 didbefore.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?
  16. 16. Jumping Mark SchemeLevel Assessment – what evidence is there? Tick What you have done well….3 Describe the mathematics used4 Explain ideas and thinking5 Identify problem solving strategies used6 Give a solution to the question7 Explain how the problem was chunked into smaller tasks8 Relate solution to the original context2 Create their own problem and follow it through3 Discuss the problem using mathematical language4 Organise work and collect mathematical information What you need to do to improve…5 Check that results are reasonable6 Justify the solution using symbols, words & diagrams7 Clearly explain solutions in writing and in spoken language8 Explore the effects of varying values and look for invariance2 Use some symbols and diagrams3 Identify and overcome difficulties4 Try out own ideas5 Draw own conclusions and explain reasoning6 Make connections to different problems with similar Level for this piece of homework… structures7 Refine or extend mathematics used giving reasons8 Reflect on your own line of enquiry examine generalisations or solutions
  17. 17. Teachers NotesLevel 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 1.2mLEVEL 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 tableLEVEL 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  3.75Level 6 Extends the task to investigate ¼ or 1/3 as much increase in jump height/length. Records results in table
  18. 18. Original Jump Increase New Jump 1m ½ 1½ 2m ½ 3 3m 4¼ Investig ate 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)
  19. 19. 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 trianglesyou 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 150triangles?Try and draw it to show if it or is not possible.
  20. 20. How many triangles? Mark SchemeLevel Assessment – what evidence is there? Tick What you have done well….3 Describe the mathematics used4 Explain ideas and thinking5 Identify problem solving strategies used6 Give a solution to the question7 Explain how the problem was chunked into smaller tasks8 Relate solution to the original context2 Create their own problem and follow it through3 Discuss the problem using mathematical language4 Organise work and collect mathematical information What you need to do to improve…5 Check that results are reasonable6 Justify the solution using symbols, words & diagrams7 Clearly explain solutions in writing and in spoken language8 Explore the effects of varying values and look for invariance2 Use some symbols and diagrams3 Identify and overcome difficulties4 Try out own ideas5 Draw own conclusions and explain reasoning6 Make connections to different problems with similar Level for this piece of homework… structures7 Refine or extend mathematics used giving reasons8 Reflect on your own line of enquiry examine generalisations or solutionsTeachers Notes
  21. 21. Answers to the questions:Triangle base of 2 trianglesSize Frequency1 52 1Total 5Triangle base of 3 trianglesSize Frequency1 92 33 1Total 13Triangle base of 4 trianglesSize Frequency1 162 73 34 1Total 27Triangle base of 5 trianglesSize Frequency1 252 133 64 35 1Total 48Triangle base of 6 trianglesSize Frequency1 362 213 114 65 36 1Total 78Triangle base of 7 trianglesSize Frequency1 49
  22. 22. 2 313 184 105 66 37 1Total 1188 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 this8Level 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.
  23. 23. Polo PatternsWhen the black tiles surround white tiles this is known as a polopattern.You are a tile designer and you have been asked to design different polopatterns (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 cansurround as many white tiles as you like)?Investigate how the number of tiles in a polo pattern depends on the number ofwhite tiles.
  24. 24. Polo Patterns Mark SchemeLevel Assessment – what evidence is there? Tick What you have done well….3 Describe the mathematics used4 Explain ideas and thinking5 Identify problem solving strategies used6 Give a solution to the question7 Explain how the problem was chunked into smaller tasks8 Relate solution to the original context2 Create their own problem and follow it through3 Discuss the problem using mathematical language4 Organise work and collect mathematical information What you need to do to improve…5 Check that results are reasonable6 Justify the solution using symbols, words & diagrams7 Clearly explain solutions in writing and in spoken language8 Explore the effects of varying values and look for invariance2 Use some symbols and diagrams3 Identify and overcome difficulties4 Try out own ideas5 Draw own conclusions and explain reasoning6 Make connections to different problems with similar Level for this piece of homework… structures7 Refine or extend mathematics used giving reasons8 Reflect on your own line of enquiry examine generalisations or solutions
  25. 25. Polo Patterns Teachers NotesLevel 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.
  26. 26. 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 100The 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 56The products of the numbers in these opposite corners are54 x 66 = 3564 and64 x 56 = 3584The difference between these products is 3584 – 3564 = 20Task: Investigate the difference between the products of the numbers in the opposite cornersof any rectangles that can be drawn on a 100 square.
  27. 27. 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 1011 12 13 14 15 16 17 18 19 20 11 12 13 14 15 16 17 18 19 2021 22 23 24 25 26 27 28 29 30 21 22 23 24 25 26 27 28 29 3031 32 33 34 35 36 37 38 39 40 31 32 33 34 35 36 37 38 39 4041 42 43 44 45 46 47 48 49 50 41 42 43 44 45 46 47 48 49 5051 52 53 54 55 56 57 58 59 60 51 52 53 54 55 56 57 58 59 6061 62 63 64 65 66 67 68 69 70 61 62 63 64 65 66 67 68 69 7071 72 73 74 75 76 77 78 79 80 71 72 73 74 75 76 77 78 79 8081 82 83 84 85 86 87 88 89 90 81 82 83 84 85 86 87 88 89 9091 92 93 94 95 96 97 98 99 100 91 92 93 94 95 96 97 98 99 1001 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 1011 12 13 14 15 16 17 18 19 20 11 12 13 14 15 16 17 18 19 2021 22 23 24 25 26 27 28 29 30 21 22 23 24 25 26 27 28 29 3031 32 33 34 35 36 37 38 39 40 31 32 33 34 35 36 37 38 39 4041 42 43 44 45 46 47 48 49 50 41 42 43 44 45 46 47 48 49 5051 52 53 54 55 56 57 58 59 60 51 52 53 54 55 56 57 58 59 6061 62 63 64 65 66 67 68 69 70 61 62 63 64 65 66 67 68 69 7071 72 73 74 75 76 77 78 79 80 71 72 73 74 75 76 77 78 79 8081 82 83 84 85 86 87 88 89 90 81 82 83 84 85 86 87 88 89 9091 92 93 94 95 96 97 98 99 100 91 92 93 94 95 96 97 98 99 100
  28. 28. Opposite Corners Mark SchemeLevel Assessment – what evidence is there? Tick What you have done well….3 Describe the mathematics used4 Explain ideas and thinking5 Identify problem solving strategies used6 Give a solution to the question7 Explain how the problem was chunked into smaller tasks8 Relate solution to the original context2 Create their own problem and follow it through3 Discuss the problem using mathematical language4 Organise work and collect mathematical information What you need to do to improve…5 Check that results are reasonable6 Justify the solution using symbols, words & diagrams7 Clearly explain solutions in writing and in spoken language8 Explore the effects of varying values and look for invariance2 Use some symbols and diagrams3 Identify and overcome difficulties4 Try out own ideas5 Draw own conclusions and explain reasoning6 Make connections to different problems with similar Level for this piece of homework… structures7 Refine or extend mathematics used giving reasons8 Reflect on your own line of enquiry examine generalisations or solutions
  29. 29. Opposite Corners Teachers NotesTeachers should introduce the task by reference to the example on the students’ sheet andsome other examples for different sized rectangles. Encourage students to develop the generalcase 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 thedifference is the same for same sized rectangles.Level 4 Students correctly work out the differences for at least five rectangles of two differentdimensions 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 fourdifferent dimensions and notice the difference is the same for same sized rectangles. Students producea 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 toand 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+12And 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 + 10mHence produce a general result for an n x m rectangle.
  30. 30. Numbers in order like 7, 8, 9 are called CONSECUTIVE numbers. 4+5=9 12 = 3 + 4 + 6=1+ 517 = 8 + 9 2+317, 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?
  31. 31. Adds in Order Mark SchemeLevel Assessment – what evidence is there? Tick What you have done well….3 Describe the mathematics used4 Explain ideas and thinking5 Identify problem solving strategies used6 Give a solution to the question7 Explain how the problem was chunked into smaller tasks8 Relate solution to the original context2 Create their own problem and follow it through3 Discuss the problem using mathematical language4 Organise work and collect mathematical information What you need to do to improve…5 Check that results are reasonable6 Justify the solution using symbols, words & diagrams7 Clearly explain solutions in writing and in spoken language8 Explore the effects of varying values and look for invariance2 Use some symbols and diagrams3 Identify and overcome difficulties4 Try out own ideas5 Draw own conclusions and explain reasoning6 Make connections to different problems with similar Level for this piece of homework… structures7 Refine or extend mathematics used giving reasons8 Reflect on your own line of enquiry examine generalisations or solutions
  32. 32. “Add On’s” Teacher NotesLevel 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.
  33. 33. Consecutivenumberspattern2Odd numbers2n -11+2=3,2+3=5,3+4=7..3Multiples of 33n+31+2+3=6,2+3+4=9..4Going up in 4’s4n+61+2+3+4=10,1+2+3+4+5=14…5Multiples of 55n+101+2+3+4+5=15,2+3+4+5+6=20..6 predicted &checkedGoing up in 6’s6n+1521,27,2277n+2128,35,42
  34. 34. 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 numbersNumbers that cannot be made 1,2,4,8,16,28,32,44
  35. 35. Match SticksLook 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 trianglesDraw 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 thereany left over?Think about different shapes you can make using matchsticks, investigate (asabove).
  36. 36. Match Sticks Mark SchemeLevel Assessment – what evidence is there? Tick What you have done well….3 Describe the mathematics used4 Explain ideas and thinking5 Identify problem solving strategies used6 Give a solution to the question7 Explain how the problem was chunked into smaller tasks8 Relate solution to the original context2 Create their own problem and follow it through3 Discuss the problem using mathematical language4 Organise work and collect mathematical information What you need to do to improve…5 Check that results are reasonable6 Justify the solution using symbols, words & diagrams7 Clearly explain solutions in writing and in spoken language8 Explore the effects of varying values and look for invariance2 Use some symbols and diagrams3 Identify and overcome difficulties4 Try out own ideas5 Draw own conclusions and explain reasoning6 Make connections to different problems with similar Level for this piece of homework… structures7 Refine or extend mathematics used giving reasons8 Reflect on your own line of enquiry examine generalisations or solutionsMatch Sticks Teacher Notes
  37. 37. 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 8 Matches 3 5 7 9 11 13 15 174 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.
  38. 38. Pattern12345678Matches47101316192225Pentagons.
  39. 39. 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.
  40. 40. 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 AppleThe only way to win on this machine is to get two apples. If you win you get 50 pence back. It costs 10pence 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 thefirst two reels of the machine. Start with lemons in both windows. Move reel 2 one space up – now youhave 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
  41. 41. . Maths Fruit Machine Cut out this window Cut out this window Only 10 pence per play. Match two apples to win 50 pence.
  42. 42. Fruit Machine Mark SchemeLevel Assessment – what evidence is there? Tick What you have done well….3 Describe the mathematics used4 Explain ideas and thinking5 Identify problem solving strategies used6 Give a solution to the question7 Explain how the problem was chunked into smaller tasks8 Relate solution to the original context2 Create their own problem and follow it through3 Discuss the problem using mathematical language4 Organise work and collect mathematical information What you need to do to improve…5 Check that results are reasonable6 Justify the solution using symbols, words & diagrams7 Clearly explain solutions in writing and in spoken language8 Explore the effects of varying values and look for invariance2 Use some symbols and diagrams3 Identify and overcome difficulties4 Try out own ideas5 Draw own conclusions and explain reasoning6 Make connections to different problems with similar Level for this piece of homework… structures7 Refine or extend mathematics used giving reasons8 Reflect on your own line of enquiry examine generalisations or solutions
  43. 43. Fruit Machine Teachers NotesLevel 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

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