Maths Coursework T Shapes

Contents

Introduction - Word Document Version

This document identifies algebra tasks used in year 7 maths lessons in the first half terms of the years 2008-2009 and 2009-2010.  It attempts to map them to the maths curriculum and suggests which could be used for the different year 7 sets in coming years.  Not all aspects of the algebra curriculum are covered.  It is intended that some aspects will be covered using other tasks and some will be covered later in the year.

The algebraic skills and techniques to learn in year 7 can be broadly categorised as shown below.  These grouping suggest a suitable ‘journey’ for the series of lessons for the year 7 sets.

  • Systematic approach, recording data and numerical methods (N*.*, A4.1)
  • Formal algebraic fundamentals
    • Need for algebra
    • Formulae (A4.2, A5.1, A7.1)
    • Symbols for unknowns (A4.3)
    • Substitution (A5.1, A7.1)
    • Collecting terms and simplifying (A5.3)
    • Terminology (A6.1, A3.2)
    • Functions (A4.3)
  • Sequences and patterns (A3.1, A6.6, A7.9)
  • Algebraic manipulation
    • Solving equations (A5.2, A6.4, A7.2, A7.4)
    • Changing the subject (A7.3)
    • Factorising/expanding (A5.3, A6.2, A7.4)
  • Graphs
    • Plotting (A4.4, A6.5, A7.8)
    • As tool to solve algebraic problems (A7.7)
    • Real life (A6.7, A7.6)
    • Properties (A7.10)
  • Trial and Improvement (A6.3)

The ‘Suggested scheme of lessons’ section at the end of this document would probably take in the order of 4 to 5 weeks to complete.  It is intended that either interlaced with these tasks or otherwise, the remaining time should be spent looking at the Number aspects of the curriculum.  Emphasis should be given to enrichment type activities.  It is suggested that the Nrich activities identified on the SOW index page be used.

Description of tasks

Magic birthday trick

This is from Jonny Heeley’s Masterclass (from Teachers’ TV  http://www.teachers.tv/videos/algebra - at about 1min 15secs in)

Good to use as a ‘hook’ – it may be better to do the trick early in the lesson sequence and later get the students to explain how it is done.  It works as follows:

Ask the students to do the following:

  • Take the month of the year you were born (1 to 12)
  • Multiply by 5
  • Add 7
  • Multiply by 4
  • Add 13
  • Multiply by 5
  • Add the day of the month

The students then give you their results and you are able to tell them their birthday by breaking the 4 digit result into two 2-digit numbers.  Subtract 2 from first to get the month and 5 from the second to get the day of the month.

Unknowns: Numbers and animals

This is a teacher lead activity based on Jonny Heeley’s Masterclass (from Teachers’ TV http://www.teachers.tv/videos/algebra - at about 3mins 20secs in).  A printable version using animals and letters instead of fruit and vegetables is in resources folder.  Give a card to each student (holding back an 8 and a 4).  Give out one rabbit, one elephant and one ‘n’.  Tell students not to tell each other what they have but get them to follow the process (and to just play along if they have something unusual).  They should all get the same answer.  You then go on to show how it works with animals and letters.

Resources:

Unknowns - animals and letters.ppt

Summing consecutive numbers

This is a simple little investigation which can be done without formal algebra.  Find a rule for summing three consecutive integers.  This can be useful to highlight the need to layout numerical ‘trials’ in a logical fashion.  Further questions could be:

How does the total vary as we move up to the next three numbers (e.g. compare 1+2+3 to 2+3+4)?  Is this always true?

What if we consider numbers that are two apart (e.g. 3+5+7)?

This is a good activity to lead onto the usefulness of symbols for unknowns.  It is also a good introduction to the concept of proof (we can show the rule is true for any three consecutives).

NRICH Task:   Consecutive Numbers        Consecutive Sums            Excel Investigation             

Me and my brother

This is a task in forming algebraic expressions and can be used at all levels.  It would make a good mini-whiteboard activity.  Get students to answer the follow kind of questions:

I am n years old.  My brother is 3 years older than me.  How old is he?

What do you get if you add our ages together?

This can then be revisited as an introduction to solving linear equations.

Grid investigation (or Opposite corners)

This is the well known investigation of taking a 100-grid and overlaying smaller (to start with, 3×3) grids.  The investigation starts by considering the sum of the numbers in the corners of the smaller grid and looks for patterns depending on its position.  This is a good task for introducing a methodical approach to spotting patterns.  When approached algebraically it can introduce ‘collecting like terms’.  This investigation can be extended by considering different size larger and smaller grids (or even unknown grid sizes).

For a set 3 taking just a numeric approach, the sequence of numbers in the larger grid could be replaced with just even numbers, odd numbers etc., use of rectangles for the smaller overlaying grid etc.

The whole investigation can be extended further to cover expansion (A6.2 and A7.4) by summing opposite pairs of corners then multiplying the results (or multiplying then summing).

Resources:

Grid investigation.doc

100 grid - printable.ppt

Smaller grids - printable.ppt

Grids.xls

100 grid - eBeam.emf

Nrich: Seven Squares (http://nrich.maths.org/2290)

Using algebra to answer problems about visual patterns (made using matchsticks).  This shows different ways to visualise the constructions of the algebraic expressions.

Nrich; What's it Worth (http://nrich.maths.org/1053)

Complete the solution to find the value of some symbols placed in a square given 6 different starts

Tile Spacers

This investigation is described in the document ‘Tile spacers.doc’.  An extract:

“In this investigation, we will start by looking at square tiles, and the three types of spacers we will use are T shaped, L shaped and + shaped.

The investigation is to see how many spacers are needed for different arrangements of tiles.”

This could be delivered over a couple of lesson.  The fist would be to teach methodical working, recording of results and identifying simple patterns.  The investigation could be revisited later to use it to derive formulae.

Resources:

Tile spacers.doc

Magic Squares

There are three strands to the work that can be done with magic squares.  The first is to find ways of constructing them (3x3 to achieve different totals).  Folens Y7 T1 U1 p13 is useful here.  The second is to practise substitution given a formula.  10Ticks L5 P5 p20 has questions.  Lastly, the students could find a, b and c (variables in the formula) given particular magic squares.  See 10Ticks L5 P5 p21 for questions.  The resources below may be useful for projecting the magic squares.

Resources:

Magic squares.ppt

Magic Squares - eBeam
Blank - eBeam

T-Shapes Investigation

This is a simplified version of the old GCSE coursework task.  It recommended that it is used for assessment for tops sets already having done a similar task (such as the Grid Investigation).  The sheet for students (‘T Shapes investigation.doc’) is left very open with some prompts for extension tasks (changing the grid size, changing the size of the T, rotating the T).  A selection of grids in emf format (suitable for eBeam) are available for class discussions in the ‘T-Shapes – eBeam’ folder.

Resources:

T Shapes investigation.doc

T-shape grids - printable.ppt

T-Shapes – eBeam

Think of a Number

These are the type of problem in which you start with a number, perform a number of functions and always end with the same result.  Nrich has a nice task as an introduction to this idea called: “Your number is…” (http://nrich.maths.org/2289).  You will need to use your web-browser’s zoom facility if you are to project these.  Exercises for students based on this idea (first numeric then algebraic) can be found in 10Ticks L6 P1 p15,16 (also available as powerpoint slides).  See also Folens 7.2 Unit 7 p108 (p126) for work on an algebraic approach to these problems (a lead in to the ‘Pots of Gold’ task below).

Resources:

Think of a Number.ppt

Pots of Gold

This is part of a lesson taken from Folens 7.2 Unit 7 p109 (p127) with worksheet available from p115 (p133).  This is a visual approach to expression building.

Calendar Investigation

This is an investigation similar to the Grid Investigation but uses calendars (so the grids are 7 squares wide and the number 1 can be in any column).  This is suitable for assessment of set 2 and 3 groups following the Grid Investigation.

Resources:

Calendar.doc

How many squares?

http://www.coolmath4kids.com/math_puzzles/s1-squares.html

This is an investigation into counting squares in a larger grid.  It starts with a 2x2 grid (which has 4+1=5 squares) then a 3x3 (which has 9+4+1=14 squares) etc.

Suggested scheme of lessons

Note that tasks in italics are not described in the section above.

Set 1

Task

Comment

A

Magic birthday trick

As a hook.  To be revisited later.

6.2

Unknowns: Numbers and animals

 

4.3

Summing consecutive numbers

Both numerically algebraically

4.1, 5.3, 6.2

Me and my brother

 

4.3, 5.3, 6.1

Algebra terminology

 

6.1

Simplifying expressions

 

5.3

Grid investigation

Include extensions

4.3, 5.3, 6.2, 7.4

Factorising

 

6.2, 7.4

Magic squares

 

5.1, 5.2, 7.1

Nrich: 7 squares

 

4.1, 5.1, 6.6

Nrich: What's it Worth   4.3, (AB.2)

Linear sequences including term-to-term and nth term

 

6.6

Think of a number

 

4.3, 5.1

Magic birthday trick

Revisited

6.2

Solving linear equations

 

5.2, 6.4

T-shapes

Individually for assessment

 

 Topics not covered above (levels 5 and 6) for later in the year:  A6.3, A6.5, A6.7

Set 2

Task

Comment

A

Tile spacers

First part

4.1

Linear sequences including term-to-term and nth term

Generating sequences only

4.1, 5.1

Tile spacers

Revisit

4.2, 5.1

Grid investigation

Summing only

4.3

Think of a number

 

4.3, 5.1

Pots of gold

 

4.3

Nrich: What's it Worth   4.3, (AB.2)

Expanding brackets

 

5.3

Grid investigation

Next part

5.3

Solving linear equations

 

5.2

Calendar Investigation

Individually for assessment

 

 Topics not covered above (levels 4 and 5) for later in the year:  A4.4

Set 3

Task

Comment

A

Supporting number work

Adding and multiplying

 

Number patterns (even, odd, multiples, squares, etc.)

 

3.1, 4.1

Grid Investigation (summing)

Numeric approach only

3.2, 4.1

Think of a number

 

4.3, 5.1

Pots of gold

 

4.3

Magic squares

Constructing and substitution only

5.1

Substitution games

 

5.1

Formulae in words

 

4.2

MyMaths: Function Machines

 

3.2, 4.3

Calendar Investigation

Individually for assessment

 

How many squares?

For assessment

 

Topics not covered above (levels 3 - 5) for later in the year:  A4.4, A5.2, A5.3


Task Mapping to National Curriculum Levels

 

Investigating the Relationship Between T-totals and T-numbers

  • Length: 701 words (2 double-spaced pages)
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Investigating the Relationship Between T-totals and T-numbers

In my maths coursework I will be focusing on T-Numbers. I will be
investigating the relationship between T-totals and T-numbers.

In my investigation I will also try and find out the relationships
between the grid size and the transformations. The T-number is always
the number at the bottom of the T shape and the T-total is always all
the numbers inside the T shape added together.

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[IMAGE]


9 By 9 Grid:

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The total of the numbers inside the T-shape is 1+2+3+11+20=37 this is
called the T-total.

The number at the bottom of the T-shape is called the T-number.

How to Cite this Page

MLA Citation:
"Investigating the Relationship Between T-totals and T-numbers." 123HelpMe.com. 11 Mar 2018
    <http://www.123HelpMe.com/view.asp?id=120906>.

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For
the t-shape in red the T-number for this T-shape is 20. When you take
the other numbers in the T-Shape away from the T-Number you get
something like this:

N-17

N-18

N-19

N-9

N

When you get the T-shape like this you can see that the centre numbers
always go up by 9's because of the table size, with the table set out
like this a formula can be worked out to find the formula to find any
T-Total on this size grid. This can be achieved by using this below:

T-Total =A

T-Number =N

T-total = N-19+N-18+N-17+N-9+N

So the formula for this grid is: 5N-63

The Testing Process for grid 9x9:

Testing the formula can be done using the variables A and N.

T-Total =A

T-Number =N

(I have used the number 5 in the formula to multiply N because there
are 5 grid boxes in one T shape on any grid)

(Example T-number = 20)

A = 5N-63

N = 20

A = 5x20

= 100-63 = 37

I will now test to see if this will work any were on the 9x9 grid.

N=43

A=5x43 24+25+26+34+43=152

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25

26

34

43

A=215-63


A=152
-----


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N=49
====

A=5x49 30+31+32+40+49=182

A=245-63

30

31

32

40

49

A=182

I have fully tested the formula for three different T-numbers and the
formula works any were correctly on a 9 by 9 grid.

The full formula for this size grid is: A=5N-63

6 By 6 Grid:

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If you take the other numbers in the T-Shape away from the T-Number
you get a T-Shape like this.

T-Total =A

T-Number =N

N-13

N-12

N-11

N-6

N

When you get the T-shape like this you can see that the centre numbers
always go up by 6's because of the table size, with the table set out
like this a formula can be worked out to find the formula to find any
T-Total on this size grid. This can be achieved by using this below:

T-Total = N-13+N-12+N-11+N-6+N

= 5N-42

The Testing Process for grid 6x6:

Testing the formula can be done using the variables A and N.

T-Total =A

T-Number =N

(I have used the number 5 in the formula to multiply N because there
are 5 grid boxes in one T shape on any grid)

A = 5N-42

N = 14

A = 5x14

= 70-42 = 28

I will now test to see if this will work any were on the 6x6 grid.

N=22

A=5x22 9+10+11+16+22=68

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16

22

A=110-42


A=68
----


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N=35
====

A=5x35 22+23+24+29+35=133

A=175-42

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35

A=113


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N=32
====

A=5x32 19+20+21+26+32=118

A=160-42

19

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21

26

32

A=118

I have fully tested the formula for three different T-numbers and the
formula works any were correctly on a 6 by 6 grid.

The full formula for this size grid is: A=5N-42



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