Algebra – Scheme of work for year 7’s first half term Draft: 31/10/2010
Algebra – Scheme of work for year 7’s first half term
Contents
Contents 1
Introduction 1
Description of tasks 2
Magic birthday trick 2
Unknowns: Numbers and animals 2
Summing consecutive numbers 3
Me and my brother 3
Grid investigation (or Opposite corners) 3
Nrich: Seven Squares (http://nrich.maths.org/2290) 4
Tile Spacers 4
Magic Squares 4
T-Shapes Investigation 4
Think of a Number 5
Pots of Gold 5
Calendar Investigation 5
How many squares? 5
Suggested scheme of lessons 5
Set 1 5
Set 2 6
Set 3 6
Task Mapping to National Curriculum Levels 7
Introduction
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
o Need for algebra
o Formulae (A4.2, A5.1, A7.1)
o Symbols for unknowns (A4.3)
o Substitution (A5.1, A7.1)
o Collecting terms and simplifying (A5.3)
o Terminology (A6.1, A3.2)
o Functions (A4.3)
· Sequences and patterns (A3.1, A6.6, A7.9)
· Algebraic manipulation
o Solving equations (A5.2, A6.4, A7.2, A7.4)
o Changing the subject (A7.3)
o Factorising/expanding (A5.3, A6.2, A7.4)
· Graphs
o Plotting (A4.4, A6.5, A7.8)
o As tool to solve algebraic problems (A7.7)
o Real life (A6.7, A7.6)
o 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).
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.
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
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 / AMagic birthday trick / As a hook. To be revisited later. / 6.2
Unknowns: Numbers and animals / 4.3
Summing consecutive numbers / Both numerically and 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
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 / ATile 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
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 / ASupporting 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
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Algebra – Scheme of work for year 7’s first half term Draft: 31/10/2010
Task Mapping to National Curriculum Levels
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