| Identifier/ No. of weeks |
Unit of Work | National Curriculum Reference |
General Learning Objective | Target Year/ Term |
| 1. / 2 | Arithmetical Symbols |
Ma1 L2 | Can use and understand the meaning of the 4 operations symbols and the equals sign | 7 / 1 |
| 2. / 1 | Sequences | Ma2 L2 | Can follow sequences involving diagrams or symbols | 7 / 3 |
| 3. / 1 | Missing Terms | Ma2 KS3/4 | Can fill in the blanks left in a sequence | 7 / 3 |
| 4. / 2 | Use Rules Expressed in Words | Ma2 L4 | Can use patterns/rules expressed in words | 8 / 1 |
| 5. / 1 | Mappings | Ma2 KS3/4 | Can complete mappings for function machines | 8 / 2 |
| 6. / 1 | Co-ordinates | Ma2 L4 | Can plot co-ordinates in the first quadrant | 8 / 2 |
| 7. / 1 | Describe a Rule in Words | Ma2 L4 | Can identify/describe rules in words | 8 / 3 |
| 8. / 1 | Generate a Graph | Ma2 KS3/4 | Can use rules to generate a graph | 9 / 1 |
| 9. / 2 | Symbolic Notation | Ma2 L5 | Can describe rules using symbolic notation | 9 / 2 |
| 10. / 1 | Evaluate Formulae | Ma2 L5 | Can substitute values in a formula | 9 / 3 |
| 11. / 1 | Simplify Equations | GCSE | Can simplify linear expressions and equations | 10 / 1 |
| 12. / 1 | Linear Equations | Ma2 L6 | Can form and solve linear equations | 10 / 2 |
| 13. / 1 | Transpose Formulae | GCSE | Can change the subject of formulae | 10 / 2 |
| 14. / 2 | General Term | Ma2 L6 | Can find and use the nth term of a series or sequence | 10 / 3 |
| 15. / 2 | Simultaneous Equations | Ma2 L7 | Can solve simultaneous equations, graphically and algebraically | 11 / 1 |
| 16. / 1 | Quadratic Equations | Ma2 L7 | Can solve quadratic equations | 11 / 2 |
| 17. / 1 | Factorise Expressions | Ma2 L8 | Can factorise expressions | 11 / 3 |
I keep a collection of about £5 in 2 pence pieces. The coins are bagged up in sandwich bags as follows (based on the Fibonacci Sequence):
1, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89
Every number in the list can be made exactly two ways from the remaining numbers.
Materials required are: the money bags and an (old-fashioned) pair of kitchen scales.
In the lesson, the teacher (or one team of pupils) chooses one bag and puts it on the scales. The other pupils must balance the scales by choosing money bags for the other scale pan. The balancing situation is written up on the board as an addition sum, using the Equals sign as a symbol of balance.
The activity is then extended to demonstrate subtraction. With 89 in the pan:
89 = 55 + 1 + 1 + 1 + 2 + 3 + 5 + 8 + 13
Replace 89 by 34, take 55 away and they balance:
34 = 1 + 1 + 1 + 2 + 3 + 5 + 8 + 13 = 89 - 55
The following square is 'magic' – all the rows and columns and the leading diagonals sum to the Magic Sum of 15:
8 1 6
3 5 7
4 9 2
Pupils are given this square and, working on their own, should find all the sums of three that equal 15, writing them down using the Equals symbol to show the equality.
Bottom set pupils will be asked to look for rows of three, as in Noughts and Crosses. Middle set will look for sums of three numbers and be told there are eight to find. Top set will be expected to find the pattern in the square.
Pupils are provided with a (photocopied) 10 x 10 grid. They are asked to show e.g. four fives by repeated addition of a row of five squares. Then show that 4 x 5 = 5 x 4 from the grid. This is repeated as often as possible, with different values. The activity is reversed for division, by asking how many rows of five can be made from these twenty.
Top set go on to tabulate the commutativity of division and the other operations.
All pupils are provided with the Picture the Pattern puzzle. This worksheet contains 10 diagram sequences (see Diagram Sequences, later). Working in pairs, pupils are asked to write down, in words, what the pattern is in each picture, i.e. how each sequence progresses. In the first one for example, the number of rings starts at 1 and increases by one each time. Afterwards they share their discoveries with the class.
Bottom set will get as far as they can. Matchsticks, dotted paper and round stickers are made available, as requested.
Middle and top sets will be expected to complete this.
All pupils are provided with the Name that Sequence game sheet. This worksheet comprises 10 number sequences (see Number Sequences, later). Working in pairs again, pupils compete to identify the sequences and to write them down in words, i.e. to state how the number sequences progress. In the first one for example, the numbers start at 9 and go down by one each time. Afterwards, pupils share their discoveries with the class.
Bottom set will get as far as they can again.
Middle and top sets will be expected to complete this too.
Some pupils are provided with the worksheet of 10 additional, more complex, sequences (see later). This is intended for top set pupils. In pairs, or on their own, they are asked to write down, in words, how these further sequences progress. In the second one for example, the number of outside rings is initially zero and it increases by six each time.
All pupils are provided with the Hidden Numbers Mystery worksheet of 15 number sequences (see Fill in the Blanks, later). Working in pairs, they are asked to crack the number sequences and describe them. Then they solve the mystery of the missing number(s) in each. In the first one for example, the numbers start at 2 and increase by two each time, and the missing number is 8.
Bottom set will get as far as they can.
Middle and top sets will be expected to complete this.
For higher ability pupils, this small additional worksheet is supplied (see later). Working on their own or in pairs, they are again asked to describe the sequences and fill in the missing number(s) in each. The first one, for example, is the Fibonacci Sequence, each number is the sum of the two previous, and the missing number is 13. Question 18 is a number chain (add the tens and units digits and double, see below), so the next number is 2.
Pupils are told that we are going to explore some number chains. Each one chooses a starting number between 1 and 48. They follow the chain rule that the next number in the sequence is given by:
Five times the current units digit plus twice the tens digit.
A typical chain might be:
|--------------<¬
1->5->25->29->49->53->25
Pupils will keep their Number chains for further work later in Year 8.
Bottom set will get as far as they can.
Middle set should exhaust one rule.
Faster pupils, including the top set, will try other rules, such as:
Four times the current units digit plus once the tens digit.
Pupils will play a game of Chess for 15 minutes. Those who are proficient will demonstrate to any who are not.
Pupils will then describe in words the moves made by: the Rook, the Knight, and both Bishops. The teacher will compile the descriptions at the board, then ask which of these pieces (if any) can visit all four corners of the board. Pupils will explore this with chess pieces or on paper.
Pupils will be given a set of rules for generating an odd order Magic Square. These are:
Begin by putting 1 in the middle of the top row. Each following number is put in the next box diagonally up one and one to the right. If this goes off the top go to the bottom row, if off to the right too far go to the left-hand column. Occasionally both these will happen at the same time, then you would go to the bottom left box. When you cannot go diagonally, because that box is already occupied, just drop vertically down one row.
Pupils then check their square is magic.
Middle set and top set can also generate the 5 x 5 Magic Square.
Top set may tabulate the Magic Sums of different Squares and look for any pattern.
Pupils work on their own to complete the mappings suggested in the worksheet (see later) (e.g. 10 + 3 -> 13; 3 + 3 -> 6).
Pupils will save their mappings for plotting next term.
The bottom set and middle set will complete the worksheet.
The top set will complete it and then pose themselves further number machines and tabulate the mappings for them.
Number Machine Puzzles are available for extension work.
All pupils will complete the worksheet for single inverse operations (see later) following the example given on the worksheet.
Middle set and top set will progress to two or more operations.
Top set will then work in pairs, one pupil proposing a function machine and demonstrating the mappings, and the second pupil developing the reverse machine. This will permit self-checking and provide social activity.
Pairs should be matched for ability.
The machines can become as complex as desired.
Pupils will develop co-ordinate pictures on TI Graphic Calculators, and animate them. They are supplied with a planning sheet (see the TI-magotchi Planning Sheet, later). Firstly, each pupil chooses and designs a (fairly simple) character as a co-ordinate picture by defining the co-ordinate pairs of grid squares to be filled in. Once the pupil is familiar with the Graphic Calculator, this is converted to two lists of values in a TI 83 (or TI 82, as available). An exemplar is shown in the TI-magotchi example sheet, which generates a dog image.
Top and middle set pupils will then write a small program to repeatedly move the Tamagotchi. There will be time for sharing achievements.
Bottom set will play Battleships. This requires the players to identify target squares in an opponent's grid where the opponent may have drawn a fighting ship.
The class is held in the computer lab. Pupils will each plot (or observe being plotted) some mappings from last term on an Excel spreadsheet (e.g. 2->7 as (2, 7), 4->11 as (4, 11) etc). They will then print out the plot.
Top set will also consider co-ordinates in other quadrants, enter them into the data and plot them.
In this exercise, pupils work in pairs, sitting back to back. They are given Multilink blocks (if available, or otherwise drawing paper). One pupil makes or draws a (fairly simple) construction and then, without showing it, describes to their partner how to make it. The partner must try to assemble a copy. Afterwards the roles are reversed.
The Further Sequences and More Blanks worksheets of Activities 6 and 8 are now given to the middle and bottom sets. These pupils are asked to identify the rules governing each of these sequences and to describe them in their own words, filling in any missing terms.
Meanwhile, the top set pupils are asked to investigate the difference of two squares, by following the worksheet given. They begin with successive squares (as in the worksheet appended) using calculators (or mental recall) to determine the squares. However, they will then move on to the squares of numbers separated by two and three, to explore the general rule – i.e. that it is the difference between the two numbers multiplied by their sum.
The rule will be expressed in each pupil's own words.
The class is taken to the computer lab. Returning to the Excel spreadsheet for the function machine, pupils suggest a range of non-integer inputs and calculate their outputs. They enter these into the data and plot them. By choice of Chart Type a straight line is plotted through the points (See example later).
If Excel is not available, a Graphic Calculator with a teacher's display could substitute.
Able pupils (including top set) will try various linear mappings, noting their straight-line characteristics.
Pupils are told the rule that the volume of a cylinder is:
Cylinder volume is (cross-sectional) area x length
And reminded that:
The area of a circle is the square of its radius x pi.
Bottom and middle sets are told that a straw has a (cross-sectional) area of half a square millimetre.
Top set pupils are given a drinking straw and asked to measure its diameter and so calculate the area of its cross-section.
Pupils then plot, on graph paper, the (straight-line) graph of the volume of the straw against its length. They need to plot at least three points.
Top set pupils then suggest other values for straw diameter and plot the corresponding graphs too.
Here, all pupils are offered this worksheet on expressing written rules in symbolic notation (see later). It should consolidate current knowledge of representing rules and build on it.
The middle and bottom sets will carry out this investigation (see later). They will consider essentially squares of successive numbers, but express their conclusions in algebraic form, i.e. a2 - b2 = a + b, where a = b +1.
Middle set will go on to consider numbers separated by 2, i.e. a2 - b2 = 2a + 2b, where a = b + 2.
This investigation is, meanwhile, given to the top set. It is introduced as in the description given (see later), pupils working on their own. Pupils are advised to start with some simple examples, say the 1 x 1 and 1 x 2, and gradually build up a table of cases. Then they can try to express the results in symbolic form.
This involves a worksheet of formulae, taken from real world applications, in which to substitute given values of the quantities. Calculators are supplied and their use encouraged.
A lesson plan and introduction for the Curly Straw experiment is included (see later). Pupils work in small groups to determine the length of their twisty drinking straw, which could not be measured directly. A time for whole class discussion of the outcomes is planned.
Before this activity, pupils are arranged in small groups and each group chooses one strand of the curriculum within which to plan a presentation. One group is allocated to this strand and is asked to prepare a series of short talks on simplifying expressions and equations. Help will be given during the term to understand and organise the material.
Lower ability and less confident pupils will need extra help at this stage.
Within the group each pupil is assigned one topic from the list: linear expressions and linear equations, expanding brackets, collecting like terms, cancelling top and bottom of a fraction and dividing through, and multiplying fractions through by their lowest common denominator.
In this activity, which may run over several periods, the presentations are made. The class may then ask questions, which may be referred to anyone else in the room, including the teacher.
For this activity, pupils are first taken through the steps of solving a linear equation with the example worksheet supplied (see later). They are then taken back to the computer lab. They are shown the DERIVE® software and use it to solve the same example equation, step by step. (Though this will not be by finding a point of intersection, as illustrated in the software description). They note the manipulations made and use the example worksheet as a memory aid.
Lower ability pupils will then solve other equations, e.g. 5x - 6 = 2x + 15.
Higher ability pupils will want to write their own linear equations and solve them.
Pupils will then all look at the second example worksheet for this unit (see later), where a linear equation is wrapped up in a real world context. Afterwards, they will tackle the Discussion Questions and Exercise 4 on Forming Equations.
The questions become progressively less straightforward, and lower ability pupils will, therefore, spend more time on the earlier ones.
All pupils are set this challenge as extended (assessed) coursework.
A number of cakes are delivered for a party. The hosts can use large plates or small plates to put them out on. If large plates are used, eleven cakes can be put on each plate, with eight left over. Using small plates, seven cakes can be put on each, with four left over. Find out:
Pupils will work on this challenge, on their own, at home. The classroom activity is to progress this coursework, looking, with individual pupils, at issues arising from it. Lower ability pupils will probably need more assistance here.
There will also be time to discuss how the coursework should be presented.
The task of transposing formulae is introduced with the example of the worksheet appended (see later).
Pupils then work in pairs, with the list of formulae in the worksheet of Activity 25, and with the example worksheet as a memory jogger. Alternately, one pupil chooses a formula and a quantity to make the subject of that formula, and the other transposes it for that quantity. The first pupil then transposes the new formula back for the original subject. The pupils then swap around and repeat.
Pupils should be matched for ability.
All the formulae should be transposed at least once, if possible.
Pupils work at their own paces, lower ability children may not tackle all the questions, which become progressively more advanced.
Higher ability pupils will be expected to complete the exercise.
Pupils revisit the Sequences worksheets of Activities 4 – 8. They will, after some explanation, determine the nth term in each sequence, working at their own pace and starting with the number sequences. Modifications to the worksheet instructions for this activity are given, as in the General Terms worksheet (see later).
The questions become progressively more advanced. Lower ability pupils will be content with answering the first two worksheets.
Pupils choose a sequence from the worksheets of Activity 30, and first produce the sequence manually. They then check this on their Graphic Calculators (say, the triangular numbers by n(n+1)/2) e.g. by using the TI 83 in ‘Seq’ mode. They will graph and tabulate the resulting sequence on the calculator.
More able pupils will repeat this for various sequences.
Top ability pupils will go on to tabulate recursive sequences such as the Fibonacci sequence.
Here pupils are all given the Mathepillar investigation (see later). They each work on their own, to tabulate the last petal and to try to identify the pattern - the first question on the worksheet only.
Lower ability pupils will get teacher input to express the pattern symbolically.
Higher ability pupils may try the second question too.
All pupils meet simultaneous equations through the Fast Food tasksheet. They work through it on their own and the teacher goes over it afterwards.
During the teacher's exposition on simultaneous equations, pupils receive an example worksheet on solving them (see later), and go through it together.
All pupils then use a (TI) Graphic Calculator to plot two linear equations, using the 'Y=' function and the 'GRAPH' facility, say:
Y1 = X + 3
and
Y2/3 = X - 1 (i.e. Y2 = 3X - 3)
so estimating their point of intersection. They then follow the example worksheet to solve the equations algebraically and check themselves.
Higher ability pupils will want to define their own pairs of linear equations, and display them.
More able pupils will then tackle the real world simultaneous equation problems given on the worksheet, Exercise 11.2.
At this time, lower ability pupils will want to consolidate the learning of the previous activity with more examples of that.
The class is first divided into groups of 5. One or two pupils are then invited to the front, and lob a tennis ball across the classroom to the teacher. In their groups, pupils sketch the path of the ball's motion. They are then given the illustrations of real world parabolas (for example - suspension bridge, satellite dish antenna, flashlight reflector, reflecting telescope etc.) and asked to identify what the pictures have in common. Pupils then use Graphic Calculators ('Y=' function again) to come up (with help) with a formula which will model this curve.
Here, pupils are given a number of quadratic equations and asked to solve them. For example, the (approximate) distance/time relationship for a ball thrown up with an initial speed of 10 ms-1:
h = 10t - 5t2
should be solved for the time, t sec, when its height, h, is 4 m. Pupils are requested to plot each parabola on their Graphic Calculators and thus estimate the solution. They use this guess to initiate a trial and improvement method for finding t more accurately. They have a copy of the discussion of the latter method from Roder (1993) to remind them of the procedure.
Faster pupils will be set extension work of a more abstract nature, e.g:
y = 2x2 - 5x
to find x when y equals 2.
After some explanation of the task, all pupils do the Factorise tasksheet. This is individual work on taking factors out of simple expressions.
The higher ability group follow that (again after explanation) with the worksheet on factorising quadratic expressions, which they again complete on their own.
Lower ability pupils will, instead, do revision of aspects of the work done in this strand during the year.
