# Activities

## Algebra

### Travel Graphs

Age 13+ Time: 1.5-2 hrs. The practical part of this activity can be completed as a first introductory lesson for distance-time graphs. No prior knowledge is required since the motion sensors provide the scaffolding and feedback to get the students to understand how the graphs are used to record position at different times. The second part of the activity gets students to consider the link between speed and distance.

### Inequality Graphs

Age 14+ Time: 1-1.5 hrs. Plotting inequalities made easy! Includes revision activities to remind students how to use equations to describe straight line graphs followed by a practical activity where students will try to recreate graphs of inequalities using Geogebra.

### Surrounded

Age: 12+ Time: 2 hrs +. This is an excellent investigation to get students to describe the pattern generated by surrounding a shape with squares. They should be able to find a formula and link the formula to what they see. The activity is well structured, but it could be set in a far more open-ended way.

### Sequences Differences Method

Age: 14+ Time: 1-2 hrs. This activity starts with a drag & drop applet requiring students to use their intuitive sense of number to categorise sequences as linear, quadratic or cubic. A spreadsheet calculator is provided as a scaffold for students in using the differences method to check their intuitions and, eventually, find the exact formula for each sequence. Paper version available also.

### Modeling Disease

Age: 14+ Time: 1-2 hrs. Students match the weekly infection figures with the corresponding disease and graph before trying to find an appropriate model to predict its spread over the coming weeks. Quarantine, Research investment, Extermination etc. – what government policy will your students recommend based on their models?

### Visual Line Equations

Age: 13+ Time: 1 h or 10min recap. This is a very kinesthetic, visual, social and hands-on activity for introducing, or practicing, the equations of lines. Working on giant laminated A3 grids, with whiteboard pens for working out, the whole class creates the line described algebraically by the teacher. Instant feedback, collaboration and assessment guaranteed!

### Loopy Polygons

Age: 13+ Time: 4 hrs. This is an investigation where students explore the link between the number of polygon tiles and the perimeters of the shapes formed by tiling them in loops. It's a great opportunity for students to find formulae in a practical situation and do some extended mathematical writing.

### Angry Birds 1

Age: 15+ Time: 1 hr. Difficulty level: Medium. This set of games asks students to find the correct equation of the parabola in order to hit the pig! This could give them an opportunity to discover the properties of graphs in the form y=a(x - p)(x - q). Great fun!

### Angry Birds 2

Age: 15+ Time 1 hr. Difficulty level: Hard. This second set of games where three set of coordinates are given and students are required to calculate the equation of the parabola. More through understanding of the equation of a quadratic will be required.

### Inequalities Introduction

Age 12+ Time 1h: Get students up and out of their seats, grouping themselves into sets of numbers according to certain conditions. Students then invent their own number line diagrams for visualising inequalities, with a quiz to discern whose is the most effective, before completing a six question applet using conventional inequality lines.

### Brand Symmetry

Age 12+ Time 1-2h: Students use their knowledge of equations of lines to rebuild well known, symmetrical logos from a small fragment of the image. They then find and/or make their own logos, providing for their partner only a small fragment (using FREE photo editors such as paint.net, gimp or others) from which to rebuild the original.

### Equations of Lines

Age 12+ Time 1-2h: Sitting back-to-back students’ define a straight line picture for their partner to draw. How do we define lines? What are the properties of a line? Student’s creative ideas are compared to find the most efficient and effective definition. Students then practice finding equations of lines on paper, using Geogebra applets for instant feedback to test, and refine, their responses.

### Transforming Functions - The Stretch

Age 15+ Time 2h. Students find transforming functions one of the most demanding topics in mathematics at this level. In particular, the effect a has on the function, f(x) for y=af(x) and y=f(ax) and understanding just what a stretch is. This activity documents the most succesful approach I've had with this topic. It approaches the idea from the angle of transforming shapes, looking at the effect on the coordinates, then applying the same transformations to graphs.

This is a very complete activity that makes use of Geogebra and includes ready made applets and help videos for teachers who may have little experience or confidence in using technology in the classroom. Watch the following video (no sound) to get an overview of the activity:

Age: 14+ Time: 2h. This is a perfect activity to discover the properties of quadratic graphs. An investigation to 1) describe the axis of symmetry, 2) find the vertex form and 3) describe the zeros of a quadratic function. It includes fun quizzes and is concluded with a firefighter game where Sam must aim the water jet correctly to put out the fire. Great fun! Click show to see short video overview of activity

### Inequality Dinosaurs

Age: 14+ Time: 1h Questions start off easy (one or two vert/horiz inequalities) to ensure all students can be engaged - will they save the eggs from the Pterodactyls? The aim of the activity is to focus student attention on how coordinates relate to the inequality and hence facilitate a better understanding of which side of an inequality students should shade. Playable on ipad etc. also.

### Inequalities Pacman

Age: 14+ Time: 1h Students have to modify the inequalities to trap each ghost in turn within their "laser fields" (no software required). Care is required because if their inequalities aren't precise they could easily burn the baby! Levels increase in difficulty, from one to three ghosts, but only using linear inequalities. Playable on ipad also.

### Even & Odd Functions

Age: 15+ Time: 1hr + This activity introduces students to the concept of even and odd functions, i.e. functions with properties f(x) = f(-x) and f(x) = -f(-x). There follows an investigation into the properties of adding, multiplying and finding composites of these functions e.g. even function + even function =?

### Formulae Substitution

Age 11+ Time: 1h Students take the orders at Luigi’s or Taj’s Waiter of the Year competition using a single letter to abbreviate each starter, main etc. Simplify the algebra and substitute in the prices to finalise the bill. This leads into letters as variables: Spin the fruit machine to select a number target before rolling a die to substitute in. Self-checking exercise to finish or for use as a homework. Lots of human interaction!

Age 13+ Time: 2h This is a very complete activity to get students factorising quadratic expressions for the first time. It includes 2 arcade games to practise expanding brackets, a product and sum puzzle and a self-checking spreadsheet for factorising. Watch the following video for an overview

### Exponential Functions

Age: 14+ Time: 1h+ This activity will challenge high achieving students to learn about the properties of exponential functions and their transformations. Interactive applets and quizzes get the students to discover the properties for themselves then there are a couple of games to challenge them to 'copy the function'. Watch the short video below for a quick overview.

### Olympic Records

Age: 14+ Time: 1-2h+. Students use Geogebra to plot, then try and find a function to fit Olympic winning data for the men and women’s 100m, High Jump, Show jumping and men’s weightlifting from 1896 to the present. What are the limits of human physical abilities? This is a great activity to develop students’ mathematical modelling: who will predict most accurately the winning times for the forthcoming Olympics?

### Renaissance Mathematics

Age: 12+ Time: 1h What changed during the Renaissance? This activity looks at the revolution in using algebra to describe geometries, graphs, 3D perspective and the introduction of decimal notation. It can be used as part of a Renaissance School Day where students make links between subjects and then present their findings in a whole school assembly. Overview of this day, lead by History department, available here.

### Trigonometric Function Family

Age: 15+ Time 1.5 hr + This activity gets students to produce families of trigonometric functions (like the one on the right) using dynamic geometry software. By exploring the effect of changing parameters they really get a deep understanding of the properties of the main transformations: translations and stretches. Get ready for "Oos and Ahhs"!

### Straight Line Graphs

{C}{C}{C} Age: 12+ Time 1 hr + A computer with internet access is required for this set of five interlinked activities where students are introduced to the equation of a straight line. A structured investigation is followed by a bowling game where students are required to enter the correct equation in order to be able to bowl over the pins and get a strike. A really entertaining way to learn about gradient and y intercept.

### Trial and Improvement

Age: 13+ Time: 1-2h. Estimation is a key skill in all areas of maths, but perhaps particularly so in Trial and Improvement. Using mini-whiteboards, paper, in pairs or teams students use what they know to estimate square and cube roots of numbers they don’t know. They then use Excel to try and hone their answers to 1, 2 or 3 d.p. accuracy. Students can also create their own Excel questions and solutions.

### Sine Cosine: Model Waves

Age: 15+ Time: 30mins-1h. Students use geometry software to model wave pictures from real-life objects and situations. In doing so, students will investigate the effects of the coefficients for sine and cosine waves e.g. y = a cos[b(x-c)]+d asking themselves: “What Changes?”, “What Stays the Same?”. No software is required

### Sine Cosine Transformations

Age: 15+ Time 1-2 hrs Using Autograph or the free Geogebra or Microsoft Maths 4.0, students investigate the functions of the sine and cosine graph. Students record the key, defining points in a pre-prepared table: coordinates of the maximum and minimum and x-intercepts, as they change different parameters using the constant controller or sliders. Without technology, students then have to predict

these key points for different functions.

### Sine Cosine: Triangle, Circle, Wave!

Age: 15+ Time 1h This activity introduces sine and cosine graphs using the video of the construction of a Ferris wheel that demonstrates the link with triangles. Students then sketch the graph of their movement on the Big Wheel. The aim is to link the sine and cosine ratios to a circle. Students use calculators to plot the graphs exactly (spotting symmetries to save them calculation time!). VM also available.

### Factorising Puzzle

Age: 11+ Time: 1hr+ This activity gets students to explore the ideas of factorising simple linear expressions. However, it does it in a way that never mentions factorising! Students will need to think critically to solve some puzzles about multiplying out number grids. By turning the questions around, students will then discover the rules for factorising. Students should be able to multiply simple algebraic expressions before they attempt this activity.

### Concentric Magic Squares

Age: 11+ Time: up to 1hr. Practise programming spreadsheets with simple formulae. Use the spreadsheet to examine the relationships and patterns between the numbers in magic squares. Do all of this while you get lost in this fantastic challenge! Create a 7 x 7 magic square with a 5 x 5 magic square inside it and a 3 x 3 one inside that!

### Exploring Expressions

Age: 12+ Time: 1h. In this activity, students are given a target and have to choose expessions that correspond to the target e.g. even number: 2n-2. They then make up their own targets and/or cards to match. In the second activity students match a series of formulae to their symbolic meaning, word meaning and physical world context. All activities focus on the concept of letter as "variable". The last activity develops students effective internet and textbook etc. research skills.

### Consecutive Proofs

Age: 12+ Time: 2 hours. In this investigation, students explore the sums of consecutive numbers and their divisors with the hope of discovering and proving that the sum of n consecutive numbers is divisible by n when n is odd. This is a gentle introduction for young students to the idea of proof!

Using algebraic terms to represent unknown numbers and very simple algebraic manipulation, students see the power of algebra. It therefore provides them with a reason and motivation to learn more about this often elusive topic. Before attempting this activity, I would expect students to have had a little exposure to adding simple algebraic expressions together.

### Solving Equations Snap

Use card games to get students practising and revising solving equations. Playing in pairs, threes or fours students roll a die, in combination with the cards, to win their partner's cards. There are many possible games using these cards, as well as a range of levels from Apprentice to Mathmagician. Age: 12+ Time: 30mins to 1h

### Which Wave

Age 14+ Time 30-40 minutes. Match the waves with their functions! The discussions and reasoning that take place during this type of activity can be incredibly valuable and effective. It is a simple idea, but so often the simple ideas can be the most effective. It is also a nice alternative to a traditional exercise. When well practised, this is not a desperately difficult concept and it can be very satisfying

to be able to quickly make the link and either deduce a function from the graph or the other way round. This simple activity lends itself to group work and presents the kind of challenge that usually engages students

### Physical World Sequences

This group activity gets students to match a physical world scenario e.g. pressure exerted by an elephant of 400kg mass, with its associated data (a number relation), the equation that defines this relationship, a graph and the nth term rule. This provokes student discussion to air and refine students’ conceptions of the relationship between these topics. Age: 14+ Time 1 hr.

### Meeting Functions

Age: 15+ Time 1 hr. Challenge students to really understand the concept of a function. Match a set of input values with a function and a corresponding set of output values. There are eight sets of three to make and only one correct solution. This activity is 'old meets new'. Students work with cut out bits of paper but can use calculators/computers to help them solve the puzzle!

### The Sliding Bus Puzzle

Age: 12+ Time: 1-2 hours. This activity is another great example of a puzzle whose solutions can be modeled by an algebraic sequence (linear). The puzzle provides an engaging introduction and an incentive to generalise, which helps students with this traditionally difficult idea. The puzzle can be modeled by a linear sequence and broken down into a series of different linear sequences that combine to form the overall model.

### The Frog Problem

Age: 12+ Time: 1-2 hours Bring life to this classic sequences problem by getting students out of their chairs and jumping around to solve the problem. This is a terrific problem for generating and investigating a quadratic sequence. It can be looked at from a number of angles and demonstrating the way they link together gives a very satisfying result. This problem has been around for a while and this activity is really about getting the most out of it.

A card game to introduce students to quadratic equations. Playing in pairs, threes or fours students roll a die, in combination with the cards, to win their partner's cards. There are many games possible using these cards. Age: 12+ Time: 10-30 minutes

This immediately absorbing and engaging activity requires students to use graphing software. The aim is to explore the basic transformations of a function, e.g. y=f(x-a), y=f(ax) for the quadratic function. However, the questions are disguised in videos of moving graphs that students are asked to reproduce. This challenge provides a great incentive to explore, experiment and share ideas. Age: 15+ Time: 1h

This activity is about linking the graphing of quadratics with the equations themselves by looking at their key features. Students match pieces of information with different graphs using logical deduction. This practical group activity leads to being able to sketch graphs from their equations. Age: 15+ Time: 1h

In one hour students should have worked out how to “factorise quadratics" for themselves using patterns in the factorisations given by CAS software, such as TiNspire, Geogebra, WolframAlpha or Derive. "How to" videos are included for those inexperienced in using these programmes. A second activity relates factorising to the "Grid Method" of multiplication including the use of an online virtual manipulative. Age: 13+ Time: 1-2h

### Human Coordinates

This is a great introductory lesson to linear graphs. Students will act as coordinates on a huge grid. Holding A3 sheets of white paper up when a rule requires it, they will plot coordinate pictures and straight line graphs following instructions such as, “Hold up your sheet if your x and y coordinates add together to make 9!” A webcam and a projector can add an extra dimension to this practical activity. Age: 9+ Time: 30m to 1hr

### The Tower of Hanoi

A real game that is fun to play and, when investigated, generates a great example of an exponential sequence. Ideal activity for exploring sequences in general and for introducing these functions. It is a practical activity that can be enhanced with access to computers. Age: 13+ Time: 1 hour

In this activity students will use a graphing package to explore the link between geometrical patterns, sequences and their graphical representations. There are 3 levels of difficulty starting with linear sequences moving on to quadratic, then other more challenging sequences. Age: 12+ Time: 1-2 hours

### Exploring Sequences with Excel

Use Excel or any other spreadsheet to explore the patterns in linear or arithmetic sequences. Students are quickly drawn to striking patterns and the teacher's role is careful questioning aimed at asking students to articulate the whats? and whys? Age: 11+ Time: 1h

Use dynamic geometry software to find the quadratic equations that model some photographs of real-life objects. This activity will get students to understand the effect of changing the parameters in the general equation y = a(x - b)² + c. Three Geogebra files are provided and are ready to use. No software is needed. Age: 14+ Time: 1h

### Factorising

By the end of the hour students should have worked out how to “factorise” for themselves by looking for patterns in the factorisations given by CAS software such as TiNspire and Derive. "How to" videos are included, for those inexperienced with the technology, to help ensure teacher and student time is focused on the mathematics. Age: 12+ Time: 1h

### Rearranging Formula

Re-arrange simple formulae with this matching pair activity. 32 cards are cut out and matched up to give 16 pairs of equivalent formulae with different subjects. This activity promotes much discussion and helps iron out fallacies. Age : 14+ Time : 1 hr

### Matching Sequences

Students get practice in finding the nth term of arithmetic and geometric sequences. Sequences are presented in a graphical form and students are required to find their nth terms. Ti Nspire calculators are recommended to get the most out of this activity and allow students to play with and test their own conjectures. Age: 15+ Time: 1hr

### OXO

How many ways to win a 3D game of three in a row? A real physical game situation that leads to algebraic sequences. There are many ways to investigate this problem and this makes a great project for Algebraic Investigation. Age: 14+ Time: 1hr to a whole week.

### Who's Fastest?

Who's the fastest in your class? How do you know they are "fast"? What does "fast" mean exactly? Student's discuss the above, then race 100m and use the distance, speed, time formulae to work out: If they maintained this speed, could they set a new marathon record?! Age : 12+ Time: 2hrs

### Using Formulae

Formulae often seem so abstract to students, expressed as they are using algebra, yet they are one of the most applied area of mathematics! Students are asked to search Google images and find one or two images to go with each formula. Students then share their pictures with the rest of the class and discuss what each letter represents and how it describes a relationship. Age: 12+ Time: 1-2hrs

### Human Transformations of Graphs

Students all too often do not realise that functions are all around them: on the dance floor, in the swaying of the branches of a tree . . . . and in people holding their hands in the air in joy! This activity gets them to use their body to feel the transformation! Age: 15+ Time: 5mins to 1h (use sections as starters/plenaries or full resource in a single lesson).

### Wire Transformations of Graphs: A Physical Feel for Graphs

Bend a wire to "feel" the shape of the different functions such as sinx, cosx, x3 etc. and their transformations e.g. sin(x-90), Cos3x, etc. Given an equation/function, can you draw the graph on a mini-whiteboard? Age: 15+ Time: 20mins to 1h (starter/plenary or full lesson)

### Giant Function Transformations

Many marks can be lost in exams because of a lack of precision in the exact coordinates of a transformation. Students are taken outside the classroom to give them a physical experience of how functions define coordinates. The discussion between students is useful for drawing out student misconceptions. Careful questioning can challenge these misconceptions. Age:15+ Time: 20mins to 1h (starter or full lesson)

### Balancing Equations

Many students find the symbols and meanings used in equations difficult to understand. This activity uses the excellent "balancing scales" manipulative to give students an intuitive understanding of how to solve equations - through experiment and discovery. Age: 11+ Time: 1h

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