designing mobile games · in an old office building in the silicon roundabout area of london, they...

A module on linear functions

Designing Mobile Games

Teacher Version

About the cover

Once upon a time, there was a company founded by an artist and a computer programmer, both graduates of theUniversity of London. In an old office building in the Silicon Roundabout area of London, they launched SandCircleMobile Games, a company dedicated to delivering the most exciting mobile phone games in the world. The name,SandCircle, is a play on silicon, which is made from sand, and roundabout, also known as a circle. SandCircle’s logoevokes the road configuration of the Old Street Roundabout, while the logo colour scheme calls to mind the LondonTube station signs.

The stories in this work are fictional. All characters and events appearing in this work are fictitious. Any resemblanceto real persons, living or dead, is purely coincidental.

Designing Mobile Games: A module on linear functionsDeveloped jointly by SRI International, Center for Technology in Learning (United States) & London Knowledge Lab,Institute of Education, University of London (UK)

U.S. contributors: Jennifer Knudsen, Ken Rafanan, Teresa Lara-Meloy, Phil Vahey, Gucci Estrella, Jeremy Roschelle,Natasha Arora, Anna Werner, Lynne Peck Theis, Meredith Ittner, George Roy, Vivian Fueyo, Rebecca Moniz,Stephen Hegedus, Jim KaputUK contributors: Bola Abiloye, Alison Clark-Wilson, Eileen Coan, Celia Hoyles, Phillip Kent, Richard Noss, TeresaSmart

These materials are provided through a grant from the Li Ka Shing Foundation and Hutchison Whampoa Limited.These materials are based upon work supported by the National Science Foundation of the United States (NSF-0437861). Any opinions, findings, and conclusions or recommendations expressed in this publication are those of theauthors and do not necessarily reflect the views of National Science Foundation, the Li Ka Shing Foundation orHutchison Whampoa Limited.Copyright © 2013 by SRI International

All graphics are by Ken Rafanan. Graphics may incorporate works by others made available through Creative Commons 2.0 licenses; as such, the derivative graphics are likewise made available under the Creative Commons 2.0 license with attribution, remix, noncommercial, and share alike conditions.Original works:Crashworks (http://www.flickr.com/photos/mrcrash/) [game designer]Jonathan Marks (http://www.flickr.com/photos/jonathanmarks/) [game designer]Robert Scoble (http://www.flickr.com/photos/scobleizer/) [game designer]SIJM MIGS (http://www.flickr.com/photos/sijm/) [game designer]Vancouverfilmschool (http://www.flickr.com/photos/vancouverfilmschool/) [game designer]Jeriaska (http://www.flickr.com/photos/jeriaska/) [game designer]Andrew Middleton (http://www.flickr.com/photos/amid/) [office building]

2nd Edition September 2015

Preface to the Teacher's Edition

I. Unit OverviewThis unit focuses on linear functions. It places them in a context of proportional and non-proportional functions. It shows how linear functions can be used to model situations, such asmotion or money, and solve problems involving a constant rate of change. Informal uses ofconcepts are introduced to compare rates visually (steeper/faster, for example). It also exploresmethods for writing equations based on situations, tables and graphs, and the connectionsbetween them.

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Investigation Key Mathematical Ideas Key TechnologyExperiences

IntroductionWelcome toSandCircleMobile Games(30 minutes)

→ Context of the unit is established: Themechanics and business of mobilephone game design are described, suchas how to design components of agame and compute salary and savings.

→ Motion can be represented on a graphof time versus distance.

→ Idealised motion on a distance-timegraph appears as a straight line(constant rate).

No technology needed.

Investigation 1Yari, the YellowSchool Bus(45 minutes)



Play and pause a video.

Investigation 2Our First MobilePhone Game(40 minutes)



Play and pause asimulation.

Investigation 3ControllingCharacters withGraphs(70 minutes)

→ Graphs are mathematicalrepresentations of relationships such asmotion.

→ Graphs of motion show characters’start position, speed (relative) andplaces and times where charactersmeet.

→ For graphs of motion (that is, timeversus distance), the steeper the line,the faster the motion.

→ Speed can be determined from differentparts of a graph and simulation.


• Edit the graph to changethe speed.

• Edit the graph to changethe final position.

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Investigation 4ControllingCharacters withEquations(120 minutes)

→ Equations are a form of mathematicalrepresentation. Graphs and tables areother forms.

→ Equations can be written based ontables or graphs.

→ You can “translate” between graphs,tables and equations.

→ Time, distance and speed arerepresented differently in these threerepresentations.

→ For equations of the form y = mx, inmotion contexts, m is the speed of amoving object.




• Edit the graph to changethe start position.

[The table and equationwindows are available toview.]

Investigation 5One to Another(40 minutes)

→ You can “translate” between graphs,tables and algebraic expressions.


Investigation 6Better Games(90 minutes)

→ Introduction to non-proportional linearfunctions (not passing through theorigin).

→ Pupils explore two ways to derive theequations of non-proportional linearfunctions: using differences of positionand time in a table; using the y-intercept and speed/gradient of agraph.

→ For equations of the form y = mx + c,in motion contexts, c is typically thestarting point and m is the speed of amoving object.




• Edit the graph to changethe start position.

• Edit the equation tochange the speed andstart position.

Investigation 7Wendella’sJourney: Movingat DifferentSpeeds(80 minutes)

→ In a position-time graph, multi-segmentgraphs can represent characters movingat different speeds.

→ Graphs tell a story. Stories can berepresented in the form of graphs. Inthis activity, pupils will learn to writestories from graphs and make graphsfor stories.

→ “Flat” or horizontal lines representstanding still.


• Edit the graph to changethe velocity.

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Investigation 8Money Matters(55 minutes)

→ Multi-segment graphs show varyingspeeds in motion contexts.

→ Multi-segment graphs can also be usedin non-motion contexts to show rate ofaccumulation, for example.

→ Graphs tell a story. Stories can berepresented in the form of graphs. Inthis activity, pupils will learn to writestories from graphs and make graphsfor stories.

→ “Flat” lines represent standing still.

→ Lines “slanting downward” representmoving backwards.


Investigation 9MathematicallySpeaking: Graphsto Know(10 minutes)

→ Graphs of rates of change are used invarious contexts.

→ Quick graph sketching helps pupils tosee general patterns.


Investigation 10Crab Velocity(65 minutes)

→ In position-time graphs, negative ratesindicate backwards motion.

→ Position can also be negative, with 0indicating some defined point such as astart line or water level.


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Investigation 11Wolf and RedRiding Hood(80 minutes)

→ Finding the velocity of a charactergiven some conditions.

→ No matter how the characters move, iftheir motion graphs have the sameendpoint, they meet at the same placeat the same time.

→ The average rate (speed or velocity) ofa character travelling at different ratesfor different times is the single speed atwhich she can make the same trip inthe same amount of time.

→ A graph can be used to find the averagerate (velocity) of a character moving atdifferent rates (velocities): Draw a linefrom beginning point to ending point ofthe character’s graph, and thendetermine its rate.


Investigation 12Problem Solving(25 minutes)

→ Apply ideas learned in the unit insimilar and different settings.


Investigation 13Problems fromthe SandCircleLunchroom(30 minutes)

→ Apply ideas learned in the unit insimilar and different settings.


Investigation 14SandCircleMobile Games:Going Full Time(15 minutes)

→ Pupils reflect on the unit as a whole,reflect on the mathematics, and notewhat they learned.

→ You may want to give feedback topupils in the form of a letter from thepotential employer, stating: “You havebeen/not been successful in thisapplication because...”


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II. National Curriculum Addressed

Key Processes

At Key Stage 3, through the mathematics content, pupils should be taught to:

Develop fluency

• use algebra to formulate mathematical relationships

• substitute values in expressions and solve equations

• move fluently between different representations such as algebra, graphs and diagrams

• develop algebraic and graphical fluency and understand linear functions

• interpret relations algebraically and graphically

• use language precisely

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Reason mathematically

• extend their understanding of the number system and make connections between numberrelationships and their algebraic and graphical representations

• extend and formalise their knowledge of ratio and proportion in formulating proportionalrelations algebraically

• identify variables and express relations between them algebraically and graphically

• develop reasoning in different areas of mathematics and begin to express their argumentsformally

Solve problems

• develop their mathematical knowledge, in part through solving problems and evaluating theoutcomes

• develop their use of formal mathematical knowledge to solve and devise problems

• begin to model situations mathematically and express the results using a range of formalmathematical representations

• select appropriate concepts, methods and techniques to apply to unfamiliar and non-routineproblems.

Subject content

Algebra: expressing relationshipsPupils should be taught to:

• read and interpret algebraic notation

• recognise, sketch and produce graphs of linear and quadratic functions of one variable withappropriate scaling, using equations in x and y and the Cartesian plane

• interpret mathematical relationships both algebraically and graphically.

Algebra: using equations and functionsPupils should be taught to:

• use formulae by substitution to calculate the value of a variable, including for scientificformulae

• begin to model simple contextual and subject-based problems algebraically

• solve linear equations in one variable in a variety of contexts, including subject-basedproblems, using algebraic methods

• calculate and interpret gradients and intercepts of linear functions numerically, graphicallyand algebraically, using y = mx + c.

Ratio, proportion and rates of changePupils should be taught to:

• use compound units such as speed, unit pricing and density to solve problems

• solve kinematic problems involving constant speed.

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III. Implementation Suggestions

Curriculum opportunitiesThe unit provides opportunities for pupils to (1) develop confidence in an increasing range ofmethods and techniques; (2) work on sequences of tasks that involve using the samemathematics in increasingly difficult or unfamiliar contexts or increasingly demandingmathematics in similar contexts; (3) work on open and closed tasks in a variety of real andabstract contexts that allow them to select the mathematics to use; (4) work on tasks that bringtogether different aspects of concepts, processes and mathematical content; (5) work onproblems that arise in contexts beyond the school; and (6) work collaboratively as well asindependently in a range of contexts.

Materials

• Pupil Workbook or Project Brief and Worksheets• Pencil and rough paper

• Straight edges and calculators

• Computers for pupils

• Whole class display: Interactive whiteboard, computer with projector, or document camera

Computer UseMost of the investigations in this module require use of the Cornerstone Mathematics software.We strongly recommend pupils use the software whenever possible.

Whole class discussions 1 whole class display visible to all

Group work 1 computer for every 3 pupils

Homework dependent on activity

Classroom OrganisationMany activities begin with whole class discussion, then pupils use what they have learnt in agroup work activity. These are our assumptions for recommending whole class, group work andindependent work.

1. Whole class discussion, teacher leads.- Whole class display is used.

- Plenaries are whole class discussions.

- Everyone can hear everything everyone says.

- Pupils take notes in their workbooks.2. Group work, teacher circulates.

- Pupils work in pairs or trios, seated so each team member can see the shared computerscreen and write in his/her own workbook.

- Pupils should work together to come up with common solutions.

- Each pupil should complete his/her own workbook.3. Homework or independent work.

- Pupils can work alone or in groups.

- Computers are not needed, so activity can be used at home or outside of classroom.

- Focused on practice of content in activity or remembering old material relevant for thenext day's activity.

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General Teaching Tips

• Encourage explanations of both correct and incorrect answers. When pupils give numericalanswers, ask, “How do you know?”

• Important instructional routine in the unit: Predict.Check. Explain.

- Pupils predict –have a go– what will happen in a simulation or in another representation.

- Pupils use software or calculations to check their predictions.

- Pupils explain any differences between what they predicted and what happened.

• Let pupils do more talking than you do. During whole class mode, let them answer questionsand challenge the answers others give. The think-pair-share routine is suggested, and you canuse your own strategies for engaging pupils in the plenary.

• Balance whole class work with individual and group work. Pupils need each.

• Allocate additional time for pupils to become familiar with the software during the firstlesson.

• Specify the time for whole class discussion so pupils know when to stop using the computers.

Tips for DifferentiationThere are a number of strategies you can use to meet the needs of learners with different levelsof achievement. Most of these strategies are inspired by the work of Carol Ann Tomlinson(2003) but rely on other resources as well.

Group work. You can arrange groups in several ways, which have different affordances for pupillearning. In the beginning of the module, you may want heterogeneous (mixed ability) groups,as pupils are beginning to grapple with new concepts. The challenge for all comes from thematerials themselves, and stronger pupils can help weaker pupils (and it may be a surprise whois which in Cornerstone modules.) Later in the module, when homing in on a specific conceptsuch as average rate, you may want homogeneous grouping so that you can help specific groupsof pupils who are likely to have difficulties with the same concept.

Flexible teacher. You can use different modalities to reach a broad group of pupils. The modulerequires the use of multiple representations—make sure to use as many as possible for eachactivity. And you may emphasize one representation for pupils who are particularly capable withit. You could, for example, provide a handout with blank tables for a pupil who prefers to usethose in an activity on graphs and stories.

Pupils with high achievement. Pupils who work at a faster pace may need supplementaryactivities. The following can be used at various times in the unit:

1. Early in the unit: Write a few sentences: A straight line on a position versus time graphrepresents motion at a constant speed. Does anything travel at a constant speed? Why orwhy not?

2. Mid-unit: Using a Cornerstone Maths activity such as Activity 7.3, make a complicatedtrip for one of the characters. Write a paragraph explaining the trip, using mathematics.

3. Any time: Make a poster illustrating what's in one of the grey boxes in the workbook.

Item 3 is particularly useful for aiding all learners. The pupil who makes the poster gets deeperknowledge of a term or concept, and his/her fellow pupils can benefit from the display of theposter.

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Pupils who struggle. You can use grouping strategies during pair work to support differentteaching activities. If you know that you will be short on time to visit every group, you can pair astronger pupil with a weaker pupil. On the other hand, if you know you want to focus on aparticular concept two pupils are having trouble with, then you can group those pupils together.In some cases, pupils who struggle can benefit from strategies designed for stronger pupils.Weaker pupils could learn a lot by making a poster about speed, for example.

Overcoming Possible Pupil MisconceptionsThese are all wonderful opportunities for teaching.

• Giving explanations. Pupils may need help in giving explanations. One issue is knowing thedifference between describing a procedure and explaining why it works. You may want tomodel some explanations for pupils in the beginning. Further help may be needed when it istime to write a story based on the representations in a Cornerstone Maths activity. Pupils mayneed help incorporating some mathematics into their creative writing, or they may need helpbeing more creative in their descriptions of what is happening to the characters.

• Confusing distance in a simulation with time on a graph. In the real-world simulation,distance is measured horizontally. On the graph, distance is represented along the y-axis. Thiscan be a point of confusion for pupils. Having them track the motion and the growing graph,perhaps with their fingers, can help.

• Rate. Pupils may be working with the concept of rate for the first time. This concept requirescoordination of two quantities instead of the single quantities they may be accustomed tofrom primary school. You will need to help pupils make the transition to this higher level ofconceptual understanding.

• Lengths of lines. Pupils may think that a shorter line on the graph always indicates a fastermotion. This is a great opportunity to explore pupils’ thinking. In fact, this is true only whenthe starting positions are the same, as one of our activities shows. Encourage pupils to find orconstruct counterexamples to show how a shorter line could actually represent a slower ratethan a longer line by considering the speeds that each line's gradient represents. Similarly,pupils may think that a longer line indicates a slower motion. Again, be ready and willing toencourage use of a counterexample to show how a longer line could represent a faster ratethan a shorter line.

• Horizontal and vertical. The software separates the horizontal (domain) and verticalmanipulations (gradient) on the graph in order to help pupils grapple with the two quantitiesthat make a rate. Pupils may try to simply relocate the endpoint of a graph wherever they like.You can focus them on the two changes they need to make.

• Functions only. If pupils have studied linear equations, they may have encountered equationssuch as x = 5. It is not possible to make such equations in the Cornerstone Maths softwarebecause they are not functions. The unit does not define function, but you can help pupils getready for this in later studies.

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Contents

iPreface to the Teacher Edition

TN-1aTeacher Notes for Introduction1Introduction: Welcome to SandCircle Mobile Games

TN-3aTeacher Notes for Investigation 13Investigation 1: Yari, the Yellow School Bus

TN-5aTeacher Notes for Investigation 25Investigation 2: Our First Mobile Phone Game

TN-7aTeacher Notes for Investigation 37Investigation 3: Controlling Characters with Graphs

TN-13aTeacher Notes for Investigation 413Investigation 4: Controlling Characters with Equations

TN-21aTeacher Notes for Investigation 521Investigation 5: One to Another

TN-25aTeacher Notes for Investigation 625Investigation 6: Better Games

TN-33aTeacher Notes for Investigation 733Investigation 7: Wendella’s Journey: Moving at Different Speeds

TN-39aTeacher Notes for Investigation 839Investigation 8: Money Matters

TN-43aTeacher Notes for Investigation 943Investigation 9: Mathematically Speaking: Graphs to Know

TN-45aTeacher Notes for Investigation 1045Investigation 10: Crab Velocity

TN-49aTeacher Notes for Investigation 1149Investigation 11: Wolf and Red Riding Hood

TN-55aTeacher Notes for Investigation 1255Investigation 12: Problem Solving

Part 1

Part 2

TN-57a57

Teacher Notes for Investigation 13Investigation 13: Problems from the SandCircle Lunchroom

TN-61a61

Teacher Notes for Investigation 14Investigation 14: SandCircle Mobile Games: Going Full Time

Teacher Notes

Introduction: Welcome to SandCircle Mobile Games

Key Ideas

→ Context of the unit is established: The mechanics and business of mobile phone gamedesign are described, such as how to design components of a game and compute salaryand savings.

→ Motion can be represented on a graph of time versus distance.

→ Idealised motion on a distance-time graph appears as a straight line (constant rate).

Whole Class | 30 minutesStarter

• Ask pupils to read Working at SandCircle Games and Mobile Phone Games and Design.

• Explain that in this unit, we do the mathematics needed to set up the games. We will not seethe final game, but pupils should imagine how it might work. Pupils also learn about howmuch money game designers make.

• Tell pupils that wherever possible, the numbers in this unit are realistic. We got them fromreal designers and design companies.

• If possible, show a real mobile phone game in the “moving character” genre. This is the kindof game we are developing in this unit.

• Introduce the Cornerstone Maths software. Open an activity such as Activity 3.3. Point out the four windows available: simulation, table, graph and equation.

TN-1aTeacher Notes for Introduction

TN-1b Teacher Notes for Introduction

Introduction: Welcome to SandCircle Mobile Games

Welcome to the headquarters ofSandCircle Mobile Games.

We make games for mobilephones—young children love them!

Other game design companies arecutting into our business, so weneed to make improvements. That’swhy you have been hired.

To make SandCircle Mobile Gamesmore competitive, you will:

• Use mathematics to analyse ourgame designs; for example,decide how to make charactersmove across the screen.

• Analyse our business.

This work will require a lot of mathematics. Now’s the time to learn it! Using this book and the software, you will learn about functions, both proportional and non-proportional. You will also learn the “real truth” about average rates. And you will understand the connections between tables, graphs and equations. Your learning will take place over time. Don’t worry, it will all be clear by the time you are finished!

The mathematics will help us in our business, but you will also use it in latermathematics classes and in your life. Keep that in mind as we move ahead.

1Introduction: Welcome to SandCircle Mobile Games

Mobile Phone Games and Design

A few facts about mobile phone game design will help you in this unit. Electronic games - on mobile phones, on consoles, on computers - must all be designed (created) by a team of people. The team works together, but there are different jobs to do. Designers come up with ideas for games and rules for how the game will work. Designers use mathematics when thinking about how realistic the game is and what rules will make the game fun and challenging. Programmers use mathematics and logic to write programs to turn the ideas into games that can be played on mobile phones and other devices. Artists make the images you see on the phone as you play. They must use mathematics to think about how big to make the images on the screen and how to make static images appear to move. Business staff make sure that the business is running smoothly. They use mathematics to handle things like employee salaries and tracking how much money the company is saving and spending.

In this module, you will mostly take on the role of a programmer, using mathematicsto make sure that the games actually work. You will have to ensure that the gamecharacters travel the correct distances, move for the right amount of time, and go atthe right speeds. While actual programming is more complicated than what you willlearn in this module, the lessons here give you a start at understanding the kind ofactivity a programmer does take on. The mobile phone game business is growing, andit’s likely that more designers, programmers, and artists will be needed in the future.Understanding mathematics helps prepare you for these sorts of jobs.

2 Introduction: Welcome to SandCircle Mobile Games

Teacher Notes

Investigation 1: Yari, the Yellow School Bus

Key Ideas



40 minutes totalMain Activity

Whole Class | 10 minutesDiscussion

• Set the context for this investigation. You may ask pupils, “If you wind up a wind-up toy andset it to roll across the floor, when will it go fastest? Slowest?”

Whole Class | 20 minutesQuestion 1 (A-D)

• After watching the video, elicit comments about the changing speed.

• Discuss (after completion of question) pupils' observations about Yari's motion.

Individual | 10 minutesQuestion 1 E

• Use photos from the video to graph the motion.

5 minutesPlenaryDiscuss with pupils the difference between providing answers and providing explanations. Aconstructive way to structure discussions is for pupils to remember to ask and answer “how” and“why” questions.

Pupil Difficulties

Pupils may have difficulty

• Relating the graph of the motion to the actual motion.

• Knowing what constitutes a good explanation.

Pupils may have misconceptions

• Interpreting the graph as a picture (e.g., a path, an incline).

TN-3aTeacher Notes for Investigation 1

TN-3b Teacher Notes for Investigation 1

Investigation 1: Yari, the Yellow School Bus

SandCircle Mobile Games is using a mechanical toy to design new games for younger children. We need to analyse the motion of the real toy, Yari, The Yellow School Bus. Then we can make our mobile phone version.

1. Use the Cornerstone software to complete this investigation. Open Investigation 1,named “Yari, the Yellow School Bus”. Click the Play button (the triangle thatpoints towards the right). Watch the video clip of the actual wind-up toy—Yari, TheYellow School Bus—moving along a wall lined with a 1-cm2 grid. Play the video clipseveral times and look through the photos to answer the following questions:

Explain

Each time you give anumerical answer,explain how you know.

A. How many centimetres (cm) did Yari travel?

Answers may vary. 100 cm.

B. How many seconds did Yari travel?

Answers may vary. 10 seconds.

C. Explain how you know how far and for how much time Yari travelled.

Answers will vary. Total run time is shown at the end of the video. There’s a metronome inthe background; a voice counting down from 10 is heard in the background. For distance, theinstructions say a centimetre grid, so pupils may assume 100 cm.

D. Describe how Yari’s speed changes throughout this trip.

Answers will vary. Yari went slowly in the beginning, went faster in the middle, and went veryslowly at the end.

3Investigation 1: Yari, the Yellow School Bus

E. Using the video clip and the photos, fill in the table and draw the graph.

Time(sec)

Distance(cm)

0 0

1 5

2 12

3 23

4 37

5 54

7 83

10 1000

x

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Time (sec)

y

10

20

30

40

50

60

70

80

90

100

110

Dis

tanc

e (c

m)

4 Investigation 1: Yari, the Yellow School Bus

Teacher Notes

Investigation 2: Our First Mobile Phone Game

Key Ideas





• Emphasise that Rita, the new character, is a SIMULATION of the real toy in the video. Wesimplify the motion so that we can work with line graphs.

Whole Class | 5 minutesQuestion 1

• Pupils should make the connection between total distance, total time and speed. Guide themto explain that speed is the relationship between distance and time expressed as a rate (usingdivision).

• Draw attention to the speed and how it indicates the unit rate.

Individual | 5 minutesQuestion 2

• Focus on the whole line, that is, a set of points. Encourage a qualitative explanation frompupils.

- Basic: As the graph builds, the y-axis shows the same number as the “ruler” in thesimulation.

- More sophisticated, optional: The line is slanting upward because, as the bus movesalong, both time and distance are increasing.


• Using the table, continue to build correspondence between overall speed and the ratedistance/time at EVERY point on the line.


• Emphasise that Rita is an idealised model of Yari and so the motion is “smoothed out”. Thisis the type of motion we will use in our games throughout the unit to simplify our work.

• Point out that Rita moves at a constant rate: She moves at the same speed at every point intime.



• Ask a few pupils to share their answers to question 4.

• Read and discuss the “About modelling” box.

10 minutesPlenaryDiscuss with pupils their experiences in providing explanations when engaging with thecollaborative activities. Remind them that they should be asking and answering “how” and“why” questions. Address any difficulties pupils may have about interpreting a position-timegraph and how a constant speed is represented.

Pupil Difficulties


• Understanding ratios and rates because these require the coordination of two quantitiesrather than working with just one.

• Using unfamiliar variables and new units (d/t).


• Interpreting the graph as a path.

• Switching axes (placing the variables on the wrong axes).


Investigation 2: Our First Mobile Phone Game

Our programmer has made our first mobile phone game. It uses the character RitaThe Red School Bus.

We want to compare the motion of Rita The Red School Bus in the game with themotion we observed of the real toy, Yari.

1. Open Activity 2.1 in the Cornrstone Maths software. Click the Play button. You will see Rita’s trip in the simulation, and the trip is represented in a graph. Play the simulation as many times as you need to answer the following questions:

100 cm

10 seconds

10 cm/sec

2. Watch how the graph builds as Rita moves. Describe how the graph relates to thesimulation.

Answers will vary. The graph and simulation both show Rita’s position. The graph and thesimulation also represent speed and time, but these are harder to see when the pupil uses thestep function.

5Investigation 2: Our First Mobile Phone Game

A. How many centimetres (cm) did Rita travel?

B. How many seconds did Rita travel?

C. What was Rita’s speed?

D. In your own words, explain what is 'speed'?

Answers will vary. Speed is a rate that coordinates (relates) time travelled with distance travelled. We derive speed by dividing the distance travelled by the time travelled. Speed measures how fast or slowly something moves.

3. Complete the table and graph below (use the Step button to help you).

Time(sec)

Distance(cm)

0 0

1 10

2 20

3 30

4 40

5 50

7 70

10 100

0x

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Time (sec)

y

10

20

30

40

50

60

70

80

90

100

110

Dis

tanc

e (c

m)

4. Compare and contrast Rita’s and Yari’s trips.

A. Describe how the trips of the mobile phone game character (Rita) and themechanical toy (Yari) are the same.

Rita and Yari travel the same distance in the same amount of time.

B. Describe how the trips of the mobile phone game character (Rita) and themechanical toy (Yari) are different.

Yari (the wind-up toy) moves at different speeds during the motion, sometimes faster andsometimes slower. Rita (the game character) goes at the same speed throughout thesimulation.

About modelling

In mathematics, we often simplify things so that we can work with them. One way we simplify is by creating models. Rita, the game character, is a model of the mechanical toy, Yari. The model simplifies the motion of the wind-up toy. Models always simplify the “real thing” in some ways.

6 Investigation 2: Our First Mobile Phone Game

Teacher Notes

Investigation 3: Controlling Characters with Graphs

Key Ideas

→ Graphs are mathematical representations of relationships such as motion.

→ Graphs of motion show characters’ start position, speed (relative) and places and timeswhere characters meet.

→ For graphs of motion (that is, time versus distance), the steeper the line, the faster themotion.

→ Speed can be determined from different parts of a graph and simulation.



• In the "Texas Road Rally" game, cars move across a map of Texas, city to city, as in a realroad rally. Explain that we are making the graphs to control the motion of the cars and settingup the mathematics that will make the game work. You may wish to reiterate that we aremodelling cars' trips. While the actual races may be hundreds of miles long and take hours,we are using mathematics and software to model the races by showing the cars moving a fewhundred pixels across a computer screen in a matter of seconds.


• Help pupils connect distance/time and “angle or steepness of line” as two ways to representthe speed of a car. Question B elicits the concept of unit rate.

• Discuss after pupils complete the question.

Group | 20 minutesQuestions 2 and 3

• “Predict, check, explain” enables pupils to come to understand that, on the same axes, steeperlines represent faster trips.

• Some pupils may believe that shorter lines represent faster trips (which is true when thedistance travelled is the same). Encourage pupils to create a counterexample in which a fastertrip is represented with a graph line that is longer than the graph line representing a slowertrip.

• Discuss with pupils after they complete the question.

Group | 20 minutesQuestion 4

• “Sketch graph” means draw the basic shapes. Showing the relative slants of the lines and/orkey positions is what is important.

• Each part should be done using “predict, check, explain”. Ask pupils to draw graphs inworkbooks first to predict and then check in the software.

- Each problem has many solutions.

- Establish a foundation for the y-intercept and intersection of lines.


Whole Class | 5 minutesPlenary

• Ask pupils to summarise what information they can derive from having only a graph ofmotion.

• Ask what kind of information is not included in a graph of motion.

• Restate the key ideas of this activity: Graphs of motion tell us starting positions, speeds andending positions; the steeper the line, the faster the motion.

• If there are any pupils still holding on to the misconception that shorter lines represent fastertrips, show a counterexample in which a faster trip is represented with a graph line that islonger than a graph line representing slower trip.

Pupil Difficulties


• Accepting that predictions can be wrong and that there is value in reflecting on andanalysing the predictions that were incorrect.

• Understanding why “on the same graph” is in the “Steeper is faster” callout box.

• Interpreting the intersection of two lines in a graph.


• Incorrectly associating the “shorter time” with simply “shorter line” to indicate fastermotion.

• Thinking that parallel lines in a graph represent parallel tracks or paths.


Investigation 3: Controlling Characters with Graphs

In America, there is a car race called the Texas RoadRally. SandCircle is building a game for sale in Americabased on the race. We need to set up the mathematicsfor a new game. In our simulation, we will use graphs tocontrol motion. Remember, we’ll make the game coollater!

In road rally races, professional drivers compete as theytravel from town to town.

Same motion, different representations

Graphs, tables, and equations are mathematical representations. Each can showthe same motion in a different way.

1. Open Activity 3.1 in the Cornerstone Maths software. The simulation shows theOrange Rose team car in the Texas Road Rally. The car starts in El Paso and stopsin Big Spring. Click the Play button and watch the simulation and graph.Remember, each time you give a numerical answer, explain how you know.

A. When the car stops, how far has it travelled and how much time has passed?

350 miles and 7 hours

B. When the car has travelled for 1 hour, how far has it gone?

50 miles

C. How fast is this car going? Explain in two ways how you know. Think about youranswers to A and B or use other ideas.

50 miles/hour• Reading from the graph or answer A, you can see that the car travelled 350 miles in 7

hours. If we simplify that, we find that the car travelled 50 miles in 1 hour or 50 mph.• Reading from the graph or answer B, you can see that the car travelled 50 miles in 1 hour

or 50 mph.

D. Edit the graph to change the speed of the car. The car still must start in El Pasoand stop in Big Spring. Use the same ideas to find the new speed from yourgraph. What is this new speed?

Answers will vary. Graphs must start at zero and end at 350 miles.

7Investigation 3: Controlling Characters with Graphs

2. Now, turn away from your computer. Below in your workbook are the graphs of theOrange Rose team car and the Longhorn team car. They are ready to travel fromSan Antonio to Beaumont.

Using the graph of the trips made by the two cars, fill in the boxes.

0x

1 2 3 4 5 6 7 8 9 10

y

50

100

150

200

250

300

350

400

450

Dis

tanc

e fr

om s

tart

(m

iles)

Orange

Longhorn

A. Predict whichcar is faster.

This is aprediction.Any answer isacceptable.

B. Check your prediction by calculating the cars' speeds.

Orange Rose:50 mph

Longhorn:35 mph

Time (hours)

Open Activity 3.2 and play the simulation to check your answers.

3. Now, edit the graph in Activity 3.2 to change the speed of each car so that the other caris faster this time. Record your graph below and complete the questions. Don’t forget to label your lines on the graph that represent each car.

0x

1 2 3 4 5 6 7 8 9 10

y

50

100

150

200

250

300

350

400

450

Dis

tanc

e fr

om s

tart

(m

iles)

Answers will vary. Correctanswers will have a greater(steeper) gradient forLonghorn's graph. Longhorn.

Answers will vary

8 Investigation 3: Controlling Characters with Graphs

C. Explain: Was your prediction correct?

Answers will vary.

A. Which car is faster this time?

B. What is the speed of the faster car?

C. What is the speed of the slower car?

Answers will vary

Time (hours)

Now play the simulation and check your answers.

D. Were you correct?

Answers will vary

The car with the steeper line wins. (Potential misconception: Pupils may think that the car with the shorter line wins. You can make a shorter line where this is not the case as a counterargument to help them arrive at the conclusion that it’s not the length of a line, but its steepness that matters, when graphed in the same coordinate system).

E. If you were given a new distance-time graph of two cars in a rally, how can you predict which car will win?.

The steeper the line, the faster the car (when it is graphed in the same coordinate system.)

4. Below are ideas to use in other games. Please help our programmers by sketchinggraphs on the axes provided. Note: There is more than one way to draw the graphfor most of these situations.

A. Predict: Sketch a graph of a race in which the Green Grass and Blue Watersteam cars start at the same position and travel the same distance but atdifferent speeds.

Don’t forget to label your graph lines.

0x

Time (hours)

y

Dis

tanc

e fr

om s

tart

(m

iles)

This is a prediction, so anyanswers are acceptable.

B. Check: Now use Activity 3.3 to create the graphs to match your drawing. Thenrun the simulation. Do not alter your prediction in the graph above.

C. Explain: Explain any changes you would have needed to make to the graph inActivity 3.3 in order to correctly represent the race described in instruction A.

Pupils should identify the critical features of correct graphs:• Start and ending positions are the same (same y-intercept and ending position).• Graphs have different gradients.


F. Explain how a car's speed and its graph are related to each other.

D. Sketch a graph of a race in which

• Green Grass starts at the starting line (0 miles),

• Blue Waters starts 50 miles ahead (50 miles),

• the two team cars finish at the same position and the same time.

Don't forget to label your lines in the graph.

0x

Time (hours)

y

Dis

tanc

e fr

om s

tart

(m

iles)

Critical features:• The y-intercept for Green

Grass is the origin.• The y-intercept for Blue

Waters is +50.• Graphs have same ending

coordinate pair (same timeand position).

E. Sketch a graph of a race in which

• Green Grass starts at the starting line,

• Blue Waters starts 50 miles ahead,

• the two team cars travel at the same speed and travel the same distance.

Don't forget to label your lines in the graph.

0x

Time (hours)

y

Dis

tanc

e fr

om s

tart

(m

iles)

Critical features:• Graph lines are parallel.• The y-intercept for Green

Grass is the origin.• The y-intercept for Blue

Waters is +50.


Steeper is faster

You may have noticed by now that steeper graph lines on the same graphrepresent faster speeds.

A line's gradient is its steepness

Gradient is a mathematical way to describe the steepness of a line on a graph.The gradient of a line is a number: it is the amount of change in y (on the verticalaxis) for one unit of x (on the horizontal axis).

Gradient and speed are related. Speed is a rate that coordinates (relates)distance and time. Distance and time are measured in units such as metres andseconds. Gradient, on the other hand, is just a number—it has no units.


Summary

Discuss the statements below.

Do you all agree with them?


Notes

Teacher Notes

Investigation 4: Controlling Characters with Equations

Key Ideas

→ Equations are a form of mathematical representation. Graphs and tables are other forms.

→ Equations can be written based on tables or graphs.

→ You can “translate” between graphs, tables and equations.

→ Time, distance and speed are represented differently in these three representations.

→ For equations of the form y = mx, in motion contexts, m is the speed of a moving object.


Whole Class | 5 minutesDiscussion“Robot Games”: Players use remote-control robots to open doors, dispose of bombs and so on.

Group | 35 minutesQuestions 1 and 2

• Pupils see an equation, a table and a graph and observe the connections between therepresentations as the simulation runs. We assume no previous exposure to equations. Elicitcomments such as the following:

- 4 cm/sec is the speed, from the graph.

- Each pair (row) in the table is related by multiplication by 4. The equation has 4 in frontof the x (explain that this means to multiply x by 4).

• Explain that y = 4x is a compact way to write the equation seen in the software, y = 4x + 0.Elicit from pupils the idea that y = mx is the same as y = mx + 0 because adding zero doesn’tchange anything.

• Pupils identify how the same information is represented in equations, tables and graphs.


• Elicit from pupils the connections they see across the representations.


• Guide pupils as they discover how to write equations from tables.

- Identify a pattern in the table, relating each pair in a row.

- Use variable names to express the pattern algebraically.

- Note that the equation links the two variables, in this case distance and time.


• Guide pupils as they learn to generate a table of values from an equation.• Discuss after pupils have completed the question to ensure that they understand how to

determine an equation for a character's motion.


Whole Class | 10 minutes

• Pupils consolidate their learning into a rule or method.

• Introduce the general form y = mx. Emphasise meaning of m.

• Discuss

Group | 15 minutes

• Pupils start from an equation and generate a table. The software table is not used; instead,pupils predict and check through the simulation, having another opportunity to translatebetween representations.

• Discuss


• Pupils apply methods in another real-world context.


• Review the key ideas in this activity: Time, distance, and speed are represented differently ingraphs, equations and tables; in motion contexts, the gradient is the speed of a moving object.

• Ask pupils to explain to one another the method for deriving equations they have learned.How does this compare to other methods they know?

Pupil Difficulties


• With tables: Focusing on one column at a time, reading down a table instead of across.

• With equations: Leaving out the variable, understanding the meaning of the coefficient; andtranslating a multiplicative relationship into an equation.

• With notation: Thinking that y = kx is not same kind of graph/function as y = ax + b.

• Knowing when to use units and which units to use.


Question 5

Investigation 4: Controlling Characters with Equations

So far, we’ve used graphs to show the relationshipbetween time and position at any time during ajourney.

Equations use numbers and symbols to show that same relationship. Give us any time, and we can find the position of the character at that time by using the equation.

We will control our characters with equations as well aswith graphs. The equations are so efficient! Remember,the players won’t see our equations.

Let’s work on a game with robots. We need to set upthe mathematics to make our robots move at differentspeeds.

1. Open Activity 4.1, which shows Shakey the Robot. Play the simulation

A. How fast is Shakey going? How do you know?

4 cm/sec

13Investigation 4: Controlling Characters with Equations

In this Investigation you will use the software to help you to identify the relationships between the different ways of representing a function, which are graphs, tables and equations. There are lots of tasks for you to complete that will help you to gain confidence in moving between these different representations.

B. How do you know?

The graph and the simulation show that Shakey travels 40 cm in 10 seconds.

The table also shows that in the first second, Shakey went 4 cm.

C. Edit the graph in Activity 4.1 so that Shakey moves slower. Draw the graph andcomplete the table and equation for the graph.

Slow Shakey

0x

1 2 3 4 5 6 7 8 9 10

Time (sec)

y

48121620242832364044

Dis

tanc

e (c

m)

For Slow Shakey, graphsshould be less steep.

x

(sec)y

(cm)

0

1

2 (Answers

3 will

4 vary.)

5

10

D. Record your equation:

Coefficientof x shouldbe lessthan 4.

E. Edit the graph in Activity 4.1 so that Shakey moves faster. Draw the graph andcomplete the table and equation for the graph.

Fast Shakey

0x

1 2 3 4 5 6 7 8 9 10

Time (sec)

y

48121620242832364044

Dis

tanc

e (c

m)

For Fast Shakey, graphsshould be steeper.

x

(sec)y

(cm)

0

1

2 (Answers

3 will

4 vary.)

5

10

F. Record your equation:

Coefficientof x shouldbe greaterthan 4.

G. Compare your equation from Slow Shakey with your equation from Fast Shakey.Describe any differences. Where are these differences shown in the graphs andthe tables?

Answers will vary. In the equation, Fast Shakey has a greater coefficient for x than SlowShakey. In the graph, the line for Fast Shakey is steeper; the line for Slow Shakey is lesssteep. In the table, the differences in positions for Fast Shakey are larger than thedifferences in position for Slow Shakey, corresponding to the same time intervals.

Pupils need not use the word “coefficient” at this point.

14 Investigation 4: Controlling Characters with Equations

2. To answer these questions, use Activity 4.1. Edit the graph and play thesimulation while noticing what is changing in the graph, table, and equation.Investigate how time, position and speed are represented in each.

A.How is time represented in each of the following?

Graph Time is on the x-axis, or the x-coordinate of any given point on theline.

Table Time is in the left column, in seconds.

Equation Time is the x variable.

B. How is distance represented in each of the following?

Graph Position is the distance from the start, which is on the y-axis, or the y-coordinate, of any given point on the line.

Table The values in the column labelled 'Shakey's distance'.

Equation Position is the y variable.

C. How is speed represented in each of the following?

Graph Speed is represented in the gradient or steepness of the line.

Table Speed is not shown but can be determined by the relationshipbetween the time and position columns. It is what you multiply timeby to get the position when the y-intercept is zero.

Equation Speed is represented by the coefficient of x (or the number by whichx is multiplied).

Key point: Time, position, and speed are sufficient to describe linear motion. Eachrepresentation (graph, equation, table, simulation) contains information about time, positionand speed.


Variables and letters

Here we use t and p as the variables in our equations instead of x and y.Sometimes it is easier to use letters for variables that relate to the units ormeasures that we are dealing with. Of course, it does not matter what we call thevariables as long as the relationship is correct.

3. SandCircle is developing a game called “Slugs Versus Snails” with characters racingacross a garden. In this activity, you will learn how to write equations from tablesto set up the races.Open Activity 4.2 and play the simulation for Roberta the Ladybird.

A. From the graph complete the column for the values of p in the table below.

Roberta

t

(time in sec)p

(position in cm)

0 0

1 8

2 16

3 24

4 32

5 40

10 80

Answers will vary. To get p, multiply t by 8. Also p = 8t. Key point: Equation is derived bystating the relationship between t and p.

C. Write your rule as an equation using p and t.

p = 8t

8 cm - so her speed is 8cm/s

E. How does this speed relate to your equation?

Answers will vary. The rate is the speed, and it’s also the multiplier or coefficient of t.


B. Look carefully at your completed table. If you choose any value of t from the column, what is the rule that gives you the corresponding value of p?Check that your rule works for each pair of values. Write your rule in words.

D. For every second that Roberta travels, how many centimetres does Roberta move?

4. Open Activity 4.3. Look carefully at the motions of Glider the Snail andHopalong the Slug. Complete the values of p in each table. Then use these to work out the equations of motion for each character.

A. Glideri) Table of values (ii) Equation of motion

t

(time in sec)p

(position in cm)

0 0

1 20

2 40

3 60

4 80

5 100

10 200

p = 20t

B. Hopalongi) Table of values (ii) Equation of motion

t

(time in sec)p

(position in cm)

0 0

1 12.5

2 25.0

3 37.5

4 50.0

5 67.5

10 125.0

p = 12.5t


C. Explain, in your own words, how to use a table to write an equation to represent acharacter’s motion.

Answers will vary.

Find the equation that relates t to p and then write the equation in symbols, where p =something times t (or p = __t ).

Answers will vary.

Critical features: Pupils’ tables should show that p = t multiplied by the coefficient of t (or x fromthe function window in the software).

A. Complete the values of p in the table.

Aesop

t

(time in sec)p

(position in cm)

0 0

1

2 (Answers

3 will

4 vary.)

5

10

B. Open Activity 4.4 and play the simulation. Check that the values of p in your table match what you observed in the simulation. You can use the Step button.

C. Explain any differences between your predictions and what you have observed.

Answers will vary.


5.

In the next question you will learn how to generate a table of values from an equation.

Now, let’s use an equation to fill in a table of values.

Aesop the Ant’s motion is represented by the equation:

p = 15t + 0

6. Use what you have learnt to help Monica work out the relationship between hersalary and the time (number of months) she works.

Monica is one of SandCircle Mobile’s best programmers. Every month, she earns asalary of £6,200.

Make a representation (graph, table, or equation) so that she can choose anymonth (where January is month 1) and find out how much she has earned so farthat year.

A. Table of values B. Equation

m

(months)s

(£ earned)

0 0

1 6,200

2 12,400

3 18,600

4 24,800

5 31,000

6 37,200

7 43,400

8 49,600

9 55,800

10 62,000

11 68,200

12 74,400

s = 6200m

where s is theaccumulated salary afterm number of months, andm is the number of monthsMonica worked.

0m

1 2 3 4 5 6 7 8 9 10 11 12

Time (months)

s

10

20

30

40

50

60

70

80

90

100

110

Bala

nce

(tho

usan

ds o

f £)

JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC


C. Graph

Teacher Notes

Investigation 5: One to Another

Key Ideas

→ You can “translate” between graphs, tables and algebraic expressions.



• This activity can be used as homework or an informal assessment/quiz, or in class if there istime.

• Pupils practise translating between graphs, tables and equations.

• Encourage pupils to use the same variables for their graphs, tables and equations.

• Pupils can use the software to check their answers.

• Optional: Cut out the different representations and ask pupils to play a matching game.


• Pupils translate from a narrative of a motion to table, graph and equation representations.


• Pupils translate from an equation of a motion to table, graph and narrative representations.


• Different contexts for rate are used.

• Pupils translate from one representation to the others.


• Pupils translate from a partially completed table to graph, equation and narrativerepresentations as well as complete the table.


• Pupils translate from a narrative to graph, equation and table representations.


• Review the translations between representations that pupils should be able to make at thispoint and address any difficulties.


Pupil Difficulties


• Translating a multiplicative relationship into an equation using variables.


Investigation 5: One to Another

You have seen how graphs, equations, and tables can each represent a character moving at a certain speed.In the questions that follow there are some new ideas for game characters and their rates. Your task is to complete all of the representations for each speed or rate. The axes of each graph are already labelled for you.In each question, you are given one key piece of information, from which you can work out the other representations. You may need to look back at the work you have done so far.Example: The beetle crawls at a constant speed of 6 centimetres every second.

Table Graph Equation

x

(sec)y

(cm)

0 0

1 6

2 12

3 18

5 30

0x

1 2 3 4 5 6 7 8 9 10

Time (sec)

y

6121824303642485460

Dis

tanc

e fr

om s

tart

(cm

)

y = 6x

1. A toy robot moves on average 8 centimetres every 1 second.

Equation

t

(sec) p

(cm)

0 0

2 16

3 24

5 40

10 80

0

Dis

tanc

e fr

om s

tart

(cm

)

Graph

Answers will vary but

the gradient should be 8.p = 8t

Time (sec)

Answers, including the variables used, will vary but must maintain the y = 8x relationship.

21Investigation 5: One to Another

Here we have been given the ‘story‛. The task is to complete the other representations.There are many ways to do to this.One way would be to complete the table first. We can choose any values for x as long aswe use the story to calculate the corresponding value of y.Plotting these values and joining them with straight line will give us the graph.Finally, noticing that each y-value is six times the corresponding x-value leads to theequation.

Table - Name your variables

2. A sloth moves on average 0.5 metres every 1 minute(s).


t

(min) p

(m)

0 0

5 2.5

3

4 2

40 20

0

Time (min)

Dis

tanc

e fr

om s

tart

(m

)

Answers will vary but thegradient should be 0.5.

p = 0.5t

Answers will vary but must maintain the y = 0.5x relationship.

22 Investigation 5: One to Another

1.5

Instead of speed, or rate of motion, we can make the same representations for other

rates, such as money earned per month.

3. Naila, a beginning game designer at our company, earns a salary of £3,000 per month.

Equation

x

(months) y

(£)

0 0

1 3,000

2 6,000

3 9,000

12 36,000

0

Time (months)

Sala

ry e

arne

d (£

)

Graph

Answers will vary but thegradient should be 3000.

y = 3000x


Table - Name your variables

4. In a quiz game, players get 3 points for every correct answer.


a

(correctanswers)

p

(points)

0 0

1 3

2 6

3 9

9 270

Correct answers

Tota

l poi

nts


p = 3a

Answers, including the variables used, will vary but must maintain the p = 3a relationship.

23Investigation 5: One to Another

5. A toy car moves at a constant speed of 0.25 metres every 1 second(s).

Table - name your variables Equation

(sec) x y

(m)

0 0

1 0.25

2 0.5

20 5

16 4

0

Time (sec)

Dis

tanc

e fr

om s

tart

(m

)

Graph

Answers will vary but thegradient should be 0.25.

y = 0.25x

Answers, including the variables used, will vary but must maintain the y = 0.25x relationship.

6. A helicopter flies up into the sky at 5 metres per second.


x

(sec) y

(m)

0 0

1 5

2 10

20 100

15 75

0

Time (seconds)

Dis

tanc

e fr

om s

tart

(m

)


y = 5x


24 Investigation 5: One to Another

Teacher Notes

Investigation 6: Better Games

Key Ideas

→ Introduction to non-proportional linear functions (not passing through the origin).

→ Pupils explore two ways to derive the equations of non-proportional linear functions:using differences of position and time in a table; using the y-intercept and speed/gradientof a graph.

→ For equations of the form y = mx + c, in motion contexts, c is typically the starting pointand m is the speed of a moving object.



• This is another game. In retro-arcade games, the characters move in a straight line, facingobstacles.

• Building on the method in the prior lessons, pupils learn to write equations with a non-zerointercept.


• Use the idea that Geneva and Reynaldo travel at the same speed to help develop equations foreach.

• After pupils have worked on their predictions, lead the class in a short 5-minute discussion tocheck the reasonableness of their predictions and check pupils’ understandings.

• Central idea: When looking down the rows, the ratio of the numbers you add to get from oneto the next is the same and is the speed of the character, as confirmed by methods in previousactivities. When the line does not start at (0,0), these methods can be used to find thecoefficient of x and then the number added onto y can be found easily from the row with 0 forx.

• Discuss after A, after C, and after E.


• Model filling out the workbook, using the interactive whiteboard.

• Central idea: The coefficient of x is the speed, and the number added (or subtracted) is thestart position. The speed and start position can be used to write equations directly from thegraph. You will need to help pupils get started.

• The equation in this context is the algebraic relationship between position and time.


• Pupils connect graphs and equations directly.

• It is OK to have pupils make tables to help, if needed.

• Using the software’s step feature may be helpful.


Review task

• In this part of the lesson, pupils create two trips (one high speed and one low speed). Pupilsfirst imagine the fast trip, choosing a (non-zero) starting position and speed for their characterand then write an equation based on those. They then sketch a graph to represent thatequation. Pupils then use the software to check that their imagined trip and their mathematicsmatch. They edit the equation and graph in Activity 6.3 to match the graph and equation theyhad written and then check for consistency among the mathematical representations and theirexpectations for the particular trip (fast or slow).

• Check that pupils’ graphs and equations are mathematically consistent.

• Check that pupils are creating trips with first a high speed and then a lower speed.


• Ask pupils to think of how to explain to another pupil all the ways they can derive equations(from graphs, from tables, from stories of motion).

• Ask pupils to discuss in pairs their methods.

• Ask a few pupils to share with the whole class. Capture as many methods as you can, giventime constraints.

Pupil Difficulties


• Sketching graphs without plotting points.

• Interpreting a non-zero starting position on the graph, in the equation and in the table.

• Interpreting a negative position and knowing how it’s related to the real world (distance isalways positive, position is not).

• Interpreting the intersection of two lines in a graph.

• Understanding the difference between the general equation of the line vs. slope-interceptequation or function form.


• Thinking that parallel lines in a graph mean parallel tracks.


Investigation 6: Better Games

We need to make mathematical controls for Reynaldo, Bommakanti, and Geneva, whowill be used in a retro arcade game for mobile phones. The equations are morecomplicated because these characters each start at a different place.

1.

0t

Time (sec)

p

Posi

tion

(cm

)

• This is a prediction; manyanswers are acceptable.Pupils may not know how torepresent Reynaldo’s headstart.

• Critical features:- Graphs are parallel.- Both characters run for

10 seconds at 10 cm/sec.- Reynaldo’s graph starts

at 5 cm.

B. Check: Now check the graph with the software.

Describe any differences between the graph you sketched above and the graphshown in the software.

Answers will vary. Pupils should identify key differences related to the critical features:Graphs are parallel. Both characters run for 10 seconds at 10 cm/sec; Reynaldo’s graphstarts at 5 cm.

C. Explain any differences between the graph you sketched above and the graphshown in the software.

Answers will vary. Pupils should identify how their sketch addressed the critical features ornot.

25Investigation 6: Better Games

A. Predict what the graph will look like for each character’s motion by sketching two lines on the axes below. Don’t forget to label your lines with each character’s name.

Table

Equation

Geneva’s time inseconds

(t)

Geneva’s positionin centimetres

(p)

0 0

1 10

2 20

3 30

4 40

5 50

10 100

p = 10 t + 0

In the equation, the speed is represented by the coefficient of t, which is thenumber that t is multiplied by.

The speed is the number you multiply by in each row to get p from t.Or speed is the ratio between the difference of two p-values and the difference of the two corresponding t-values.The speed (10cm/sec) is the number you add to each value of p to get the number in the next row if the time interval is one second.

26 Investigation 6: Better Games

D. For each character, complete the table and write the equation.

Geneva Reynaldo

Table

Reynaldo’s timein seconds

(t)

Reynaldo’sposition in

centimetres(p)

0 5

1 15

2 25

3 35

4 45

5 55

10 105

Equation p = 10 t + 5

E. Look at Geneva’s table of values. What are two different ways that the table shows you Geneva’s speed?

(i)

(ii)

F. Look at Reynaldo’s table of values. How can you work out Reynaldo’s speed?

The speed (10cm/sec) is the number you add to each value of p to get the number in the next row if the time interval is one second.


Coefficients

In algebra, the coefficient of a variable is the number that the variable is multiplied by.

Now look at Geneva’s and Reynaldos’s equations. In each equation, the speed is represented by the ‘coefficient’ of t.

Constant term

In a linear function, the constant term is the value of y, when x is zero. For a position-time graph, it is the starting position when time is zero.

Reynaldo starts at a different place from Geneva. Reynaldo has a headstart. The headstart must be represented in the graph, the table, and the equation

G. How is the different starting position shown in:

(i) Reynaldo’s graph?

(ii) Reynaldo’s table?

(iii) Reynaldo’s equation?

The line starts at (0, 5)

When t=1 the value of p is 5 cm

The constant term is 5


2. Now we will create equations directly from graphs. Open Activity 6.2.

A. Predict: Play the simulation. Predict what the graph will look like for

each character’s motion by sketching three lines on the axes below. Don’t

forget to label your lines.

0t

1 2 3 4 5 6 7 8 9 10

Time (sec)

p

-20

-10

10

20

30

40

50

60

70

80

90

100

Posi

tion

(cm

)

Answers will vary. Noticestarting position and speed.

B. Check: Open the graph window. Run the file several times. Pay attention to thegraph and the simulation and how they work together. Describe any differencesbetween the graph you sketched above (in A) and the graph shown in thesoftware.

Answers will vary.

C. Explain: Explain how your sketch was correct or not.

Answers will vary. Pupils should note information about the y-intercept, the speed orgradient, and the length of time each character moved.


D. Using the graph in Activity 6.2, which shows the simulation for Geneva,Reynaldo, and Bommakanti, fill in the speed and starting position for eachcharacter. Then use that information to determine the equation that links theposition and time for each robot.

Speed Starting Position Character’s

4.5 cm/s 10 cm

Equation

p = 4.5t + 10

6 cm/s 5 cm p = 6t + 5

Geneva

Reynaldo

Bommakanti 8 cm/s −10 cm p = 8t − 10

E. Explain how to write an equation directly from a graph.

A linear graph has an equation in the form of p = __ t + __. To find the coefficient of t,calculate the speed. To find the constant, determine the start position or where the graphintersects the y-axis (the y-intercept).

Review task

In this activity, you will imagine two trips (one when the character moves fast andone when the character moves slowly) and describe those trips mathematically.Then you will use the Cornerstone software to make sure that the trips youimagined match up with the mathematics you used to describe them.Below, use mathematics to describe the trip when the character moves fast andthen the trip when the character moves slowly.

A. For the fast trip, do the following:

i. Use mathematics and imagine the whole trip. Make sure to use a non-zerostarting position and a fast constant speed.

ii. Record the (non-zero) starting position and speed of the character below.

iii. Write the equation that goes with that start position and speed.

iv. Sketch the graph and label your axes.

v. Open Activity 6.3 and edit the equation and graph to match what you wrotefor the fast trip below. Check that the trip of the character in the softwarematches the trip that you had imagined in 4Ai.

B. Now, repeat the process from part A for the slow trip but instead of using a fastspeed, you should use a slow speed.

Fast Trip Slow Trip

Start position: Start position:

Speed: Speed:

Equation: Equation:

0t

p

0t

p

Answers will vary, but the gradient of the Fast Trip should be greater than the gradient of the Slow Trip.


Teacher Notes

Investigation 7: Wendella’s Journey: Moving at DifferentSpeeds

Key Ideas

→ In a position-time graph, multi-segment graphs can represent characters moving atdifferent speeds.

→ Graphs tell a story. Stories can be represented in the form of graphs. In this activity, pupilswill learn to write stories from graphs and make graphs for stories.

→ “Flat” or horizontal lines represent standing still.



• The mobile phone game is “Wendella’s Journey”, which will be about a little dog travellingin a magic forest, slower or faster depending on where she is.

• Make sure pupils see how the three modes of motion (forwards slowly, forwards faster,stopped) are represented differently on the graph.


• Predict, check, explain requires pupils to write story before running the file.

• The story need not contain any numbers; encourage creativity.

• Ask pupils to share stories and discuss how well each prediction describes the graph beforerunning the simulation.

• Discuss after pupils finish.


• Pupils find speeds for each segment in a trip.

• Watch for pupils to develop the following ideas in order to share them with the whole class:

- For the second segment, the speed is 0 because she journeyed for 2 minutes and did notget anywhere (her position did not change)

- For the third segment, reading the coordinates at the endpoint will not work—you mustfind distance and time by counting boxes or subtracting.




• Ask pupils to predict the motion before running the file.

• Help pupils build connection between “slanting downward” and moving backwards.

• It is OK for pupils to use everyday language (such as slanting, backwards, steeper) in theirstories. This may provide for interesting debates among pupils. Use of everyday languagemay leave stories open to different interpretations. Encourage pupils to be as specific as theycan, but allow them to explore the affordances and limitations of everyday language (asopposed to mathematically precise language).

• Speed is calculated for each direction travelled, but the connection to negative velocity comesin another activity.



• In a position-time graph, multi-segment graphs can represent characters moving at differentspeeds.

• Graphs tell a story. Stories can be represented in the form of graphs. In this activity, pupilswill learn to write stories from graphs and make graphs for stories.

• Lines “slanting downward” represent moving backwards.


• Pupils practise connecting stories and graphs and how moving backwards is represented.


• Emphasise the meaning of flat (horizontal) and “downward slanting” segments.

Pupil Difficulties


• Representing backwards motion; often pupils draw a line segment that is “backwards”along the x-axis.

• Interpreting standing still (or no change in the position value) on the graph.


• Thinking that a flat or horizontal segment on the graph means moving at a constant (non-zero) rate.

• Interpreting the graph as a picture (e.g., a path, an uphill).

• Confusing explanation with description or with an account of procedures.


Investigation 7: Wendella’s Journey: Moving at DifferentSpeeds

In our game, “Lost in the Pines”, Wendella the dogmakes many journeys through the magical Lost PinesWetlands.

On her journeys, Wendella moves

•Forwards slowly when she is in the swamp.

•Forwards faster when she is on the path.

• Wendella stops and barks for help when she is inquicksand.

We need many journeys and stories to use in our game. Help us set up themathematics and tell the story for each journey. Your work is very important.Remember, you are doing the mathematics that will make the game work.

1. Here is the graph for Wendella’s first journey.

0x

1 2 3 4 5 6 7 8 9 10

Time (min)

y

50100150200250300350400450500550600650700750

Posi

tion

(m

)

A. Predict: Using the graph, predict howWendella will move for this journey. Finishthe story below.

Wendella started out quickly on the path.

She was happy to be on her journey.

Then...

Answers will vary. Sequence is path, thenswamp, then quicksand, then path.

B. Check: Open Activity 7.1, run the simulation, and check your prediction.

C. Explain: Explain any differences between your prediction and what youobserved in the simulation.

Answers will vary.

33Investigation 7: Wendella’s Journey: Moving at Different Speeds

2. Here is a different journey that Wendella made.

A. Label the graph segments below to show when Wendella was in the swamp, inthe quicksand, and on the path.

0x

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Time (min)

y

10020030040050060070080090010001100

Posi

tion

(m

)

Path

QuicksandSwamp

Path

Swamp

B. For each line segment in the graph above, find the number of minutes Wendellatravelled, the number of metres she travelled, and her speed. Use the tablebelow to keep the information all organised.

Segment Minutes Metres Speed

1st 1 300 300 m/min

2nd 2 0 0 m/min

3rd 4 100 25 m/min

4th 2 600 300 m/min

5th 4 100 25 m/min

C. Choose one line segment in the graph—not the first one—and explain how youfound Wendella’s speed for that segment.

Answers will vary.

D. Open Activity 7.2, and run the simulation to check your work.

34 Investigation 7: Wendella’s Journey: Moving at Different Speeds

3. Wendella takes another journey. The graph is shown below.

0x

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Time (min)

y

100200300400500600700800

Posi

tion

(m

)

A. Look at the graph. Wendella does something new here! Predict her motion inthe simulation.

This is a prediction, many answers are acceptable. Pay attention to how pupils explain thethird segment.

B. Check: Now open Activity 7.3. Run the simulation. Did Wendella do what youpredicted?

Answers will vary.

C. Why or why not?

Answers will vary. Pupils may not have predicted that the third segment representsbackwards motion.


D. What did Wendella do 6 minutes after starting this journey?

Wendella turns around or Wendella goes back or Wendella goes backwards.

E. How does the graph show this motion?

The line slants downward (negative gradient).

F. What is her speed between 4 and 6 minutes?

300 m/min forwards

G. What is her speed between 6 and 8 minutes?

300 m/min backwards

H. Compare these two parts of Wendella’s journey.

In both segments, Wendella is moving at a rate of 300 m/min. First she is moving forwards,and then she is moving back towards the start.


4. Open Activity 7.4 and do the following:

A. Change the graph so that Wendella goes forwards and backwards at least twicein her journey. Run the simulation to make sure it works.

B. Record your graph on the axes provided.

0x

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Time (min)

y

50

100

150

200

250

300

350

400

450

500

550

600

650

700

750

800

Posi

tion

(m

)

Answers will vary. Graphs should have atleast two “peaks” and two negative gradients.

C. Write a story to go with this journey of Wendella.

Answers will vary. Stories should reflect the Wendella’s changes in direction.

D. One of the programmers doesn’t understand what is happening in the graph.Explain how to use a graph to make Wendella go back towards her startingpoint.

Graphs must have a downward gradient to make Wendella go back towards her startingposition. Graphs with a downward gradient represent a decrease in position (a backwardsmotion) as time goes on.


5. We need more journeys and stories. Open Activity 7.5 and create your own graphshowing Wendella’s next journey.

A. Sketch the graph on the axes:

0x

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Time (min)

y

50

100

150

200

250

300

350

400

450

500

550

600

650

700

750

Posi

tion

(m

)

Answers will vary.

B. Write a story to match your graph:

Answers will vary. Stories and graphs should match. The story should contain all the criticalfeatures of the graph, such as the starting point (y-intercept), numbers of segments andmotions that are appropriately faster and slower and forwards and backwards incorrespondence with graph lines that slant upward or downward.


Teacher Notes

Investigation 8: Money Matters

Key Ideas

→ Multi-segment graphs show varying speeds in motion contexts.

→ Multi-segment graphs can also be used in non-motion contexts to show rate ofaccumulation, for example.

→ Graphs tell a story. Stories can be represented in the form of graphs. In this activity, pupilswill learn to write stories from graphs and make graphs for stories.

→ “Flat” lines represent standing still.

→ Lines “slanting downward” represent moving backwards.

Graph Wendella’s Journey Money Matters

x-axis Time Time

y-axis Position or distance Money accumulated

Steepnessof line

Rate of motion: speed Rate of accumulation: How much money went into(or was taken out of) the account for a particular

period of time.



• The new context here is the bank account balance of SandCircle Mobile Games company.

• We use line graphs to represent the changing balance of an account from the end of onemonth to the end of the next month. The lines connecting the points for the monthly balanceare basically meaningless. But it makes sense to draw the line to show the shape of the graph.This is a common practice among both mathematicians and mathematics users such asaccountants.


• Make sure pupils understand that balance refers to the amount in the bank account.

• Help pupils interpret the graph and table, highlighting the connections between them.

• Running the file will show the balance changing month by month, which can help pupilsmake sense of the graph.

• Remind pupils that rate refers to “pounds per month”.

• Keep asking: What is this part like in Wendella’s Journey?

- Increasing balance is like moving forwards.

- Decreasing balance is like moving backwards.

- Rate of change is like speed.




• Contrast a straight-line graph with the multi-segment graph in Question 1.



• Pupils apply lesson concepts to a new problem.

- Make sure pupils understand that the graphs show how often the two games are played.

- For Question C: Either game can be chosen, but the mathematical argument has to makesense.

Note: There is a little extra time in this lesson that you can use to review.


• Ask pupils to compare the motion (Wendella) and non-motion (Money Matters) meaning ofthe various line segments.

• You can provide them with a table to compare the information on the axes and the meaning ofthe y-intercept and gradient of the lines.

Pupil Difficulties


• Thinking that longer segments (longer time) mean larger increases in the balance.

• Understanding that graphs can be used to predict the future, not simply to describe whathas happened.


• Confusing rate with accumulation.


Investigation 8: Money Matters

We’re glad you helped us with some great designs for our mobile phone games. ButSandCircle Mobile Games has to make money too! Please help us with some businessproblems.

We keep a record of how much money we have in the bank. When we make money,we add it to our bank account. When we spend money, we take it out of our bankaccount.

• The amount of money (in £) that we have in the bank on any day is called our bank

account balance.

• On the last day of each month, we check our bank account balance and we graphit.

Help us understand what happened to our bank account this year.

1. Open activity 8.1, which shows our bank account balance over the past year. Runthe simulation to see how the bank account balance changed from month tomonth. Using the graph, simulation, and table, answer the questions below.

• You can also use this graph to answer the questions below.

0x

1 2 3 4 5 6 7 8 9 10 11 12

Month

y

10

20

30

40

50

60

70

80

90

100

110

Bala

nce

(tho

usan

ds o

f £)


39Investigation 8: Money Matters

A. Fill in the table below and explain how you know. Use evidence from the graph,table, or simulation.

Beginning balance

£0.00

Ending balance

£50,000

Months in which our balanceincreased.

January, March, June, July, October,November, December

Months in which our balancedecreased.

February, August, September

Month(s) when the balance increasedat the fastest rate.

March

Month(s) when the balance decreasedat the fastest rate.

February

Month in which the balance was thehighest.

December

Months during which the balance didnot change at all.

April, May

B. Write a sentence or two that describes what happened to our bank accountbalance over the year. Make up your own explanations for what might havehappened. For example: The balance decreased between the end of Januaryand the end of February because we had to pay for a lot of marketing.

Answers will vary.

Stories should reflect the following changes: Increase – decrease – faster increase – nochange – increase – slight decrease – increase.

Critical features: Rate change and length of time.

C. Our goal was to have £60,000 in our bank account by the end of the year. Byhow much did we miss our goal?

£10,000

40 Investigation 8: Money Matters

2. Our goal for next year is to end the year with £110,000 in our bank account. Wewould like to increase our balance by an equal amount of money each month.Remember we are starting next year with a balance of £50,000. Complete thegraph below and use the graph to answer the questions below.

0x

1 2 3 4 5 6 7 8 9 10 11 12

Month

y

10

20

30

40

50

60

70

80

90

100

110

Bala

nce

(in

thou

sand

s of

£)


A. How much money do we need to put in the bank at the end of each month toreach our goal?

£5,000

B. How can we predict what our balance will be at the end of each month usingequations or tables?

We can use the equation y = 5000x + 50000or build a table showing each month’s balance.

41Investigation 8: Money Matters

3. We have two new games, “Texas Road Rally” and “Robots Gone Wrong”. Nextmonth, we want to start selling just one of them. To help us decide which one tosell, we let children in the Electronic Gaming Afterschool Club use our games for14 days. At the end of each day, we graphed the total number of times each gamehad been played altogether (since Day 0).

0x

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Days

y

10

20

30

40

50

60

70

80

90

Acc

umul

ated

pla

ys

Robots Gone Wrong

Texas Road Rally

A. Fill in the table to show how many times each game had been played altogether(since Day 0).

Texas Road Rally Robots Gone Wrong

1 day 5 10

4 days 20 40

10 days 50 58

14 days 70 70

B. Compare the children’s use of the two games over the 14 days.

Children played “Texas Road Rally” the same number of times each day (5 plays per day).They played “Robots Gone Wrong” a lot more each day for the first 4 days (10 plays perday), but after 4 days they played fewer times per day (slowed down to 3 plays per day).At the end, they had played both games the same number of times total (70 plays).

C. We care about selling games children really like to play. Using the information inthe graph, advise us: Which game should we start selling next month? Make yourown decision, and defend it with your mathematical investigations from A and Babove. (There is no one right answer.)

Answers will vary. Pupils should justify their answers. Possible reasonings include thefollowing:

• SandCircle should choose “Texas Road Rally” because it was being played more at theend (5 times per day); or

• SandCircle should choose “Robots Gone Wrong” to get money fast because it was morepopular initially.

42 Investigation 8: Money Matters

Teacher Notes

Investigation 9: Mathematically Speaking: Graphs toKnow

Key Ideas

→ Graphs of rates of change are used in various contexts.

→ Quick graph sketching helps pupils to see general patterns.

Group | 10 minutesMain Activity

• This activity can be used as homework or an informal assessment/quiz, or in class if there istime.

• Pupils build fluency in graph interpretation skills.

• Help pupils make the connections between speed in motion contexts and rate of accumulationin non-motion contexts.

• Each row should have a similarly shaped graph, but the specific starting point, lengths oftime, and gradients may be different.

Pupil Difficulties


• Tying steepness to speed and rate of increase or decrease.


• Switching axes (placing the variables on the wrong axes).


Investigation 9: Mathematically Speaking: Graphs toKnow

1. In order to understand graphs, it helps to know some “by sight”. Sketch a graph foreach of the following situations. Use what you learnt in the Wendella game and theBank Account activity to think about these more general situations.

A. An object moving forward at aconstant speed, then moving forwardat a greater constant speed.

0x

Time

y

Posi

tion

B. A bank account balance increasing ata constant rate, and then increasingat a higher constant rate.

0x

Time

y

Bala

nce

C. An object moving forward at aconstant speed, then movingbackward at a constant speed.

0x

Time

y

Posi

tion

D. A bank account balance increasing ata constant rate, and then decreasingat a constant rate.

0x

Time

y

Bala

nce

43Investigation 9: Mathematically Speaking: Graphs to Know

E. An object standing still (not moving).

0x

Time

y

Posi

tion

F. A bank account balance that is notchanging.

0x

Time

y

Bala

nce

Answers will vary. Each of the graphs has critical features that pupils’ graphs should reflect. This is agood time to prompt pupils to reflect by asking, “What new mathematical ideas have you learnt aboutin this unit?”

44 Investigation 9: Mathematically Speaking: Graphs to Know

Teacher Notes

Investigation 10: Crab Velocity

Key Ideas

→ In position-time graphs, negative rates indicate backwards motion.

→ Position can also be negative, with 0 indicating some defined point such as a start line orwater level.



• The crab mobile phone game will have crabs moving up and down out of the water whileplayers try to catch them. We are creating the graphs and equations to control the motion ofthe crabs.

• Velocity is described in the box. The idea will be developed in the activity.

• Revisit equations from “Controlling Characters with Equations” and “Better Games”, ifneeded.


• Comparing two crabs: Highlight that the speeds are the same, but the velocities are differentbecause of sign/direction.



• Activity 10.2 shows five crabs: Some are going up, some are going down, some have negativestart positions.

• Guide pupils in placing negative start positions in equations. Ask pupils to count squaresacross the 0 line to get the rate of distance per unit of time—velocity—or m in y = mx + c.



• Pupils practise in class or homework. If used for homework, pupils need to work withoutsoftware.



• Discuss how speed and velocity are related.

• Review all the new ideas presented in this activity:

- Velocity is speed with direction.

- Graphs that have negative starting positions can still have positive velocities (and vice-versa).

Pupil Difficulties


• Understanding the difference between velocity and speed.

• Making the conceptual switch; the “world” is now vertical.

• Understanding the importance of signs of rate (slope).

• Keeping track of the number of crabs and their graphs.


Investigation 10: Crab Velocity

In our other games, characters can moveforwards and backwards, or up and down.We already know how to calculate theirspeed. But there’s a bit more to it thanthat. Help us learn how to use velocity inour games.

Velocity?

It’s just speed—but with direction!+ (positive) for up or forwards− (negative) for down or backwards

Our crab characters, Bettina (blue) and Rory (red), move up and down—above thewater (on the rocks) and below it. The water level is at 0 metres.

• Positive positions are above the water line.

• Negative positions are below the water line.

1. Open activity 10.1, and run the simulation.

A. Find

Bettina’s speed: 12 metres in 12 seconds, or 1 m/s

Rory’s speed: 12 metres in 12 seconds, or 1 m/s

Bettina’s velocity: −12 metres in 12 seconds, or −1 m/s

Rory’s velocity: 12 metres in 12 seconds, or 1 m/s

B. Explain how Bettina and Rory have the same speed but have different velocities.

During their motions, they covered the same distance in the same amount of time but movedin opposite directions.

C. Explain what it means for Bettina’s and Rory’s velocity to have different signs(negative and positive).

The different signs just tell you in which direction the crab is going. Here, negative velocitymeans the crab is going down. Positive velocity means the crab is going up.

45Investigation 10: Crab Velocity

2. Open Activity 10.2. The graph shows the motion of Rory (red), Bettina (blue), andtheir friends, Galileo (green), Penelope (purple), and Orlando (orange).

A. Find the velocity of each crab in the group.

Bettina: −12 metres in 12 seconds, or −1 m/s

Rory: 12 metres in 12 seconds, or 1 m/s

Galileo: −4 metres in 10 seconds, or −2/5 m/s or −0.4 m/s

Penelope: 4 metres in 10 seconds, or 2/5 m/s or 0.4 m/s

Orlando: 4 metres in 10 seconds, or 2/5 m/s or 0.4 m/s

B. Using what you already learnt, complete the table below for motions ofPenelope’s, Galileo’s, and Orlando’s motions.

Penelope Galileo Orlando

Speed 0.40 or 2/5 m/s 0.40 or 2/5 m/s 0.40 or 2/5 m/s

Velocity +0.40 or 2/5 m/s −0.40 or −2/5 m/s +0.40 or 2/5 m/s

Equation(giving position in

term of time)

y = 0.40x y = −0.40x − 5 y = 0.40x + 2

C. Open the function window to see if your equations are correct.

D. Compare the equations for Penelope’s, Galileo’s, and Orlando’s motions.

Penelope, Galileo, and Orlando all move at the same speed (0.4 m/s). Penelope andOrlando have a positive velocity, while Galileo has a negative velocity.

Their starting positions are all different. Galileo starts at (0, −5) or 5 metres below waterlevel, Penelope starts at water level (0, 0), and Orlando starts at (0, 2) or 2 metres abovewater level.

46 Investigation 10: Crab Velocity

3. Open Activity 10.3 and use it to create a graph for each of the situations below.Then sketch the graph and write an equation for your graph. Remember, the waterlevel is at position 0.

A. A crab starts at the water level andgoes up.

0x

Time

y

Posi

tion

Equation:

Gradient should be positive,y-intercept should be zero.

B. A crab starts at the water level andgoes down.

0x

Time

y

Posi

tion

Equation:

Gradient should be negative,y-intercept should be zero.

C. A crab starts from above water leveland goes up.

0x

Time

y

Posi

tion

Equation:

Gradient should be positive,y-intercept should be positive.

D. A crab starts from below water leveland goes down.

0x

Time

y

Posi

tion

Equation:

Gradient should be negative,y-intercept should be negative.

47Investigation 10: Crab Velocity

E. A crab starts from above waterlevel, goes down, and ends belowwater level.

Graph:

0x

Time

y

Posi

tion

Equation:

Gradient should be negative,y-intercept should be positive,graph should cross the x-axis.

F. A crab starts from below waterlevel, goes up, and ends above waterlevel.

Graph:

0x

Time

y

Posi

tion

Equation:

Gradient should be positive,y-intercept should be negative,graph should cross the x-axis.

Note: Answers shown are example graphs; actual answers may vary in gradient and length of time.

48 Investigation 10: Crab Velocity

Teacher Notes

Investigation 11: Wolf and Red Riding Hood

Key Ideas

→ Finding the velocity of a character given some conditions.

→ No matter how the characters move, if their motion graphs have the same endpoint, theymeet at the same place at the same time.

→ The average rate (speed or velocity) of a character travelling at different rates for differenttimes is the single speed at which she can make the same trip in the same amount of time.

→ A graph can be used to find the average rate (velocity) of a character moving at differentrates (velocities): Draw a line from beginning point to ending point of the character’sgraph, and then determine its rate.



• Explain that in this activity, pupils will assume the role of the game tester.

- We are figuring out in advance how to know what the right answer is.

- Players will not have access to our mathematical representations, but they will make iteasy for us to do our work.

- Doing mathematics enables players to make strategic decisions that will help them win.

• Read the introductory text that explains the rules of the game.

- Red Riding Hood (RRH) and Wolf move across the screen to Grandma’s House.

- RRH’s speed changes.

- To win, players have to pick a speed for Wolf so that he arrives at Grandma’s House atexactly the same time as RRH.


• Collect different predictions (and explanations) for the Wolf’s speed. Do not identify the rightanswer yet.

• Try different predictions out by setting the speed on the graph and verifying this with thesimulation.

• Pupils may notice features of the graph that help: The two lines must intersect at the positionof Grandma’s house order to make a winning game.

• In this case, the Wolf’s velocity is the mean of RRH’s two velocities, but this is because shetravelled an equal time for each segment.


• If you think pupils would need additional scaffolding, you can try this sequence of questions(these would replace B in the pupil workbook):

- Bi. If the Wolf continues along the same path and at the constant velocity as shown inActivity 11.1, do you predict that the Wolf and RRH will arrive at Grandma’s house at thesame time?

- Bii. Check your prediction by extending the Wolf’s graph to so that his trip ends at thesame time as RRH’s.

- Biii. If necessary, predict at what velocity the Wolf must travel so as to arrive atGrandma’s house at the same time as RRH.




• Try different predictions out by setting the speed on the graph and verifying it with thesimulation.

• In this case, the Wolf’s velocity is the mean of RRH’s two velocities, but this is because shetravelled an equal time for each segment.



• Try different predictions out by setting the speed on the graph and verifying it with thesimulation.

• In this case, the Wolf’s speed is NOT the mean of the two velocities because each velocitycontributes to the average rate differently—different amounts of time. No need to quantifythis, but make sure pupils understand it.

• Discuss after pupils finish and ensure pupils understand what is in the “About AverageVelocity” information box.


• The answer is more straightforward: The graphs must begin and end at the same point.


• There are many, many—an infinite number—of ways to make multi-segment trips for anyone straight-line graph.


• Here pupils have to “aim” the graph at the end point using an equation.


• Ask pupils to explain in their own words what average velocity means and how theydetermined average velocities during this activity (by using the graph to align the start and


ending positions).

• If necessary, restate the main idea: The average rate (speed or velocity) of a charactertravelling at different rates for different times is the single speed at which she can make thesame trip in the same amount of time.

Pupil Difficulties


• Understanding why the average rate for a motion with several segments of possiblydifferent constant rates (corresponding to a piecewise graph) is not necessarily the averageof the rates of the segments.

TN-49cTeacher Notes for Investigation 11

TN-49d Teacher Notes for Investigation 11

Investigation 11: Wolf and Red Riding Hood

We need you to test a new game based on the fairy taleRed Riding Hood.

In this game, Red Riding Hood (RRH) and the Wolf startfrom Home and move toward Grandma’s House. RRHmoves at two velocities during the journey. The Wolfmoves at a constant velocity the entire time.

The Wolf must arrive at Grandma’s House at exactly thesame time as RRH. Only then can he devour her.

Help us find the “trick” so that the Wolf always arrivesat exactly the same time as RRH.

1. Open Activity 11.1.

A. The graph shows RRH’s complete trip, which ends at Grandma’s house (600m).Use the graph to describe RRH’s trip. Include how long it took her to get thereand estimate her two velocities.

RRH travelled 600m in 10 minutes.Her first segment was 450m in 5 min, so her velocity was +90 m/min.Her second segment was 150m in 5 min, so her velocity was +30 m/min.

B. Predict the Wolf’s velocity. He must arrive at Grandma’s at exactly the sametime as RRH. Remember, the Wolf moves at a constant velocity the entire time.

This is a prediction, so many answers are acceptable. Pupils should notice the start andending points of RRH’s graph to calculate Wolf’s velocity.

C. Check your prediction. Edit the Wolf’s graph to match your prediction and runthe simulation. Was your prediction right? Explain why or why not. Revise untilyou get it right.

Pupils should extend the Wolf graph to end at 600 m in 10 min, the same place that RRHends. Wolf’s velocity is 60 m/min.

D. How is the Wolf’s velocity related to the two velocities that RRH travelled?

Wolf’s velocity is the same as RRH’s average velocity.Or Wolf’s velocity is the same as if RRH travelled at a constant velocity instead of changingrates in the middle.

49Investigation 11: Wolf and Red Riding Hood

2. Open Activity 11.2.

A. The graph shows RRH’s complete trip, which ends at Grandma’s house. Use thegraph to describe RRH’s trip. Include how long it took her to get there andcalculate her two velocities.

RRH travelled 600 metres in 10 minutes.Her first segment was 50m in 5 minutes, so her velocity was +10 m/min.Her second segment was 550m in 5 minutes, so her velocity was +110 m/min.

B. Predict the Wolf’s velocity. He must arrive at Grandma’s at exactly the sametime as RRH. Remember, the Wolf moves at a constant velocity the entire time.

This is a prediction, so many answers are acceptable. Pupils should notice the start andending points of RRH’s graph to calculate Wolf’s velocity.

C. Check your prediction. Edit the Wolf’s graph and run the simulation. Were youright? Explain why or why not. Revise until you get it right.

Pupils should extend the Wolf graph to end at 600m in 10 min, the same place that RRHends. Wolf’s velocity is 60 m/min.

D. How is the Wolf’s velocity related to the two velocities that RRH travelled?

Wolf’s velocity is the same as RRH’s average velocity.

50 Investigation 11: Wolf and Red Riding Hood

3. In the next round of the game, RRH still travels at two different velocities. Shetravels 100 metres per minute for 5 minutes and then 40 metres per minute for 10minutes.

A. Predict the Wolf’s velocity. The Wolf must arrive at Grandma’s at exactly thesame time as RRH. Remember, the Wolf moves at a constant velocity the entiretime.

This is a prediction, so many answers are acceptable.

B. Explain how you predicted the Wolf’s velocity.

Answers will vary. Pupils can sketch the graph to see the ending point or they can use thegiven rates to calculate the end point of RRH’s journey to determine Wolf’s velocity.

C. Open Activity 11.3. Edit the Wolf’s graph so that he arrives at Grandma’s at thesame exact time as RRH. Run the simulation and check your prediction. Wereyou right? Explain why or why not. Revise until you get it right.

Answers will vary.


About average velocity

For the Wolf to arrive at Grandma’s house at exactly the same time as RRH, hisvelocity must match RRH’s average velocity. RRH’s average rate is the single rateat which she could complete the same journey in the same amount of time. Butwe must pay close attention because rates are not like simple numbers—theyrelate two numbers to each other. Velocity is a rate that relates time andposition. Just adding the numbers and dividing by the number of items will leadyou astray.

4. In future mathematics, you will learn how to calculate average rates. For now, youhave seen how powerful a graph is. You can use a graph to find an average rate, asyou did when you figured out the Wolf’s velocity.

A. Explain how to use a graph to find an average rate when given a trip with two ormore different rates.

The start point and the endpoint of a line representing an average rate (like the Wolf’s)should match the start point of the first segment and endpoint of the last segmentrepresenting a trip at two or more different rates (like RRH’s). A line representing an averagerate will be a single line segment connecting the two points.

B. In these three trips, RRH’s velocities change a great deal. How is it that theWolf’s velocity can be the same each trip?

RRH can go as fast and as slowly as she wants, but her average velocity can be the same,because we use the total distance divided by the total time to calculate her average velocity.In the first two trips, she travels the same distance in the same amount of time. In the thirdtrip she travels 1.5 times as far but takes 1.5 times the amount of time. So, her averagevelocity is the same as in the first two trips.


5. Open Activity 11.4. Create a trip where RRH travels at three different velocities.

Graphs will vary. Pupils can include negative velocities as well.

A. Using the Function window, set the Wolf’s velocity so that he arrives atGrandma’s house at the same exact time as RRH. Check and revise until you getit right.What is the Wolf’s velocity?

Answers will vary.

B. Write a story describing RRH’s trip.

Answers will vary.

C. Explain how you used the equation to set the Wolf’s velocity so that he arrivedat Grandma’s house at the same exact time as RRH, even though you had onlythe graph and the animation of RRH’s journey.

Answers will vary. Look at the total distance RRH travelled and divide that by the total timeshe travelled. Use this number (gradient) to set the coefficient of x in the Function window.


6. RRH and the Wolf can help us think about average rate, using graphs. RRH’saverage rate is the single rate at which she could make the same journey in thesame amount of time.

A. You’ve seen graphs like these before. Each graph shows RRH moving at twodifferent rates. Use each graph to fill in the blanks below it.

0x

1 2 3 4 5 6 7 8 9 10

Time (min)

y

50

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Posi

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(m

)

0x

1 2 3 4 5 6 7 8 9 10

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y

50

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(m

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First, RRH moves at a rate of 75 m/min for 4 minutes.

Then, she moves at a rate of 50 m/min for 6 minutes.

So, she travelled 600 metres in 10 minutes.

What is her average rate, in metresper minute (m/min)?

60 m/min

(Hint: What was the Wolf’s rate, inm/min?)

First, RRH moves at a rate of 10 m/min for 5 minutes.

Then, she moves at a rate of 100 m/min for 4 minutes.

So, she travelled 450 metres in 9 minutes.

What is her average rate, in metresper minute (m/min)?

50 m/min

(Hint: What was the Wolf’s rate, inm/min?)


Teacher Notes

Investigation 12: Problem Solving

Key Ideas

→ Apply ideas learned in the unit in similar and different settings.


Whole Class | 5 minutesDiscussionThis activity can be used as homework or an informal assessment/quiz, or in class if there istime.


• Pupils practise finding average rates from graphs.


• Pupils use average rate in a problem solving/modelling setting.

Whole Class | 5 minutesPlenaryThis investigation is a review. Discuss any pupil difficulties.


Investigation 12: Problem Solving

We still have other problems to solve at SandCircle Mobile Games! Use all you havelearnt to help us with the following problems.

1. Think back to Wendella. Help us to analyse Wendella’s journeys and average rates.

A. What is Wendella’saverage rate of motion(her average speed) onthis journey?

800/12 m/min = 66.66…m/min ≈ 66.7 m/min orapproximately 67 m/min.

0x

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Time (min)

y

100

200

300

400

500

600

700

800

Posi

tion

(m

)

B. What is Wendella’saverage rate of motion(her average speed) onthis journey?

500/9 m/min = 55.55…m/min ≈ 55.6 m/min orapproximately 56 m/min.Answer will depend onrounding.

0x

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Time (min)

y

100

200

300

400

500

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Posi

tion

(m

)

C. Wendella went 25 metresper minute through theswamp for 4 minutes, then100 metres per minute onthe path for 2 minutes.Using the graph to helpyou, what was her averagerate, in metres perminute?

50 m/min

0x

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Time (min)

y

100

200

300

400

500

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700

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(m

)

55Investigation 12: Problem Solving

2. Marissa has worked for SandCircle Mobile Games for 5 years. For the first 2 years,her salary was £50,000 per year. Then she got a big raise. For the last 3 years, hersalary has been £70,000 per year.

• Marissa says, “So my average yearly salary for these 5 years has been £60,000.”

• Her supervisor, Tanya, disagrees, “Actually, Marissa’s average salary has been£62,000.”

The £2,000 difference matters to them! Explain who is right and why (use thegraph if it will help you).

0x

y

Key features of the graph:• Pupils label axes.• There are two rates represented.• Important coordinates (2, 100000), endpoint of first rate; and (5, 310000), endpoint of second

rate.

The supervisor is correct.£310,000/5 years = £62,000/year

56 Investigation 12: Problem Solving

Teacher Notes

Investigation 13: Problems from the SandCircleLunchroom

Key Ideas

→ Apply ideas learned in the unit in similar and different settings.


Whole Class | 5 minutesDiscussionThis activity can be used as homework or an informal assessment/quiz, or in class if there istime.


• Pupils practise filling in values in a table representing a proportional relationship. This workcan be surprisingly difficult for some pupils.


• Pupils use a unit rate to solve problems for particular values and then create a graph thatgeneralises.

• Help pupils see that the graph provides answers for any amount of coffee or cups.


• Pupils practise translating from algebraic symbols to words.


• Pupils fill in values for a proportional relationship, which is given in a somewhat unusualway, and pupils must evaluate each time value to solve the problem.

Whole Class | 5 minutesPlenaryThis investigation is a review. Discuss any pupil difficulties.


Investigation 13: Problems from the SandCircleLunchroom

We still have other problems to solve at SandCircle Mobile Games! Use all you havelearnt to help us with the following problems.

1. Darrell, from the business office, is helping his son sell raffle tickets to raisemoney for his school. He’s making a chart so that his son can quickly check that hehas the right amount of money for the number of tickets he has sold. Fill in themissing numbers for him. There is no discount for buying a lot of tickets.

Tickets sold Money Collected (£)

5 12.50

10 25.00

25 62.50

40 100.00

2. Our SandCircle Mobile employees drink a lot of strong coffee while creating ourgreat mobile phone games. In our office, with each 1.5 pounds of coffee, we make36 cups.

A. How many cups can we make with one pound?

24 cups of coffee36 cups/1.5 pounds = 24 cups per pound

B. How many pounds do we need to make 72 cups? How about 120 cups?

For 72 cups, we need 3 pounds.For 120 cups, we need 5 pounds.

C. If we have 3.5 pounds of coffee left, how many cups can we make with it?

84 cups of coffee84 = 24 * 3.5

57Investigation 13: Problems from the SandCircle Lunchroom

D. Draw a graph that allows us todetermine how many pounds ofcoffee we need no matter how manycups we plan to make that day.

0p

c Graphs will varydepending on the scalepupils use.

Key points:• graph starts at origin• pounds of coffee are

represented on x-axis• number of cups is

graphed ony-axis

E. Write the equation that shows thelink between the number of poundsof coffee (p) and the number of cupsof coffee (c).

c = 24pwhere c is the number of cups of coffeemade, and p is the pounds of coffeeused.

3. Shanae, a graphic artist, often writes in mathematical symbols instead of words.She put up this advert on the company bulletin board:

• Babysitter wanted! Will pay y pounds, where y = 6x + 10, and x is the number ofhours you work.

Explain what Shanae means in words:

The babysitter receives £6.00/hr plus a £10.00 bonus (perhaps for cab fare).

58 Investigation 13: Problems from the SandCircle Lunchroom

4. SandCircle Mobile’s company plane, the SuperNova, can fly at an average speed of225 mph. The SuperNova can hold enough fuel for only 3½ hours of flight.

City Distance fromHeathrow

Travel Time Non-stop Flight?(yes or no)

Dublin, Ireland 280 1.24 yes

Berlin, Germany 600 2.67 yes

Gothenburg, Sweden 675 3.00 yes

Vienna, Austria 780 3.47 yes

Krakow, Poland 900 4.00 no

To which cities can the SuperNova fly non-stop (with only a single tank of fuel) if itstarts at Heathrow?

Dublin, Berlin, Gothenburg, and Vienna. Or any other city within 787.5 (or so) miles of Heathrow,not accounting for headwinds and tailwinds or other factors that could affect how far the planemight fly on a single tank of fuel.

59Investigation 13: Problems from the SandCircle Lunchroom

60 Investigation 13: Problems from the SandCircle Lunchroom

Teacher Notes

Investigation 14: SandCircle Mobile Games: Going FullTime

Key Ideas

→ Pupils reflect on the unit as a whole, reflect on the mathematics, and note what theylearned.

→ You may want to give feedback to pupils in the form of a letter from the potentialemployer, stating: “You have been/not been successful in this application because...”

Individual | 15 minutesMain ActivityPupils’ answers will vary. Encourage pupils to provide specific examples in their responses.


Investigation 14: SandCircle Mobile Games: Going FullTime

1. Your co-workers at SandCircle Mobile Games are pleased with what you have done.You’ve helped them improve their work, you are such a hard worker, and you havealso learnt to apply the mathematics that is important to future success.

There are several openings at SandCircle Mobile Games, and many people thinkyou would be an asset to the company. Fill in the letter of application below sothat Ms. Bahey, our Hiring Manager, will know your strengths and interests.

61Investigation 14: SandCircle Mobile Games: Going Full Time

62 Investigation 14: SandCircle Mobile Games: Going Full Time

Designing Mobile Games

Cornerstone Maths is a collaboration among:

designing mobile games · in an old office building in the silicon roundabout area of london, they...

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