Teacher Sourcebook - ORIGO Education

Teacher Sourcebook

MAL SHIELD

AuthorMal Shield, Ph.D.

ConsultantKathy Blum, Ph.D.

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Queensland GO Maths Teacher SourcebookLevel 5/6 (Year 9)

Copyright 2008 ORIGO Education

Author: Mal Shield

Consultant: Kathy Blum

Shield, M. J. (Malcolm John).

Go maths : level 5/6. Teacher sourcebook.

For secondary school students.

ISBN 9 78192102 3651 (pbk.).

1. Mathematics - Study and teaching (Secondary).

I. Title.

510.712

For more information, email [email protected]

or visit www.origo.com.au for other contact details.

All rights reserved. Unless specifically stated, no part of this

publication may be reproduced, copied into or stored in a

retrieval system, or transmitted in any form or by any means,

electronic, mechanical, photocopying, recording or otherwise,

without the prior permission of ORIGO Education.

ISBN: 978 1 921023 65 1

10 9 8 7 6 5 4 3 2 1

The author and project team would like to acknowledge the

Queensland Studies Authority, Brisbane, for its permission to

reproduce extracts from the Mathematics: Years 1 to 10 Syllabus.

Graphics calculator instructions in the Student Textbook are

based on the models CASIO CFX-9850GC PLUS and TI-83 Plus,

and may therefore vary for different models. When completing

the exercises and explorations, students may also need to refer

to the instruction manual for the particular graphics calculator

they are using.

Every effort was made to correctly publish any information

regarding computer software, calculators and websites. Any

instructions for computer software and calculators are based

on the specific version or model that was used to compile the

instructions, and may differ for other versions and models. Any

instructions for websites are based on the websites at the time

of publication. ORIGO Education does not endorse any brand of

computer software or calculator.


Contents

Unit Number Unit Topic Outcomes

1 Working with Data CD 5.2/CD 6.2 PA 5.1time trends, cross-sectional data, census and sample, location, spread, tables, graphs

2 Numbers and Operations N 5.1/N 5.2/N 5.3/N 6.1/N 6.2/N 6.3operations with whole numbers, decimal fractions, common fractions and integers, including mental computation, scientific notation

3 Modelling with Functions PA 5.1/PA 5.2N 5.3non-linear from data, linear functions, gradient, equation solving

4 Proportions N 5.1/N 5.3/N 6.3 PA 5.1/PA 5.2solving problems in percentage, ratio and rate, inverse proportion

5 Working with 2D and 3D Shapes S 5.1/S 5.2/S 6.2transformations, scale, plans of 3D objects, position and distance on a map, 3D on a map

6 Perimeter, Area, Volume and Capacity M 5.1/M 6.1areas of standard shapes, circumference and area of a circle, sector of a circle, volume and capacity of prisms and cylinders, surface area

7 Time and Global Position M 5.2/M 6.2/S 6.2N 6.3durations of events, time management decisions, synchronisation, position on

the earth, international time zones and longitude

8 Working with Chance CD 5.1/CD 6.1addition rule, compound events, two-way tables and tree diagrams, conditional probability

9 Living with Money N 6.1earning money, budgets, bank accounts, cards, rates and fees, financial planning

10 Lines, Angles and Shapes S 5.1/S 6.1naming lines and angles, angles in a triangle — exterior angles, parallel lines cut by a transversal, deductive reasoning

11 Working with Maps S 5.2/S 6.2N 6.3maps, scale, co-ordinates, distance, bearings, drawing maps with survey lines

12 Chance, Data and Relationships CD 6.1/CD 6.2PA 6.1addition and multiplication rules, tree diagrams, data and chance, time plots,

scatterplots, two-way tables

13 Working with Triangles S 5.1/S 6.1/M 6.1similarity, congruence, Pythagoras’ theorem, tangent ratio

14 Using and Interpreting Mathematical Models PA 6.1/PA 6.2CD 6.2modelling with linear functions, lines of best fit, trends, quadratic models

15 Surface Area, Volume and Capacity M 6.1/S 6.1cylinders, cones, pyramids, surface area, volume and capacity

16 Working With Algebraic Symbols PA 6.1/PA 6.2distributive property, linear functions, proportion equations, quadratic functions


141414

UNIT14

Mathematical BackgroundIn this unit, students build on their earlier learning about linear

functions. They continue to develop linear models for real-world

situations involving constant rates of change. Students find the

equation of a linear function from the gradient and y-intercept

and, having drawn linear lines of best fit for two-variable data

in Unit 12, they now find the equations of linear lines of best fit.

Students work with points and lines, finding the gradient of a

straight line through two points, and finding co-ordinates of

points by equation solving. They work with linear functions in

the complete Cartesian plane (four quadrants). Students are

introduced to inequalities. They are also introduced to quadratic

functions through graphing. In this unit, students are encouraged

to use graphics calculators as one method for drawing the graphs

of functions, including for finding a linear line of best fit.

Unit OverviewPreparation Linear Equations

14.1 Modelling with Linear Functions

14.2 Working with the Equations of Lines

14.3 Lines of Best Fit

14.4 Points and Gradients

14.5 Developing Quadratic Functions

14.6 Working with Inequalities

Extension More Functions

Major Learning Outcomes

OUTCOME

PA 6.1

Students create mathematical models of realistic situations and use interpretations of the models to draw conclusions or make decisions.

OUTCOME

PA 6.2

Students interpret and solve mathematical models of realistic situations by using algebraic, graphical and electronic methods.

Minor Learning Outcome

OUTCOME

CD 6.2

Students use and interpret cross-sectional data and data collected over time to identify the nature of variations and relationships.

Using and Interpreting Mathematical Models

Level 5/6

PA 6.1 14 16

PA 6.2 14 16

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14.114.1

14.214.2

14.314.3 Lines of Best FitLines of Best Fit

14.414.4 Points and GradientsPoints and Gradients

Developing Quadratic FunctionsDeveloping Quadratic Functions

Working with InequalitiesWorking with Inequalities

Linear EquationsLinear Equations

Modelling with Linear FunctionsModelling with Linear Functions

Working with the Equations of LinesWorking with the Equations of Lines

Lines of Best FitLines of Best Fit

Points and GradientsPoints and Gradients

points by equation solving. They work with linear functions in points by equation solving. They work with linear functions in

the complete Cartesian plane (four quadrants). Students are the complete Cartesian plane (four quadrants). Students are

introduced to inequalities. They are also introduced to quadratic introduced to inequalities. They are also introduced to quadratic

functions through graphing. In this unit, students are encouraged functions through graphing. In this unit, students are encouraged

to use graphics calculators as one method for drawing the graphs to use graphics calculators as one method for drawing the graphs

of functions, including for finding a linear line of best fit.of functions, including for finding a linear line of best fit.

Students work with points and lines, finding the gradient of a Students work with points and lines, finding the gradient of a

straight line through two points, and finding co-ordinates of straight line through two points, and finding co-ordinates of

points by equation solving. They work with linear functions in points by equation solving. They work with linear functions in OUTCOME

CD 6.2

PA 6.2PA 6.2

Minor Learning Outcome

Students use and interpret cross-sectional data and data collected over

Students create mathematical models of realistic situations and use interpretations of the models to draw conclusions or make decisions.

Students interpret and solve mathematical models of realistic situations by using algebraic, graphical and electronic methods.

Formative Assessment1414CriteriaOn completion of this unit, the students should be able to

A model a real-world situation with a linear function

B find the equation of a linear function from a given gradient and y-intercept

C find the equation of a linear line of best fit for two-variable data

D draw the graph of a quadratic function by finding the co-ordinates of points

E use a graphics calculator to draw graphs and find gradients and equations of lines

Ffind one co-ordinate of a point on a straight line when given the other co-ordinate and the equation of the line (equation solving by balance)

G find the gradient of a straight line through two given points

H find values of a variable that satisfy an inequality

Techniques1. Topic Check

The Topic Check on pages 102–103 of the Student Portfolio assesses prerequisites for Unit 14 (GO Maths Level 5/6, Unit 3). Depending on students’ success you may wish to complete the Unit 14 Preparation in the Student Textbook.

2. Unit Check A B C D F G H Questions in the Unit Check on pages 104–109 of the Student Portfolio relate to the following criteria.

Question 1 2 3 4 5 6 7

Criteria A B F C G D H

3. Final Investigation D F The Final Investigation on pages 110–111 of the Student Portfolio focuses on using a quadratic function to graph and describe a situation.

4. Observation E

Observe students using graphics calculators in Exploration 3.

RecordingA reproducible student Progress Record is provided in the introduction to the Teacher Sourcebook. Record each student’s achievement of the criteria by shading the boxes for Unit 14 in PA 6.1 and PA 6.2.

OUTCOME

PA 6.2

OUTCOME

PA 6.1

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Questions in the Unit Check on pages 104–109 of the Student Portfolio relate to the following criteria

Question

Criteria A B

3. Final Investigation D

The Final Investigation on pages 110–111 of the Student Portfolio focuses on using a quadratic function to graph and describe a situation.

Level 5/6, Unit 3). Depending on students’ success you may wish to complete the Unit 14 Preparationin the Student Textbook.

C D F G H


3 4 5

F

find values of a variable that satisfy an inequalityfind values of a variable that satisfy an inequality

The Topic Check on pages 102–103 of the Student Portfolio assesses prerequisites for Unit 14 (Level 5/6, Unit 3). Depending on students’ success you may wish to complete the Unit 14 Preparation


find one co-ordinate of a point on a straight line when given the other co-ordinate find one co-ordinate of a point on a straight line when given the other co-ordinate and the equation of the line (equation solving by balance)and the equation of the line (equation solving by balance)

find the gradient of a straight line through two given pointsfind the gradient of a straight line through two given points

find values of a variable that satisfy an inequalityfind values of a variable that satisfy an inequality

draw the graph of a quadratic function by finding the co-ordinates of pointsdraw the graph of a quadratic function by finding the co-ordinates of points

use a graphics calculator to draw graphs and find gradients and equations of linesuse a graphics calculator to draw graphs and find gradients and equations of lines

find one co-ordinate of a point on a straight line when given the other co-ordinate find one co-ordinate of a point on a straight line when given the other co-ordinate

-intercept-intercept

draw the graph of a quadratic function by finding the co-ordinates of pointsdraw the graph of a quadratic function by finding the co-ordinates of points

use a graphics calculator to draw graphs and find gradients and equations of linesuse a graphics calculator to draw graphs and find gradients and equations of lines

find one co-ordinate of a point on a straight line when given the other co-ordinate find one co-ordinate of a point on a straight line when given the other co-ordinate

Preparation

PA 6.1 PA 6.2 Level 5/6, Unit 14, Using and Interpreting Mathematical Models

Linear Equations

Materials• GO Maths student

textbook, Unit 14 Preparation, pages 266–268

• 1 sheet of grid paper for each student

Teaching NoteHighlight that rise and run are in proportion for a linear function (e.g. if the run is doubled, the rise is doubled).

Key Ideas• Using the gradient and y-intercept to draw a graph of a linear function

• Solving linear equations

Number Sense DiscussionWrite on the board: 5 × 4 – 7 6 × 0.5 + 8 25 – 3 × 6

Ask the students to calculate the answers mentally. Question them on the order of operations. Repeat for 24 ÷ 6 + 7 and 15 + 5 × 3.

Activity 1. Introduce Example 1. Remind the students that this is a graph of a linear function.

Ask: What is the meaning of gradient? What does it tell us? (The gradient expresses how quickly the dependent variable — vertical axis — changes as the independent variable — horizontal axis — changes.) Work through the example with the students to revise finding a gradient from the graph of a linear function. While the two most convenient points have been used, show the students that other points could be used to achieve the same result.

2. Guide the students through Example 2 to revise drawing the graph of a given linear function. Explain that a run greater than 1 can be used for greater accuracy.

3. Have the students complete Exercise 1. In this exercise, they work in the first quadrant only of the Cartesian plane.

4. Work through Example 3 with the students to revise using the balance method. Although some students may be more comfortable with the backtracking method, encourage them to try solving some equations by balance.

5. The students can then complete Exercise 2 to practise solving linear equations.

ReflectionWrite the equation y = 2x + 3 on the board. Ask: What will the graph of this function look like? What is the gradient? What is the y-intercept? Have the students find the value of y when x = 2, and the value of x when y = 9. Say: Equation solving is equivalent to finding an unknown co-ordinate of a point when one of the co-ordinates is known.

Formative AssessmentGO Maths student portfolio, Topic Check, pages 102–103

The unit preparation covers key concepts that students will build on in this unit. If students successfully complete the Topic Check for this unit they may not need to complete the unit preparation.

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Have the students complete Exercise 1. In this exercise, they work in the first quadrant only of the Cartesian plane.

Work through Example 3 with the students to revise using the balance method. Although some students may be more comfortable with the backtracking method, encourage them to try solving some equations by balance.

The students can then complete Exercise 2 to practise solving linear equations.

Reflection

students that other points could be used to achieve the same result.

Guide the students through Example 2 to revise drawing the graph of a given linear function. Explain that a run greater than 1 can be used for greater accuracy.

Have the students complete Exercise 1. In this exercise, they work in the first quadrant only of the Cartesian plane.

Work through Example 3 with the students to revise using the balance method. Although some students may be more comfortable with the backtracking method, encourage them to try solving some equations by balance.

The students can then complete Exercise 2 to practise solving linear equations.

Introduce Example 1. Remind the students that this is a graph of a linear function. What is the meaning of gradient? What does it tell us?What is the meaning of gradient? What does it tell us?

expresses how quickly the dependent variable — vertical axis — changes as the independent variable — horizontal axis — changes.) Work through the example with the students to revise finding a gradient from the graph of a linear function. While the two most convenient points have been used, show the students that other points could be used to achieve the same result.

Guide the students through Example 2 to revise drawing the graph of a given linear function. Explain that a run greater than 1 can be used for greater accuracy.

Have the students complete Exercise 1. In this exercise, they work in the first

Ask the students to calculate the answers mentally. Question them on the order Ask the students to calculate the answers mentally. Question them on the order

Introduce Example 1. Remind the students that this is a graph of a linear function. (The gradient

expresses how quickly the dependent variable — vertical axis — changes as the independent variable — horizontal axis — changes.) Work through the

doubled).doubled).Ask the students to calculate the answers mentally. Question them on the order Ask the students to calculate the answers mentally. Question them on the order

sheetsheet ofofforfor eacheach

Teaching NoteTeaching NoteHighlightHighlight thatthat andand

areare inin proportionproportion forforfunctionfunction (e.g.(e.g. ifif

doubled,doubled,doubled).doubled).

14.114.1


Modelling with Linear Functions

Key Ideas• Modelling real-world situations with linear functions

• Representing linear models with graphs and equations

Number Sense DiscussionWrite on the board: 3 × 8 + 4 20 ÷ 4 – 2 30 – 6 × 4

Ask the students to calculate the answers mentally. Question them on the order of operations. Repeat for 25 – 18 ÷ 3 and 3 × 4 + 2 × 5.

Activity1. Use an example, such as walking at a constant speed of 4 km/h, to discuss

a function being used to model a situation. In this case, the linear function is d = 4t, and it models the relationship between distance and time. Question the students on the terms dependent variable and independent variable and rate of change.

2. Have the students work through Exercise 3. These questions on modelling situations with linear functions are similar to those in Unit 3.

3. Invite some students to present their solutions to Exercise 3. Reinforce the idea that a relationship can be represented using an equation and a graph, and that either can be used to find particular points (cases) in the relationship.

4. Introduce Exploration 1. In this exploration, the students develop linear models for simple interest investments, linking with ideas from Unit 9.

5. The students can then complete Exploration 1.

ReflectionHave some students show their graph from Exploration 1. Ask: What does the graph show? Why do the 8% and 15% lines differ the way they do? Reinforce the significance of the values for gradient and y-intercept in the symbolic rules.


textbook, Section 14.1, pages 268–270

• 2 sheets of grid paper for each student

Teaching NoteReinforce that we use functions to model real-world situations and that these models help us make predictions.

Formative AssessmentGO Maths student portfolio, Unit Check, Question 1, page 104

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graph show? Why do the 8% and 15% lines differ the way they do? graph show? Why do the 8% and 15% lines differ the way they do? the significance of the values for gradient and

4. Introduce Exploration 1. In this exploration, the students develop linear models for simple interest investments, linking with ideas from Unit 9.

5. The students can then complete Exploration 1.

ReflectionHave some students show their graph from Exploration 1. Ask: graph show? Why do the 8% and 15% lines differ the way they do? graph show? Why do the 8% and 15% lines differ the way they do? the significance of the values for gradient and

, and it models the relationship between distance and time. Question the students on the terms dependent variable

Have the students work through Exercise 3. These questions on modelling situations with linear functions are similar to those in Unit 3.

Invite some students to present their solutions to Exercise 3. Reinforce the idea that a relationship can be represented using an equation and a graph, and that either can be used to find particular points (cases) in the relationship.

Introduce Exploration 1. In this exploration, the students develop linear models for simple interest investments, linking with ideas from Unit 9.

The students can then complete Exploration 1.

Use an example, such as walking at a constant speed of 4 km/h, to discuss a function being used to model a situation. In this case, the linear function

, and it models the relationship between distance and time. Question dependent variable and

Have the students work through Exercise 3. These questions on modelling situations with linear functions are similar to those in Unit 3.

3 × 4 + 2 × 53 × 4 + 2 × 5..

Use an example, such as walking at a constant speed of 4 km/h, to discuss a function being used to model a situation. In this case, the linear function

, and it models the relationship between distance and time. Question independent variable

Ask the students to calculate the answers mentally. Question them on the order Ask the students to calculate the answers mentally. Question them on the order

14.214.2


Working with the Equations of Lines

Key Ideas• Drawing graphs of linear functions on the Cartesian plane

• Finding co-ordinates of points on graphs of linear functions

Number Sense DiscussionWrite on the board: 8 × 1000 32 ÷ 100 4.6 × 1000 0.52 × 10 000

Ask the students to calculate the answers mentally. Invite some students to explain their reasoning. Encourage them to think in terms of the digits changing place and therefore their values changing, rather than the decimal point moving or ‘adding’ zeros. For example, the students should think: ‘8 ones become 8 thousands when multiplied by 1000’.

Activity1. Draw a four-quadrant Cartesian plane on the board. Ask the students to give the

co-ordinates of various points in the four quadrants. Reinforce that the axes are similar to number lines extending in both the positive and negative directions from 0 (the origin).

2. Work through Example 4 with the students. After working through finding the co-ordinates in Parts c and d, demonstrate how they can be substituted into the equation and how the equation balances.

3. Have the students complete Exercise 4 to practise drawing graphs of linear functions and finding co-ordinates of points.

4. Question the students on their knowledge of operations with positive and negative integers. Revise addition and subtraction operations as moves on a number line. Revise the rules for multiplication and division operations.

5. The students can then complete Exercise 5 to practise operations with positive and negative integers.

6. Guide the students through Example 5, which shows how the given value of one variable in a function is used to find the unknown value of the other variable.

7. Direct the students to complete Exercise 6 to practise finding unknown values for variables in linear functions.

ReflectionUse the function y = 2x + 1 to question the students on linear functions. Ask a student to sketch the graph of the function on the board. Ask: What is the value of y when x = 2? (5). Show how this is a point on the line (2, 5). Then ask: What is the value of x when y = 3? (1) Show how this is another point on the line (1, 3).




Teaching NoteHighlight that solving a linear equation for an unknown involves finding the value of one variable in a function when the value of the other variable is known.

Formative AssessmentGO Maths student portfolio, Unit Check, Questions 2 and 3, page 105

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integers. Revise addition and subtraction operations as moves on a number line. Revise the rules for multiplication and division operations.

5. The students can then complete Exercise 5 to practise operations with positive and negative integers.

Guide the students through Example 5, which shows how the given value of one variable in a function is used to find the unknown value of the other variable.

Direct the students to complete Exercise 6 to practise finding unknown values for variables in linear functions.

Have the students complete Exercise 4 to practise drawing graphs of linear functions and finding co-ordinates of points.

Question the students on their knowledge of operations with positive and negative integers. Revise addition and subtraction operations as moves on a number line. Revise the rules for multiplication and division operations.

The students can then complete Exercise 5 to practise operations with positive and negative integers.

Guide the students through Example 5, which shows how the given value of one variable in a function is used to find the unknown value of the other variable.

Draw a four-quadrant Cartesian plane on the board. Ask the students to give the co-ordinates of various points in the four quadrants. Reinforce that the axes are similar to number lines extending in both the positive and negative directions

Work through Example 4 with the students. After working through finding the co-ordinates in Parts c and d, demonstrate how they can be substituted into the equation and how the equation balances.

Have the students complete Exercise 4 to practise drawing graphs of linear functions and finding co-ordinates of points.

Question the students on their knowledge of operations with positive and negative integers. Revise addition and subtraction operations as moves on a number line.

Draw a four-quadrant Cartesian plane on the board. Ask the students to give the co-ordinates of various points in the four quadrants. Reinforce that the axes are similar to number lines extending in both the positive and negative directions

Work through Example 4 with the students. After working through finding the

their reasoning. Encourage them to think in terms of the digits changing place and their reasoning. Encourage them to think in terms of the digits changing place and therefore their values changing, rather than the decimal point moving or ‘adding’ therefore their values changing, rather than the decimal point moving or ‘adding’ zeros. For example, the students should think: ‘8 ones become 8 thousands when zeros. For example, the students should think: ‘8 ones become 8 thousands when

variablevariable isis

sheetsheet ofofeacheach studentstudent

Teaching NoteTeaching NoteHighlightHighlight thatthat solvingsolvingequationequation forfor unknownunknowninvolvesinvolves findingfinding thethe valuevalue

variablevariable inin aa functionfunctionvaluevalue ofof thetheknown.known.

14.314.3


Lines of Best Fit

Key Idea• Finding the equation of a linear line of best fit

Number Sense DiscussionWrite on the board: 25 × 1000 9.5 ÷ 100 0.64 × 100 0.52 ÷ 10

Ask the students to calculate the answers mentally. Invite some students to explain their reasoning. Encourage them to think in terms of place value.

Activity1. Refer to some situations from Unit 12 (Sections 12.5, 12.6) where the students

looked at two-variable data and scatterplots, for example, height and weight. Revise drawing a line of best fit through the data to represent the possible relationship between the two variables.

2. Work through Example 6 with the students. A line of best fit is drawn as in Unit 12 and then the gradient and y-intercept are read from the graph and used to write the equation for the function. Ask: Will the sales continue to increase with experience?

3. Direct the students to work through Exercise 7 to practise finding equations of lines of best fit for two-variable data.

4. Review the students’ answers for Exercise 7. Compare the validity of the two predictions. Have the students justify the water prediction given the likely variation in the rate of rainfall. Ask if the 400-m time prediction in Question 2 is at all valid given the likely levelling-off of times in this situation.

5. The students can then complete Exploration 2 in which they use data gathered in Unit 12, Explorations 4 and 5. If the data is not available, first measure the hand spans and foot lengths for everyone in the class, represent the data in a scatterplot and draw a linear line of best fit.

ReflectionAsk some students to present their equations from Exploration 2 and describe the way the data was scattered in the scatterplots they drew. Ask them how reliable the predictions would be. Emphasise that while we can draw a line of best fit and derive an equation, the relationship between the variables may be weak and the predictions unreliable.




• Scatterplot of class foot length and hand span from Exploration 5 in Unit 12

Teaching NoteHighlight the link between solving an equation and finding a point on the graph of a function.

Formative AssessmentGO Maths student portfolio, Unit Check, Question 4, pages 106–107

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Reflection

predictions. Have the students justify the water prediction given the likely variation in the rate of rainfall. Ask if the 400-m time prediction in Question 2 is at all valid given the likely levelling-off of times in this situation.

5. The students can then complete Exploration 2 in which they use data gathered in Unit 12, Explorations 4 and 5. If the data is not available, first measure the hand spans and foot lengths for everyone in the class, represent the data in a scatterplot and draw a linear line of best fit.

Reflection

Work through Example 6 with the students. A line of best fit is drawn as in Unit 12 and then the gradient and to write the equation for the function. Ask:

Direct the students to work through Exercise 7 to practise finding equations of lines of best fit for two-variable data.

Review the students’ answers for Exercise 7. Compare the validity of the two predictions. Have the students justify the water prediction given the likely variation in the rate of rainfall. Ask if the 400-m time prediction in Question 2 is at all valid given the likely levelling-off of times in this situation.

The students can then complete Exploration 2 in which they use data gathered

Refer to some situations from Unit 12 (Sections 12.5, 12.6) where the students looked at two-variable data and scatterplots, for example, height and weight. Revise drawing a line of best fit through the data to represent the possible relationship between the two variables.

Work through Example 6 with the students. A line of best fit is drawn as in -intercept are read from the graph and used

to write the equation for the function. Ask: Will the sales continue to increase Will the sales continue to increase

Direct the students to work through Exercise 7 to practise finding equations

Refer to some situations from Unit 12 (Sections 12.5, 12.6) where the students looked at two-variable data and scatterplots, for example, height and weight. Revise drawing a line of best fit through the data to represent the possible

Work through Example 6 with the students. A line of best fit is drawn as in

0.52 ÷ 100.52 ÷ 10

Ask the students to calculate the answers mentally. Invite some students to Ask the students to calculate the answers mentally. Invite some students to explain their reasoning. Encourage them to think in terms of place value.explain their reasoning. Encourage them to think in terms of place value.

Refer to some situations from Unit 12 (Sections 12.5, 12.6) where the students looked at two-variable data and scatterplots, for example, height and weight.

14.414.4


Points and Gradients

Key Ideas• Understanding gradient as change in y divided by change in x

• Using a graphics calculator to find equations of lines

Number Sense DiscussionWrite on the board: 5 – 8 -6 + 10 -2 × 5 -8 ÷ 2

Ask the students to calculate the answers mentally. Invite some students to explain in general terms how they add and subtract integers and how they multiply and divide integers. Encourage thinking in terms of movement along a number line for addition and subtraction and the rules for multiplication and division.

Activity1. Ask: How is the gradient of a line calculated? (rise ÷ run or rise

___ run ) Explain how rise can be found by subtracting the y co-ordinates of two points on the line and how run can be found by subtracting the x co-ordinates of two points on the line. Define gradient as change in y divided by change in x.

2. Work through Example 7 with the students to demonstrate calculating the gradient of a straight line using the co-ordinates of two points.

3. Have the students complete Exercise 8 to practise finding gradients for straight lines from two given points on the line.

4. Introduce Exploration 3. In this exploration, the students use a graphics calculator to find gradients and equations of straight lines from given points. They have had some experience using a graphics calculator in Units 2 and 3. Students may need help in interpreting the information about the equations of straight lines as it is presented on their calculators. To find the equation of a straight line they need to select a regression model, but at this stage it is only necessary for them to know that in selecting X on the CASIO and LinReg (linear regression) on the TI they are asking the calculator for a linear model. Direct the students to complete the exploration.

ReflectionQuestion the students on their use of a graphics calculator to find gradients and equations. Ask how their equations for the lines of best fit that they found earlier in Exercise 7 compare with those they found with their graphics calculator in Part B of Exploration 3.




• 1 graphics calculator for each student

Teaching NoteHelp students interpret the information on their graphics calculators when using them to find a linear model.


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help in interpreting the information about the equations of straight lines as it is presented on their calculators. To find the equation of a straight line they need to select a regression model, but at this stage it is only necessary for them to know that in selecting they are asking the calculator for a linear model. Direct the students to complete the exploration.

Reflection

Introduce Exploration 3. In this exploration, the students use a graphics calculator to find gradients and equations of straight lines from given points. They have had some experience using a graphics calculator in Units 2 and 3. Students may need help in interpreting the information about the equations of straight lines as it is presented on their calculators. To find the equation of a straight line they need to select a regression model, but at this stage it is only necessary for them to know that in selecting X on the CASIO and they are asking the calculator for a linear model. Direct the students to complete

co-ordinates of two points on the line and co-ordinates of two points on the

divided by change in x

Work through Example 7 with the students to demonstrate calculating the gradient of a straight line using the co-ordinates of two points.

Have the students complete Exercise 8 to practise finding gradients for straight lines from two given points on the line.

Introduce Exploration 3. In this exploration, the students use a graphics calculator to find gradients and equations of straight lines from given points. They have had some experience using a graphics calculator in Units 2 and 3. Students may need help in interpreting the information about the equations of straight lines as it is

(rise ÷ run or rise___run ) Explain how rise

co-ordinates of two points on the line and co-ordinates of two points on the

Work through Example 7 with the students to demonstrate calculating the

in general terms how they add and subtract integers and how they multiply and in general terms how they add and subtract integers and how they multiply and divide integers. Encourage thinking in terms of movement along a number line for divide integers. Encourage thinking in terms of movement along a number line for

findfind aa linearlinear


11 graphicsgraphicsforfor eacheach studentstudent

Teaching NoteTeaching Notestudentsstudents interpretinterpret thethe

informationinformation onon theirtheir graphicsgraphicswhenwhen usingusing

linearlinear

14.514.5


Developing Quadratic Functions

Key Ideas• Recognising the general form of a quadratic function as y = ax2 + bx + c

• Evaluating quadratic functions for values of x

• Drawing a graph of a quadratic function

Number Sense DiscussionWrite on the board: 5 – 8 -6 + 10 -2 × 5 -8 ÷ 2

Ask the students to calculate the answers mentally. Ask some students to explain in general terms how they add and subtract integers and how they multiply and divide integers.

Activity1. Ask the students what happens when a ball is thrown straight up in the air. Discuss

how it slows down then speeds up again as it falls back down. Ask: What would a height versus time graph look like for the flight of the ball?

2. Introduce the term quadratic function to describe functions that have a term with the independent variable squared, for example, y = x2. Write the general form of a quadratic function, y = ax2 + bx + c, on the board.

3. Work through Example 8 with the students to demonstrate substituting values for x to find values of y for a quadratic function.

4. The students can complete Exercise 9 to practise evaluating quadratic functions.

5. Introduce Exploration 4. In Part A the students investigate some of the properties of quadratic functions, looking particularly at the symmetrical shape of the graphs of quadratic functions and the effect of the constant c. Have the students work through Part A.

6. Invite some students to report their findings for Part A of Exploration 4. Emphasise how the value of the constant term c moves the graph up or down relative to the graph of y = x2. Compare this with the effect of c in y = mx + c, which is the same. Also point out how the lines of each graph do not cross.

7. If time permits, the students can work through Part B of Exploration 4 in which they use a graphics calculator to draw graphs of quadratic functions and then further explore their properties.

ReflectionHave the students sketch the graphs of y = x2, y = x2 + 3 and y = x2 – 2. Ask them to describe the way the gradient changes in each graph. Reinforce the continuous change and the increase in gradient going from left to right.





Teaching NoteAsk students to describe the behaviour of quadratic functions for different values of the constant term.


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4. The students can complete Exercise 9 to practise evaluating quadratic functions.

5. Introduce Exploration 4. In Part A the students investigate some of the properties of quadratic functions, looking particularly at the symmetrical shape of the graphs of quadratic functions and the effect of the constant through Part A.

6. Invite some students to report their findings for Part A of Exploration 4. Emphasise how the value of the constant term graph of

Ask the students what happens when a ball is thrown straight up in the air. Discuss how it slows down then speeds up again as it falls back down. Ask: a height versus time graph look like for the flight of the ball?a height versus time graph look like for the flight of the ball?

quadratic functionthe independent variable squared, for example, a quadratic function, y = axy = axy = a 2 + b2 + b2 + c, on the board.

Work through Example 8 with the students to demonstrate substituting values for to find values of for a quadratic function.y for a quadratic function.y

The students can complete Exercise 9 to practise evaluating quadratic functions.

Introduce Exploration 4. In Part A the students investigate some of the properties of quadratic functions, looking particularly at the symmetrical shape of the graphs

Ask the students what happens when a ball is thrown straight up in the air. Discuss how it slows down then speeds up again as it falls back down. Ask: a height versus time graph look like for the flight of the ball?a height versus time graph look like for the flight of the ball?

to describe functions that have a term with the independent variable squared, for example, y = y = y

, on the board.

Ask the students to calculate the answers mentally. Ask some students to explain Ask the students to calculate the answers mentally. Ask some students to explain in general terms how they add and subtract integers and how they multiply and in general terms how they add and subtract integers and how they multiply and

Ask the students what happens when a ball is thrown straight up in the air. Discuss how it slows down then speeds up again as it falls back down. Ask:

Ask the students to calculate the answers mentally. Ask some students to explain Ask the students to calculate the answers mentally. Ask some students to explain in general terms how they add and subtract integers and how they multiply and in general terms how they add and subtract integers and how they multiply and

14.614.6


Working with Inequalities

Key Idea• Understanding that inequalities have many possible solutions

Number Sense DiscussionWrite on the board: -5 + 3 -6 – 6 -4 × -5 -10 ÷ 4

Ask the students to calculate the answers mentally. Invite some students to explain in general terms how they add and subtract integers and how they multiply and divide integers.

Activity1. Ask: What are some numbers that fit the statement x > 3? Encourage the

students to think about whole numbers and fractional values. Conclude that there are many possible numbers.

2. Introduce the term inequality to describe statements such as x > 3, and revise the symbols < (less than), > (greater than), ≤ (less than or equal to), ≥ (greater than or equal to).

3. Direct the students to work through Exercise 10 to practise interpreting situations involving inequalities and writing possible solutions. They need to consider whether the variable is discrete or continuous in each situation.

4. Invite some students to present their solutions to Exercise 10. There are two inequalities involved in Question 4 (n ≥ 1 and n ≤ 22) and Question 5 (n ≥ 10 and n ≤ 24).

5. Lead a discussion about a two-variable situation that has multiple combinations, such as the example on page 282 of the Student Textbook with boys and girls sitting around a six-place table where all the chairs do not need to be occupied. Discuss how this situation can be recorded as a two-variable inequality: g + b ≤ 6.

6. Introduce Exploration 5 in which the students use points on a graph to model two-variable situations involving inequalities. The aim is to see that the solutions fall within a defined area of the graph. Have the students complete the exploration.

ReflectionHave some students present their findings for Exploration 5. For each graph, ask the students to look at the lines that define the area in which the points lie. Emphasise that for these situations, only positive or 0 values are possible.




Teaching NoteEncourage students to consider context when finding solutions to inequalities.

Formative AssessmentGO Maths student portfolio, Unit Check, Question 7, page 109

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such as the example on page 282 of the Student Textbook with boys and girls sitting around a six-place table where all the chairs do not need to be occupied. Discuss how this situation can be recorded as a two-variable inequality:

6. Introduce Exploration 5 in which the students use points on a graph to model two-variable situations involving inequalities. The aim is to see that the solutions fall within a defined area of the graph. Have the students complete the exploration.

Reflection

inequalities involved in Question 4 (

Lead a discussion about a two-variable situation that has multiple combinations, such as the example on page 282 of the Student Textbook with boys and girls sitting around a six-place table where all the chairs do not need to be occupied. Discuss how this situation can be recorded as a two-variable inequality:

Introduce Exploration 5 in which the students use points on a graph to model two-variable situations involving inequalities. The aim is to see that the solutions fall within a defined area of the graph. Have the students complete the exploration.

to describe statements such as ≤ (less than or equal to),

Direct the students to work through Exercise 10 to practise interpreting situations involving inequalities and writing possible solutions. They need to consider whether the variable is discrete or continuous in each situation.

Invite some students to present their solutions to Exercise 10. There are two inequalities involved in Question 4 (n ≥ 1 and n ≤ 22) and Question 5 (

Lead a discussion about a two-variable situation that has multiple combinations, such as the example on page 282 of the Student Textbook with boys and girls

> 3? > 3? Encourage the students to think about whole numbers and fractional values. Conclude that

to describe statements such as x > 3, and revise (less than or equal to),

Direct the students to work through Exercise 10 to practise interpreting situations involving inequalities and writing possible solutions. They need to consider whether


Teaching NoteTeaching NoteEncourageEncourage studentsstudents

considerconsiderfindingfinding solutionssolutions

inequalities.inequalities.

Extension


More Functions

Key Ideas• Investigating further properties of the quadratic function y = ax2 + bx + c

• Investigating properties of simple cubic functions of the form y = x3 + c

Number Sense DiscussionWrite on the board: 5 + -2 × 3 -6 × 3 – 3 -2 × 6 + 2

Ask the students to calculate the answers mentally. Remind them of the rules for order of operations with integers.

Activity1. Introduce Exploration 6. Ask the students to work through the exploration to

investigate the effects of the coefficient b in quadratic functions. They work with the simplified form y = x2 + bx. Little direction is given, and the students are expected to create several examples and use their graphics calculator to see what happens to the shape of the graph, and to generalise.

2. Ask some students to report their findings for Exploration 6. The students should have tried positive and negative values for b. Check that they all reached the same generalisation about the shift of the graph (left for positive b and right for negative b) and that they mentioned that the shape does not change compared with y = x2.

3. Direct the students to work through Exploration 7 to investigate properties of the cubic function y = x3 + c and the effect of the value of c. Again, little direction is given about how to go about this investigation.

4. Ask some students to report their findings for Exploration 7. They should describe the general shape of the graphs and the way the lines are in the first and third quadrants of the Cartesian plane. Students should have tried positive and negative values for c and seen that the shifts up and down were similar to the shifts for quadratic functions. Point out that, as for quadratic functions, the lines of the different graphs do not cross.

ReflectionQuestion the students on the general classes of functions they have been investigating: linear (y = mx + c), quadratic (y = ax2 + bx + c), cubic (y = x3 + c). Review how the value of the constant term c had a similar effect in all cases.


textbook, Unit 14 Extension, page 284



Teaching NoteDiscuss the similar effect of the constant term in linear, quadratic and cubic functions.

Formative AssessmentThere are no questions for this section in the Student Portfolio

TSB_Unit_14.indd 10 30/7/07 1:30:47 PM

3. Direct the students to work through Exploration 7 to investigate properties of the cubic function is given about how to go about this investigation.

4. Ask some students to report their findings for Exploration 7. They should describe the general shape of the graphs and the way the lines are in the first and third quadrants of the Cartesian plane. Students should have tried positive and negative values for quadratic functions. Point out that, as for quadratic functions, the lines of the different graphs do not cross.

expected to create several examples and use their graphics calculator to see what happens to the shape of the graph, and to generalise.

Ask some students to report their findings for Exploration 6. The students should have tried positive and negative values for same generalisation about the shift of the graph (left for positive negative b) and that they mentioned that the shape does not change compared

x2.

Direct the students to work through Exploration 7 to investigate properties of the cubic function y = y = y c and the effect of the value of c and the effect of the value of cis given about how to go about this investigation.

Ask some students to report their findings for Exploration 7. They should describe

Introduce Exploration 6. Ask the students to work through the exploration to investigate the effects of the coefficient in quadratic functions. They work with

. Little direction is given, and the students are expected to create several examples and use their graphics calculator to see what happens to the shape of the graph, and to generalise.

Ask some students to report their findings for Exploration 6. The students should have tried positive and negative values for b. Check that they all reached the same generalisation about the shift of the graph (left for positive

Introduce Exploration 6. Ask the students to work through the exploration to in quadratic functions. They work with

. Little direction is given, and the students are expected to create several examples and use their graphics calculator to see

Ask the students to calculate the answers mentally. Remind them of the rules for Ask the students to calculate the answers mentally. Remind them of the rules for

UNIT14

Final Investigation

In this investigation, the students are given a function for vertical motion under gravity and are asked to investigate the motion described by the function. They are given little direction for completing this task. While some students may need some assistance to get started, their experiences with the explorations in this unit and in earlier algebra units should provide a basis for their investigation. The students are asked to accompany the report with a graph. They should generate data to plot a graph of height versus time, or use technology (graphics calculator or spreadsheet) to produce the graph. In their report, they should describe the ball’s motion in terms of height above the ground, and explain how the graph shows when the ball is going up and when the ball is coming down. They should mention the maximum height reached by the ball and the time it takes for the ball to land.

Formative AssessmentGO Maths student portfolio, Final Investigation, pages 110–111

Teacher’s Notes

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Materials and Resources

© 2008 ORIGO Education

1414Preparation 14.1

14.314.2

14.4 14.5

Extension14.6

Linear Equations

Materials• GO Maths student textbook, Section 14.1, pages 268–270• 2 sheets of grid paper for each student

Materials• GO Maths student textbook, Unit 14 Preparation, pages 266–268• 1 sheet of grid paper for each student

Materials• GO Maths student textbook, Section 14.2, pages 270–272• 1 sheet of grid paper for each student

Materials• GO Maths student textbook, Section 14.4, pages 276–278• 1 sheet of grid paper for each student• 1 graphics calculator for each student

Materials• GO Maths student textbook, Section 14.6, pages 281–283• 1 sheet of grid paper for each student

Materials• GO Maths student textbook, Section 14.3, pages 273–276• 3 sheets of grid paper for each student• Scatterplot of class foot length and hand span from

Exploration 5 in Unit 12

Materials• GO Maths student textbook, Section 14.5, pages 279–281• 1 sheet of grid paper for each student• 1 graphics calculator for each student

Materials• GO Maths student textbook, Unit 14 Extension, page 284• 2 sheets of grid paper for each student• 1 graphics calculator for each student

Modelling with Linear Functions

Lines of Best Fit

Developing Quadratic Functions

More Functions

Working with the Equations of Lines

Points and Gradients

Working with Inequalities

TSB_Unit_14.indd 12 30/7/07 1:30:48 PM

student textbook, Section 14.4, pages 276–2781 sheet of grid paper for each student1 graphics calculator for each student

student textbook, Section 14.4, pages 276–2781 sheet of grid paper for each student1 graphics calculator for each student

Points and GradientsPoints and Gradients 14.514.5

Materials• GO Maths student textbook, Section 14.3, pages 273–276

3 sheets of grid paper for each studentScatterplot of class foot length and hand span from Exploration 5 in Unit 12

student textbook, Section 14.3, pages 273–2763 sheets of grid paper for each studentScatterplot of class foot length and hand span from

Lines of Best FitLines of Best Fit

Documents

Teacher Sourcebook - ORIGO Education