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Course Profile Principles of Mathematics, Grade 9 academic, Pulbic

Unit 2

Course Profiles are professional development materials designed to help teachers implement the new Grade 9 secondary school curriculum. These materials were created by writing partnerships of school boards and subject associations. The development of these resources was funded by the Ontario Ministry of Education. This document reflects the views of the developers and not necessarily those of the Ministry. Permission is given to reproduce these materials for any purpose except profit. Teachers are encouraged to amend, revise, edit, cut, paste, and otherwise adapt this material for educational purposes.

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Acknowledgments

Public District School Board Writing Teams – Principles of Mathematics

Course Profile Writing Team

Myrna Ingalls, Lead Writer, York Region District School Board

Shirley Dalrymple, York Region District School Board

Carolyn Gallagher, Kawartha Pine Ridge District School Board

Mary Howe, Ontario Association for Mathematics Education

Irene McEvoy, Peel District School Board

Lionel LaCroix, Peel District School Board

Christine Surtamm, Peel District School Board

Reviewers

Bill Clarke, Ottawa Carleton DSB; Angela Con, Kawartha Pine Ridge DSB; Donna Del Re, Peel DSB; Sandra Emms Jones, Waterloo Region DSB; Ron Lewis, Rainbow DSB; Bob McRoberts, York Region DSB

Lead Board

Peel District School Board

Allan Smith, Project Manager

Partner Boards

Kawartha Pine Ridge District School Board, Ottawa Carleton District School Board, Rainbow District School Board, Waterloo Region District School Board, York Region District School Board

Associations

Ontario Association for Mathematics Education (OAME)

Ontario Mathematics Co-ordinators Association (OMCA)

Unit 2: Algebraic Models and Rates of Change

Activity 1 | Activity 2 | Activity 3 | Activity 4 | Activity 5 |

Activity 6 | Activity 7 | Activity 8 | Activity 9 | Activity 10

Time: 30 hours

Unit Description

Students use linear equations with variables x and y to algebraically summarize patterns derived from real-life contexts and to communicate using graphing technology. Working from real-life contexts, students develop a “common-sense” understanding of slope as unit rate of change prior to the algebraic definitions. They also explore connections between initial conditions and the y-intercepts of lines. The intent is for students to understand slope, equation, and line concepts in a manner which lends itself to application when problem solving. Properties and equations of lines are investigated and algebraic manipulations are taught and practised as needed.

Strand(s) and Expectations

Some specific expectations from the Number Sense and Algebra, and Relationships Strands have been combined with overall expectations from the Analytic Geometry Strand. Weaving together the expectations of the strands in this way helps students make connections.

Analytic Geometry Strand Specific Expectations: AG1.01, 02, 03, 04; AG2.01, 02, 03, 04, 05; AG3.01, 02, 03, 04, 05, 06, 07, 08.

Number Sense and Algebra Specific Expectations: NA1.01, 02, 03, 04, 05, 06; NA2.06; NA3.01, 02, 03, 04, 05, 06; NA 4.01, 02, 03.

Relationships: RE1.01, 03, 04, 05, 06, 07; RE2.01, 03; RE3.01, 02, 04.

Activity Titles

What follows is a suggested sequence for teaching Unit 2. The timing for activities and skill development is included. This Profile develops the mathematics in a sequence that may be different from the sequence used in previous courses of study, and weaves expectations from all strands together.

Since there are specific times when it would be best to introduce certain vocabulary and notation, and to develop certain skills, the outline for Unit 2 details when specific algebraic skills are required. Some of the activities include a large amount of skill development as indicated in the [square brackets]; other activities may require additional time for skills identified as Follow-Up Skills. Time has been allotted, in the table below, for the skill development within or following each activity. There is an additional 150 minutes of asterisked * time in this unit for skill building needs, as identified by the teacher.

Activity 2.1 & Follow-Up	Match Me Up! [y = mx in contexts, where m is unit rate. No x’s and y’s until algebraic models are summarized. Then, use x and y notation so that graphing calculators can be instructed to create graphical models using “y =” form.] Follow-Up Skills: plot points in all 4 quadrants; use an equation in “y =” form to create a table of values to plot on the Cartesian plane	150 minutes
Activity 2.2 & Follow-Up	Ramps 'R Us [slope =] Follow-Up Skills: reviewing Pythagorean theorem; solving equations from knowing 2 of the variables in slope =	150 minutes
Activity 2.3 & Follow-Up	Slippery Slope [slope = developed through motion of students; slope = from contexts and tables; numeracy skills] Follow-up Skills, as homework: graphing lines, given slope and a point	75 minutes
Activity 2.4 & Follow-Up	Programs for Sale! [Use y = mx + b form when (x1, y1) is a point on the y-axis and represents initial conditions in a partial variation context; solve simple linear equations by inspection; use list features of graphing calculators to enter formulas, as with spreadsheets] Follow-Up Skills: combining like terms; using distributive property; common factoring; exponent laws; solving linear equations involving some algebraic manipulation; reading and interpreting intersections of lines from graphs	300 minutes
Activity 2.5	What’s My Spring? Stretching a Penny [Apply y = mx + b form in a context that yields somewhat messy data; form and solve equations]	75 minutes
Activity 2.6 & Follow-Up	Sunshine, Whiskers, and Windmill [Investigate slopes of parallel and perpendicular lines and lines that are reflections in either the x-axis or y-axis] Follow-Up Skills: graphing lines, given slope and a special point, the y-intercept; forming the equation of a line, given slope and y-intercept; forming the equations of families of lines that share a slope, or a y-intercept	150 minutes
Activity 2.7 & Follow-Up	Break the Bank! [Re-arrange from Ax + By + C = 0 to y = mx + b form to graph; determine the x- and y-intercepts of a line, given Ax + By + C = 0 or y = mx + b forms; solve linear equations] Follow-Up Skills: rearrange linear equations; graph lines, given the intercepts, any two properties of the line, or the equation in any form	225 minutes
Activity 2.8 & Follow-Up	Fireworks and Twinkle, Twinkle [Develop y - y1 = m(x – x1) and y = y0 + m(x – x0) forms of the equation of a line] Follow-up Skills: graphing lines, given slope and a point, or given two points; forming equations of lines having properties like those of another line (e.g., parallel to one line and having the same y-intercept as another)	225 minutes
Activity 2.9 & Follow-Up	All in the Family [Graph lines and curves from equations] Follow-Up Skills: recognize linear vs non-linear relations from tables of values and equations; numeracy skills; graph y = b and x = a	75 minutes
* Time to practise skills		150 minutes
Activity 2.10	Planning for a Trip: a Summative Assessment Activity Sample questions for a pencil & paper test: Confirm or Deny; Roof Trusses	225 minutes

Prior Knowledge Required

Unit 1

Unit Planning Notes

· The first activity is intended to introduce the use of x and y notation, with x’s representing independent variables, and y’s representing the dependent variables. Until now, letters having meaning in specific contexts have been used. To compare or summarize relations, or to instruct graphing technology to draw a graph, x’s and y’s are used.

· The word “slope” is introduced as a measure of inclination of a line through the context of wheelchair ramps. At this time, rates of change are connected to the abstraction of slopes of lines.

· The summative assessment activity in Unit 2 is intended to help prepare students for the type of activities in Unit 4.

Teaching/Learning Strategies

Small group organization of students works well as the comparisons of equation and graphical models for relationships are explored. Issue each student a graphing calculator, but arrange students in pairs as they learn new techniques so that they can help each other. Independent work is important in developing skills with algebraic manipulation and graphing calculators.

Using graphing calculators to reproduce Kitty’s Whiskers, Rays of Sunshine, and other designs using sets of lines creates an interesting context for practice with equations of lines in y = mx + b and y - y₁= m (x -x₁) forms. Games like ‘Battleship’ make the learning of co-ordinate graphing fun.

Direct teaching of algebraic manipulation skills can be moved from concrete to pictorial to abstract symbolic stages through the use of algebra tiles. Time has been allotted for this in the Follow-up skills part of many activities.

Assessment/Evaluation Techniques

As in Unit 1, a variety of assessment tools and strategies is recommended. Performance assessments may be used to effectively assess Thinking/Inquiry/Problem Solving, Communication, and Application categories of the Achievement Chart when students do open-ended tasks. Learning Skills can be assessed using teacher and peer observation, and self-reflection. Rubrics and rating scales are useful when a wide range of performance is expected and when many complex criteria are to be judged. Whereas, checklists and marking schemes can still be used for more traditional tasks with predictable solutions.

Resources

Visualized Geometry: A van Hiele Level Approach. Portland, Maine: J. Weston Walch, 1990.

www.kings.k12.ca.us/math/lessons/ti83tutorial.html for instructions and sample data for using the LIST features of the TI-83+

Classroom Activities & Teacher Resources from Texas Instruments

www.ti.com/calc/docs/activities.htm

Ramp criteria

http://calder.med.miami.edu/pointis/ramp.html

Activity 1: Match Me Up!

Time: 150 minutes

Description

Students construct tables of values for direct variation from a variety of scenarios. Lines of best fit are drawn by hand and an equation for each line is developed. Students recognize that the multiplier or co-efficient in the equation relates to steepness, that several scenarios could produce lines with the same steepness, and that algebraic model can represent a variety of situations. The students then replace the independent and dependent variables in their equations with x and y to enable them to graph their relations with a graphing calculator. The follow-up graphing skills include introducing the four quadrants of the Cartesian plane and graphing equations of the form “y =” without technology by creating tables of values.

Strand(s) and Expectations

Strand(s): Number Sense and Algebra, Relationships, Analytic Geometry

Specific Expectations: NA1.01, .03, .04; RE1.04, 05, 06; RE2.01, 03, AG2.02, .03, .04, AG3.01, .02, .03, 07.

Planning Notes

· The teacher must have identical, quadrant I grids photocopied onto acetate sheets for each of the groups. See the Appendix for a template. A washable overhead marker is required for each group of three students.

· A photocopy or overhead of the Class Example is needed so that students see the questions that need to be answered in their groups.

· A photocopy of the scenarios should be made and cut into strips so that each group receives one scenario on a strip of paper.

· Lead a full class summary of the results from groups, drawing out the comparisons between the graphs, and introducing the usefulness of x and y notation.

· Provide a class set of graphing calculators, an overhead graphing calculator and LCD panel, and grid paper.

Prior Learning Required

From Unit 1: working with integers and rationals; discrete vs connected points; dependent vs independent variables; first differences; determining trends and patterns by making inferences from graphs; identifying and discussing patterns in algebraic terms; substituting into and simplifying an expression; entering lists and graphing scatter plots using the graphing calculator.

Teaching/Learning Strategies

This activity is broken into two parts. The first part begins with the teacher working through the scenario below with the full class. The students then work in groups, examining different scenarios that are similar to the worked scenario. Graphs are then compared and generalized which leads into the introduction of graphing calculators to graph equations using x’s and y’s. The second part of the activity begins with a full class exploration of using the graphing calculators to graph lines in the form of “y =”. This includes moving from data and lines in the first quadrant to relationships in all four quadrants.

Part I

Teacher Facilitation: The teacher introduces a scenario like the one below, and outlines the questions to be considered in later scenarios. This example serves to review the concepts from Unit 1 that are needed for the Activity that follows. It is important to review with students how to determine dependent and independent variables using clues like unit rates. In expressing relations as equations, students should be using meaningful variables. DO NOT use x’s and y’s yet.

Class Example

A crystal growing kit contains enough material to perform two experiments. Complete the following steps to explore the relationship between the number kits and number of experiments.

i) The number of experiments performed depends on the number of kits purchased. Construct a table of values that shows the number of experiments that can be performed if up to five separate kits are purchased.

ii) Add a column in your table for first differences for the number of experiments that can be performed

iii) What is the independent variable? The dependent variable? How did you decide?

iv) Should the points be connected or not? Why?

v) Construct a scatter plot by hand.

vi) Write an equation that would describe this relation using letters appropriate for your scenario.

vii) What units are associated with the number in your equation? What does this number represent?

viii) Describe the connection between your equation and the first differences.

Teacher Facilitation: Place the students in groups of three and provide each group with a scenario and an acetate for constructing the graph of their scenario. The acetate should have a set of horizontal and vertical axes with appropriate scales photocopied onto it in order to facilitate the comparison of graphs later on. Students carry out the eight steps outlined in the previous example for the scenario that they are given.

The teacher should observe groups as they work to make sure that dependent and independent variables are being identified correctly and that correct decisions are being made regarding graphing of discrete and continuous data.

Student Activity

Scenario 1: A radio-controlled model car travels at a speed of 2 m/s. Graph the relationship between time (in seconds) and distance (in metres).

Scenario 2: Luke is purchasing packages of peanut butter cups to sell at a school dance. Complete the table of values for the numbers of packages given. Graph the relationship between the number of peanut butter cups and the number of packages between 0 and 50.

Number of packages	10	20	30	40	50
Number of peanut butter cups	30

Scenario 3: In the tricycle department of a toy store, the number of wheels depends on the number of tricycles. Graph the relationship between the number of wheels and the number of tricycles.

Scenario 4: Bob’s sock drawer is a mess. All of his socks are in it, but none of them have been put together in matched pairs. The number of pairs of socks depends on how many socks are in the drawer. Graph the relationship between the number of pairs and the number of socks.

Scenario 5: Photocopying on both sides of a piece of paper saves money and space. The number of pieces of paper required for a copy of a document depends on the number of pages to be photocopied. Graph the relationship between the number pieces of paper required and the number of pages being copied.

Scenario 6: Chocolate bars are often sold in packages of two. The number of bars you have depends on the number of packages you buy. Graph the relationship between the number of chocolate bars and the number of packages purchased.

Scenario 7: The number of participants in a chess tournament depends on the number of chessboards available. Graph the relationship between the number of chessboards and number of participants.

Scenario 8: Legal documents are often produced in triplicate. The number of copies depends on the number of documents prepared. Graph the relationship between the total number of copies and the number of documents prepared if the documents are produced in triplicate.

Teacher Facilitation: Once the groups have followed the eight steps and have their graphs on the acetates, the teacher brings closure to the activity by summarizing findings and connections with the full class. For example:

i) Ask a member of the group with Scenario 3 to bring their acetate to the overhead projector and explain the graph.

ii) Ask which other group has a graph that matches Scenario 3 in some way. (Scenarios 2 and 8 should match in steepness; Scenario 4 matches for disconnectedness)

iii) Focus on the steepness comparison. Ask students from group 2 and 8 to come up to the overhead and explain their graphs.

iv) Pile the three acetates on top of each other to show that they match identically for steepness.

v) Ask the following questions and draw out all of the mathematics:

1. Why did the graphs match? (identical tables of values, same first differences, always tripling, same type of equation)

2. What is different about the three graphs? (discreteness, title on graph, labels on axes, units on axes, units on the number in the equation (the unit rate))

3. What is the real life meaning of the numerical multiplier in each equation?

vi) ***Introduce the idea that all three relationships could be summarized as y = 3x where x represents the independent variable and y represents the dependent variable.***

vii) Repeat using Scenarios 1, 6, and 7, then Scenarios 4 and 5, introducing direct variation vocabulary, and xy notation.

Follow-up: Before starting, discuss the need to use the variables x and y when using a calculator, and then replace the independent variable with x and the dependent variable with y in the specific scenario that you are using.

Using a full class presentation, the teacher chooses one of the above Scenarios and coaches the students through the steps for constructing a scatter plot and graphing the equation of the line using the graphing calculators.

Students:

· clear all lists and previous graphs

· enter the data into two lists

· create a scatter plot using those two lists and view it using the ZOOM, ZoomStat feature

After examining the scatter plot, check the correctness of the equation that was determined for the relationship.

Does the line pass through the points of the scatter plot? Does the equation produce a fitting model?

The teacher needs to discuss the fact that we are representing discrete data with a continuous line. However, this is similar to talking about lines of best fit for discrete data.

Student Activity or Homework:

The students are given further scenarios of direct variation to graph and develop equations. These scenarios should include decreasing relations (e.g., the distance below sea level as a submarine descends at 5 m/s), relations that have fractional slopes (e.g., currency exchange rates) and relations that require different scales (e.g., distance driven vs gas consumed).

Ask students to describe a situation that would explain the events illustrated by an equation for a direct variation.

This would also be a good place to take some time to review skills regarding ratios and rates.

Part II

Teacher Facilitation: When students have completed the above activity or homework, the teacher should use three sets of data from their work to “take up” and extend the activity in a whole class setting using the overhead graphing calculator. Students can be working either in pairs or individually with graphing calculators. The class enters the table of values in the lists, creates a scatter plot and enters the “y =” equation for each of the three sets chosen by the teacher. The teacher should choose sets of data that include positive, negative, and fractional slopes and discuss: What is the same about these graphs? Different? What happens to the line when the multiplier or co-efficient is negative or positive?

Follow-up Graphing Skills:

After the discussion of the previous work concludes, the teacher should introduce students to graphs in four quadrants by showing students how to move from Zoom-stat to Zoom-standard. This leads into a formal discussion and labelling of the axes for the four quadrants including:

· Locating ordered pairs in each quadrant;

· Playing a short game of Battleship to familiarize students with plotting points in all quadrants;

· Graphing linear equations of the form “y =” by hand by creating a table of values. This helps students understand the process of creating a graph and reduce the “magic” of a calculator and understand different ways of modelling a relation. Use integer and fractional values in the equations and independent variables.

· Students may check the hand-drawn graphs by inputting the equations in the graphing calculator.

Assessment/Evaluation Techniques

While students work in pairs or groups, the teacher can gather data on Learning Skills such as teamwork, working independently, and work habits, using Appendix 1 (Phase 1). This activity also provides opportunities for formative assessment to determine areas where students need more assistance. This includes recognizing gaps in knowledge or understanding about relationships from Unit 1; difficulties communicating with appropriate terminology; or weak technology skills. This informal formative assessment guides the teacher in terms of the degree of review or prompting that is required.

Activity 2.2: Ramps ’R Us

Time: 150 minutes

Description

In this activity, students use the specifications for designing wheelchair ramps to investigate slope in the “rise over run” form.

Strand(s) and Expectations

Strand(s): Number Sense and Algebra, Analytic Geometry

Specific Expectations: NA1.03, .05, NA3.05, AG2.01, .05.

Planning Notes

· The teacher may wish to show a brief movie clip that shows a person using a wheelchair. (e.g., Coming Home - the scene where John Voight is learning how to navigate a ramp; Forrest Gump - the scene where Lt. Dan uses the ramp to his boat; or use slides of ramps from local surroundings.)

· Provide students with rulers and graph paper.

Prior Learning Required

Pythagorean theorem; ratios; converting between fractions, decimals and ratios; drawing to scale

Teaching/Learning Strategies

Teacher Facilitation: Teachers need to review the Pythagorean theorem, working with ratios, converting between fraction, decimals, and ratios, and the use of scale drawings before the activity or as the need arises. Students work in pairs for this activity.

Student Activity: Ramps 'R Us

If a family member becomes confined to a wheelchair, the home must be outfitted with ramps to provide access for that person.

Here are a few criteria adapted from suggestions by rehabilitation specialists (from http://calder.med.miami.edu/pointis/ramp.html):

· The maximum incline recommended for wheelchair users is 1:12, (i.e., for each centimetre in height, the ramp must extend 12 centimetres).

· For exterior ramps in climates where ice and snow are common, the incline should be more gradual, at 1:20.

· For unusually strong wheelchair users, for extra-powerful motorized chairs, and if the person is lightweight but the pusher is strong, the ramp can have an incline of 1:7. The steepest ramp should not have an incline exceeding 1:5.

· There should be at least 150 cm of straight clearance at the bottom of the ramp.

PART A

Refer to the descriptions of the following three clients as you design wheelchair ramps.

1. Client A lives in a split-level house. He owns a very powerful motorized chair. He wishes to build a ramp that leads from his sunken living room to his kitchen on the next level. The height of the ramp must be 60 cm.

2. Client B requires a ramp that leads from her back deck to a patio. She is of average strength and operates a manual wheel chair. The deck is 25 cm above the patio.

3. Client C lives in Sudbury where ice and snow are a factor. She is healthy, but not particularly strong. Her house is a single level bungalow but the front door is 0.45 m above ground level. The path that leads directly to the front door from the street is 60 m long. Keep in mind that the ramp does not need to be 60 m long.

On a piece of graph paper design a ramp that would meet the criteria listed above and that would conserve ground space.

Construct a scale drawing for each design. Justify your reasoning for each design. Watch the units!

PART B

Mathematicians call the steepness of an incline slope. The slope can be calculated by dividing the height of a ramp (called rise) by the horizontal length (called run). This is often written as m = where m is the variable used to identify slope.

· Determine the rise and run for each ramp from part A by forming a right angle triangle under each ramp. If your design includes resting points, use only one of the inclined sections for the chart.

· Sketch and label each ramp in the space provided.

· Complete the following table.

Wheelchair Ramps
Ramp #	Diagram	Rise, Vertical Distance (cm)	Run, Horizontal Distance (cm)	Slope, m =

Questions

1. Compare the slopes of your ramps. If all of the ramps were of equal lengths, which incline would be the easiest one to push a wheelchair up? The most difficult?

2. The following is a scale drawing of a ramp. Measure the sides and determine the slope. Does this design fall within the acceptable range for wheelchair ramps?

3. How long is each of the ramps in your chart?

4. A building code requires a slope of 1:12 for a wheel chair ramp. If the length of a wheel chair ramp is 13 m and its horizontal distance is 12 m, is it safe?

5. What would be a good slope for a ski jump or skate board ramp? Explain your reasoning.

6. What is the slope of a rest platform? Explain.

7. What is the slope of a vertical wall? Explain.

Extension Question

Locate a wheel chair ramp in your community. Measure the rise and the run and calculate the slope. Does it fall within the suggested range?

Challenge Questions

1. A safe wheel chair ramp has a slope in the range from 1:5 to 1:20. If a ramp has a length of 20 m, what are the ranges of values for the rise and run of the ramp?

2. On a long ramp of any steepness or on a steep ramp, level rest platforms are needed every 3m of horizontal distance and should be 2 m long. Assume that such a ramp is a 2 m square. Given this information, how would you change your design for client C in Part A?

Teacher Facilitation: Bring closure to the Ramps inquiry by connecting steepness, developed in Unit 1- Activities 6 and 10, to slope. It might be advisable to re-visit some of the scenarios in Unit 1 and discuss them using slope vocabulary. Some of the questions in this activity provide beginning points for discussion surrounding such concepts as zero and undefined slopes.

Homework:

Students can be assigned other “rise over run” and Pythagorean theorem practice from their textbooks. Equations resulting from knowing 2 of the variables in the formula m = should also be practiced.

Assessment/Evaluation Techniques

The teacher may use this activity as an opportunity for informal, formative assessment of numeracy skills and facility with the Pythagorean theorem. This could be as simple as making observations to identify students who require remediation.

As this activity is inquiry-based, the teacher could assess students’ problem-solving skills, such as risk-taking or testing out a variety of solutions, as they design ramps to satisfy the given criteria in ramp design.

A journal entry with students reflecting on other situations where differences in slope make a significant difference (roofs, highway ramps, skateboard ramps) may also be appropriate. This journal entry would help the teacher assess the students’ application of knowledge and communication skills.

Activity 3: Slippery Slope

Time: 150 minutes

Description

Using the Slippery Slope Activity worksheet, students investigate slope as the change in y divided by the change in x. They observe and explain when and why the slope is positive or negative and compare the value of the slope of a line to its steepness and its direction. A follow-up exercise focusses on the calculation of slope using a formula, which sets the stage for determining the equation of a line given two points.

Strand(s) and Expectations

Strands: Analytic Geometry, Number Sense and Algebra, Relationships

Specific Expectations: AG2.01, .02, .05; AG3.01, .02; NA1.01, .02, .03; RE1.01, .02, .03, .04, .05, .06, .07; RE3.02.

Planning Notes

· Collect equipment needed: masking tape, metre sticks, graphing calculators, CBR™’s (Computer-Based Ranger), graphing calculator overhead display, class set of Slippery Slope Activity worksheets.

· Students work in groups of three to collect the data for the “Slippery Slope Activity.” One student walks, one is responsible for the CBR™, and one is responsible for the calculator. They may change roles for different trials.

· Students need floor space for setting up this activity.

Prior Learning Required

Activities 1.8 and 1.9 (distance/time relationships, analysing and interpreting relationships), Activities 2.1 and 2.2 (using x and y notation, using m = )

Teaching/Learning Strategies

Teacher Facilitation: For the first activity, Slippery Slope, group work should be interspersed with whole class discussion as needed. The teacher may want to refer to an overhead of the chart for this activity. Once the instructions to the students have been given, circulate and help groups as necessary. The teacher should make sure that the students are familiar with the procedure for setting up the CBR™. Students collect data and record their observations using the chart provided.

Student Activity: Slippery Slope Activity

(Modified from “Slippery Slope”, Math and Science in Motion, TI Inc.)

In this activity, you will create Distance-Time plots by moving in front of a CBR™, find slopes on the plots, and determine the formula for the slope of a line.

You will need the following materials: CBR™ unit, TI-83+ and calculator-to-CBR™ cable, metre stick, and masking tape

Collecting the Data:

1. Three students work together to collect the data. One is the walker, one controls the CBR™, and one controls the calculator. You change roles for subsequent trials.

2. Place the CBR™ on a table or desk so that the sensor is aimed at or above the walker's waist. The height can be adjusted using textbooks, if necessary

3. Put a masking tape marker on the floor at a distance of 0.5 m from the CBR™ and at a distance of 3.0 m.

4. The walker stands at the 0.5 m mark and prepares to move away from the CBR™ at a slow and steady rate. When the walker is ready the calculator person presses [ENTER] to begin the walk. The partner presses the Trigger button on the CBR™ to stop the recording when the walker passes the 3 m mark. The walker must try to keep a steady pace for the whole walk.

The plot should look like a straight line that rises gently to the right.

5. If your line is reasonably straight, sketch it in your notebook and label the graph, Trial 1, and go to question 6. If not, press [ENTER] select 3: REPEAT SAMPLE from the PLOT MENU, and try again.

6. Using the [}] and [|] keys, find and record the co-ordinates of two points on the line in the chart below. Choose points near the beginning and end, so that the line between them is the straight line that best models the plot.

7. Repeat steps 3-5 for the following motions, sketching your results for each in your notebook.

Trial 2 - moderate, steady walking away from the CBR™

Trial 3 - slow, steady walking towards the CBR™

Trial 4 - moderate, steady walking towards the CBR™.

Observations:

Answer the following questions, recording your results in the chart.

1. Refer to the graphs in your notebook and compare the four plots in terms of steepness of the line (slope), and the direction of the line (is it going up or down, from left to right?). Enter this information into the chart.

The D (delta) symbol is shorthand to represent change and is often used in physics and other disciplines. In general,

D = final condition - initial condition

Dy (delta y) and Dx (delta x) denote the vertical change and the horizontal change from starting point to end point.

2. Calculate the vertical change (Dy) by calculating the change in y from starting point to ending point (y_ending - y_starting). Enter this amount into the table.

3. Calculate the horizontal change (Dx) by calculating the difference in x from the starting point to ending point (x_ending - x_starting). Enter this amount into the table.

4. Calculate the slope by finding the .

Collecting Data						Calculations
Trial	Starting Distance	Type of Motion	Steepness and Direction of Line	Starting Point	Ending Point	Vertical Change Dy	Horizontal Change Dx	Slope (m/s)
1	0.5 m	Slow, steady away from CBR™
2	0.5 m	Moderate, steady away from CBR™
3	3.0 m	Slow, steady toward CBR™
4	3.0 m	Moderate, steady toward CBR™

Questions:

1. What pattern do you notice in the slope values in the trials where you were walking away from the CBR™? Towards the CBR™?

2. What pattern do you notice relating the steepness of the lines and the speed that you walked away/towards the CBR™?

3. What pattern do you notice relating the steepness of the line with the values of the slopes?

4. What pattern do you notice relating the direction of the line with the slope values?

5. Hypothesize the slope of the line and draw a sketch for the following situations:

(a) run away from the CBR™ (0.5 m - 3.0 m)

(b) run toward the CBR™ (3.0 m - 0.5 m)

(c) move to a point 2 m away from the motion detector, and then stop

6. Check your hypotheses using the CBR™.

Further Questions:

1. Look at the plots of the four trials. Come up with a general rule about where the graph starts on the vertical axis (distance axis).

2. What are the units for distance in this activity? For time?

3. What are the units for horizontal change? For vertical change?

4. Explain why the units for slope are in m/s.

5. Draw a sketch if a walker starts at 2.5 m away from the motion detector and walks at a rate of

(a) 2 m in 3 sec;

(b) -1 m in 2 sec.

6. Describe in words the motion indicated by this distance-time plot.

7. Create an interesting distance-time plot of your own and then describe in words the motion indicated by it.

Follow-Up Activity

Teacher Facilitation: Before beginning the follow-up activity, involve the class in a discussion of how to approximate and guess how the slope of a line segment will look. Students should be advised to consider both the steepness and the direction of the line segment. For example: ask the students to stand up so they can use their arms to approximate line segments. Ask them to imagine two points on the plane say (3, -4) and (6, 1) and then use their arms to estimate what the slope of the line segment might look like. Ask them to compare with two or three others and come to a consensus for how they should hold their arms. Repeat this process with other points that include line segments that have a variety of slopes from 0 to infinite in both directions. The ideas could be summarized in a chart that shows angles like 0°, 25°, 45°, 70°, and 90° so that students connect slope with the relative steepness or approximate angle.

Provide instructions for the pencil and paper activity below and introduce the representations of points using x and y with subscripts. Substituting into formulas might need to be reviewed, as well. The important idea here is to help students realize that the subscripts are used to designate coordinates of specific points; it doesn’t matter which is the first point and which is the second and, when substituting into the formula, it is essential that students not mix up the coordinates of the points. Discuss the different forms that could be used for m values: rational numbers in or decimal form, if both are equivalent. Include a review of when the decimal forms of fractions are approximations as opposed to exact values.

Students may also need help determining how to use a point and the slope to extend the line segment in both directions.

Student Activity

1. Construct a grid on graph paper where the points on the x-axis and y-axis are laid out as follows:

-10 #x # 10, -10 # y # 10

2. a) Plot the points A(-8, -9) and B(-3, -6). Join the points to form line segment AB.

b) Construct a right angle triangle under the line segment AB.

c) Determine the rise and the run for the line segment.

d) Calculate and record the slope using m = .

3. Let A(-8, -9) be represented by (x₁, y₁) and B(-3, -6) be represented by (x₂, y₂).

a) Calculate and record the slope using the formula m =

b) Now let B(-3, -6) be represented by (x₁, y₁) and A(-8, -9) be represented by (x₂, y₂). Calculate the slope using the formula.

c) Record your results in the table provided.

4. Repeat the process (steps 3b and c) for the other sets of points in the table.

Points	rise	run	m =	x1	y1	x2	y2	m =
(2, 6), (-10, -4)
(-10, -4), (2, 6)
(-10, 5), (-6, 2)
(-6, 2), (-10, 5)

In your notes, answer the following questions:

1. Identify any differences between the slope when it is calculated using rise over run and when it is calculated using the formula m =. Explain.

2. Identify any differences in the value of slope when point A is (x₁, y₁) and B is (x₂, y₂) and when the points are switched. Account for your observations.

3. For each pair of points, describe how you would start at one point and get to the other point using the slope. (Hint: Starting at the point (x₁, y₁) move up or down so many units and then left or right so many units to get to (x₂, y₂)).

4. Describe how you could find a new point on the line if your line segment needed to be extended.

5. You have now worked with three ways to calculate the slope of a line segment. Explain which method is the most convenient and use it to calculate the slope for each of the following:

a) b)

c) Line through the points (-3, 1) and (4, -5)

Homework:

Include practice using the formula and extending lines using the slope and a point. Students could be assigned a journal entry asking them to relate the three methods of calculating the slope and describe situations or problems where they would use each form.

Assessment/Evaluation Techniques

The teacher could make observations on Learning Skills, such as teamwork and initiative, while the students are performing this activity.

If students submit their answers to the questions associated with this activity, then teachers could assess Communication Skills by focusing on: the clarity of the students’ descriptions of the patterns that they recognized, and the use of appropriate mathematics terminology and appropriate units on axes. Reviewing students’ journal entries also aids in the assessment of Communication.

A summative quiz could be given to ensure that students are able to calculate the slope of a line segment using various formulas; identify properties of line segments and graph line segments using the slope and a point.

Accommodations

· The teacher may have to lead parts of the instruction using a CBR™ and an overhead graphing calculator. Having students that benefit from kinesthetic activities model the activity may be helpful. Some individual instruction and prompting may be required.

· Some students may have difficulty describing how to graph line segments in both directions using a starting point and a slope. Some instruction/prompting may be necessary for some students.

Activity 2.4: Programs for Sale!

Time: 300 minutes

Description

In this activity, students develop an understanding of the formula y = mx + b by making a connection between the initial conditions and the y-intercept, b. Instructions guide students to use the TI-83+ as a spreadsheet to generate tables of values, thereby offering an alternative way to answer questions where one of the variables is known. Points of intersection are found graphically and interpreted in context. As a follow-up, students develop their algebraic skills of collecting like terms, using the distributive property, and common factoring, and apply these new algebraic skills to solving linear equations. Students also learn to form the equation and graph the line whose slope and y-intercept are known.

Strand(s) and Expectations

Strand(s): Analytic Geometry, Number Sense and Algebra, Relationships

Specific Expectations: AG2.02, .04, .05; AG3.02, .03, .04, .05, .06; NA2.06; NA3.01, .02, .03, .04; NA4.01, .02, .03; RE3.01.

Planning Notes

· Graphing calculators are required for Activity 2.4.

Prior Learning Required

Students should know how to make a table of values for a linear relation from a description of a situation, graph from a table of values, compute slope of a line, form the equation of a linear relation, solve a linear equation by inspection or systematic trial, and use graphing technology to graph an equation in y = form.

Teaching/Learning Strategies

Teacher Facilitation: Students should be organized in pairs for this activity. The teacher intersperses students working in pairs with whole class discussion, as needed. Students should begin pair work on Programs for Sale! and Time out for Technology. Some class discussion should follow and then pairs should continue their work, moving to the activities: Ringaling, Reliable Appliance Repairs, and Decisions, Decisions.

Students are expected to write equations for each situation using x for the independent variable and y for the dependent variable. Intersperse pair work with whole class discussion and instruction. Some of these questions could be assigned as homework.

Students generate intersecting lines in Decisions, Decisions, Decisions! They are expected to find points of intersection graphically. Algebraic techniques are not expected until Grade 10.

Students are expected to use inspection or informal methods for solving the equations in the problems. Instructions on formal methods for solving equations should be scheduled after the worksheets are completed and before the 2.4 homework sheet.

Student Activity: Programs for Sale!

Carolyn gets a job at the Sky Dome selling programs. She is paid $10 per game and she is given 25 cents for every program she sells.

1. Create a table of values showing the pay she can expect for a game where the number of programs she sells varies from 0 to 100. Use increments of 10 programs in your table.

2. Calculate the first differences. What do they tell you about the relation?

3. What is the rate Carolyn is paid per program?

4. Write an equation.

5. Graph the relation. Should you connect the points? Why or why not?

6. What is the slope of your line? Compare your answer to #2 and #3?

7. Examine your table. What is her pay for selling 0 programs? Where does this information show on your graph?

8. A common method of writing equations of linear models is y = mx + b, where b is the initial value of the dependent variable and m is still ___________ (Fill the blank.)

9. How much would Carolyn be paid if she sold 45 programs? How could you use your graph to answer the question? Show how to use your equation to answer the question.

10. How many programs would Carolyn need to sell to earn $26.50 for a game? How would you use your graph to answer the question? Show how you would use your equation to answer the question.

11. What would be the equation if Carolyn were paid:

a) $15 per game, plus 25 cents per program sold?

b) $20 per game? $18 per game plus 30 cents per program?

c) $15 per game and 50 cents per program?

Student Activity: Time Out for Technology

Graphing calculators can be programmed much like a spreadsheet on a computer to give up-datable tables of values. You can choose the increments for x. Follow the example below on your TI-83+.

The table of values you generated in Programs for Sale! could be done on a TI-83+ as follows:

a) Clear lists 1 and 2

STAT

1:Edit

Up arrow to highlight L1

CLEAR

ENTER

Repeat for L2

b) Enter the independent variable values in L1

STAT

1: Edit

Up arrow to highlight L1

LIST (or 2^nd STAT)

Right arrow to OPS

Down arrow to 5:seq(

ENTER

X,X,0,100,10)

{This sets 0 as the lowest, 100 as the highest, and 10 as the increment for X}

ENTER

c) Enter the dependent variable values in L2

Right and Up arrows to highlight L2

“10+.25L1” {This

ENTER

d) Changing values in L1 automatically updates L2 since you have put quotation marks around your formula for L2. You might want to try to do some updates to see how to use your calculator to answer questions 9 and 10. For example, if you guessed that the answers to 9 and 10 involved between 40 and 80 programs, change L1 as follows:

Left and Up arrows to highlight L1

CLEAR

LIST

Right arrow to OPS

Down arrow to 5:seq(

ENTER

X,X,40,80,1)

ENTER

e) You can now use the Down arrow to scroll down the lists to read the amount Carolyn was paid for selling 45 programs, or how many she has to sell to earn $26.50.

Carolyn is paid $21.25 for selling 45 programs. Carolyn has to sell 66 programs to earn $26.50.

Alternatively, you can form the equation of a relation with x as the independent variable and graph it using Y =. You can read the lists that the calculator has generated by going to TABLE (2^nd GRAPH). Use the arrow keys to scroll through the lists to read off values. The table starts at x = 0 and show increments of 1 unless you go into TBLSET (2 ^nd WINDOW). On this screen you can set TblStart to start the table with whatever x value you want, and êTbl to set whatever increment you want.

Student Activity: Ringaling!

A phone company charges $15 per month for an emergency cell phone service, plus a call charge of $1 per minute.

1. Use your TI-83+ to create a table of values showing the total charges for one month where the calls vary from 0 to 60 minutes.

2. Graph the relation.

3. Identify the slope and y-intercepts of your line. How do these relate to the phone charges?

4. Describe 3 different ways that you could find out how much the company would charge for a month when 45 minutes of calls are made.

5. Describe 3 different ways that you could find how many minutes were used for calls if the monthly charge was $43.

6. How would your graph be different if the company charged the minute rate for every full minute or portion of a minute used? [The graph you should have guessed is called a step graph. Do you see why?] Using this step model, what would the phone company charge you for a 30 second phone call? for 40 minutes and 2 seconds of calling during the month?

Student Activity: Reliable Appliance Repairs

The Reliable Repair Company charges $25 per visit plus $60 per hour for labour.

1. Use your TI-83+ to create a table of values showing the total charges for a repair that could take from no time at all to 8 hours.

2. Graph the relation.

3. Identify the slope and y-intercepts of your line. How do these relate to the repair charges?

4. How much would be charged for a repair that takes 3 hours and 20 minutes of labour?

5. If this company submitted an invoice for $160 for a repair at your home, how much time should have been spent on labour?

6. If the Reliable Repair Company changed their hourly rate to $65, but kept the fixed rate of a house call at $25, what would be the equation for their new charge structure? How would this change affect the graph?

7. If the Reliable Repair Company changed its fee per visit to $30 but kept their hourly rate at $60, how would the equation and graph compare to the original?

Student Activity: Decisions, Decisions, Decisions!

You are helping a friend decide which cellular phone package is best for her to buy. The cost of purchasing the phone is the same in all cases. Crystal Net charges a flat fee of $27 for up to 200 minutes per month, while Rover charges $12 per month, plus $0.15 per minute, and Ring Mobility charges $15 per month, plus $0.10 per minute.

1. Use your calculator to graph each relation, using different markings for each one.

2. For which time value are the costs the same for the Rover and Crystal Net options?

3. For which time value are the costs the same for the Rover and Ring Mobility options?

4. For which time values is Ring Mobility a better deal than both Rover and Crystal Net?

5. If your friend expects to use her cell phone between 40 and 100 minutes per month, explain to her which service you think she should choose.

6. If you were in charge of marketing for Ring Mobility and it was your goal to increase your market share, how could you adjust your pricing formula to attract customers from the other two companies? State the equation for your pricing structure. Explain the decisions you made in forming your equation, remembering that you have to make money for the company as well as attract as many customers as possible.

Follow-Up Skills Lessons:

Teacher Facilitation: Lead students to graphing equations of the form y = mx + b without a table of values. Practice these.

Use a situation like the following one to illustrate to students that 10x + 20x = 30x since the graph of y = 10x + 20x and y = 30x are the same.

Student Activity:

Conrad delivers 2 different newspapers to the same set of homes. His pay is 10 cents per paper from the Oakville Beaver and 20 cents per paper from the Oakville North News.

1. Using x to represent the number of homes and y to represent his pay, form equations to represent his weekly pay from each newspaper and his total combined pay.

2. Create tables of values for these 3 relations, including values from 0 to 40 homes.

3. What is the relationship between the pay scales for each of the two papers and the combined total? How can this be shown algebraically?

4. Enter your equations in a graphing calculator where y₁= 10x + 20x and y₂ =30x.

Teacher Facilitation: Since these two lines are on top of each other, instruct the students as to how to get different markings for the two functions. Arrow to the left of y₂ and press enter 5 times until the circle option is displayed.

5. Compare the two displays.

6. Go into the table menu to view the values in the two tables. How do the values compare?

Teacher Facilitation: Lead students to conclude that 10x + 20x = 30x and explaining why this is so, within the context.

Use algebra tiles or Virtualtiles for concrete or visual manipulation to extend the algebraic skills of combining like terms. Students may well have used integers tiles in elementary school. To build on their use of colour and area from integer tiles, use the same two-colour distinction to differentiate between positives and negatives. To model x, use a tile that is x units long and 1 unit wide (the same unit size as used for each side of an integer square). To model x², use an x by x tile.

Introduce the distributive property and common factoring using an example like the following.

Student Activity:

A taxi charges a flat rate of $2 plus 16 cents per kilometre traveled.

1. Construct a table of values for a single trip that range from 0 to 100 km.

2. Identify the slope and y-intercept, and use them to form the equation for this relation, using x as the distance traveled and y as the charge for the ride.

3. Suppose a person is using the taxi daily to get to and from work. Construct a table of values for their weekly charges (10 trips)

4. Identify the slope and y-intercept, and use them to form the equation for the relation in #3.

5. Realizing that your first equation represents the cost for 1 trip, what would you expect to have to determine the cost for 10 trips? Write an equation with this idea in mind.

6. Graph your relations in #2 and #5 on the same axe, allowing up to $200 in charges. How do your graphs compare?

Teacher Facilitation: Lead students to the realization that since the graphs of the relations

20 + 10 x 0.16x = 10(2 + 0.16x) since the graphs of the relations y = 20 + 10 ´ 0.16x and

y = 10(2 + 0.16x) are identical.

Another situation that illustrates the distributive property or common factoring is: One pizza costs $6, plus $1 per topping. Find the cost of 4 pizzas if each pizza has x toppings. [4(6+1x) and 24+4x are the two ways that you could form the expression, so these must be equivalent] Introduce the concept that distribution and common factoring are inverse operations.

Use algebra tiles to further students’ understanding of the algebraic operations of expanding [given the length times width form, find the area of the rectangular region], and common factoring [given the area of a rectangular region, find the length and width]. For example, this pictorial representation of algebra tiles shows that x(2x + 3) = 2x² + 3x. Tie this statement to the formula: length ´ width = area of a rectangle.

Teach students how to apply patterns to develop the exponent rules for multiplying and dividing monomials, then practise these rules. Solve linear equations involving integral and fractional coefficients. For this work, teachers should use discretion in choosing the upper level of difficulty they assign to specific groups of students.

Follow-up Homework: (Note that you may wish to use #7 as a pairs assessment task)

1. A salesman is paid $600 per week, plus 10% on sales.

a) Create a table of values showing his pay in a week where his sales might vary from $0 to $2 000.

b) Write an equation for the relation and sketch the graph.

c) How much is the salesman paid if he makes $1 500 worth of sales?

d) What sales need to be made to earn $850 in a week?

2. A waitress is hired at Lionel’s Lunch Spot. She must pay $50 for a uniform (which is deducted from her pay) and she earns $6 per hour, before tips are added.

a) Create a table of values showing her pay without tips for her first week, where she might work 0 to 40 hours.

b) Write an equation and sketch a graph.

c) How long would she have to work to earn $142 before tips?

d) How many hours does she have to work to pay off her uniform? How is this information shown on your graph?

3. A car’s gas tank holds 60 L of gasoline and the car can travel a total of 510 km on a full tank of gas.

a) What does this information tell you about the graph of the relationship between fuel remaining and distance travelled? Draw the graph.

b) How much gas is left in the tank when the car has travelled 100 km?

c) How far has the car travelled when 38 L of gas remains?

d) The “low level” warning light appears when there are 5 L of gas remaining in the tank. How much farther could the car go, once the warning light comes on, before running out of gas?

e) What is the rate of fuel consumption in kilometres per litre? Write an equation which represents the amount of gas remaining in terms of the distance travelled.

f) If the car gets a tune up, the fuel efficiency will increase to 9 kilometres per litre. How will this change the equation? How many more kilometres will the car be able to travel on a full tank of gas?

4. The cost to manufacture 200 baseball caps is $1 400. The cost for 500 is $2 600.

a) Assuming that it is a linear relation, graph it.

b) What is the initial cost if no hats are manufactured? Explain the answer.

c) What is the cost for each hat added? How does this answer compare to the graph?

d) Form an equation that expresses the cost in terms of the number of hats that are purchased.

5. Describe a situation that could be modelled by the following equation:

a) d = 2t + 3, where d represents distance and t represents time

b) c = 50b + 100, where c represents cost and b represents number of books

6. Ali is moving to Ottawa as a bicycle salesman. He is offered two methods of payment. Method A is a salary of $400 per week plus 5% commission on sales made. Method B is a straight commission of 15% on sales made.

a) Form the equation for each method of payment.

b) Graph both equations on the same grid.

c) Under what conditions will the two methods of payment result in the same earnings? Explain how you arrived at your answer.

d) What advice would you give to Ali about his choice on method of payment?

e) If Ali could change his payment method on a monthly basis, what advice would you give him? Explain.

7. Create a situation of your own and the equation(s) which models it. Pose different types of questions. Show or explain how to answer the questions you posed.

Teacher Facilitation: Ask the following types of questions as homework is taken up:

i) How is the slope identified in each question? [the unit rate]

ii) How is the b value identified in the situations? [the fixed initial quantity. Now is a good time to introduce the terms “y-intercept” and partial variation]

iii) In what type of situation will the b be negative? Slope be negative?

iv) When answering questions that require interpolation, which method do you prefer - using the equation, or looking at the graph, or reading from the lists on the calculator? Why?

Assessment/Evaluation Techniques

Question #7 in the Follow-Up Homework could be used as an in-class pairs assignment. The teacher could assess the Communication and Application Categories of the Achievement Chart by having students present orally or in written form.

As students work in pairs, the teacher could assess Learning Skills such as Team Work and Organization using rubrics provided earlier.

After this activity has been completed and concepts from Activities 2.1 to 2.4 consolidated, it would be appropriate to give a cumulative test on concepts learned through these activities. Questions on the test could cover the four categories of the Achievement Chart and assess the following expectations:

Knowledge: Calculations of slope using a variety of slope definitions; identifying slope and y-intercepts in y = mx + b form of the equation of a line; solution of equations

Thinking/Inquiry: Justifying reasoning within the solution to a problem

Communication: Describing the meaning of slope and y-intercept in context; describing situations that could be modeled by a linear equation; using appropriate mathematics symbols and terminology: slope, y-intercept, labeling axes, constant rate, fixed cost

Application: Relating abstract concepts of slope and y-intercept to realistic contexts; recognizing the purpose and value of measures of slope in a variety of contexts (highway ramps, skateboard ramps, roofs)

Include questions that allow students to demonstrate Level 4 achievement of the expectations need to be included. Do this by asking extending questions and providing opportunities for students to explain their reasoning or show a more efficient or creative solution. Open-ended questions provide opportunities for students to demonstrate all four Categories at all four Levels of Performance.

Accommodations

The use of algebra tiles helps students who benefit from work at the concrete and pictorial stages before they abstract. They appeal to visual and kinesthetic learners. The use of Virtualtiles also helps visual learners.

Resources

Lesage, J., B. Scully, and J. Scully, “Alge-Tile” Resource Binder, Exclusive Educational Products.

Activity 2.5: What's My Spring? Stretching a Penny

Time: 75 minutes

Description

Students collect data using a CBR™, a slinky, and pennies (or similar experiment using springs or elastics and a ruler) to determine the relationship between the distance a spring is stretched and the mass (number of pennies) attached to it. Students use the slope/y-intercept form to determine the equation which models the data. They also determine the equations of graphs which represent other spring relationships and be expected to explain the physical meaning of the slopes and the y-intercepts in the contexts provided and to use their equations to answer other questions.

Strand(s) and Expectations

Strand(s): Analytic Geometry, Number Sense and Algebra, Relationships

Specific Expectations: AG2.04, .05; AG3.02, .03, .04, .05, .06, .07; NA3.02; NA4.01, .03; RE1.01, .02, .03, .04, .06, .07; RE2.02, .03; RE3.03.

Planning Notes

· For each group the teacher sets up a slinky suspended from the ceiling (or see your Science department for springs, retort stands, and clamps), a paper plate attached to it (to hold pennies), 30 pennies, a CBR™, and TI-83+ calculator to collect the data. See CBR™ Explorations book for more detailed information.

· The plates may need to be secured to the slinkies with ribbon and taped on (so that they do not slip off).

· A worksheet needs to be prepared to record their data, determine the equation of the line of best fit, give meaning to the slope and y-intercept in the given context and include extending questions. See Appendix 2.5A - Stretching a Penny Worksheet.

· Students who did this experiment in Activity 1.10 with a CBR™ could recall their data at this time and answer the questions from the work sheet.

Prior Learning Required

· How to determine the equation of a line by identifying the slope and y-intercept;

· How to use a graphing calculator to enter data into lists, graph an equation, and set appropriate parameters in the WINDOW.

Teaching/Learning Strategies

Student Activity

Students work in groups of three on the Stretching A Penny Worksheet (Appendix 2.5A), recording distances of the plate from the CBR™. They then answer the worksheet questions about the relation.

Teacher Facilitation: To develop a context before starting the activity, the teacher could ask the students where the relationship between spring length and weight might be used. Some possible answers might include a bungie jump, a trampoline, or a mattress. Discuss possible sources of error and emphasize that the CBR™ should not record until the plate has stopped moving. Circulate around the room helping students as they collect their data and analyse it. Students may need to be reminded to find the slope and y-intercept as a way to find the equation. Data collected can be inserted into their calculators as lists. They could then verify the correctness of their equation by entering it into the calculator.

Note: The equation is to be adjusted if it is not a fitting model for the data. Help may be needed with the sign of the slope.

Suggested Homework: Refer to Appendix 2.5B – What’s My Spring? The first two grids show graphs of the length of a spring when a mass is attached to it. The third grid shows the graph of the length of a mattress spring when a mass is placed on it.

Follow-up: Appendix 2.5C applies student knowledge from the first two activities to the context of temperature.

Teacher Facilitation: Note that the temperature intervals given in the charts are not equal to 1°.

Students cannot use First Differences to calculate the slope. When the solutions are taken up from the Frozen Balloon activity, question #5 can be expanded further by bringing out the fact that according to the kinetic molecular theory, a sample of gas would exert zero pressure only when the molecules are not in motion. This would be the case when the temperature is at absolute zero. Absolute zero is the coldest possible temperature. The accepted value for absolute zero is -273.18°C.

Assessment/Evaluation Techniques

As students work in groups of three on the investigation Stretching a Penny (Appendix 2.5A), the teacher can circulate, observe, and assess: Learning Skills such as Teamwork and Thinking/Inquiry Skills by examining student’s ability to pose and answer questions, and to make and verify predictions.

The two problems posed in Appendix 2.5C could be used as assessment activities. Students could work individually or in pairs and either present their solutions or submit a written solution. If class presentations are used then peers could assess the organization of data into graphs as well as the students’ use of correct mathematical words. For either written submission or presentation, the teacher could assess the students’ ability to create the equation of a linear relation from a context, and the effective use of technology.

Resources

Coxford, A., et al. Contemporary Mathematics in Context. P.O. Box 812960, Chicago, Il 60681: Everyday Learning Corp., 1997. ISBN 1-57039-475-X

MCTM (Montana Council of Teachers of Mathematics)/SIMMS. Integrated Mathematics: A Modelling Approach Using Technology. 401 Linfield Hall, Bozeman, MT 59717-2810: Simon & Schuster Custom Publishing.

TI. Explorations with the CBR™.

Appendix 2.5A: Stretching a Penny

Calculate the first differences by filling in the following table.

Number of pennies	Distance to the Plate (m)	First Differences (m)	Stretch per Penny (m)
0
5
10
15
20

Questions:

1. Is this data linear? Justify your answer?

Enter your data from your table to Lists in your calculator.

Turn STAT PLOT on and graph the data. Does it appear linear?

2. From your table, what do the First Differences represent?

3. Calculate the stretch per penny for each First Difference.

4. Calculate the average stretch per penny.

5. What is the average rate of change in this relation (the steepness of the line)?

6. Using d for distance and p for number of pennies, predict the equation for this relation.

Test your hypothesis by replacing y for d and x for p in your equation and entering this in your calculator using the Y₁= key. Does your equation produce a fitting model? If not, modify your equation.

The calculator has a built-in feature that allows it to compute best-fitting line through a set of data. This procedure is called a linear regression. To perform a linear regression on the data you have entered in your lists, press 'STAT', 'CALC'. Select LinReg, L₁, L₂, Y₂ (VARS,YVARS, FUNCTION,Y₂). Press ENTER.

How close is the regression equation in Y₂ to your equation in Y₁?

Press GRAPH to display the data, your line in Y₁ and the regression line in Y₂, all on the same screen.

7. Suppose you had 100 pennies on the plate. Calculate the amount of stretch for this number.

8. a) Grab a big handful of pennies and place them gently on the plate. Trigger the CBR™ and collect the distance as you did earlier. Record this distance.

b) Calculate the number of pennies.

c) Count the number of pennies on your plate. How close was your prediction?

9. Suppose that the stretch was 0.5 m. How many pennies would have been on the plate? Confirm your prediction.

10. Suppose you had $12.52 worth of pennies (assuming a slinky could stretch indefinitely), how long would they stretch the slinky?

11. How would using quarters affect the steepness of the line?

Appendix 2.5B: What’s My Spring

For the following spring graphs determine the equation of each relation by identifying the slope and y-intercept.

1. Three Coil Springs

Identify the independent and dependent variables. Identify the slopes and y-intercepts. Why are the slopes the same? Why do the lines start at different points? Find the slopes then write equations for the relations. What is the physical meaning of the slopes and the y-intercepts?

Using your equations determine the lengths for the three springs if a 5 g mass is attached to each of the springs. Check your answer using the graph. Determine the length of each spring if 30 g is attached to it.

2. Four Coil Springs

Identify the independent and dependent variables. Identify the slopes and y-intercepts. Why are the slopes different? Why are the y-intercepts the same? Find the equations for the relations. What is the physical meaning of the slopes and the y-intercepts?

Using your equations determine the lengths for the four springs if a 10 g mass is attached to each spring. Check your answer using the graph. Determine the length of the springs if 50 g is attached to each of them.

Appendix 2.5B: What’s My Spring (Continued)

3. Mattress Springs

Identify the independent and dependent variables. Why is the length of the spring getting shorter? How will this be identified in the value of the slope? Identify the slope and y-intercept. Why are the slopes different? Why is the y-intercept the same? Find the equations for the relations. What is the physical meaning of the slope and y-intercept?

Using your equations determine the lengths if each spring has a mass of 20 kg placed on them. Check your answers using the graph. How long would the springs be if each had an 80 kg mass on it?

Appendix 2.5C: Temperature Applications

Crickets Beat the Heat

Crickets make chirping sounds by rubbing their wings together. For some crickets, the relationship between the number of chirps per minute and the air temperature is very close to being linear. Use the following table of values to answer the given questions.

Temp (°C)	18	20	22	25	27	30
Number of chirps	152	124	181	204	172	240

1. Graph this relation on a scatter plot. Let x represent the temperature in degrees Celsius and y represent the number of chirps per minute.

2. Draw a line that fits the data as closely as possible.

3. Identify the y-intercept of the line. What does this intercept suggest? Is it reasonable to extrapolate the data to this point?

4. Write an equation of the line in the form y = mx + b.

5. Enter your data points onto a graphing calculator or computer. Graph the equation of the line. Does the equation produce a fitting model? If not, go back and make the necessary adjustments.

6. Predict the temperature at which these crickets make 160 chirps per minute.

7. How many chirps would they make if the temperature is 23°C?

A Frozen Balloon

When an inflated balloon is placed in a freezer, its volume decreases as the air inside it grows colder. When the balloon is removed from the freezer, its volume increases as it warms. The following table shows some data comparing the temperature of air in the balloon to its volume.

Temperature (°C)	Volume (mL)
5	513
10	520
15	532
20	544
25	550
30	559
35	569
40	578

1. Make a scatter plot of this data. Let y represent the volume in millilitres and x represent the temperature in degrees Celsius.

2. Draw a line that closely approximates the data.

3. Write an equation of the line in slope/intercept form.

4. Use your equation to predict the volume of the balloon at an air temperature of 100°C.

5. Extrapolate back to find the x-intercept. (You may need to adjust the values in your window to do this.) What is the physical meaning of this?

Activity 2.6: Sunshine, Whiskers, and Windmill

Time: 150 minutes

Description

Students investigate the slopes of parallel and perpendicular lines by graphing equations in the form y = mx + b using a graphing calculator or computer. They also investigate lines with the same y-intercept, lines symmetric about the x and y axis, and vertical and horizontal lines.

Strand(s) and Expectations

Strand(s): Number Sense and Algebra, Relationships, Analytic Geometry

Specific Expectations: NA3.05; RE1.05; AG2.02, .04; AG3.02, .03, .04, .05.

Planning Notes

· This activity could be done in pairs or small groups.

· Graphing calculators or computers are needed for each pair or group of students

Prior Learning Required

Graphing lines using technology; Identifying the slope and y-intercept from an equation

Teaching/Learning Strategies

Teacher Facilitation: The teacher may wish to have students write all they know about equations and graphs of lines in about two minutes of individual work. Gather students’ ideas and organize them on the blackboard. This full class review of everything students know about equations and graphs of lines is followed by small group investigation.

Student Activity: What’s My Property?

As you work through the following investigations make notes of what you have discovered.

TI-83+ Instructions:

· Turn on grid points: FORMAT (2nd; Zoom); highlight GridOn; ENTER

· Set the WINDOW parameters as follows: Xmin = -5; Xmax = 5; Xscl = 1; Ymin = -5; Ymax = 5; Yscl = 1

· Make the appearance of the X and Y scales the same on the calculator screen. Press Zoom; press 5 (to select the Zsquare feature)

· Now, go back to the Window screen to see what is new. Why are the X max and min values different?

Activity Instructions

As you work through the following investigations make notes of what you discover

1. a) For each group (column) of equations identify what is the same and what is different.

Using a graphing calculator or computer sketch the groups of graphs in your notebooks, graphing each group on its own set of axes. Clear the equations from each group before beginning on the next.

b) How are the lines for each group related?

A	B	C
y = 3x	y = -2x	y = ½x
y = 3x + 3	y = -2x +3	y = ½x + 3
y = 3x -1	y = -2x - 1	y = ½x - 1
y = 3x - 3	y = -2x - 3	y = ½x - 3

2. For each pair of given lines identify the slopes, graph them on the same set of axes and determine the relationship between the lines. Clear the equations from each pair before beginning on the next.

A	B	C
y = x	y = 4x - 2	y = ⅔x
y = -x	y = -4x -2	y = -⅔x

3. a) Graph the following groups of lines by creating a table of values first. Note that y = 2 means that the value of y is 2 for every value of x. Graph each group on its own set of axes.

b) How are the lines in column A related? Pick any 2 points on a line from column A and calculate the slope. How does the slope relate to the graph?

c) How are the lines in column B related? Pick any 2 points on a line from column B and calculate the slope. How does the slope relate to the graph?

A	B
y = 2	x = 2
y = 5	x = 5
y = -2	x = -2
y = -5	x = -5

4. a) Sketch the graphs of the following pairs of lines on the same set of axes. How are the pairs of lines related? How could you verify this?

b) Express the slopes for each pair of lines in fraction form. How are the numerators and denominators related? What is the product of the slopes in each pair?

A	B	C
y = 2x + 1	y = ¼x - 2	y = ⅔x - 4
y = -½x + 3	y = - 4x +5	y = - x + 2

5. a) For the following lines describe what the graphs look like without graphing the line. Then, if possible, confirm your conclusions by graphing using a calculator or computer.

b) Identify pairs of lines that are related, (i.e., parallel, perpendicular, same y-intercept, reflections). For each pair of lines, explain how the equations indicate the relationship.

y = -1

x = 4

y = x - 1

y = -x + 4

y = -x - 1

6. Summarize what you now know about the graphs of lines and their equations. Submit your conclusions to your teacher.

Teacher Facilitation: Introduce and discuss the “negative reciprocal” relationship. Help pairs as needed. A comparison to transformations could be made (shift up/down, reflection in the x or y axis). Discuss slopes of vertical lines and x-intercepts with the whole class. This would also be an excellent opportunity to reconnect slope as a rate of change and y-intercept as an initial condition.

Follow-Up Practice and Assignment:

Teacher Facilitation: Have students practice (i) Determining equations of lines given two characteristics about the line (slope and y-intercept; y-intercept and parallel or perpendicular to a line, vertical or horizontal line and a point or intercept, given a graph containing the preceding information).

(ii) Graphing lines given equations in the form y = mx + b;

Have pairs of students sketch the graphs of linear relations, one student sketching by hand, while the partner uses technology. Compare results, then exchange roles.

Have students work on questions like the following to identify the role of slope and y-intercept from real life situations.

Student Activity:

1. Each equation given below models a stretching or compression experiment with a spring where L is the spring length in centimetres and m is the mass acting on the spring in grams.

Identify the following:

· the initial length of spring

· the rate of change of length

· whether the experiment was designed to measure spring stretch or spring compression

(i) L = 5m + 2

(ii) L = 3.6m + 4

(iii) L = -2.5m + 1

(iv) L = -0.3m + 6

2. The student athletic council at Valley High School operates a bottled water machine near the gymnasium. The council is paid $50 per month plus $4 per case sold.

(i) Find an equation showing the monthly income as a function of the number of cases sold. Use your equation to calculate the monthly income if 15 cases are sold. Find the number of cases that need to be sold for an income of $138.

(ii) What would the equation be if the council was paid $4 per case plus $60 per month? Is the new situation better or worse for the council? Explain.

(iii) What would the equation be if the council was paid $50 per month plus $5 per case? Is the new situation better or worse than the original for the council? Explain.

Assessment/Evaluation Techniques

Student submissions of the investigation could be assessed for:

· communication and justification of conclusions about relationships between graphs

· recognition of slope and y-intercept in context

· identification of patterns, properties, and relationships of slopes

· ability to determine the equation of a line

The following assignment Line Designs also aids in the assessment of the student’s ability to determine equations of lines. The student’s engagement in designing their own creation could serve as an indicator of initiative.

Line Designs

RAYS OF SUNSHINE

a) Write the equations of the lines that create the following pictures on a graphing calculator or computer screen and hand in your solutions.

Rays of Sunshine

Kitty’s Whiskers

Windmill

b) Design a creation of your own. Include your picture and equations. Give your picture an appropriate title.

Activity 2.7: Break the Bank!

Time: 225 minutes

Description

Students investigate linear equations in y = mx + b form and Ax + By + C = 0 form to determine that both forms of the linear equation graph the same line. This activity provides a context for revisiting the solution and rearrangement of linear equations. Students rearrange linear equations of various forms, (e.g., y = mx + b, Ax + By + C = 0, x = a, and y = b). As a follow-up, students determine the x- and y-intercepts of a line given its equation in Ax + By + C = 0 or Ax + By = D form and graph lines given their equations in the various forms listed above.

Strand(s) and Expectations

Strand(s): Number Sense and Algebra, Analytic Geometry

Specific Expectations: NA1.01, .05; NA3.01, .03, .04; NA4.01; AG3.01, .02, .03.

Planning Notes

· Students use graphing calculators for this activity.

· Overhead transparencies with the same grid markings and intervals must be prepared ahead of time. Water-soluble overhead markers are also needed.

· Prepare a variety of sets of equations (two or three in each set) that represent the same linear graph:

e.g., x - y = 4, x - y - 4 = 0, and y = x - 4

or 2x + y = 4, y = -2x + 4, and -2x - y + 4 = 0

Put each equation on a strip of paper for distribution to pairs later in the lesson.

· Teachers should explain to their students that Ax + By + C = 0 and Ax + By = D are both common forms for a linear equation. Both forms should be included in examples discussed in class and assigned to students.

Prior Learning Required

Students should be competent in graphing equations in “y =” form using graphing technology and in creating and using a table of values to graph equations not in that form. Students are also familiar with rearranging and solving equations from Activity 2.4.

Teaching/Learning Strategies

Teacher Facilitation: Begin this activity by building on the students’ intuitive sense of relationships between numbers. For instance, ask students to list, as ordered pairs, five sets of numbers that have a sum of 15. (Or use a scenario such as: there are 15 coins in a piggy bank, made up of nickels and dimes.) Ask a few students to name their ordered pairs and then in a full class discussion ask how to write the equation if one number is x and the other is y. You should receive the response x + y = 15 but you may also receive y = 15 - x. If not, then ask students what the second number of an ordered pair would be if the first number was ten. Then ask how they calculated the number. Hopefully someone says something like: “You just subtract the number from 15.” Talk to students about how to write this as an equation (y = 15 - x). Discuss with students whether the two equations are the same. These could be checked in a variety of ways: compare hand-drawn graphs; enter y = 15 - x into the graphing calculator, look at the table and compare it to the original table of values; re-arrange one equation to a form similar to the second. Similar scenarios could be pursued (e.g., how many of each of two kinds of donuts make up a dozen; how many boys and girls there are in a class of 30; the length and width of a rectangle if its perimeter is 24; the difference in age between John and his brother is five years;)

Using the prepared sets of equations, give one equation to each pair of students. Include equations like y - 2 = 0 and y = 2. Note that some of the equations can be done easily with a graphing calculator, while others cannot.

Student Activity:

Each pair of students graph their equation on the pre-gridded acetate using a graphing calculator or a table of values as necessary. Using the overhead, students from one pair display their graph and ask if another group has a graph that matches their graph. Place the match the graph on top to confirm. When the matching graphs are found, the presenting pair write the equations on the board.

Teacher Facilitation: Discuss patterns and similarities in each set of equations before beginning the next. Discuss which form of the equation is easiest to graph. When students choose y = mx + b as the easier form for graphing, use this as a lead in to practice how to re-arrange equations. Build on their intuitive sense from the first example and from their knowledge of solving equations from Activity 2.4. Include examples with fractional coefficients as in x + 2y = 6 as well as examples set in context such as; converting Fahrenheit temperatures to Celsius; relating the speed of sound to temperature, v = 330 + 0.6T, where v is speed in m/s, 330 is the speed of sound in m/s at 0EC, and T is temperature in degrees Celsius; the perimeter of a rectangle 2(l + w) = P, where l is length in metres, w is width in metres, and P is perimeter in metres.

Homework/Extension

Assign textbook practice on re-arranging linear equations into the forms Ax + By + C = 0, y = mx + b,

x - a = 0, x = a, y - a = 0 and y = a. The focus should be on re-arranging Ax + By + C = 0 into y = mx + b with limited practice on converting the other way.

Find slope and y-intercept of equations in the form Ax + By + C = 0 by writing the equation in the form y = mx + b.

Follow-up Algebra Lessons:

· Determine the x- and y-intercepts of a linear equation in the form Ax + By + C = 0 or Ax + By = D, and use them to graph the line.

· Graph a line given its equation in Ax + By + C = 0, y = mx + b, x = a or y = b form.

Assessment/Evaluation Techniques

· A rubric for observation could be used as the students complete the graphing activity, to assess students’ ability to work in small and large group situations, solve problems, graph lines using a table of values, recognize patterns, and make conclusions.

· Communication skills could be assessed through journal writing, as students explain how and why they used graphing calculators or tables to graph lines, which method they prefer, and why, and when it is appropriate/necessary to graph using a table or calculator.

· Paper and pencil tasks or a short quiz could assess student’s ability to re-arrange linear equations, solve equations in one variable and graph a line using x- and y-intercepts.

Activity 2.8: Fireworks and Twinkle, Twinkle

Time: 75 minutes

Description

Students find the equation of a line using two points on the line. They recognize that any two points on a line lead to the same equation. Using this new skill they then determine the equations for distance-time data collected using a CBR™ or CBL. This activity also offers an extension leading to piecewise functions.

Strand(s) and Expectations

Strands: Analytic Geometry, Number Sense and Algebra, Relations

Specific Expectations: AG2.01, .04; AG3.03, .04, .06, .07; NA1.03; NA3.04; RE1.03, .04, .05.

Planning Notes

· Graphing calculators are required for this activity.

· Photocopies of Canada Day Fireworks and Twinkle, Twinkle for each pair of students.

· Each group needs access to at least one graphing calculator and a CBR™.

· If using a CBL, then Match It, Graph It program from the Real World Math Book from TI needs to be down loaded into their calculators.

Prior Learning Required

Students need to know how to calculate slope, enter lists and graph using “y =” in a graphing calculator, isolate a variable in an equation, or substitute into a formula, then solve.

Teaching/Learning Strategies

Teacher Facilitation: The activity begins with a full-class discussion interspersed with individual or pair student work on the following three examples.

In Example 1 Canada Day Fireworks, students can find the equation of the lines by using the skills developed in Activity 2.6. Some are simply calculating slope and reading the y-intercept from the graph. This gives them the y = mx + b form. Tell students that the interval on each axis is one unit.

In Example 2, Twinkle, Twinkle, it is difficult to read the y-intercept and therefore another method for finding the equation of the line must be used. At this point, the teacher should interrupt the work on Example 2 and direct the students to work through Example 3, which leads students through the development of finding the equation of a line, given two points on it.

In Example 3, each pair should be assigned a specific point as their starting point. After students complete step 3 of Example 3, the teacher should interrupt the class and have students put their groups' solutions on the board or chart paper and compare the slopes. Since all groups get the same result the class can now generalize this by letting P(x,y) be any point on the line. Students should then be directed back to the example and complete the rest of the steps.

After students have obtained the equation of the line using their point, groups should put their solutions on the board or chart paper and compare the initial equations and the final equations. Students may need help simplifying their equations. The purpose of the activity is for students to see that any two points on a line will lead to the same equation. After completing Example 3, teachers and students should summarize finding the equation of a line given two points. Students should then revisit Example 2 and find the equations of all of the lines in the Twinkle, Twinkle grid.

Student Activity:

Example 1 (Canada Day Fireworks)

The picture shows the graphs of several different lines. Find the equations of the lines and graph them on a graphing calculator, if possible. Make sure that your window matches the diagram. Graph them on your calculator.

Example 2 (Twinkle, Twinkle)

Find the equation of each line. Notice that although it is easy to locate two points on each line, it is difficult to locate the y-intercept. Do you have some ideas as to how to find the equation?

Example 3 (Publishing Yearbooks)

The Mustang Publishing Company produces yearbooks. Their charge is based on the number of books ordered. The following chart is an example of the costs.

Number of books	200	400	600	800	1 000	1 200
Cost	3 500	6 500	9 500	12 500	15 500	18 500

1. Enter this data into lists on a graphing calculator and determine if the relation is linear.

2. Take the starting point assigned to you by the teacher and call it (x₁, y₁). Name the point to the right of your point (x₂, y₂). Find the slope of the line using: = m.

3. Change the order of the subtraction i.e., = m and determine if it will give you a different result.

4. Repeat step 2 from above by replacing (x₂, y₂) with (x, y). In other words, substitute your slope value for m and your point for (x₂, y₂) in = m.

5. If you wanted to graph this you would need to put it in the form of “y =”. Re-arrange the equation to isolate y.

6. Graph the equation on your graphing calculator. Does it match your original relation created from lists?

Follow-up Activities:

Student Activity:

Students collect real-life distance-time data using CBL or CBR™. They determine the equation of the line describing their motion using the previous method, (i.e., they should find the slope of their line and let (x₁,y₁) be any point on the line). If (x, y) is any other point on the line, then the equation of the line would be: (y – y₁)/(x – x₁) = m.

Their equation can now be entered into the calculator and graphed. They need to identify the domain for each section. This verifies the correctness of their model. Students give the meaning of the slope and y-intercept if appropriate. Using their equation they are asked to find the distance walked for several different time periods. This could be checked on their calculators using the trace key. They should also answer questions that require them to find the time taken to travel different distances using their equation. Once again their answers can be confirmed using the trace key.

Extension:

Teacher Facilitation: For some students ready for extensions, they can investigate how to graph piecewise functions through the following example.

Example: Speedy Copy charges the following rates for photocopying:

· 3 cents each copy up to and including 500 copies

· 2 cents for each additional copy beyond 500.

1. Draw a table of values and graph the relationship between the number of copies and the cost.

2. Notice that the graph is broken into two sections. Calculate the slope of each section of your graph.

3. Calculate the equation for each section of the graph.

4. Notice that we could obtain the equation by using the re-arrangement of = m

into y = y₁ + m(x - x₁) form where y₁ is the initial value of y and x₁ is the initial value of x for each section.

{

This approach is useful when defining the equations of segments of piecewise linear functions which do not pass through the y-axis (e.g., for the Speedy Copy charges in the example above).

_{y = 3x, 0 £ x £ 500}

1500 + 2(x - 500), x < 500

Student Activity:

Describe a scenario that would produce a piecewise graph similar to the photocopy charge graph. Find the equation for each piece of the graph.

Homework/Practice Suggestions:

1. Create a linear design on a grid and determine equations for the design. Ask another student to graph the equations to see if it produces the suggested linear design.

2. Using the data from Activity 2.5: Crickets Chirp to the Beat of the Temperature, determine the equation of the line by choosing two data points and compare it to your earlier model.

3. Investigate a situation of your own choice that would be modelled by a linear equation. Submit a report that includes the following:

a) Describe your data. How was it collected? (Using probes; taking measurements (e.g., distance, height, temperature, etc.) found in magazines, reference books, the Internet, newspapers.)

b) Sketch a graph of your data. Explain why the data is linear. Draw a line of best fit. Find its equation.

c) Interpret the real-life meaning of the slope, y-intercept (and x-intercept, if there is one).

d) Use your equation to predict the value of a point between two given values and beyond the given value.

e) Are there any restrictions or limitations to your data values? If so, explain why?

f) Suggest a possible use for your equation.

Assessment/Evaluation Techniques

Teachers could assess students’ communication in mathematics and problem-solving skills through observation while students are working on pair activities. Students could submit their solutions to some of the above activities for assessment of the relevant expectations such as:

· determining the equation of a line given two points

· graphing lines by hand or using a graphing calculator

· communicating solutions to multi-step problems

· describing the meaning of slope and y-intercept in context

· describing a situation that can be modelled by a liner relation.

References

Real-World Math with the CBL System

Texas Instruments. Explorations using the CBR™.

Data Analysis and Statistics. NCTM Addenda Series, p. 36.

Zap-A-Graph Tutorials

Green Globs

Math Trek

Activity 2.9: All in the Family

Time: 75 minutes

Description

Students complete a worksheet activity investigating the characteristics that distinguish a linear relation from a non-linear relation. They use calculators or graphing software to obtain the graphs of a variety of linear and non-linear relations from their equations; classify the relations according to the shape of their graphs, and determine the characteristics of the linear equations that differ from the non-linear equations.

Strand(s) and Expectations

Strand(s): Analytic Geometry, Number Sense and Algebra, Relationships

Specific Expectations: AG1.01, .02, .03; AG3.03; NA2.01, .02; RE2.06; RE3.03.

Planning Notes

· Reserve a class set of graphing calculators or time in the computer lab.

· Prepare individual copies of the worksheet found in the Appendix. Provide each student with a sheet of graph paper to fold into four sections for graphing the four types of equations.

· Help students to suitably set the domain and range of the calculator.

· Prompt students in questions #4 and #5 to rewrite the equations in the form y = mx + b to facilitate the use of a graphing calculator.

· Prompt students in question #4b to use a table (or prior knowledge) to graph equations of the form Ax + C = 0 as these lines cannot be graphed on a graphing calculator.

Prior Learning Required

Students already have competence graphing linear and non-linear relations from data gathering and from descriptions of realistic situations. Finite differences from a table of values have been examined for both linear and non-linear relations in Unit 1 (Phase 1).

Teaching/Learning Strategies

Teacher Facilitation: As a brief introduction and review, ask students to draw, in the air, one of the shapes of the graph of a relationship in Unit 1 or 2. The teacher could name activities like Activities 1.5 A Cagey Problem (parabolic), 1.7 Fold It! (exponential), and scenarios from Activities 2.1 or 2.4 (linear) to remind students of the different types of relationships they have encountered. Draw these shapes on the blackboard and label them. Tell students that graphs that share a property are referred to as a family of graphs.

Student Activity:

Using the student worksheet All in the Family found in Appendix 2.9, students use graphing calculators or software to classify the graphs and make conclusions about the difference between the linear and non-linear equations.

Teacher Facilitation: Ensure that the students have determined the characteristics of the equation of a line and can recognize the linear equations from a list of equations. Using the calculator, have students examine the table of values for both a curve and a line, then calculate first differences for a linear relation, and further differences for other types of relationships (2^nd differences are constant for parabolas, 3^rd differences constant for cubics, no differences constant for exponentials). Give students a table of values and ask them to calculate finite difference to determine if the relation is linear or non-linear.

Homework/Extension:

Give textbook practice in recognition and classification of linear and non-linear equations. Many students are able to recognize the equations as parabolas, cubics, and exponentials, although that level of specificity is not required.

Assessment/Evaluation Techniques

· Teachers could observe student performance during the investigation and pay attention to students’ perseverance to the task as an indicator of initiative.

· Paper and pencil tasks or a short quiz could assess students’ ability to recognize linear equations and classify equations and graphs.

· Communication and the application of new knowledge about classifying relations could be assessed through journal entries describing the characteristics of a linear equation that make it differ from non-linear equations. Students could also verbalize or write about the characteristics of a non-linear graph determined by second degree, third degree and exponential equations.

Appendix 2.9: All in the Family Worksheet

1. The equations below represent 4 different shapes/families of graphs. Use a computer or a graphing calculator to draw the graph of each relation. Fold a piece of graph paper into 4 sections. Place graphs of the same family on the same grid. Write the equation on top or beside each graph as you draw it.

y = x² y = x³ + 4 y = x + 4 y = -2x² y = (x - 3)³

y = -3x + 1 y = 2^x y = ⅓x – 2 y = 3^x y = (x + 3)²

y = ⅓x y = (½)^x y = (x - 12) ³ - 13 y = -x - 3 y = 2x² - 1

y = x³ + 2 y = (x - 2)² - 3 y = -2 + x y = (2x)³ y = ½x – 3

2. Examine the 4 families of graphs you sketched. Write the equations of any 6 graphs that are linear. Write the equations of any 6 graphs that are not linear. Compare the equations of the lines with the equations of the others. How do the equations of the lines differ from the equations of the others?

3. Complete this statement about how you would recognize an equation that graphs a line:

“An equation graphs a line if...”

4. a) Graph these equations that contain a y variable only:

y = 5 y + 2 = 0 2y = 1 y = 0 3y + 6 = 0

Are the graphs linear? What special characteristic do each of the lines possess? Make a statement about the appearance of a graph if its equation has a y variable only.

b) Repeat the questions in #4a above for these equations that contain an x variable only:

x = -1 x + 3 = 0 3x - 1 = 0 x = 0 2x + 1 = 5

5. Put checkmarks on the equations that graph lines. Use a graphing calculator to graph the equations that you checked to verify that they are linear. Predict the shapes of the other graphs and check using a graphing calculator.

o y – x = 0 o y = 1 - x² o y - 2x² = 3 o y + x + 1 = 0 o y – x³ = 5

o y = (x - 1)² o y = 5^x o 0 = 2x + y + 3 o y = -3x³ o y = x – 2

6. Make up 5 equations of your own that graph lines. Graph your 5 equations to ensure that they are linear equations. Graph a variety of lines that have different directions.

7. Examine the exponents of the different types of equations that graph non-linear relations. Is there a link between the value of the exponent and the shape of the graph? Make a prediction about the link and test your prediction by making up and then graphing several examples. Did you verify or refute your prediction? If you refuted the prediction, make another prediction and test it.

8. Make up some equations that graph curves that do not belong to any of the types examined today. Sketch their graphs.

Activity 2.10: Planning a Trip - a Set of Summative Assessment Activities

Time: 225 minutes

Description

In this activity, students use given data and information to create mathematical models, analyse choices, and make decisions. They use the skills developed in Unit 2 to answer a series of questions that connect to the theme of planning a trip. The organization of these summative assessment tasks is similar to those in Unit 4. These questions help prepare students for the type of questioning that can be expected at the end of the Grade 9 course. Time is also set aside for students to learn, practise, and take pencil and paper tests of algebraic skills.

Strand(s) and Expectations

Strand(s): Analytic Geometry, Number Sense and Algebra, Relationships

Specific expectations: AG1.03; AG2.03, .04; AG3.01, .02, .03, .04, .05; .05, .06; .04; NA4.01, .03; RE2.01; RE3.03, .04.

Planning Notes

· The Planning a Trip activities could be used over a week, where half of the class time is spent reviewing key skills and the other half is spent applying knowledge to these activities. Or, this set of summative assessment tasks could be concentrated in the last three days of the Unit. It is estimated that Planning a Trip Phases 1 and 2 requires about 30 minutes each and Phases 3 and 4 about 15 minutes each. A traditional pencil and paper test could follow these performance assessment and review sessions.

· It is suggested that students gather their written analyses in a portfolio. All students should become familiar with the quality of work that is judged to be Level 3 and 4. This could be achieved through discussion of the characteristics of student performance and written submissions that would be Level 3 or 4, before the activities are undertaken. Or, there could be some full class presentations and discussion of students’ findings after each activity is assessed. Teachers should make specific suggestions to each student as to how their work in various categories could be improved.

Prior Knowledge Required

The students have completed the work of Unit 2 of the Profile.

Teaching/Learning Strategies

Student Activity: Planning a Trip

Mr. And Mrs. Lee and their three children, Todd, Chris, and Vikki are planning to take an extended holiday (perhaps four - six weeks) a year from now. They want to drive their van from <your town here> to Vancouver, visiting many points of interest along the way. The Lees know that they have to budget and save all year to be able to afford the type of trip they want. Over the next few classes, you will be given details that the Lees want to analyze to develop their budget for the trip. Keep your analyses together in a “trip portfolio”.

Planning a Trip Phase 1: Kennel Costs

The Lees need to board their pet cat, Bailey, while they are on their holiday. It is Chris’s job to research the fee structures of local kennels. He clips the following ads from his local newspaper.

1. Create a graphical model for each of the kennels that is appropriate for Bailey. Show all graphs on the same grid and label them clearly.

2. Form the equations for the fee structures at the kennels you think the Lees should consider for Bailey. Define the variables you use.

3. Use your equations to contrast the kennel charges if the Lees think that they will be leaving Bailey for either 4 or 5 weeks. Label your computations clearly.

4. Where do you think Chris should recommend the family board their cat? Explain your reasoning.

5. Later, Chris sees an ad for a new kennel in their neighbourhood. It charges $5 for an initial flea check, plus $10/day. Chris rejects this kennel without creating a mathematical model or computing costs. Explain Chris’ reasoning for rejecting this kennel so quickly. Use the vocabulary of this Unit as you make your points.

Teacher Facilitation:

· Students work individually on this activity.

· The teacher circulates around the classroom as students engage in this activity, giving prompts with a coloured pen if a student is having trouble engaging the problem, or making a misinterpretation. Teacher prompts are taken into consideration when the student’s work is graded.

· As students work on this activity, ensure that they do not spend much time on the Bow Wow Weekend ad, since that kennel accepts only dogs.

Planning a Trip, Phase 2: Gas Up!

On a 3-day visit to their grandparents in New York, the Lees drive their van, and gather data to analyse the rate of fuel consumption of the van and the accuracy of its fuel gauge. They fill the tank up with gas and set the trip odometer to zero as they set out. It is Todd’s job to record the trip odometer readings as the gas gauge reaches each of the eighth markings on the gas gauge. Todd records his readings below.

1. Create a table of values using the odometer reading as the independent variable.

2. Plot the data on a carefully labeled and scaled grid.

3. What does this data suggest about the rate of fuel consumption of the van?

4. Based on this data, what comment could you make about the accuracy of the fuel gauge?

5. What would you expect the gas gauge to read at the instant the Lee family had driven the following distances from their last gas up? Show your work in a format that is easy to follow.

a) 100 km b) 415 km

6. How far from their last gas up would you expect the Lee family to be when their gas gauge reads 1/3 full? Explain your reasoning.

7. The route the Lee family plans to take for their trip is 7500 km on the way out to Vancouver and 7300 km on the way back. How many tanks of gas might they expect to need for their trip? Explain your reasoning.

8. The low gas warning light comes on when the van’s gas gauge reads 1/16 full. The Lees always gas up just before or just after the warning light comes on. How many fill ups will the Lees need for their holiday?

9. Pose other questions that would fit the context of a trip in your family vehicle.

Teacher Facilitation:

· Ensure that students know that an odometer measures distance traveled and that the needle on a gas gauge rotates counterclockwise as the gas tank goes from full to empty.

· Students work individually on this activity.

· The teacher should circulate around the classroom giving prompts with a coloured pen if a student is having trouble engaging the problem or is misinterpreting a question.

Planning a Trip Phase 3: Lawn Care

The Lees usually tend their lawn and gardens themselves. However, during their holiday, they plan to engage either the Green Grass Company or the Careful Cutters Company because they notice that both companies work in their neighbourhood.

When they called Green Grass for a price quote they found out that the costs are:

$90 start-up, plus $20/hr for lawn care

They tried phoning Careful Cutters several times but were only ever able to get an answering machine and no one called them back. They were persistent because two of their neighbours claimed that Careful Cutters gave the best deal. Todd was asked to investigate. He went to one neighbour who used Careful Cutters and found out that after Careful Cutters had worked for them for 22 hours, the cost was $575. Another neighbour claimed that after their first 11 hours they were charged $300.

1. Graph the relationship between the number of hours and cost for the two companies on the same grid.

2. Form an equation for each lawn care company, using h as the number of weeks for the contract to run and c as the cost of the service. What assumptions are you making about the Careful Cutters price structure (relationship)?

3. Write an advertisement for the Careful Cutters Company and state the start-up cost and the cost per hour in the advertisement.

4. The Lees estimate that their lawn requires approximately 1.5 hours of care per week. If they are gone for 6 weeks, how much would each lawn care service cost? 4 weeks? 1 week?

5. Which company is less expensive? Explain your answer.

Teacher Facilitation:

As students work on this activity, remind them to use h as the independent variable, and to identify an initial point and rate for each company. Indicate prompts using your colourful pen.

Planning a Trip Phase 4:

1. Vikki knows that her father prefers to see graphs of data when he wants to make comparisons. He gives her a file folder of ads that he has gathered and asks her to do quick sketches for him. Describe the graphs that Vikki should draw for each ad by referring to as many of the following descriptors as apply:

a) linear b) non-linear c) increasing d) decreasing

e) a set of parallel lines f) a set of lines that all pass through the same point

i) Security Guard - a Home Protection Service

Plan A: $5/day; Plan B: $50 labels for windows, plus $5/day; Plan C: $100 permanent registration fee and labels, plus $5/day

Descriptors that apply: _______________________

ii) Pool Cleaning Companies

Company A: $50 for the first 2 weeks, plus $25/week thereafter

Company B: $50 for the first 2 weeks, plus $30/week thereafter

Company C: $50 for the first 2 weeks, plus $20/week thereafter

Descriptors that apply: _______________________

iii) Reliable Power Battery Company

Our 9-volt batteries maintain their voltage extremely well for the first 3 weeks of constant use, then quickly drop off, resulting in a dead battery after 4 weeks.

Descriptors that apply: _______________________

2. Describe the type of relationship that would model:

i) the distance from Vancouver vs the number of hours away from <your home town>( as the Lees drive out to Vancouver).

______________________________________________________________

ii) the amount of money spent by the Lee family as the number of days of their holiday increases

______________________________________________________________

Assessment/Evaluation Techniques

Teachers could assess the students’ trip portfolios and could concentrate on any of the following areas:

Phase 1

Knowledge: correct graphs, correct equations, recognizes slope and y-intercept and uses them to form equations

Communication: clearly explains recommendations

Phase 2

Thinking/Inquiry - reasoning, justifying answers

Knowledge - unit rates, meaning of slope

Phase 3

Knowledge: forming equations given two points

Application: creating equations from given scenario and applying equation to answer questions.

Phase 4

Communication: used descriptors/terminology appropriately.

Accommodations

· Break activities into smaller parts, increase timelines, help student to organize each task

Appendix 2.10: Suggestions for Questions for Summative Assessment

The following questions are suggestions for inclusion in a summative assessment. They require that students extend and apply their knowledge and offer opportunities for Level 4 performance. Teachers may want to choose one question to be included in a test or may want to choose one question for students to work on in groups as a performance task.

1. Relations are often represented by equations, tables of values, and graphs of lines or curves. It is possible to use equations and graphing to construct geometric patterns with lines or curves as well.

For each set of lines given, determine if the lines form the sides of a square, rectangle, rhombus or parallelogram. Before you graph the lines, make a guess as to what shape the lines form. Record this guess as well as your reasons for guessing that figure. Then draw the graphs and make a statement as to whether your guess was accurate. If you guessed incorrectly, suggest what you overlooked in your first analysis.

Case a) y = - 2, y = -3x + 18, y = ⅓x + 18, y = -3x - 12

Case b) 2x - 3y + 15 = 0, x + 3y - 24 = 0, 4x - 6y - 6 = 0, x + 3y + 3 = 0

· The diagram below is of a roof truss for a cottage. Apply your knowledge of slope and the equations of lines to reproduce the identical pattern on a graphing calculator and/or graph paper. What are the equations of the lines that are involved?

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