Course Profile Foundations of Mathematics, Grade 9, Applied, Public

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 and Training. 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.

Any references in this document to particular commercial resources, learning materials, equipment, or technology reflect only the opinions of the writers of this sample Course Profile, and do not reflect any official endorsement by the Ministry of Education and Training or by the Partnership of school Boards that supported the production of the document.

Acknowledgements

Public District School Board Writing Teams - 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, Mark Pankratz, Kelly Searle, Ottawa Carleton DSB: Angela Con, Kawartha Pine Ridge DSB; Donna Del Re, Peel DSB; Sandra Emms Jones, Waterloo Region DSB; Gary Flewelling, Ontario Mathematics Co-ordinators Association; 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 1: Constructing Graphical Models Through Investigation

Time: 35 hours

Unit Description

Students will gather, analyze, manipulate, and display data from primary and secondary sources to model and communicate results about both linear and non‑linear situations. Many contextual problems will be studied to ensure that students gain depth of understanding through meeting the same specific expectations in different contexts. Students will conduct investigations to verify or refute their own conjectures, using lines or curves of best fit, tables and pattern descriptions. They will communicate their findings and describe trends. A rich contextual foundation for subsequent algebraic studies will be built in this unit. Several different types of technologies will be introduced for gathering, analyzing and displaying data.

Strands and Expectations

Some specific expectations from the Number Sense and Algebra, and Measurement and Geometry Strands have been combined with overall expectations REV.01, REV.02, REV.03 from the Relationships Strand. Weaving together the expectations of the strands in this way will help students make connections.

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

Number Sense and Algebra Specific Expectations: NA 1.01, 03, 04; NA2.01, 02, 03, 04, 05, 06; NA 3.06; NA 4.01, 02, 03

Measurement and Geometry Specific Expectations: MG1.01,04; MG 2.01, 02, 03, 04

Activity Titles

What follows is a suggested sequence, with timing, for teaching Unit 1. This profile develops only the activities that depart from traditional pencil and paper skill development. These activities are designed to help students make sense of mathematics by working through concrete experiences to develop their understanding of various mathematics concepts. The need for remediation and further development of skills will arise from the activities.* Up to 960 of the 2100 minutes have been allotted for work, as needed, on Cartesian graphing, integers, percents, exponents, basic algebra, graphing calculator and spreadsheet skills, to name a few. As the new elementary curriculum becomes fully implemented, the use of this time will change.

Activity 1:	What's My Style? Gather, Organize & Display Learning Styles Data (1 variable)	75 minutes
Activity 2:	What's Our Class Profile? Gather, Organize & Display 1 Variable Data	75 minutes
Activity 3:	Is There a Relationship Here? Searching for Two Variable Relationships	150 minutes
Activity 4:	What Type of Relationship Is This? Interpret and Analyze Two Variable Data	75 minutes

*Time for: activity completion, other activities that address the same expectations, building graphing calculator skills, other Skill Building, quizzing (distribute as needed throughout activities) 200 minutes

Activity 5:	A Cagey Problem! Discovering Linear & Quadratic Relationships between Geometric Measures	90 minutes
Activity 6:	A Design Problem. Factors Affecting Steepness of a Linear Graph	150 minutes
Activity 7:	Fold It! Discovering an Exponential Relationship Practising and Extending Exponent Skills	75 minutes

*Practise and extend exponent skills 225 minutes

Activity 8:	Walk This Way! Describing Constant & Non‑Constant Speeds in Distance/Time Relationships	75 minutes
Activity 9:	Tell Me a Story: Interpreting and Analyzing Graphs in Contexts	75 minutes
Activity 10:	What Does It Mean? Steepness as Rate of Change	75 minutes

*Time for: activity completion, other activities that address the same expectations, building graphing calculator skills, other Skill Building, quizzing (distribute as needed throughout activities) 160 minutes

*Sample Assessment activities (practise performance tasks form EQAO, OAME, NCTM) or further skill development 150 minutes

*Time to Consolidate Skills 225 minutes

Activity 11:	Testing Roofs: A Summative Assessment Activity	150 minutes
Activity 12:	Testing Skills: Sample Questions for a Summative Pencil and Paper Test	75 minutes

Prior Knowledge Required

See Appendix.

Unit Planning Notes

• The first activity is intended to help the teacher discover the range of learning styles in the class, and to help students better understand themselves as learners. Teachers might use this information to balance and adjust the types of learning and assessment activities that they use.

• Introduce linear and non‑linear relationships using concrete materials, and use words, units, and representative letters from a wide variety of contexts, with no x's and y's until Unit 2.

• Practise using spreadsheets to record data and draw graphs, graphing calculators to enter data into lists and display graphs using the appropriate lists, Calculator Based Laboratory (CBL) and Calculator Based Ranger (CBR) equipment to gather and display data.

Note:

Only activity 8 requires the use of graphing calculators and CBL or CBR equipment. It is expected that this activity will be used some time after September, giving teachers time to become comfortable with the introductory applications of technology that this activity requires. Should it happen that graphing technology is not available to the teacher during Unit 1, it is recommended that the use of activities 8 & 9 be postponed until later in the course. Together, these activities present a powerful way for students to develop a concrete understanding of rate of change and spacial sense involving interpretation of graphs.

There are many more opportunities for the use of technology than those suggested in this profile, and it is possible to find many activities that require no technology. This profile shows many more sample activities requiring no technology than requiring it. As teachers get more of the equipment in their schools and become more comfortable with its use, they may gather more activities that benefit from the use of technology.

• Gather examples of other activities, in different contexts that address the same expectations as each activity outlined. You may wish to use these as followups, warmups, or substitutes. New textbooks and publications from OAME/OMCA and others will contain appropriate examples.

• Develop a plan and gather materials appropriate to diagnose and develop prior learning skills as they are identified throughout the unit. Some students will need remedial work to perform at the expected standard. 16 hours out of 35 hours have been earmarked in Unit 1 for skill development. It is to be used throughout the unit as the need for skills arises within the contexts of the activities.

• Involve students whose prior learning skills are not in need of remediation, in extending questions and alternate extending activities.

• The relative need for remedial vs extending experiences may vary according to the group of students in a class.

• The concrete experiences presented in unit 1 are important preparation for the abstract algebraic treatment of linear relations that will occur in unit 2. Both units give students opportunities to develop higher level thinking and communication skills, an expectation prominent in the curriculum policy document.

Teaching/Learning Strategies

There are many types of skills required of students as they engage in the activities of this Unit. In the teacher facilitation section of each activity, skills of the following types have been identified, in bold, as they are needed: numeric skills, measurement skills, communication skills, analytic skills, use of technology skills, and collaborative skills. A teacher may find that direct instruction is needed for any of the skills identified. It is intended that repeated, short, direct instruction in these skills, within a wide variety of contexts, will improve student skills in an incremental and natural way.

Teachers will be working diagnostically with students to determine which students do and which do not need remedial support on items identified in the Prior Learning section of each activity. Direct teaching of necessary skills for one group can happen at the same time that extending and enriching activities are used by another group of students. Flexibility of timing and structure are needed so that all students are engaged in meaningful tasks. Time has been allotted for this in the *asterisked time periods.

Students will often be working in pairs or small groups, but growing independence is also a goal.

Activities that are suggested as teaching tools could be used as assessment tools, and vice versa, since assessment activities should be learning activities.

Assessment/Evaluation Techniques

When students do open‑ended, multi‑dimensional work that requires them to perform in a situation which calls for mathematics, it is not useful to score their work on the basis of right or wrong, alone. Inviting students to show what they know and explain their reasoning means assessing a piece of work, or an entire performance, as opposed to a correct or incorrect solution. Teachers will need to look at the strengths and weaknesses of the whole piece of work or an entire performance, as it pertains to the specific Expectations conveyed in the purpose of the task. Work can be scored holistically, with consistent standards, using a rubric. Rubrics are required when there is a range of student responses possible and when there is a need for teachers to be much more precise about criteria for assessment. Several examples of assessment activities and their scoring rubrics have been included in this Profile. Rubrics will be the most effective means of measuring student performance on the Thinking/Inquiry/Problem Solving and Communication and Applications in unfamiliar settings categories of the Achievement Chart.

Most traditional pencil and paper tests do not offer students opportunities to demonstrate Level 4 performances. This Profile includes sample questions for pencil and paper tests that do allow students to demonstrate Level 4 work.

Resources

Heid, M., Algebra in a Technological World. Addenda Series. 1995 NCTM

Asp, G. et. al., Graphic Algebra : Explorations with a Graphing Calculator. Key Curriculum Press

Meridian Creative Group, CBL Explorations in Algebra

Murdock, J. and Kamischke, E., Advanced Algebra Through Data Exploration. Key Curriculum Press

MCTM/SIMMS Integrated Mathematics A Modeling Approach Using Technology. Simon & Schuster Custom Publishing, 401 Linfield Hall, Bozeman, MT 59717‑2810

Specht, Jim, More Than Graphs: Activities for TI Graphics Calculators. 1996 Key Curriculum Press

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

Texas Instruments Real-world Math with the CBL System; 25 Activities Using the CBL and TI-82

Texas Instruments Explorations-Modeling Motions: High School Math Activities with the CBR, 1997

MARS - Mathematics Assessment Resource Service.

Appendix ‑ Prior Mathematics Knowledge That Students Bring to Unit 1

Patterning and Algebra

• find patterns and describe them using words and algebraic expressions

• write an algebraic expression for the nth term in a numeric sequence

• complete a table of values and write words to explain the pattern (from grade 7)

• use variables to write equations and algebraic expressions from patterns and complex statements

Data Management

• collect primary data using both a whole population (census) and a sample of classmates

• assess bias in data collection methods

• manipulate and present data using spreadsheets or search databases for information and use the quantitative data to solve problems

• construct frequency tables, stem-and-leaf plots, line graphs, comparative bar graphs, circle graphs, and histograms, with and without the use of technology, and use information to solve problems (e.g. extrapolate..., predict...)

• read and report information about data presented on the graphs listed above

• understand the difference between a bar graph and a histogram

• know that a pattern on a graph may indicate a trend

• from grade 8 geography: construct a variety of graphs, charts, diagrams, and models to organize information (e.g. graphs that demonstrate correlations between two population characteristics, such as literacy and birth rates)

• understand and apply the concept of the best measure of central tendency and determine the effect on a measure of central tendency of adding or removing a value

• make inferences and convincing arguments that are based on data analysis

Number Sense and Numeration

• compare and order fractions, decimals and integers

• arithmetic operations including order of operations involving fractions, decimals and integers

• solve and explain multi‑step problems involving simple fractions, decimals, integers, percents, ratios and unit rates

• express repeated multiplication as powers

• express whole numbers in expanded form using powers and scientific notation

Unit 1 ‑ Constructing Graphical Models Through Investigation

Activity 1 ‑ What's My Style?

Time: 75 minutes

Description

In this introductory activity students will gather information about their own learning styles using a learning styles inventory (a website reference is provided) and organize their results using graphs. The data from this activity will be used for analysis in a number of activities which follow. In addition, information on students' learning styles will give mathematics teachers pedagogically useful information about their grade nines.

Strand(s) and Expectations

Strands: Relationships; Number Sense and Algebra

Expectations: This activity reviews expectations RE1.0, RE1.06 and NA1.03 in the context of one variable data in preparation for the work with two variable data that follows.

Planning Notes

The activity described here makes use of the learning styles inventory “Assessing Your Learning Style”, which can be found at <. This activity can be adapted for use with another learning styles inventory, if the teacher so chooses. Copies of the learning styles inventory are needed for each student in the class. A class list chart with columns to record each student's scores in each of the two learning styles categories is needed. A blank copy of this chart could be prepared on an acetate for use with the overhead projector, or in the form of a wall chart to be posted in class.

The student's scores would be compiled and computed by the student as follows, in this sample chart:

Students	List A scores	List B scores
Andrew	14	9
Thansha	6	12
Iris	7	15
Class total scores	27/60	36/63
Class total scores as percentages	45%	57%

A second chart to tally students’ preferred learning styles should also be prepared ahead of time. A sample is provided.

	Tally	Total Number of Students	Percentage of Students
stronger auditory
stronger visual
similar strengths
class totals

The teacher should become familiar with the learning styles inventory ahead of time in preparation for any questions which may arise.

Prior Knowledge Required

Percent calculations, construction of circle and bar graphs.

Teaching/Learning Strategies

Student Activity: Students will complete the learning styles inventory and tally their scores for each of the different learning styles. These results will be copied onto the class list. Once the class results are compiled, the students will discuss ways to organize the learning styles data for the entire class (e.g. bar graph, circle graph) and together graph them.

Teacher Facilitation:

• introduce learning styles and how information about learning styles can be used by students and their teacher.

C introduce the activity and inform the class that this information will be used in subsequent activities. Also provide numeric skills instructions for students to evaluate and score their own learning styles inventory results and assign this task to the students. (5 minutes),

• circulate around the classroom, assisting students in completing their learning styles inventories as needed (20 minutes),

• compile the learning styles scores on the class list chart (10 minutes)

• review/demonstrate numeric skills of percent calculation, and graphical communication skills of how to construct bar and circle graphs by constructing graphs of the entire class’s results as follows: (20 minutes)

1)find the class's total scores in each category

2) calculate each total as a percentage of the maximum number of responses possible in each category

3) tally the numbers of students whose learning style preferences lie in each of the three categories (auditory - higher on list A, visual - higher on list B, tactile-kinesthetic - approximately the same on lists A and B)

4) calculate each score in 3) as a percentage of the entire class in preparation for constructing a circle graph

5 )point out the difference between the class total scores as percentages and the most prominent styles as percentages. Relate these to mean and mode and point out that in this case mean takes into account all of the data whereas mode takes into account only part of the data.

6) construct a bar graph to show the relationship between the response rates (as percentages) for the list A and B items within the class. Then construct a circle graph to show the distribution of students amongst the three learning styles categories.

• assign students the following task for homework (5 minutes) Add whatever skills questions that seem appropriate. Emphasize to students that they should demonstrate their numeric, communication, and analytic skills in completing homework.

Student Activity:

1) Construct a bar graph to compare your response rates for the list A and B items and answer the following question: “How does your learning style profile fit that of the class as a whole?”

2) Form a hypothesis about the similarity of your own learning style profile with that of a family member or friend, and test your hypothesis by administering the learning styles inventory to that person so that the results can be compared.

Assessment/Evaluation Techniques

None.

Accommodations

The teacher should be prepared to assist individual students who have difficulty performing these tasks. Parts of the inventory may be completed as a whole class exercise to get the students started and individuals who have difficulty with the text may be grouped with other students for assistance. If a student has difficulty with a question it may be left blank.

Resources

The teacher may wish to do some research on learning styles and how to incorporate this into their teaching. The website cited above is a good start. Another useful reference is Teaching and Learning Stlyes: Celebrating Differences, OSSTF, 1986.

Activity 2 - What's our class profile?

Time: 75 minutes

Description

In this activity, students will continue to gather information about themselves and the class. Following a brief review of graphing, the students will graph one item from the class data and draw an inference about the class from this.

Strand(s) and Expectations

Strand: Relationships

Specific expectations: The activity relates to expectations RE1.04, RE1.05, and RE1.06, in the context of one variable data, in preparation for the work with two variable data that follows.

Planning Notes

Needed are copies of the personal traits questionnaire found at the end of this activity, and a class list chart with ten columns, to record students' responses for each of the eight questions from the questionnaire. Part A of the questionnaire will be completed during activity 2 and part B, during activity 3. As in the first activity, this chart could be prepared on an acetate, or in the form of a wall chart. Prepare examples of a comparative bar graph, and a histogram, and calculations for measures of central tendency. The teacher should assist in making available the materials students will need to present their group graph to the class (e.g., chart paper, markers, overhead acetates, compasses, protractors, and calculators). Prepare a homework task where students will interpret information from various types of graphs.

Prior Knowledge Required

Using mode, organizing and graphing data.

Teaching/Learning Strategies

Student Activity: Discuss responses from the previous day's homework. Students will complete part A of the personal traits questionnaire and have their answers recorded on the class list by the teacher. Working in groups, students will graph class data and look for trends.

Teacher Facilitation:

• Lead the class discussion of the previous day's homework on learning styles, helping students with their communication skills. Conclude this by having students relate the different learning styles to specific behavior patterns that are familiar to them. (e.g., an auditory learner may prefer radio to TV) (10 minutes)

• Get the students started on part A of the personal traits questionnaire.

• Record students' answers for the personal traits questionnaire as soon as they have completed it (15 minutes).

• Provide a review of the communication skills, of the use and construction of graphs, and the numeric skills of calculation of mean, median, and mode from the raw data, and the analytic skill of the appropriate use of each of these measures. (This should be a quick review of grade 8, not a lesson.) (10 minutes)

Brainstorm ways in which to demonstrate collaborative skills of working effectively in a group. The following guidelines are suggested:

1) assign specific roles to each member of the group. For the current activity these could be coordinator, person who calculates, graphics designer, and writer. (Assign roles to special needs students first to encourage their participation)

2) each person contributes to their group's work

3) listen carefully to what other group members say and ask questions when needed

4) help and encourage other members of your group

5) keep working until everyone in your group understands your results and can explain them fully.

• Organize the class into ten groups and assign each group the task of graphing the class results for one of the questionnaire items. Direct students to use numeric skills to calculate an appropriate measure of central tendency for their data, and to use communication and analytic skills to prepare a description of any trends or suggestions about the class indicated by their graph and to justify their conclusions.

• Circulate within the class and help individual students as needed. (15 minutes)

• Have the groups use their communication skills to present their completed graph to the class and share their conclusions. (15 minutes)

• Have each group use their analytic skills to complete a single evaluation of their teamwork

• Introduce the homework (5 minutes).

Assessment/Evaluation Techniques

Observations of student's team work and organizational skills could be recorded at this time. Students could be asked to reflect on their group’s work using the following questions.

Group Evaluation

1. What made your group effective as a team?

2. What could you do differently to work more effectively as a team in the future?

3. Assess your group’s efforts under the following criteria. Place a checkmark in the appropriate space for each.

Rarely Sometimes Usually Always

a) Every person contributed to the group’s work. ____ ____ ____ ____

b) Every person listened carefully to other group members. ____ ____ ____ ____

c) We asked questions when needed. ____ ____ ____ ____

d) We helped and encouraged each other. ____ ____ ____ ____

e) We kept working until everybody in the group understood ____ ____ ____ ____

the results and could explain them fully.

Accommodations

Assist individuals as needed and provide examples that students can use as a reference. The inventory can be read to students and responses can be scribed.

Resources

There are many examples of the different types of graphs in current textbooks, magazines, and newspapers for students to find and bring in.

Appendix

Personal Traits Questionnaire

Part A: Answer the following questions in the spaces provided.

1. What is your gender, male or female? __________

1. What is your age in months? Round down to the nearest whole month. __________

1. On average, how much time in hours do you spend watching TV per day? __________

1. How many children (including yourself) live at your home? __________

1. On average, how many hours to you sleep each night? ___________

1. How much time in minutes, on average, did you spend on your homework each school night during your previous semester at school? __________

1. On average, how much time in minutes do you spend listening to music each day? __________

1. On average, how many minutes do you spend reading per week for your own enjoyment? __________

1. How much time in hours, on average, do you spend per week doing physical activities such as exercise, active games, sports, etc. Round off to the nearest half hour. __________

1. On average, how many hours do you spend per week doing chores or other household tasks? __________

Part B: You will complete this section with the help of a partner.

1. Hand span is the distance from the tip of your thumb to the tip of your baby finger when your hand is stretched out as far as possible. What is your span of your right hand in centimetres? __________

1. Forearm length is measured from the crease on the inside of your elbow to the crease on the inside of your wrist. What is length of your left forearm in centimetres? __________

1. Arm span is the distance from finger tips to finger tips when your arms are stretched out horizontally as far as possible. What is your arm span in centimetres? __________

1. What is the length of your right foot in centimetres? __________

1. Stride length is the distance from the tip of your toe on one foot to the tip of your toe on the other when you take a normal walking step. What is your stride length in centimetres? __________

1. What is your height in centimetres? _________

Activity 3: Is There a Relationship Here? Searching for Two Variable Relationships

Time: 150 minutes

Description

This activity introduces the students to two variable relationships and the possibility that some sets of data will not yield a relationship. Students will use data collected in the previous activity and newly collected data to form an hypothesis, find relationships between variables, make scatter plots, choose appropriate axes and draw a line or curve of best fit. The graphing calculator projection panel can be used as a demonstration to display data and underline the need for decisions concerning scale.

Strand: Relationships

Specific Expectations: RE1.01, RE1.04, RE1.05, RE1.06, RE2.02, RE2.0, RE2.05

Planning Notes

The teacher will supply metre sticks, measuring tapes and the activity outline, "Is There a Relationship Here?" Each student should have a copy of the Personal Traits Questionnaire for the previous activity.

Emphasis will be placed on students exploring and constructing relationships between sets of data. The data from activities 1 and 2 should be displayed on a bulletin board or wall or organized using a spreadsheet and made available to the students. The class set of data must not identify individual students by name to avoid any discomfort that may stem from a discussion of physical characteristics.

It is appropriate to introduce the notion that there are formal, mathematical ways to find a line of best fit. This would be an excellent springboard for discussion about criteria for such a line and the need to communicate these criteria clearly. Teachers may wish to use Fit-ness from the Harvard Balanced Assessment before or during this activity to introduce students to the criteria for choosing a line of best fit.

The view screen for the graphing calculator is a useful device to illustrate the scatter plot of the data and generate discussion around the ‘viewing window' using technology and by hand with the class. Teachers will instruct students on how to use the list, stat plot and window functions on a graphing calculator.

Prior Knowledge Required

Patterning and Algebra Grade 7 and 8: Recognize patterns and use them to make predictions.

Data Management and Probability Grade 7 and 8: Manipulate and present data using spreadsheets; identify and describe trends in graphs, using informal language to identify growth, clustering, and simple attributes; know that a pattern on a graph may indicate a trend.

Grade 8 Geography: Constructs a variety of graphs, charts, diagrams and models to organize information (including correlations between two population characteristics).

Teaching/Learning Strategies

Student Activity: Students will work in pairs to measure their hand spans, forearm lengths, arm spans, foot lengths, stride lengths and heights, to complete part B of the personal traits questionnaire. Students should record their data using the data base or wall chart started earlier.

Teacher Facilitation: The teacher will group the students in same gender pairs and distribute the materials for this measurement activity. While students are working, the teacher will circulate to assist students with the measurement and collaborative skills needed to complete the task.

Student Activity: Students will work with a partner to carry out the following, “Is there a Relationship Here?” Upon completion of the activity, students will present their findings to the class, reporting on questions 1, 3, and 5.

Is There a Relationship Here?

A relationship between two sets of data can help you to make predictions that lead to informed decisions about important matters such as setting up a small business or the effectiveness of a new medicine. You will be looking for connections between two of the data sets that you have collected. For example, can foot size be used to estimate a person's height? Is the amount of time that you spend watching TV related to the amount of time that you spend on homework?

1. What question about the sets of data that have been collected would you like to answer? Write this question down and then form a hypothesis. The hypothesis is your best guess, it answers the question , "What do you think the relationship will be?"

2. Create a scatter plot of the relationship in #1. Answer these questions:

a. What are your variables?

b. On which axis will you place each variable?

c. What scale will you use for each axis?

3. Does your scatter plot show a definite relationship between the data sets? If so, describe the relationship. Consider these questions:

a. Does the scatter plot reveal an obvious pattern?

b. Do the points seem to describe a straight line or a curve?

c. Does the value of one variable increase or decrease as the value of the other increases?

4. Make a list of the criteria for drawing a line of best fit that were identified by the class. Using the list of criteria, draw a line or curve of best fit through the points on your scatter plot.

5. Describe the relationship between your data sets. Consider these questions:

a. What are the variables that are represented in this relationship?

b. Is this relationship linear or non linear?

c. Does the value of one variable increase or decrease as the value of the other increases?

d. Is the rate of increase or decrease low? high?

e. Does the relationship support your hypothesis? If not, create a new hypothesis. Explain your reasoning.

f. Can you think of an equation that would describe this data?

6. a. Plot your data using the list and stat plot functions of your graphing calculator.

b. Set the window and explain your choices.

c. Describe and explain differences between your hand scatter plot and the calculator’s image.

Teacher Facilitation: The teacher will set the stage for this activity by describing the following scenario:

Marketing analysts study data gathered from a population to predict social trends so that they can decide whether or not a product or service will be in demand in a certain area. It is often necessary to know if there is a relationship between two variables or not.

Brainstorm with the whole class to develop analysis skills. Ask students to identify sets of data that may be related (e.g., relationship between hand span and foot size, amount of time spent watching TV and score for a specific learning style, arm span vs height). Choose one example, formulate a hypothesis and discuss how you would prove or disprove it using the class data. Discuss ways to represent the data visually. Review the graphical communication skills involved in plotting points on a grid to create a scatter plot. The class should also consider the decisions to be made about scale and whether or not it is appropriate to use a break in an axis. An arbitrary decision regarding the placement of variables on the axes must also be made by the students since dependence and independence will not be obvious at this stage.

The goal of this activity is to identify both linear and non linear patterns, and to realize that some sets of data are not related in an obvious way. The students' communications and analysis skills will dictate how much guidance is needed in formulating an hypothesis. The teacher will direct students as needed, to use a scatter plot.

During the activity, the teacher will circulate and prompt. When most students have worked on question 3, the teacher will lead a whole class discussion to further develop analysis skills. Students will be asked to list criteria for a line of best fit (e.g., distance from the points, number of points above and below the line, number of points on the line, whether or not the origin is on the line, ignoring outliers). The students should not draw a line of best fit before the class discussion that identifies the criteria for this task. Students can judge each others work to determine the line of best fit. Some students may extend their communication skills by writing an equation to describe their line, but this is not necessary at this point.

As students use their communication skills to present their findings, the teacher will ask probing questions as needed to ensure that the key concepts described in the activity description emerge, using analysis skills (e.g., How does the trend shown in your graph compare to your hypothesis?).

Assessment/Evaluation Techniques

The teacher should make observations regarding the students' independence and teamwork skills using the rubric in Appendix 1. Observational assessment can also be used together data on students’ knowledge, and problem solving skills. This would be the ideal time to provide formative assessment feedback to students regarding their communication skills using “Rubric for Assessing Student Presentations”.

Appendix

Rubric for Assessing Student Presentations

Formulate a hypothesis associated with a relationship between two variables

RE 1.01

C iidentifies the variables, with help

C Fformulates a hypotheses that does not describe the relationship

C Iidentifies the variables, with help

C Fformulates a hypothesis that describes the relationship, with some help

C Iidentifies the variables

C Fformulates a hypothesis that describes the relationship

C Iidentifies the variables

C Fformulates with confidence a hypothesis that describes the relationship

Describe trends and relationships observed in data

RE 1.05

C Mmakes correct inferences from data, with much teacher support

C Eexplanations of the differences between the inferences and the hypothesis unclear and incomplete

C Hhas difficulty discussing any relationship that exists in the data

C Mmakes correct inferences from data, with some prompting

C Eexplanations of te differences between the inferences and the hypothesis unclear

C Ddiscusses only one relationship that exists in the data with minor errors

C Mmakes correct inferences from data with minor errors

C Eexplains the differences between the inferences and the data

C Ddiscusses only one relationship that exists in the data

C Mmakes correct inferences from data

C Cclearly explains the differences between the inferences and the hypothesis

C Ddiscusses other relationships that might exist in the data

Communicate the findings of an experiment

RE 1.06

C Ccommunicates unclearly

C Uuses little or no justification

C Ccommunicates results with some inappropriate forms

C Uuses faulty logic to justify conclusions

C Ccommunicates clearly using appropriate forms

C Jjustifies relationship with respect to the class

C Ccommunicates and justifies clearly and concisely

C Ggeneralizes relationship beyond the context of the classroom

Activity 4: Interpreting and Analyzing Two Variable Data

Time: 75 minutes

Description

This activity builds on the students' ability to identify linear and non linear relationships using tables and graphs based on primary and secondary data. Data will be selected by the teacher to allow for the use of finite differences (particularly the first differences) in the classification of relationships as linear or non linear. Dependence and independence, and the four quadrants of the Cartesian plane will be introduced in context as needed.

Strands: Number Sense and Algebra, Relationships, Analytic Geometry

Specific Expectations: NA1.01, RE1.04, RE2.01B, RE2.02B, RE2.04B, RE2.05B, RE3.03B, AG1.01

Planning Notes

A variety of data tables that represent linear (with positive and negative slopes) and non linear relationships must be prepared for this activity. Current (secondary) data could be gathered from the Internet (for example, at www.statcan.ca ), but keep in mind that this activity requires data with a readily apparent relationship between the variables. Data can be generated easily using simple formulas from science (distance vs time, speed vs time, volume of a gas vs temperature, voltage vs current, mass vs volume, conversion between Fahrenheit and Celsius temperature scales) and other everyday applications (loan balance, bank account balance, sales tax gas consumption, perimeter vs area, metric conversions).

The calculation of first differences will require the subtraction of 2^nd minus 1^st, 3^rd minus 2^nd, etc, values of the dependent variable. The subtraction of integers should be reviewed here to facilitate finding finite differences when negative integers are used. A number of the tables should contain data with negative values and/or a negative slope.

Linear and non linear graphs are needed for the second half of this activity. The graphs should include grid lines or dots and the scales must be easy to read.

The concepts of dependence and independence may be introduced when the students are choosing a variable to start constructing their tables of values. A class discussion about whether or not it matters which variable is chosen in order to read the value of the other variable, and when it matters in the design of an experiment could clarify these issues for the students (e.g., If a person was traveling at 100 km/h on a highway, the distance traveled would depend on the time traveled).

Prior Learning

Number Sense and Numeration Grade 8: Add and subtract integers

Patterning and Algebra Grade 7 and 8: Recognize patterns and use them to make predictions

Data Management and Probability Grade 7 and 8: Identify and describe trends in graphs, using informal language to identify growth, clustering, and simple attributes; know that a pattern on a graph may indicate a trend.

Teaching/Learning Strategies

Student Activity: Students will carry out the following activity in small, heterogeneous ability groups.

Interpreting and Analyzing Two Variable Data

1. A stone is dropped off the top of a tall building. Two sets of data were recorded and are shown below.

Time (s)	Speed (m/s)	First Difference in the speed (new value - previous value)
0	0	<10-0 = < < < <
1	10
2	20
3	30
4	40
5	50

ii)

Time (s)	Distance (m)	First Difference in the distance fallen
0	0	< < < < <
1	5
2	20
3	45
4	80
5	125

a) Using the entries in the first two columns of each table, graph each relationship on a separate grid. When choosing your axes, use time as the independent variable.

b) What is the speed of the stone and the distance fallen when the time is 2.5 seconds? Explain your reasoning, including the assumptions you made.

c) When would the speed of the stone be 80 m/s? Explain your reasoning.

d) When would the distance travelled by the stone be 245 m? Explain your reasoning.

e) Which relation is linear?

f) For the linear relation, form an equation that models the data.

g) Complete the third column for each table. Contrast the differences formed in the two tables.

h) How does the type of differences in the table relate to the shape of the graph?

i) How does the constant first difference for table i) relate to the graph and to the equation for this

relationship?

2. The air temperature on a cold, clear night is measured at the beginning of each hour.

Time (hours)	Temperature ( °C)	First Difference in the temperature
0	‑2	< < < <
1	‑5
2	‑8
3	‑11
4	‑14

a) Calculate the first difference for the temperature.

b) Do you think that this data describes a straight line or a curve? Explain your reasoning.

c) Create a scatter plot of the data and draw a line or curve of best fit. Does the graph support your answer to part b?

d) Find an equation to model the data.

e) When would the temperature be ‑20 °C? Explain how you arrived at your answer.

f) What would be the temperature if the time were 2.5 hours? Explain two different ways to answer

this question.

3. A cold drink is left out in the sun on a warm day. The graph of its temperature vs time follows.

a) Is this graph linear or non-linear?

b) What pattern do you think will exist in the first difference for this relationship? Explain your reasoning.

c) Use the table to record the values of time and temperature for five points on the graph and calculate the first difference for temperature.

Time (minutes)	Temperature (EC)	First Difference
		< < < <

d) Do the first differences support your answer to part b? Explain why or why not.

4. Tara wishes to keep track of the number of kilometres that she travels on a basketball team road trip. She forgets to begin making observations until she is 50 km from home. She begins timing the trip (time equals zero hours) at the 50 km point. The graph below provides distance vs time data for Tara's trip.

a) Is this graph linear or non linear?

b) What pattern do you think will exist in the first differences for this relationship? Explain your reasoning.

c) Use the table to record the values of time and distance for five points of your choosing on the graph and calculate the first differences for distance.

Time (hours)	Distance (km)	Finite Difference
		< < < <

d) Do the first differences support your answer to part b? Explain.

e) Estimate the amount of time that had elapsed before Tara began to record observations. Explain your reasoning and state any assumptions that you have made.

Teacher Facilitation: First differences (a subset of finite differences), dependence and independence, and discrete vs continuous functions are introduced in this activity. In question 2a, lead students to the understanding that they have been given a discrete sample of a continuous function. This means that points on the graph could be connected, and interpolation using a fractional time value would be valid. Averaging values of temperature would also be valid since the relationship is linear. Finish this activity with a class discussion that summarizes strategies for students to use when answering the questions including: patterns of first differences in linear and non-linear relationships (constant and not); averaging to interpolate; previous history and prediction for extrapolation. Allow for a brief discussion at the end of the class to introduce the homework assignment. This assignment should include questions that review essential integer numeric skills, as well as new tabular and graphical communication skills.

Assessment/Evaluation Techniques

Collect and assess individual student work for accuracy of calculations, quality of communication and completeness. The following questions could be incorporated into a quiz in which the teacher also tests skills.

Accommodations

Provide or post a chart outlining integer skills. Leave examples and guidelines, that were ‘talked about’, posted in the classroom.

Sample Questions to Quiz Interpreting and Analyzing Data

Name ____________________________________________

1. After the winter season, a swimming pool needs to be filled with water. A hose is left on for several days. The height of the water in the pool is given in the table below.

Time, d, (days)	Height,h, (Metres)	First Difference
0	1.5
1	1.75
2	2
3	2.25
4	2.5
5	2.75

b) Calculate the first differences for the height.

c) Do you think that this data describes a linear relation? explain your reasoning.

d) Create a scatter plot of the data and draw a line or curve of best fit. Does your graph support your answer to part b?

e) Find an equation to model the relation.

f) How many days did it take the hose to fill the pool to 2.1 m?

g) After 5.5 days, what is the water level in the pool?

2. Angela wonders if there is a relationship between the total volume of beverages she consumes and the amount of time spent watching television. She decides to keep a log for 5 days, then analyzes the data that she collects.

Day	Beverages Consumed (ml)	Amount of TV Watched (hours)
1	500	2
2	400	1
3	585	4
4	550	3
5	600	5

a) Create the first differences for the volume.

b) Does there seem to be a relationship between the volume of beverages consumed and the amount of time spent watching television. Is it linear? Explain your reasoning.

c) Draw a graph for the relation between time and volume.

d) Predict the amount of beverage she would consume if she watched TV for 3.5 hours.

e) Predict the number of hours she watched TV if she consumed 225 ml of beverage.

3. Angela reconsiders the variables and decides to examine the relationship between the number of hours that she watches television and the amount of orange juice that she drinks. Once again, she keeps a log for five days and then analyzes the data that she collects.

a) Does there seem to be a correlation between the amount of orange juice that Angela drinks and the amount of time that she spends watching television? Justify your reasoning.

b) Create a scatter plot for this data.

c) Does your graph support your answer to part a)?

Day	Orange Juice Consumed (ml)	Amount of T.V. Watched (hours)
1	210	1
2	500	4
3	0	3
4	700	5
5	100	2

Activity 5: A Cagey Problem ‑ Searching for a Relationship Between Geometric Measures

Time: 90 minutes

Description

Students continue to explore relationships between two variables by pursuing expectations from the Measurement and Geometry strand. Since students will be working with formulas, the data will be clean (no outliers and fitted exactly by a smooth curve). This unit also introduces the concept of optimization.

Strands: Relationships, Measurement and Geometry, Number Sense and Algebra

Specific Expectations: NA1.01,NA 1.02, NA1.03, NA1.04, NA1.05, NA1.06, RE1.04, RE1.05, RE 1.06, RE 1.07, RE2.04, RE2.05, MG1.04

Planning Notes

This activity might best be done in pairs so that students could share ideas and materials. However, each student should write out his or her own solution as outlined in the student activity description below. The teacher may want to design a response form to help students structure their written responses to the problem.

Prior Learning

Patterning and Algebra Grade 7 and 8: Recognize patterns and use them to make predictions; use the concept of variable to write equations.

Measurement Grade 7 and 8: Apply the area formula to problem‑solving situations.

Teaching/Learning Strategies

Student Activity: Students will investigate the relationship between the perimeter and area of a rectangle. Student instructions:

A Cagey Problem

You are working on the design of a home for a small animal. You have to determine the best dimensions of a rectangular base. The perimeter of the base is limited to 225 cm due to the high price of the fasteners that secure the top to the base. You are to investigate this problem and:

1. Form a hypothesis about the best dimensions for the cage.

2. Collect the data needed to support your hypothesis. Include well-labelled diagrams and computations.

3. Organize the data and predict the shape of the graph. Support your prediction.

4. Graph your data and describe how well it fits your prediction in step 3.

5. Prepare a recommendation for the best dimensions and include your supporting evidence. Ensure that tables and graphs have appropriate titles and units.

Teacher Facilitation:

The teacher could provide graph paper and geo‑boards, and lead a whole class brainstorming discussion, or visit individual groups, to guide analysis skills as students generate and critique criteria to use for the ‘best’ dimensions. The teacher may need to lead a short activity to review use of technology skills in showing how geo-boards can be used. It is intended that maximum area will be explored. Use technology skills through the use of spreadsheets or graphing calculators where possible to organize data for this activity. Students may have to be helped with analysis skills and led to the idea that as the length varies, the width is determined and the area changes. This discussion may be addressed with the class as a whole at the beginning, or with small groups during the activity. The teacher may have to remind individual students to consider the use of numeric skills of finite differences and to analyze their data in sequence for step 3.

Assessment/Evaluation Techniques

Teachers may use the “Rubric for Assessing Student Presentations” from Activity3, with the category “collects and organizes data” added. (See Unit 4 Appendix, A Rubric for Observing Students, for ideas.)

Alternatively, a good strategy to help students see the kinds of things that they can do with a solution of an investigative problem, is to have students move into groups of 4 and receive photocopies of their classmate's work (remove names or use work from another class if there is more than one grade 9 class doing the same activity). Students then should sort the solutions into four piles (or 3 depending on the quality of the solutions the teacher has available). Each group should discuss the criteria they used to sort into piles and write these down so that they are able to discuss as a whole class. The whole class discussion should focus on building descriptors of how students sorted the piles. This could be done on the chalkboard. Through this discussion, students will start to understand the levels of performance on solving problems, and communicating methods and findings.

The teacher could then generate a rubric which includes four levels and incorporates the student suggestions. The teacher could use it to assess student work by circling the appropriate descriptors. The categories for assessment could include: clarity of communication of their hypothesis, predictions and recommendations; correctness of computations; applications of previously-learned skills. The purpose of this formative assessment is to provide feedback and suggestions for improvements.

Activity 6: A Design Problem; Algebraic and Graphical Models of a Relationship

Time: 150 minutes

Description

In this activity, students work with available, discrete, clean data gathered through explorations of volume, base area and height of buildings. Through construction of both algebraic and graphical models, the students investigate concepts such as steepness of graphs, appropriateness of scale and the creation and rearrangement of formulas. Extensions allow for further work on optimization problems.

Strands: Number Sense and Algebra, Relationships, Measurement and Geometry

Overall Expectations: NAV.01, NAV.03, NAV.04, REV.01, REV.02, REV.03, MGV.01, MGV.02

Specific Expectations: NA1.01, NA1.02, NA1.03, NA 1.04, NA3.06, NA4.01, NA4.02, NA4.03, RE 1.04, RE1.05, RE 1.06, RE1.07, RE2.02, RE2.03, RE3.04, MG1.01, MG1.04, MG2.01, MG2.02, MG2.03, MG2.04

Planning Notes

Students may need to review ratios (while creating scale drawings) and formulas for area and volume. The teacher may wish to introduce this activity by brainstorming a list of criteria for a building’s base shape (foot print) that would be important to a designer. These could include: size of building lot, height restrictions, by-laws re: distance of the building from the lot line, any natural obstructions on the lot, etc. Before students work on the problem, they will need a discussion of the appropriate scale on their graphs for the large numbers that will come out of the activity.

Prior Learning

Patterning and Algebra Grade 7 and 8: Recognize patterns and use them to make predictions; use the concept of variable to write equations.

Measurement Grade 7 and 8: Apply the volume formula to problem‑solving situations involving rectangular prisms.

Teaching/Learning Strategies

Student Activity: Students will work in heterogeneous groups of three but will submit individual reports to the following problem.

A Design Problem

A designer is hired to create a multistoried building for a client who wants to open a business for people to play laser tag. The lot is 100 m × 100 m and has a large old tree exactly in the middle. This tree is to be preserved with an open space of at least 5 m in all directions from the centre of it. Town bylaws require a fire route of at least 10 m width along the two sides and back of the property. Each storey is to be 4 m high, have a flat roof, and have the same footprint as the base. The client wants the footprint of the building to have an area of 4750 m².

1. Create scale drawings of 3 building footprints which satisfy all of the constraints.

2. For each of the three footprints, how does the volume of the building change as the number of storeys increases? Explain the relationship between the height and volume of each building using a table of values, finite differences, a graph, in words, and using an equation with variables H and V.

3. Should you connect the points on your graphs? Explain your reasoning.

4. How would the graph of volume versus height change if the area of the footprint was 2000 m² instead of 4750 m²? What would be the equation of this new relationship? Can you suggest an area of footprint that would create a graph steeper than your original graph?

Teacher Facilitation: The teacher will show students configurations that are allowed and not allowed, as shown below. The teacher should ensure that discussion about discrete vs continuous functions happens while students are working on question #3. The teacher will circulate around the room and provide prompts as needed, or ask students from one group to give quick hints to another group. Each student in each group of three should take responsibility for creating the written submission of one of the footprints for their group. However, encourage the three students to keep collaborating and comparing their results.

Configurations allowed: Configurations not allowed:

Student Activity Extension: Students will discuss the following extension to the building design problem in a large or small group, then work independently to prepare a solution.

Suppose that the client can afford the climate control systems necessary for the volume of a 4-storey building with the footprints that you designed. However, the city passes a new bylaw that restricts the maximum height to 3 storeys. Design a building that has the same volume, obeys the new bylaw, and fits on the property with all of the original restrictions, if it is possible. If it is not possible to preserve all of the original restrictions, explain why and suggest a way around the problem.

Teacher Facilitation: The teacher may have to lead students to the idea that, as the height of the building decreases, the footprint area must increase to preserve the volume. Students will find that they cannot redesign the building to fit all of the constraints. They may have creative solutions like cutting down the tree, petitioning the town for an exemption from the fire law restriction if they add enough sprinkler systems, etc. Encourage analytic skills, supported by appropriate numeric and communication skills.

Go to the Max

Extension 2SStudent Instructions: The owner has received suggestions from previous clients that a more interesting game would have a floor plan in which the walls have a maximum surface area and would be only 3 storeys high. To satisfy these customers, what would the shape and dimensions of your new footprint be if its area must be 3600 m²and the walls must be at least 15 m long?

You still have the constraint of the 10 m fire route on three sides, but no longer need to worry about the tree (it has been removed). There should be no blocked paths so that customers can travel freely throughout the building.

Justify your recommendation to the owner with diagrams and calculations of all the models you have tried.

Teacher Facilitation: Teachers may provide each group of students with blocks to build their floor plan models. Instruct students that each block will represent 15 m × 15 m. When blocks are put together their sides must completely overlap. They may not overlap partially or simply at vertices (as shown).

Students will need to use analysis skills to realize that in order to maximize surface area they will need to maximize the perimeter. They may discover that increasing the number of corners will improve the perimeter.

Assessment/Evaluation Techniques

Use the accompanying rubric to assess student work. The criteria in the rubric elaborate on the criteria in the Achievement Chart.

Accommodations

Provide a summary of the key points from the ratio review. Provide instructions and requirements in different formats.

Rubric for Activity 6

Categories

50-59%

(Level 1)

60-69%

(Level 2)

70-79%

(Level 3)

80-100%

(Level 4)

Construction of Graphical Model (RE1.04)

• organize and analyze data

• sets up axes but labels are missing or scale is incorrect

• table or graph is inaccurate or doesn’t match

• sets up axes, uses labels but scale is inappropriate

• sets up table or graph but may contain minor errors and may forget about discreteness of data

• sets up axes, uses labels and scale which eventually approaches an ideal

• sets up appropriate tables and graphs, but may forget about discreteness of data

• sets up axes, uses labels and readily chooses an ideal scale

• sets up well-organized tables, graphs data appropriately and recognizes discreteness of data

Construction of Algebraic Model (NA4.01,NA3.06)

• use algebraic modelling

• rearrange formulas

• needs assistance to develop equation of a relationship

• depends on group to explain how to get new base area

• develops equation of relationship given graph or statement of relationship using words or symbols

• calculates new base area using trial and error

• develops equation of relationship using words and symbols

• calculates new base area by inspection

• accurately and consistently develops equation of relationship and explains the steepness factor in this context

• uses informal algebra to solve formula for new base area

Construction of Geometric Model (MG1.01, MG2.01, MG2.02)

• understands relationship between volume, base area and height once it has been shown

• is able to calculate area, volume and perimeter with assistance; weak understanding of concepts

• recognizes some relationship between volume, base area and height but not fully developed

• relies solely on simple methods such as counting to calculate area, volume and perimeter

• recognizes relationship between base area, volume and height after investigation

• calculates volume, area and perimeter

• quickly recognizes relationships between base area, volume and height

• calculates base area, volume and perimeter using efficient methods

Solve and Pose Problems (RE1.07)

• needs coaching for each step

• solves the problem through periodic reassurance from group leader

• can solve the problem after group discussion

• quickly decides on procedures and solves the problem

Compare Algebraic Model with Other Strategies Used for Solving the Same Problem (NA.4.02)

• cannot see the connection between the two models

• can make connections between algebraic and graphical model with assistance

• sees the connections between algebraic and graphical model but uses trial and error to check extensions

• makes connections between algebraic model and graphical model including how changes in one affect the other

Describe Effect on Graph (RE3.04)

• is unable to identify components of equations and factors that affect steepness

• states components of equations and factors that affect steepness but cannot explain them

• explain components of equations and factors that affect steepness but lacks details and/or has trouble generalizing

• can easily explain all components of equations and factors that affect steepness in context with detail; can generalize

Communicate Findings from an Experiment (RE1.06)

• has difficulty following or incomplete solution

• lacks description of solution but most mathematical forms are present

• combines some description of solution with mathematical forms (tables and graphs); not all connections are evident

• combines description of solution with appropriate mathematical forms; logical flow is evident

Continue to Unit 1 (Activities 7-12)