Essentials of Mathematics 11 -
Relations and Formulas

This sub-organizer contains the following sections:
Prescribed Learning Outcomes
Suggested Instructional Strategies
Suggested Assessment Strategies
Recommended Learning Resources

PRESCRIBED LEARNING OUTCOMES

It is expected that students will represent and interpret relations in a variety of contexts

It is expected that students will:

• express a linear relation of the form
  • in words
  • as a formula
  • with a table of values
  • as a graph
• interpolate and extrapolate values from the graph of a linear relation
• determine the slope of a linear relation and describe it in words and interpret its meaning in a problem context
• interpret the graph of a relation and describe it in words
• construct a graph of a relation from its description in words
• evaluate formulas

SUGGESTED INSTRUCTIONAL STRATEGIES

Using graphs of linear relations to represent real-life situations, and interpreting graphs of relations are key components of many problem solving situations. It is important for students to be able to make appropriate conclusions and interpretations based on graphical data.

• Provide students with graphs showing the wages earned versus hours worked for jobs typically held by young temporary workers (e.g., cashier, server). Ask students the following questions:
  • What is the hourly wage?
  • Can the wages earned be predicted for a point off the graph?
  • How would the graph change if a raise were incorporated into the hourly wage?
  • What type of job would result in wages earned for 0 hours of work?
• Have students research in newspapers, trade reports, and the Internet to collect tables of data for graphing. Ask them to define the variables, determine the appropriate scaling of axes, plot the data, and draw conclusions. Then have students identify the linear relationships, identify the slope and y-intercepts, and develop equations to describe the relations.
• Give pairs of students two sets of data showing the relationship between two different base salaries and two different rates of commission. One partner graphs the first set of data, and the other graphs the second. Have students discuss their graphs in relation to slope and y-intercepts. Ask them to generate an equation for each graph to predict each income for a given value of sales. Ask students:
  • What effect will the change in commission rate have on the graph and equation?
  • Can you suggest other applications for this type of relationship?
• Have one group of students find car rental advertisements in newspapers or magazines, and graph the costs of renting a vehicle for one day for several distances. Have another group predict the costs of renting the same vehicle driven for longer distances or for distances in between the plotted points.

SUGGESTED ASSESSMENT STRATEGIES

By extracting meaning from linear relations, graphical representations and using graphs to represent data, students can use these skills throughout their lives. Assessment in this area should focus on the real-world applications of these skills.

Observe

• When students are solving problems involving real-life applications, note whether they:
  • choose appropriate axes for given quantities
  • scale and label axes appropriately, and title the graph
  • plot data correctly and determine the slopes of linear relationships
  • can make predictions (e.g. extrapolate or interpolate) about other values based on their equations or graphs
  • correctly interpret the graph of a relation and describe its intent in words
  • make appropriate conclusions and interpretations of slope and y intercept in problem situations (e.g. car rental problems where there is an initial cost and a rate charge)

Collect

• Give students various linear graphs taken from newspapers, business magazines, the Internet and various government publications. Have students write the equations of the lines, defining the variables and identifying the units involved. Have the students provide feedback: can they provide reasonable interpretations of the data and make interpolations or extrapolations with the data?

Peer Assessment

• Have students develop criteria for evaluating graphs and then mark each others graphs.

Question

• Have students examine a prepared graph and explain why some information is not shown.
• Have students determine missing data on a graph (e.g., labels, y-intercept, slope).

RECOMMENDED LEARNING RESOURCES

Comprehensive learning resources for this course are currently under development. As an interim measure, schools are encouraged to use the teacher-developed learning resources distributed to schools (student and teacher resources). Please note that the student materials require photocopying for student use.

Print Materials

• Models and Patterns: Experimenting With Linear Equations
• What If ...?: The Straight Line: Investigations with the TI-81 Graphics Calculator
• What If ...?: The Straight Line: Investigations with the TI-81 Graphics Calculator

Software

• Green Globs & Graphing Equations
• Math Tools

© 2000 Copyright. All Rights Reserved. Curriculum Branch.
Maintained by: Mathematics Coordinator
Revised: November 22, 2000

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