Applications of Mathematics 10 -
Patterns and Relations (Relations and Functions)

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 :

• examine the nature of relations with an emphasis on functions.
• represent data, using function models.

It is expected that students will:

• plot linear and nonlinear data, using appropriate scales
• represent data, using function models
• use a graphing tool to draw the graph of a function from its equation
• describe a function in terms of:
  • ordered pairs
  • a rule, in word or equation form
  • a graph
• use function notation to evaluate and represent functions
• determine the domain and range of a relation from its graph
• determine the following characteristics of the graph of a linear function, given its equation:
  • intercepts
  • slope
  • domain
  • range
• use partial variation and arithmetic sequences as applications of linear functions

SUGGESTED INSTRUCTIONAL STRATEGIES

Understanding and describing functional relationships is essential to interpreting and predicting the behavior of the world around us. Students will explore how to translate between algebraic and graphical representations and use these skills to make inferences and solve problems.

• Ask students to explore non-linear functions with graphing technology (calculators or software) and discuss the similarities and differences in the graphs and equations.
• Have students use graphing calculators or computers to graph and investigate the graphic changes as a number is added (e.g., ) and when x is multiplied by a constant (e.g., ) to develop the concept . Have them record their findings in their journals.
• Ask students to work in groups to determine techniques that will generate linear equations, given information other than slope and intercept.
• To explore the relationship of linear equations in computer programming, have students work in co-operative groups to generate short programs that produce linear drawings, then share these with the class.
• Organize a carousel activity in which students move in groups around various stations set up with activities to generate data. Students decide as a group how best to graph or display the data. Discuss as a class the similarities and differences in the solutions.
• Use a bakery model (ingredients in pastry out) or a sawmill/pulp model (logs in lumber pulp/paper out) to illustrate the concept of domain and range.
• Have students use a function machine to calculate the range of a function from its domain.
• Have students generate two data points in a linear function (e.g., kilometres driven and car rental cost) and graph them. Ask them to generate model functions to answer questions about the intercepts (fixed costs) and slope (variable costs).
• Ask students to use maps and scale diagrams to calculate distances and sizes.

SUGGESTED ASSESSMENT STRATEGIES

Plotting data, and the subsequent analysis of relations and functions, are key to students’ understanding of patterns. Assessment in this area should focus on students’ ability to perform individual outcomes (such as graphing and analyzing functions derived from the physical world), and reflect students’ understanding of the uses of graphs.

Collect

• Have students collect data on relations between pizza costs and diameters. This data could be represented as a list of ordered pairs, expressed as a rule, described in equation form, as a hand sketched graph, and finally drawn using a graphing calculator. Check particularly for students’ ability to select scale, find slope in appropriate units, and select window parameters.

Question

• As students work on using function notation, evaluating functions and determining domain, range, intercepts and slope, check for their ability to work competently using both sketches and graphing tools. Note students’ abilities to translate their graphing skills to understand the real-world implications of the data.

Presentation

• Give students ordered pairs such as (0,32), (100,212), and (-40,-40) that show converted temperatures in both Fahrenheit and Celsius scales. Ask students to graph the data to see if it is linear, then determine the slope and y-intercept. Have students then reverse the axes and re-graph the data. Circulate through the classroom to ensure that students’ graphs are correct. Ask students to determine where the two temperature scales meet. Have them generate a conversion formula for Fahrenheit and Celsius.

Peer Assessment

• Ask students to work in pairs using graphing calculators. Students can display graphs on their calculators, then challenge their partners to reproduce the graphs on their own calculators.
• Have one student generate an equation in slope-intercept form and a partner change it to general form and vice versa. Students can check each other’s work for accuracy.

RECOMMENDED LEARNING RESOURCES

Print Materials

• Applied Mathematics 10, Wester Canadian Edition
  Ch.3 (Sections 3.1 - 3.6)
  Ch. 6 (Sections 6.1 - 6.6)
  Projects: Safety Technology, Scuba Diving Tours, hair Products, Anthropology, Space Technology
  
  
• Exploring Functions with the TI-82 Graphics Calculator
• Graphing Calculator Activities for Enriching Middle School Mathematics
• An Introductions to the TI-82 Graphing Calculator
• Model Motion: High School Math Activities with the CBR
• Using the TI-81 Graphics Calculator to Explore Functions
• A Visual Approach to Algebra
• What if...?: The Straight Line: Investigations with the TI-81 Graphics Calculator
• What If...?: The Straight Line: Investigations with the TI-82 Graphics Calculator

Software

• GrafEq (Macintosh & Windows Version 2.09)
• Green Globs and Graphing Equations

Games/Manipulatives

• Radical Math: Math Games Using Cards and Dice (Volume VII)

© 2000 Copyright. All Rights Reserved. Curriculum Branch.
Maintained by: Mathematics Coordinator
Revised: January 11, 2001

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