Developing Algebraic Reasoning Through a Real Context—Myrna Manly

Slide 2 What is Algebra? Functions and Relations Modeling Language and Representations Structure NRC (1998)

Slide 3 What is Algebra? Functions and Relations GEDTS Analyze and use functional relationships to explain how a change in one quantity results in change in the other quantity, including linear, quadratic, and exponential functions. Recognize and use direct and indirect variation

Slide 4 What is Algebra? Modeling GEDTS Analyze and represent situations involving variable quantities with tables, graphs, verbal descriptions and equations. Create and use algebraic expressions and equations to model situations and solve problems.

Slide 5 What is Algebra? Language and Representations GEDTS Convert between different representations such as tables, graphs, verbal descriptions, and equations. Analyze tables and graphs to identify and generalize patterns and relationships.

Slide 6 What is Algebra? Structure GEDTS Create and use algebraic expressions and equations to solve problems Evaluate formulas

Slide 7 Other Relevant GED Specifications Measurement and Geometry Find, use, and interpret the slope of a line, the y-intercept of a line, and the intersection of two lines Number, Number Sense, Operations Select the appropriate operation to represent problem situations (e.g., when do you divide?)

Slide 8 Examples from Practice Tests Form PA #19: Write an equation from a verbal description of the situation. Note the fixed cost and the cost that varies. Form PD #11: What element of the equation changes when the starting distance increases?

Slide 9 Test Preparation, Life Skills and Deep Understanding A dynamic activity that –Requires the use of high school math –Reflects the kinds of problems that are prevalent on the GED practice tests –Builds on a context that concerns students today –Is rich with possibilities for extensions.

Slide 10 Comparing Vehicle Costs Situation: High Gas Prices demand that we try to cut back on our use. Hybrid Vehicles use a combination of gas and electricity to save fuel. Hybrid Vehicles are more expensive to buy than standard vehicles.

Slide 11 Comparing Vehicle Costs Data for Cost Comparison 2006 Honda Civic 2006 Honda Civic Hybrid MSRP (Manufacturer’s suggested retail price) $18,500*$20,000* Miles per gallon (average driving) 3450 Average annual fuel cost (15,000 annual miles and gasoline prices of $3 per gal, ) $1300*$900* Tank size13.2 gal12.3 gal * Costs include available incentives and are rounded to the nearest hundred Sources:

Slide 12 Comparing Vehicle Costs

Slide 13 Comparing Vehicle Costs Imagine that the graph and table make up the stimulus for an item set on the GED Math Test. What questions could be asked? 1.Discuss with a partner 2.Share with your table 3.Write one question per table on a transparency and select a presenter. (5 minutes)

Slide 14 Comparing Vehicle Costs Imagine that the table and graph appeared in a newspaper article that you brought to class. What questions can you ask that will encourage the students to explore the mathematics of the situation further? -“How did they get…” -“What if…” -“How would that affect…”

Slide 15 Comparing Vehicle Costs Focus on Annual Fuel Costs. How did they get that number? How would you figure your own costs? A = ?? General Strategy What elements are involved? What operations apply? How do you know?

Slide 16 Comparing Vehicle Costs In your groups: Analyze the process of arriving at the formula for annual fuel cost. Share techniques and questions that can be used to foster understanding of modeling this situation.

Slide 17 Comparing Vehicle Costs Variations and Extensions: - Go to the web sites referenced - Explore a personal scenario – vary the parameters to fit own habits or compare costs when selling own vehicle - Reverse the order – build up to the equation and graph - ??

Slide 18 Comparing Vehicle Costs Reflect: What are the prerequisite skills that students should have in advance of participating in this activity? What reservations do you have about implementing an activity like this in your classrooms? How does this activity embody the ANN Principles of Teaching and Learning? Did we address both test preparation and deep understanding?

Slide 19 References Pendleton, Kenn, Item Writers’ Manual, Test 5, Mathematics. 1999, GED Testing Service, Washington, DC National Research Council, The Nature and Role of Algebra in the K-14 Curriculum: Proceeds of a National Symposium, 1998, National Council of Teachers of Mathematics and Mathematical Sciences Education Board,