Process Standards
Communication, Reasoning, Connections

School: San Lorenzo High School; 1,500 students
Location: San Lorenzo, California
Teacher: Carlos Cabana
Years Teaching: 5
Students in Classroom: 34
Grade: 10 – 12

Group Test

Video Overview

Mr. Cabana distributes a semester review test on mathematical modeling and functions to small groups of students. As they work together to solve the test problems, the students use graphing calculators and yellow pages (resource sheets created by each student). Mr. Cabana circulates through the classroom and observes students' knowledge and cooperation. He intervenes only when he recognizes that a group's thinking has strayed too far from the correct solution.

The Class Challenge

Students focus on the following two of the group test's four problems:
1 From the given sequence of numbers 11.25, 10, 9.25, 9, 9.25, 10, 11.25, 13, 15.25, 18, . . . ,

• a Graph the sequence.
• b Find the equation of the curve.
• c Determine the 100th term of the sequence.

2 Using data about college costs in a problem about linear and exponential growth, a Write equations to model the cost of private school tuition. b Predict the cost of tuition in 1996. c Predict in what year a private school would cost $100,000. d Choose which model is more realistic and explain why.

Year Average Yearly Tuition 1978 $4,960 1979 5,510 1980 6,060 1981 6,845 1982 7,600 1983 8,435 1984 9,000 1985 9,659

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Course and Curriculum

This elective Algebra 2/Trigonometry course is the third course in the College Preparatory Mathematics (CPM) curriculum. In this class, students learn how to use and apply functions in various settings. The group test covered units on mathematics as modeling, functions, and families of functions. Students wrote their own resource sheets (yellow pages) to include information they could consult during the test.

Mr. Cabana gave each group one grade, which he determined by grading a different problem on each student's test. Following this test, students took an individual semester exam containing more specific questions in which they had to calculate, conjecture, and explain their reasoning. The group test grade was worth half the individual exam grade.

During the group test, Mr. Cabana assessed student thinking and collaboration. Throughout the year, he also assesses students through group presentations, homework, portfolio problems, and individual tests. Mr. Cabana groups his students randomly and changes the groups after every unit. In each unit, students learn new concepts through group work. In this and most other San Lorenzo High School mathematics courses, students are taught to discuss questions in groups before asking their teacher.

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A Pre–Viewing Exploration for Teacher Workshops

Solving a Problem in Groups

Objective: To experience group problem solving using symbolic and graphic representations of a quadratic function.
Materials: One graphing calculator for each small group.

Activity
Solve this problem in small groups.

1. Use the graphing calculator to investigate the flight of a golf ball, hit vertically into the air with an initial velocity of 32 meters per second. The height h in meters after t seconds is approximated by the function: h(t) = 32t – 5t².
2. Use the intercepts, axis of symmetry, and turning point to sketch a curve that represents the height of the golf ball over time. Graph the equation on the graphing calculator and compare your sketch to the graph on the calculator.

Exploring Content

1. How much does the height increase during the first, second, and third seconds of flight?
2. After how many seconds does the ball reach its maximum height? After how many seconds does it hit the ground?
3. The initial upward velocity of the ball is 32 meters per second. What happens if the upward velocity in the original equation is less than 32? More than 32?
4. How did your sketch of the graph compare to the calculator graph?
5. What role do the graphic and symbolic representations of the function play in your investigation of the function?

Exploring Teaching Issues

1. How important to your understanding were the group discussions of the problem?
2. If you had completed this activity alone, would you have completed it in a shorter or longer time? Explain.
3. What prior experiences are necessary for students to work together in a group test situation?

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Exploration Answers

Exploring Content

1. 27 meters, 17 meters, 7 meters.
2. The ball reaches its maximum height after 3.2 seconds and hits the ground after 6.4 seconds.
3. If the initial upward velocity was less than 32 meters per second, the maximum height would be less and the ball would land sooner, which would be represented by a flatter curve and a smaller intercept. If the initial upward velocity was greater than 32 meters per second, the maximum height would be greater and the ball would land later, which would be represented by a steeper curve and a larger intercept.
4. The graphs should be similar.
5. Answers will vary. One possible answer is that the graphic shows the relationship between time and the height of the ball, and the symbolic shows the relationship as parabolic.

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Topics for Discussion

What can students and Mr. Cabana learn from group tests?

• What is the purpose of a group test? What different roles could a teacher play in this kind of test setting?
• To what degree should a test be novel so that it evaluates how students would use current knowledge to solve a new problem?
• How would you assess these students' understanding of the mathematics based on their group discussions?
• How would you gather information about the knowledge and skills of students who do not appear to be participating in group work? How would you structure the group assignment so that each group member has an opportunity and is expected to make a contribution?
• How would you score a group test? Would different members of the group receive different grades? How would you determine the differences between the grades?

How does Mr. Cabana empower students by having them use multiple representations?

• Discuss the students' facility with solving the problem by using one of the representations (data, equation, graph) and then figuring out the other two representations.
• How does the use of multiple representations enable students to understand the mathematical and practical aspects of graphing parabolas?
• Mr. Cabana and the students use different representations to communicate ideas. Share your students' discussions of key ideas and your understanding of their range of representations and perspectives on a concept.

How did Mr. Cabana promote student reasoning?

• How would you have intervened in similar or different ways compared to Mr. Cabana? Explain.
• Discuss some of the students' arguments during the group work. What criteria did they use to judge the validity of their own arguments?
• Respond to Mr. Cabana's statement that not requiring students to memorize information "that ultimately is not important" allows him to offer a more challenging curriculum.
• Is there a set of definitions, formulas, or procedures that students must memorize in one of your courses? If so, why is it important to memorize this information? Is there any information that students have memorized but don't fully understand? How would you increase their understanding?

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