A number of the lessons address particular topics in algebra. The following lesson summaries cover a selection of these, listed here in the same order in which a teacher might present similar material in class. The first two lessons, S2 and US2, introduce students to the concept of variable. Additional resources listed at the end offer more detail on these topics.
Note: Users may also wish to examine: N4, which focuses on factoring; S1, which deals with factoring and illustrates how one teacher introduces students to factoring quadratic expressions; C3, which has students working on raising numbers and algebraic expressions to certain powers; and US3, in which students calculate with exponents.
S2 Introducing terms and variables
In this first lesson on algebra, the teacher starts by discussing the meanings of “terms” and “variable.” He gives an overview: “We are going to calculate with variables . . . in the next couple of math lessons.” He shows an example of how to represent the number 2 using lengths of 3 and 5, and then asks students to use lengths of 3 and 5 to represent other numbers (17, 15, 12, 13, 3). Students come to the board individually to show their work using blue strips of paper to represent lengths of 5 and yellow strips for lengths of 3. He then assigns pairs of students to different tasks, all requiring that they represent terms in x, y, and z using lengths they draw on the board and label. As each pair presents its work at the board, the other students evaluate it. The lesson ends with an expression written in terms of variables that represent “real-world” costs. The teacher asks students to explain what cost the expression represents.
US2 Introducing variable, variable expression, evaluating expression
The lesson begins with warm-up problems on a variety of topics (one geometry problem and computations). The class then goes over solutions to the warm-up problems and homework. Halfway through the lesson, the teacher presents new material: the meaning of variable, variable expression, evaluating expressions, and translating words to expressions. Students then represent situations with expressions in public class work. The class ends with a game of “24,” in which students use different combinations of numbers and strategies to arrive at the number 24.
US1 Graphing linear equations
This lesson reviews a unit on graphing linear equations. The teacher mentions two methods of graphing linear equations in a whole-class discussion: (1) using slope and intercepts, and (2) using a table and plotting points. Students work in groups of three or four to graph two sets of five linear equations of the form y = mx + b; m is positive for the first set of equations and negative for the second set of equations. The teacher circulates, helping students and checking their work. Toward the end of class students write about what they have learned.
N1 Graphing linear equations, writing an equation for the graph of a linear function, and identifying linear and non-linear relationships in real-world contexts
Using an overhead projector, the teacher begins by going over problems that the students have worked, requesting answers from students:
- (Showing a graph of a line:) “Write an equation for the graph.” (The students give two equivalent equations, and the class discusses the equivalence.)
- “In the graph above, calculate the value of y if x = 10.”
- (Showing four equations and four graphs of lines:) “Determine which equation describes which graph.”
Students watch a 7.5-minute video. (This is not shown on the lesson video.) The video illustrates examples of relationships between two quantities, such as: automobile’s braking distance and its speed; the number of alcoholic drinks a driver consumes and that driver’s automobile accidents; the number of days below freezing and the thickness of ice on a lake; the depth of scuba diving and the water pressure; and the number of cell phone calls and the phone bill. The teacher’s comments during the video concern whether these relationships are linear or non-linear. After the video there is a brief whole-class discussion, and then the teacher circulates as students work individually on problems concerning equations and graphs.
S4 Solving linear equations in one variable; a brief example of graphing linear equations
The lesson begins with a whole-class discussion of homework: solutions of linear equations in one variable. Students are to justify solutions using Theorems of Equivalence (if they add or subtract the same quantity from both sides of an equation, or if they multiply both sides of an equation by a non-zero constant, the resulting equation is the same as the original equation). The remainder of the lesson alternates between a whole-class discussion of problems and individual work as the teacher circulates. As in HK4 (see below), one equation (4x = 4x — 1) leads to a result (—1 = 0) that requires interpretation from the teacher. In contrast to HK4, there is a brief whole-class discussion of graphing the two sides of the equation: graphing 4x = y and 4x — 1 = y.
J3 Solving linear inequalities in one variable (the first lesson of a unit on inequalities)
Students individually solve a word problem in various ways, and then the teacher selects individuals to come to the board to present solutions to the class (the teacher determines the order of presentation). Solution methods include: counting objects, using a table, solving the problem arithmetically, using a linear equation, and using a linear inequality. Students then use linear inequalities to solve another problem.
J4 Solving linear inequalities in one variable (the seventh lesson of a unit on inequalities)
Students come to the board to write homework solutions. The teacher poses a word problem, students work on it individually, and then two students and the teacher present solutions (the teacher determines the students and the order). Solution methods are: guessing and checking, using arithmetic, and representing the problem as an inequality. The teacher then poses two related problems. Students work on these individually, and then two students come to the board to write their solutions, which use inequalities.
HK4 Introducing identities
The lesson begins with two students solving linear equations in one variable at the board:
2x + 4 = x + 6 and 2x + 10 = 2(x + 5)
The second equation does not have a unique solution. This leads to a whole-class discussion of this situation: what is an identity and how to determine if an equation is an identity. All examples in the lesson are linear equations in one variable. The students expand products and combine like terms to determine whether the right and left sides of a given equation are the same.
HK2 Solving a system of linear equations in two variables using elimination
The lesson begins with two students going to the board to solve (using substitution) systems of linear equations in two unknowns, as other students work individually and the teacher circulates. The teacher introduces the method of elimination using two examples. The teacher assigns four problems on solving systems of equations. Students work the problems individually, at the blackboard, or at their desks as the teacher circulates. The teacher uses the method of elimination to solve another problem with the class. There is a brief discussion of slope at the end of the lesson; the teacher writes the formulas for slope and for distance on the board.
Additional Resources
Chiu, M., Kessel, C., Moschkovich, J., & Muñoz-Nuñez, A. (2001). Learning to graph linear functions: A case study of conceptual change. Cognition and Instruction, 19(2), 215—251.
This is a case study of how a student who saw a change in b as moving a line diagonally or horizontally learned to see it as moving a line b units vertically. The student did not erase his original conception. Instead he refined his horizontal view and the associated strategy for graphing a line as well as developing a new strategy.
Kodaira, K. (Ed.). (1992). Japanese Grade 7 Mathematics (H. Nagata, trans.). Chicago: University of Chicago School Mathematics Project (UCSMP). Available from UCSMP.
Chapter 3 discusses letters and expressions. Chapter 4 discusses equations, and solving and applying linear equations, including “properties of equalities” that are similar in content to the theorems of equivalence of S4. The technique used in HK4 of evaluating the left and right sides of the equation to show that a particular number is a solution is shown in two places: example 1 on pp. 81—82, and example 2 on p. 85). Chapter 5 discusses variables, graphing, functions, including graphs of y = ax. Note that chapter 3 discusses “letters,” and chapter 5 discusses “variables,” suggesting a subtle difference in meaning.
Kodaira, K. (Ed.). (1992). Japanese Grade 8 Mathematics (H. Nagata, trans.). Chicago: University of Chicago School Mathematics Project (UCSMP).
Chapter 3 discusses linear equations in two variables. Chapter 4 discusses linear functions, graphs of linear functions, and solving simultaneous linear equations in two variables by graphing. Available from UCSMP.
Schoenfeld, A., Smith, J. & Arcavi, A. (1993). Learning: The microgenetic analysis of one student's understanding of a complex subject matter domain. In R. Glaser (Ed.), Advances in instructional psychology (Vol. 4, pp. 55—175). Hillsdale, NJ: Lawrence Erlbaum Associates.
This paper describes a detailed study of one student’s understanding of linear equations and their graphs.
Shannon, A. (1999). Keeping score. Washington, DC: National Academy Press.
See pp. 39—48 for a discussion of student responses for “Supermarket Carts,” its scaffolded version “Shopping Carts,” and other assessment tasks that deal with representing a situation with a linear function.
Shannon, A., & Zawojewski, J. (1995). Mathematics performance assessment: A new game for students. Mathematics Teacher, 88(9), 752—757.
This paper deals with how to educate students to understand and benefit from new forms of assessment and discusses (in more detail than in Shannon’s Keeping Score) differences in student responses to two similar tasks that use linear functions to represent situations: “Supermarket Carts” and “Shopping Carts.