This lesson focuses on linear equations in one variable that are identities. It is the first of an eight-lesson unit on identities. The lesson is taught in English and is 32 minutes long. There are 42 students in the class.
| 00:34–5:52 | (Whole class) The class solves 2x + 4 = x + 6 and 2x + 10 = 2(x + 5). |
| 00:34–2:50 | Two students come to the board to solve the equations, using methods the teacher says they had learned previously (1:43). One student makes a mistake and another student comes to the board to correct it. |
| 2:50–5:52 | The teacher discusses the equations and their solutions. Equation 1 has the solution x = 2, but the same solution methods produced 0 = 0 for equation 2. The teacher asks, “What does this mean?” |
| 6:04–12:31 | The class checks the left and right sides of equation 2 for particular values of x. |
| 6:04–9:35 | (Whole class) The teacher checks the values of the left and right sides of equation 2 for x = 2 and x = 3. |
| 9:35–11:18 | The students work individually to see if x = 0, –1, and –1/2 are solutions for equation 2 by evaluating each side and comparing these values. |
| 11:18–12:31 | (Whole class) The teacher asks individual students to report their results. |
| 12:31–17:40 | (Whole class) The class compares the left and right sides of the equation by expanding each side. The teacher introduces identity to denote equations for which both sides are the same in expanded form. |
| 12:31–13:20 | The teacher mentions two possible forms for expressions—expanded form and factored form—and introduces the idea of putting each side of the equation in expanded form and comparing the two sides. The class does this for equation 2. |
| 16:25–17:40 | The teacher explains that if the two sides are the same when put in expanded form the equation is called an identity. |
| 17:40–27:46 | (Whole class) On the board, the teacher writes: 5(x – 3) – 3(x – 1) = 2(x – 6) and 4(2x – 1) – 3(x + 2) = 5(2 – x) |
| 17:40–24:21 | The teacher asks how to expand both sides of the first equation to determine whether or not it is an identity. |
| 24:21–27:46 | A student comes to the board to determine whether or not the second equation is an identity. The teacher discusses the student’s solution. |
| 27:46–31:42 | The teacher assigns problems from the textbook for students to work on individually. Students work as the teacher circulates. |
Few (if any) U.S. schools teach the topic of identity in the middle grades. In high school students often learn about identities in the context of trigonometric identities and how to show that an equation is an identity by leaving one side fixed and transforming the other side. However, identity is an associative property of addition—a concept that is taught in the elementary grades.
This lesson compares values of the left and right sides of an equation for particular values of x (6:04–12:31). Lesson J3 also uses this technique in context of a linear inequality (31:39–46:02).
The lesson introduces the idea of counterexample, though the teacher does not use this term, but says that one example may disprove a universal statement (18:45). The class considers the statement “for any value of x, 2x + 10 = 2(x + 5).” For further discussion, see Logic, below.
The use of letters as variables can cause problems in school mathematics. It is best to think of and work with the letter x as if it were a number. The discussion moves (12:31–17:40) from checking some possible values of x to how to check for all possible values of x. The teacher suggests writing both sides in expanded form. Then, if the terms on both sides are the same, they will be the same regardless of the numerical value of x. The teacher does not make this last point explicitly, perhaps because he expects that the class would know that, if the two sides have the same terms, they really are the same and will be equal for each value of x.
The researcher comments: “These first two problems both have problem statements that imply students can apply a set of procedures to reach the solution (i.e., ‘Find the solution for these two equations’). However, as they discuss the problems the class uses mathematical reasoning and examines the mathematical relationships involved. Therefore both are coded as having a ‘using procedures’ problem statement and a ‘making connections’ implementation.”
The use of the word “therefore” helps to make clear the flow of the argument (5:25). The teacher illustrates the use of “identity” and “identically equal” (17:14–17:27, 17:56, 27:34–27:46). The teacher also repeatedly asks, “What does it mean?”, both in the case of equation 1 (for which the student obtains x = 2) and equation 2 (for which the student obtains 0 = 0). For equation 1, the teacher illustrates what it means to be a solution (using x = 2) and not to be a solution (using x = 3). For equation 2, he illustrates the idea that any value of x is a solution—five different values of x are shown to be solutions. However, he states this does not prove every value of x is a solution.
The lesson is in English, which is not the students’ first language. The National Research Coordinator comments (17:37) that “This is important for teachers in the U.S. who teach classes with students whose native language is not English.”
The teacher’s actions contribute to the slow and deliberate pace of the lesson, even though it is only 32 minutes long.
The teacher illustrates the meaning of expanded and factorized form with empty boxes and parentheses (see Lesson Graph and 13:50–14:14).
Technical detailFile name: NB10311008.
Students should be familiar with:
Students will learn the following:
Terminology
Some early algebra textbooks call equations that are not identities “conditional equations” (Chrystal, 1952; first published in 1886). Fisher and Schwatt (1898, Part 1) classify equations as either “identical equations” or “conditional equations,” with the following descriptions:
“An Identical Equation, or simply an Identity, is an equation one of whose members can be reduced to the other, or both of whose members can be reduced to a common form, by performing the indicated operations.” (p. 153)
“A Conditional Equation is an equation one of whose members can be reduced to the other only for certain definite values of one or more letters contained within it.” (p. 154)
Comparing the left and right sides of an equation
Comparing the left and right sides for particular values of x is used also in J3 (31:39–46:02) in the case of a linear inequality.
There are conceptual differences among the three statements “the left and right sides of an equation are equal for every value of the variable,” “the two sides of the equation have the same expanded form,” and “one side of the equation can be transformed into the other side using laws of algebra.” The lesson does not explicitly address this matter, but the first two ideas underlie checking that the right and left sides of the equation have the same value for x = 2, 3, 0, –1, –1/2 (6:29–13:06) versus comparing the expanded form of each side of the equation.
The Cartesian Connection
Schoenfeld, Smith, and Arcavi (1993) have described what they call the Cartesian Connection—a detailed specification of what constitutes understanding of linear equations and graphs of lines in the Cartesian plane. In the lesson that she taught, Tanya DeGroot (see Appendix 3 in Susanna Epp’s report ) focused on an important piece of this connection, namely, “a point (x, y) lies on a line if and only if (x, y) satisfies the equation of the line” (see Schoenfeld et al., figure 24). In her lesson, the phrase “(x, y) satisfies the equation of the line” means that “(x, y) makes the equation true.” She further elaborated by asking students to check the values of the left and right sides of the equation for particular values of x and y in a manner similar to HK4.
Logic
The teacher says that one example may disprove a universal statement (18:45). However, several examples do not prove a universal statement, such as, in this case, “for any number x, 2x + 10 = 2(x + 5).” The teacher makes this explicit after checking for five values of x by asking, “Then I can conclude they are an identity?” No, as the statement may fail at the sixth trial (18:45). For further discussion of the idea of counterexample and that several examples do not prove a universal statement, see Epp (1998).
While it is usually true that checking a finite number of cases does not prove something that holds for infinitely many, there are important cases when checking a finite number of cases is sufficient. The situation in this lesson is one example. The teacher was right not to mention this to the students, but teachers should know some examples:
If a polynomial of degree one is zero at two different points, then it is identically zero. In the present situation, having two polynomials of degree one agree at two points forces them to be identically equal. When the linear equations are graphed, one sees that is equivalent to saying that two points determine a straight line. The simple geometric interpretation of this type of result breaks down for higher degree polynomials, but there are similar results. Two polynomials of degree two that are the same at three points are identically equal.
Making connections
The researcher comments that these first two problems both have problem statements that imply students can apply a set of procedures to reach the solution (i.e., “Find the solution for these two equations”). However, as they discuss the problems the class uses mathematical reasoning and examines the mathematical relationships involved. Therefore both are coded as having a “using procedures” problem statement and a “making connections” implementation.
James Hiebert discusses this “making connections” implementation in HK4 in his 2003 presentation about the TIMSS Video Study.
“Motivating” identity
The segment 00:34–17:40 is an example of motivating the concept of identity—developing the concept (using Stigler and Hiebert’s distinction in The Teaching Gap, 1999) or developing a rationale (U.S. Dept. of Ed., 2003, p. 194). Here it makes sense to single out an object or class of objects by giving them a special name. In this case some equations have every number as a solution, while others do not. The former get the name “identity” and have a special notation (17:56). For example, to indicate that the equation 2x + 10 = 2(x + 5) is an identity, one writes 2x + 10 2(x + 5).
Instructional decisions
(6:27) In response to the question “What will be the solution?,” the teacher hears, “Any number.” Rather than moving on to the next topic, the teacher asks the students to check five values of x.
(29:27–30:05) The teacher (like the textbook) requires students not simply to answer whether both sides of the equation are the same with a “yes” or “no,” but to write down their conclusions. These and other actions contribute to the slow, deliberate pace of the lesson.
The teacher introduces the idea of identity using linear equations rather than later with trigonometric equations (as is often done in the U.S.).
Representations
The teacher illustrates the meaning of expanded and factorized form with empty boxes and parentheses (see the lesson graph and 13:50–14:14). The teacher denotes that an equation is not an identity by writing “L.H.S. ≠ R.H.S.” He introduced the sign for “identically equal to” nine minutes earlier (17:56), so this would have been a good time to remind students of this notation and to have introduced the notation for “not identically equal to” [
]. He again introduces the identically equal sign a minute later (27:34) but never mentions a symbol for not identically equal.
Susanna Epp used several clips from this lesson (00:34–08:48, 11:18–19:13, and 27:34–31:53) in a course for teachers. The students were prepared to view the clips by first working carefully chosen and worded problems. One of Epp’s students used the idea of thinking that an equation held if and only if the left and right sides are equal in a lesson that she taught to high school students on graphing linear equations. (For a description of this lesson, see Appendix 3 in Epp’s report.
Another option is to summarize an early segment of the lesson (00:34–2:50) and then show a middle segment (2:50–17:40). Showing one long segment rather than several shorter clips allows viewers to focus on teacher actions and motivation for the concept of identity.
Chrystal, G. (1952). Algebra: An elementary text-book, part 1 (6th Ed., 1st Ed. published 1886), New York: Chelsea Publishing Company.
Epp, S. (1998). A unified framework for proof and disproof. Mathematics Teacher 91(8), 708–713 (available from Susanna Epp at sepp@condor.depaul.edu).
Fisher, G.E. & Schwatt, I.J. (1898) Text-book of algebra with exercises for secondary schools and colleges (Part 1). Norwood, MA: Norwood Press.
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.
Stigler, J. W. & Hiebert, J. (1999). The teaching gap: Best ideas from the world’s best teachers for improving education in the classroom. New York: Free Press.
U.S. Department of Education, National Center for Education Statistics (March 2003). Teaching Mathematics in Seven Countries: Results from the TIMSS 1999 Video Study, NCES (2003-013), by A. M.-Y. Chiu, W. Etterbeek, R. Gallimore, H. Garnier, K. Bogard Givvin, P. Gonzales, J. Hiebert, H. Hollingsworth, J. Jacobs, N. Kersting, A. Manaster, C. Manaster, M. Smith, J. Stigler, E. Tseng, and D. Wearne. Washington, DC: Author.
| School Improvement |
Research and Evaluation |
International Studies |
RBS Publications |
Education Resources |
About RBS |
| Contact Us | Home |
