This is the third in a series of lessons on factoring and the first to deal with solving quadratic equations. The lesson is 46 minutes long, and is in Swiss German. There are 24 students in the class. Of the three possible tracks (with basic, extended, or highest demands), it is a “track three” class with highest demands.
The previous lesson dealt with simplification of rational functions to show the usefulness of factoring (see Overhead Transparency 2: “Factoring” in the Resources section on the CD).
| 00:35–11:18 | (Whole class) Review of homework. See the lesson graphs for the problems and solutions. |
| 11:20–16:46 | Students work on the first problem as the teacher circulates and talks individually with students. |
| 16:47–19:15 | (Whole class) The teacher discusses the first problem (x² + 10x + 25 = 0) with the class. Students give the factorization and answer the question of how to use the factorization to solve the equation. |
| 19:16–22:30 | Students work on the second (x² + 22x + 121 = 0) and third problems. |
| 22:35–25:05 | (Whole class) The teacher discusses the third problem. Before starting the discussion the teacher makes a correction on the overhead, which has the problems and solutions, since she solved x²– 169 = 0 rather than x²– 196 = 0. |
| 25:05–29:00 | The teacher presents a more complicated problem (6x² – 7x – 3 = 0). |
| 29:04–35:31 | (Whole class) Discussion of how to attack this problem, with the teacher using some of the student work she had observed while walking around the room. The whole class work culminates in a solution. |
| 35:31–46:00 | (Individual work) The teacher introduces the problems by saying: “but for today you have got enough new material. Now I would like you to practice a little on these techniques.” |
A team member noted that the first three problems are instances of identities, which the class had derived earlier but they now examine from a different perspective:
(a + b)² = a² + 2ab + b²
(a – b)² = a²– 2ab + b²
(a – b)(a + b) = a²– b²
Students are given the right side and need to find the left side. The teacher may or may not have led up to this in the previous lesson (see 14:42–15:27).
| 25:05–25:48 | The teacher introduces the last problem, solving 6x²– 7x –3 = 0, by saying: “It’s just that quadratic equations are not always of this kind. And the next one is a rather tricky problem.” She provides a hint: to write 6x²–7x – 3 = 0 as ( )( ) = 0, and suggests guessing to try to fill in the brackets. |
| 39:34–40:00 | A student is writing very large and carelessly. The teacher says: "…and you're using that much space, then you will need a whole page only for one problem and won't be able to keep the overview anymore." To "keep an overview", students need to write all the problems on one page, so they can see the progression from problem to problem. |
Technical detailFile Name: NB10311049.
The students should know:
Students will learn to:
The problem of solving 6x² – 7x – 3 is two steps removed from the first three equations. An intermediate step would have been to ask students to solve an equation like
x² + 7x + 6 = 0
where there is an obvious way to decompose x² as x times x rather than to decompose 6x² as 6x times x or 3x times 2x. Teachers should ask themselves: What are some reasons for using or not using an intermediate step? Does the fact that this is the most advanced class influence your reasons?
It is possible to reduce the problem of solving 6x² – 7x – 3 = 0 to a problem like solving x² + 7x + 6 = 0.
Let x = y/6. The equation becomes
(y²)/6 – 7y/6 – 3 = 0, or y² – 7y – 18 = 0.
Teachers should ask themselves: Does this method generalize, and could it be used just for factoring?
| School Improvement |
Research and Evaluation |
International Studies |
RBS Publications |
Education Resources |
About RBS |
| Contact Us | Home |
