Strategy: Compare and Contrast
Compare and contrast would work particularly well for a situation where multiple approaches are possible to solve a problem. This is the case in solving quadratics. There are four possible methods: factoring, completing the square, graphing and the quadratic formula. Each has benefits depending on the type of equation to be solved. First ask the students to solve four equations using the four different methods to be sure they remember how each method works. Then, introduce the rating scale (created by samjshah.com).
Show students a new equation and ask them to first place each method on the scale, then use the most efficient one to solve the equation. This gets students to consider the options before jumping to the quadratic formula (an effective but not efficient method). Have several students present their method, defending their choices. Personal opinion such as “I hate completing the square.” is a valid reason - it will not be as efficient for that student to complete the square if they are not confident using it. Discuss a few more problems as a class, highlighting the situations where each method is most efficient, then allow students to continue solving problems at their own pace. Remind them to continue to compare the methods before solving.
Strategy: Reading for Meaning
- The product of two consecutive integers is 182. What are the numbers?
- Given 40 feet of fencing, surround the largest rectangular dog park possible. Prove that the dimensions you chose give the maximum area.
- The width of a poster is 2/3 the length. The area is 384 square inches. What are the dimensions of the poster?
Word problems appear regularly in the mathematics classroom, yet students frequently shy away from them, expressing hatred of word problems. This is largely due to the nature of a word problem; it requires interpretation rather than giving students the information directly. For example, “The product of two consecutive integers is 182. What are the numbers?” In order to solve this problem students must know the mathematical definition of product, consecutive and integer. The word product may be familiar to students but the mathematical definition is much more specific than the general definition. Once students have recognized they are multiplying numbers that come one after the other, they still have to write an equation. The majority of problems students encounter in class start at the next step, that is, students are given n(n+1)=182 and are merely asked to solve the equation. Even though this word problem is only two sentences, it requires close analysis and a few extra steps beyond what students are accustomed to. This means that I need to be careful to use precise mathematical language regularly to familiarize students with the vocabulary they will need when they solve word problems. I also need to recognize the extra effort required to solve a word problem and support students appropriately.
Learning Activity: Review Solving Quadratics
Grade Level: 11
Topic: Solving Quadratics
Learning Objective: Students will solve quadratic equations accurately and efficiently.
The goal of this lesson is to review the methods of solving quadratics before introducing complex numbers. Students will have seen all four methods (factoring, completing the square, graphing and the quadratic formula) in previous courses. As students enter the room project four equations and the four methods along with instructions “Using each method once, solve the equation below.” Have students share their process to be sure every student has an example of each method. Then, introduce the rating scale (created by samjshah.com).
Show students a new equation and ask them to first place each method on the scale, then use the most efficient one to solve the equation. After a few minutes have several students present their ratings, defending their choices. Discuss a few more problems as a class, highlighting the situations where each method is most efficient, then allow students to continue solving problems at their own pace. Remind them to continue to compare the methods before solving.
- Slides with a mixture of quadratic equations (factorable and not)
During class I will circulate the room, checking in with each student to make sure that they are not using the same method for every problem and that they are solving the problems accurately. Students will have more mixed problems for homework, which I will check at the beginning of the next class. At the beginning of the following class students will complete a short (3 question) quiz with three different types of quadratics (factorable, non-factorable no leading coefficient, leading coefficient). They will check their own answers at correcting stations around the room when they are done. This gives students immediate feedback and enables me to flip through the quizzes quickly to see how the class did and which students need remediation.
Introducing the rating scale before setting students loose on the problems forces students to consider the differences between the methods of solving quadratic equations. This comparison encourages them to consider what type of equation they are solving before blindly following the steps of a procedure. Mixing different types of problems together results in a deeper analysis than when students complete an algorithm without understanding. When students are tuned in to the characteristics of the problems they are solving it results in deeper meaning, not to mention more efficient calculations.