Download Solving Quadratic Equations Student Probe Lesson Description

Solving Quadratic Equations Student Probe Probe 1 Solve
Answer: Probe 2 Solve
Answer: Probe 3 Solve
Answer: . (Refer to Part 1 of the lesson.) . (Refer to Part 2 of the lesson.) . (Refer to Part 3 of the lesson.) Lesson Description At a Glance What: Solving quadratic equations Common Core State Standard: CC.9-­‐
12.A.REI.4 Solve equations and inequalities in one variable. Solve quadratic equations in one variable. Matched Arkansas Standard: AR.9-­‐
12.QEF.AII.3.3 (QEF.3.AII.3) Analyze and solve quadratic equations with and without appropriate technology by: -­‐-­‐ factoring, -­‐-­‐ graphing, -­‐-­‐ extracting the square root, -­‐-­‐ completing the square, -­‐-­‐ using the quadratic formula Mathematical Practices: Make sense of problems and persevere in solving them.
Look for and make use of structure. Who: Students who cannot solve quadratic equations Grade Level: Algebra 1 Prerequisite Vocabulary: square root, absolute value Prerequisite Skills: solve linear equations in one variable, factor quadratic trinomials Delivery Format: individual, small group, whole group Lesson Length: 30 minutes for each part Materials, Resources, Technology: graphing calculator (optional) Student Worksheets: none In this three part lesson, students will learn to solve quadratic equations by extracting square roots, by factoring, and by using the quadratic formula. Each part of the lesson can be taught independently, based upon the needs of the student. Rationale Quadratic equations are the simplest form of polynomial equations to solve and are usually encountered once students are proficient in solving linear equations in one variable. Solving quadratic equations by extracting square roots is efficient, but is dependent upon the form of the equation. Solutions by factoring are efficient and the factors determine the zeros of the function. However, many quadratic equations are difficult or impossible to factor. The quadratic formula will solve all quadratic equations provided the equation is in the form . Preparation None Lesson The teacher says or does… Expect students to say or do… If students do not, then the teacher says or does… Part 1: Solve Quadratic Equations by Extracting Square Roots 1. We are going to solve quadratic equations. In some ways solving quadratic equations is like solving linear equations. 2. When we solve equations Taking the square root of the Prompt students. we must “undo” number. Refer to Solving Equations. operations. What “undoes” squaring a number? 3. In some ways solving It is a parabola. Prompt students. quadratic equations is Refer to Relating Quadratic different than solving Functions and Graphs. linear equations. What does the graph of a quadratic function look like? 4. Since the graph of a 0, 1, or 2 Sketch a graph of a parabola quadratic function is a showing the three parabola, how many possibilities. solutions can a quadratic (See Teacher Notes.) equation have? What number times itself 5. Solve . equals 4? (See Teacher Notes.) Is 2 the only number times itself that is equal to 4? What about ? 6. Solve . 7. Repeat Steps 5-­‐6 with additional equations in the form , as necessary. The teacher says or does… 8. Solve . Expect students to say or do… If students do not, then the teacher says or does… 9. Repeat Step 8 with additional equations in the form
, as necessary. Part 2: Solve Quadratic Equations by Factoring 10. We are going to solve quadratic equations. In some ways solving quadratic equations is like solving linear equations. 11. When we solve equations Taking the square root of the we must “undo” number. operations. What “undoes” squaring a number? 12. In some ways solving It is a parabola. quadratic equations is different than solving linear equations. What does the graph of a quadratic function look like? 13. Since the graph of a 0, 1, or 2 quadratic function is a parabola, how many solutions can a quadratic equation have? Prompt students. Refer to Solving Equations. Prompt students. Refer to Relating Quadratic Functions and Graphs. Sketch a graph of a parabola showing the three possibilities. (See Teacher Notes.) The teacher says or does… Expect students to say or do… If students do not, then the teacher says or does… 14. What do we know if the One or both of the numbers If , what is x? product of two numbers is must be 0. How do you know? 0? This is called the Zero Product Theorem. It is stated as: If , then We use this theorem when we solve quadratic equations by factoring. Either or Refer to Step 14. 15. Solve . We have the product of The Zero Product Theorem two quantities (or tells us. numbers) equal to 0. What can we say? How do you know? 16. Now we have easy equations to solve: and Solve them. 17. Solve . Can we factor the quadratic expression 18. Now we can use the Zero Product Theorem to solve it. Refer to Solving Equations. Yes. Refer to Factoring Quadratic Trinomials. Model for students. 19. Repeat Steps 17-­‐18 with additional equations, as necessary. Part 3: Solve Quadratic Equations using the Quadratic Formula 20. We are going to solve quadratic equations. In some ways solving quadratic equations is like solving linear equations. The teacher says or does… 21. When we solve equations we must “undo” operations. What “undoes” squaring a number? 22. In some ways solving quadratic equations is different than solving linear equations. What does the graph of a quadratic function look like? 23. Since the graph of a quadratic function is a parabola, how many solutions can a quadratic equation have? 24. We are going to use a method of solving quadratic equations that always works—the Quadratic Formula. In order for us to use the formula, the equation must be written in the form . The formula is Expect students to say or do… If students do not, then the teacher says or does… Taking the square root of the Prompt students. number. Refer to Solving Equations. It is a parabola. Prompt students. Refer to Relating Quadratic Functions and Graphs. 0, 1, or 2 Sketch a graph of a parabola showing the three possibilities. (See Teacher Notes.) For this equation, Students may forget that the implied coefficient of and is 1. . 25. Let’s solve using the formula. What is a? What is b? What is c? (See Teacher Notes.) 26. Substitute the values into the formula. Monitor students. Model for students, if necessary. The teacher says or does… Expect students to say or do… If students do not, then the teacher says or does… 27. Now we just need to Monitor students. simplify this expression. Students may need to review What are the values for x? order of operations. See Teacher Note 4 for questions concerning . Monitor students. 28. Solve using the quadratic formula. 29. Repeat Steps 25-­‐28 with additional equations, as necessary. Teacher Notes: 1. Quadratic functions may have 0, 1, or 2 zeros as shown. No Solutions One Solution Two Solutions 2. Most students will solve simple quadratic equations such as by inspection. 3. A common misconception is . By definition, the square root of a number is positive. Thus, and . The two solutions arise from the absolute value of x, rather than from the . 4. The symbol is read “plus or minus”. It is a shorthand notation indicating positive 2 and negative 2 (or some other quantity. If students find it confusing, they may write instead. 5. Make sure that students understand the quadratic equation must be equal to zero to solve using both factoring or the quadratic formula. 6. To avoid incorrect application of the quadratic formula, have students explicitly the values of a, b, and c. 7. Once students have correctly substituted values into the quadratic formula, you may wish to allow them to use a calculator to obtain the solutions. 8. The quadratic formula may be derived by solving the general quadratic by completing the square. Many texts provide the derivation. 9. Students should be encouraged to check their answers, either by substitution or by graphing. 10. Every effort should be made for students to have the opportunity to connect the solutions to quadratic equations to the graphs of the corresponding quadratic function. Variations 1. Encourage students to use the graphing calculator to determine the number of solutions and the type of solutions (rational, irrational, etc.) for quadratic equations. 2. Once students are confident in their ability to solve quadratic equations, provide them with a variety of equations that may or may not be solved by extracting square roots or factoring. Have students determine the best method for solving each equation. Formative Assessment Part 1 Solve Answer: Part 2 Solve Answer: Part 3 Solve Answer: . by factoring. using the quadratic formula. References
Mathematics Intervention at the Secondary Prevention Level of a Multi-­‐Tier Prevention System: Six Key Principles. (n.d.). Retrieved May 13, 2011, from rtinetwork: http://www.rtinetwork.org/essential/tieredinstruction/tier2/mathintervention Paulsen, K., & the IRIS Center. (n.d.). Algebra (part 2): Applying learning strategies to intermediate algebra. Retrieved on October 18, 2011 from http://iris.peabody.vanderbilt.edu/case_studies/ICS-­‐
010.pdfhttp://iris.peabody.vanderbilt.edu/case_studies/ICS-­‐010.pdf