Marcos Dafaee
SED 595JG
2/7/07
Unit Plan for Chapter 9: Quadratic Equations and Functions
California Standards:19.0 Students know the quadratic formula and are familiar with its proof by completing the square. 20.0 Students use the quadratic formula to find the roots of a second-degree polynomial and to solve quadratic equations. 21.0 Students graph quadratic functions and know that their roots are the x- intercepts. 22.0 Students use the quadratic formula or factoring techniques or both to determine whether the graph of a quadratic function will intersect the x-axis in zero, one, or two points. 23.0 Students apply quadratic equations to physical problems, such as the motion of an object under the force of gravity.
Learning Objectives:
-Students will be able to simplify radicals, and then use this skill to solve simple quadratic equations. –Students will be able to use prior knowledge about the Cartesian coordinate system and derivation of ordered pairs to graph quadratic functions. –Students will be able to derive the roots (or solutions) of a quadratic equation by drawing and analyzing its graph. –Students will be able to use the quadratic formula to solve simple and more complicated quadratic equations. –Students will be able to use the discriminant to determine the number of solutions of a quadratic equation. –Students will be able to apply their acquired knowledge of quadratics in solving near to real-life situations.
Day 1 / Day 2 / Day 3
Warm-up: Give problems that review simplifying exponential expressions. Review afterwards.
Introduce basic square roots that yield whole number answers. Explain that the terminology for “square roots” implies tracing the number under the radical to its “root”, or, the number that, when squared, will equal it.
Have students practice, and then lead them into imperfect square roots, and ask students how they think they might be able to simplify.
Introduce splitting an imperfect radical into factors that can be taken out of the radical.
Ask students what other aspects of math involve the concept of “doing” and “undoing”.
Practice: Give students a series of practice problems with perfect and imperfect radicals to simplify.
Activity: Explain how a model for the speed at which a tsunami moves involves a radical. Present students with varying ocean depths for which they will derive the moving speed of a tsunami. Ask students to reflect on why the answers to the conditions might be similar or different.
Homework: Have students make 10 of their own radicals (both perfect and imperfect) and derive the answers for each. Tell them to use their notes and books as sources for help and ideas. / Warm-up: Give students a linear equation.
Ask students to work in pairs to write the steps for deriving two ordered pairs from the equation. Have teams volunteer to list steps one at a time on the board (or overhead). Explain that volunteering teams must write the next step to the one before from the previous team. Discuss with class afterwards.
Question: Ask students if they think the procedure they listed would work for any function of x equation. Briefly discuss why or why not.
Intro: Tell students to get ordered pairs for the basic quadratic equation y=x² using values of x ranging from 0 to 3, and their opposites, after which they are to plot their points. Ask them if they think a single line could be drawn through all of them. Why or why not? After properly connecting the points, have students write a list that describes as many features of the parabola shape they observe. Share and discuss. Incorporate ideas of symmetry, vertex, and x-intercepts.
Activity: Divide the class in half, and each half into pair teams. Each gets a half-sheet grid transparency. First half graphs a given parabola when “a” is negative, and the other half when “a” is positive. T circulates to check for correctness. T chooses one pair team from each to display results simultaneously on overhead. Whole class discussion on similarities and differences; guided questioning to have students arrive at what makes them inverted.
(If Time Allows): Have pair teams on each side brainstorm a list of things (big or small) in real-life that incorporate a parabola shape.
Homework: Have students search through magazines or the internet for pictures from their list that incorporate a parabola shape. In what way do you think the shape is useful for the object? (ex: functionality, looks, side-effect of another design, etc.) Bring findings w/answers to class to share. / Warm-up: Students simplify imperfect square roots, and graph a parabola from a given equation.
Review through volunteered explanations and any needed clarifications.
Activity: Students pass their homework around to share findings w/ each other. Collect afterwards.
Discussion: Show students a picture of a bridge w/suspension cables. Ask what part of the resulting parabola would indicate the maximum height of the bridge’s structure? What effect might changing the height have on the shape of the parabola? What effect might changing the distance between the bridge posts have on the parabola? Would it also mean that the equation for the parabola would also change?
Activity: Divide class in half; tell everyone on one side to graph y=1/2x², and the other side y=2x².
T circulates around class to check for correctness. T chooses one student from each side to present on the overhead. Take observations from class on similarities and differences. T asks guided questions to get students to realize that the “a” value makes the shapes differ.
Instruction: Teacher provides instruction on solving quadratic equations by graphing. Checkpoints allowed for students to practice w/problems.
3-Team Activity: Put students into teams of 3. Each team is given a different picture of a bridge and its corresponding quadratic equation for the parabola involved. A coordinate plane transparency overlay is also provided, that indicates the axis of symmetry and the height measurements of two towers. Students are to use graphing calculators to graph the parabola of their equation, and then determine the distance between the two towers given the information provided and their calculators, as well as supporting their answers by solving the equation mathematically.
Quiz: Students simplify radicals, solve quadratic equations involving using radicals, and graph parabolas by first getting at least five ordered pairs.
Homework: Students derive the x-intercepts of quadratic equations by graphing and solving mathematically.
Day 4 / Day 5 / Day 6
Warm-up: Students do problems that reinforce previously learned concepts of simplifying radicals, determining x-intercepts, and graphing a parabola.
Take volunteers to present answers w/explanations.
Question: Do you think the methods discussed so far for solving quadratic equations might apply to one such as y=2x²-2x+3? Ask guiding questions to get students to realize that it would be much more difficult to find the x-intercepts that way.
Instruction: Introduce the quadratic formula. How does this formula relate to the quadratic equations we have been working with so far? Do you see any elements inside the formula that you might have seen before? What do you think the “±” sign means you will have to do when simplifying the formula? Students practice by using the formula to find the x-intercepts of given quadratic equations.
Activity: Break students into pair-teams. Students experiment with a tennis ball and stopwatch to gauge how long it takes the ball to fall varying distances. Predictions are made prior to this by using the quadratic formula for a dropped object.
They then compare this time prediction to the actual time gauged in the experiment. Students record their data along with predictions and solutions on a table.
Question: What factors might have affected the outcome of your experiment? What difference in the data do you think there might have been if the ball had been thrown instead of being dropped? What would have caused this difference? Would it make sense to have a negative value for time? Why or why not?
Activity (continued): Students work backwards to derive the initial height of a tennis ball before it was thrown vertically downwards to hit the floor (or ground). Students compare their answers to the previous questions to the data they collected from throwing the ball. They then make comparisons of both sets of data on a poster.
Homework: Students work on problems that incorporate all types of quadratic equations and methods of solving them. Work involves graphing by first deriving ordered pairs for points. / Activity: Students present their posters and explain their findings. They then post their work around the classroom and do a gallery walk, critiquing each others’ work according to set criteria and questions, established the prior day. Teams then return to their posters and write reflections on their peers’ comments and on their experience.
Activity 2: Students are put into teams of 3 and are given a picture of a real-life object that infuses the parabola. Also given one transparency grid w/coordinate plane. Teams experiment w/grid, picture, and what they know to derive a probable equation for the parabola. All parabolas are assumed to have a vertex at the origin. Selected groups are chosen to present.
Semi-Long Quiz: Students find x-intercepts by solving quadratic equations, and by using the quadratic formula when solving the equation is impractical.
Review quiz with class if time allows.
Homework (3 days): Students create their own word problem involving a dropped or thrown down object. Factors such as velocity and choice of metric units must be considered when needed, as well as creating the scenario itself. Illustrations are required to enhance the realism of the scenario as well as to heighten the understandability of the problem. Students are to write explanations of their answer as it relates to the problem, as well as explain any answers that didn’t make sense to keep. / Warm-up: Students work on review problems dealing with finding x-intercepts by solving an equation, using the quadratic formula, and supporting answers by graphing. Review w/class.
Instruction: Tell students that although the quadratic formula is good for reaching a definite numerical answer should there be one, a short cut to going through all the work is being able to tell ahead of time if there will even be an answer waiting at the end. Introduce the discriminant as a means to this.
Application – Use the discriminant to determine whether a person who can jump 12 f/s will be able to dunk a basketball, if the minimum height needed to do this is 2.2 feet.
Question – How do you think the velocity of a person’s jump can be measured? Students brainstorm ideas.
Activity: Teams of 3 students work together using a meter stick, marker, and stopwatch to gauge and record each others’ jumping velocity. They use this information and the vertical motion equation to determine whether or not they would be able to dunk a basketball according to the minimum height stipulation in the application problem.
Questions to consider: Was your answer reasonable or unreasonable? Given your present height, how high do you think the rim would have to be in order for you to be able to dunk if your answer says you should or shouldn’t? Explain. Comparing the initial velocity of the person in the applications problem to your own, might there be factors that are not being considered in deciding whether or not you would be able to dunk? Explain.
Homework: Practice problems w/getting the discriminant. Students continue working on the creation of their word problem.