Unit Plan for Chapter 9: Quadratic Equations and Functions



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 |Warm-up: Give students a linear equation. |Warm-up: Students simplify imperfect square |

|simplifying exponential expressions. |Ask students to work in pairs to write the |roots, and graph a parabola from a given |

|Review afterwards. |steps for deriving two ordered pairs from |equation. |

| |the equation. Have teams volunteer to list|Review through volunteered explanations and |

|Introduce basic square roots that yield |steps one at a time on the board (or |any needed clarifications. |

|whole number answers. Explain that the |overhead). Explain that volunteering teams| |

|terminology for “square roots” implies |must write the next step to the one before |Activity: Students pass their homework |

|tracing the number under the radical to |from the previous team. Discuss with class|around to share findings w/ each other. |

|its “root”, or, the number that, when |afterwards. |Collect afterwards. |

|squared, will equal it. | | |

| |Question: Ask students if they think the |Discussion: Show students a picture of a |

|Have students practice, and then lead them|procedure they listed would work for any |bridge w/suspension cables. Ask what part |

|into imperfect square roots, and ask |function of x equation. Briefly discuss |of the resulting parabola would indicate the|

|students how they think they might be able|why or why not. |maximum height of the bridge’s structure? |

|to simplify. | |What effect might changing the height have |

| |Intro: Tell students to get ordered pairs |on the shape of the parabola? What effect |

|Introduce splitting an imperfect radical |for the basic quadratic equation y=x² using|might changing the distance between the |

|into factors that can be taken out of the |values of x ranging from 0 to 3, and their |bridge posts have on the parabola? Would it |

|radical. |opposites, after which they are to plot |also mean that the equation for the parabola|

|Ask students what other aspects of math |their points. Ask them if they think a |would also change? |

|involve the concept of “doing” and |single line could be drawn through all of | |

|“undoing”. |them. Why or why not? After properly |Activity: Divide class in half; tell |

| |connecting the points, have students write |everyone on one side to graph y=1/2x², and |

|Practice: Give students a series of |a list that describes as many features of |the other side y=2x². |

|practice problems with perfect and |the parabola shape they observe. Share and|T circulates around class to check for |

|imperfect radicals to simplify. |discuss. Incorporate ideas of symmetry, |correctness. T chooses one student from each|

| |vertex, and x-intercepts. |side to present on the overhead. Take |

|Activity: Explain how a model for the | |observations from class on similarities and |

|speed at which a tsunami moves involves a |Activity: Divide the class in half, and |differences. T asks guided questions to get|

|radical. Present students with varying |each half into pair teams. Each gets a |students to realize that the “a” value makes|

|ocean depths for which they will derive |half-sheet grid transparency. First half |the shapes differ. |

|the moving speed of a tsunami. Ask |graphs a given parabola when “a” is | |

|students to reflect on why the answers to |negative, and the other half when “a” is |Instruction: Teacher provides instruction on|

|the conditions might be similar or |positive. T circulates to check for |solving quadratic equations by graphing. |

|different. |correctness. T chooses one pair team from |Checkpoints allowed for students to practice|

| |each to display results simultaneously on |w/problems. |

|Homework: Have students make 10 of their |overhead. Whole class discussion on | |

|own radicals (both perfect and imperfect) |similarities and differences; guided |3-Team Activity: Put students into teams of |

|and derive the answers for each. Tell |questioning to have students arrive at what|3. Each team is given a different picture of|

|them to use their notes and books as |makes them inverted. |a bridge and its corresponding quadratic |

|sources for help and ideas. | |equation for the parabola involved. A |

| |(If Time Allows): Have pair teams on each |coordinate plane transparency overlay is |

| |side brainstorm a list of things (big or |also provided, that indicates the axis of |

| |small) in real-life that incorporate a |symmetry and the height measurements of two |

| |parabola shape. |towers. Students are to use graphing |

| | |calculators to graph the parabola of their |

| |Homework: Have students search through |equation, and then determine the distance |

| |magazines or the internet for pictures from|between the two towers given the information|

| |their list that incorporate a parabola |provided and their calculators, as well as |

| |shape. In what way do you think the shape |supporting their answers by solving the |

| |is useful for the object? (ex: |equation mathematically. |

| |functionality, looks, side-effect of | |

| |another design, etc.) Bring findings |Quiz: Students simplify radicals, solve |

| |w/answers to class to share. |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 |Activity: Students present their posters and|Warm-up: Students work on review problems |

|reinforce previously learned concepts of |explain their findings. They then post |dealing with finding x-intercepts by |

|simplifying radicals, determining |their work around the classroom and do a |solving an equation, using the quadratic |

|x-intercepts, and graphing a parabola. |gallery walk, critiquing each others’ work |formula, and supporting answers by |

|Take volunteers to present answers |according to set criteria and questions, |graphing. Review w/class. |

|w/explanations. |established the prior day. Teams then | |

| |return to their posters and write |Instruction: Tell students that although |

|Question: Do you think the methods |reflections on their peers’ comments and on |the quadratic formula is good for reaching|

|discussed so far for solving quadratic |their experience. |a definite numerical answer should there |

|equations might apply to one such as | |be one, a short cut to going through all |

|y=2x²-2x+3? Ask guiding questions to get |Activity 2: Students are put into teams of 3|the work is being able to tell ahead of |

|students to realize that it would be much |and are given a picture of a real-life |time if there will even be an answer |

|more difficult to find the x-intercepts |object that infuses the parabola. Also given|waiting at the end. Introduce the |

|that way. |one transparency grid w/coordinate plane. |discriminant as a means to this. |

| |Teams experiment w/grid, picture, and what | |

|Instruction: Introduce the quadratic |they know to derive a probable equation for |Application – Use the discriminant to |

|formula. How does this formula relate to |the parabola. All parabolas are assumed to |determine whether a person who can jump 12|

|the quadratic equations we have been |have a vertex at the origin. Selected |f/s will be able to dunk a basketball, if |

|working with so far? Do you see any |groups are chosen to present. |the minimum height needed to do this is |

|elements inside the formula that you might | |2.2 feet. |

|have seen before? What do you think the |Semi-Long Quiz: Students find x-intercepts | |

|“±” sign means you will have to do when |by solving quadratic equations, and by using|Question – How do you think the velocity |

|simplifying the formula? Students practice|the quadratic formula when solving the |of a person’s jump can be measured? |

|by using the formula to find the |equation is impractical. |Students brainstorm ideas. |

|x-intercepts of given quadratic equations. | | |

| |Review quiz with class if time allows. |Activity: Teams of 3 students work |

|Activity: Break students into pair-teams. | |together using a meter stick, marker, and |

|Students experiment with a tennis ball and |Homework (3 days): Students create their own|stopwatch to gauge and record each others’|

|stopwatch to gauge how long it takes the |word problem involving a dropped or thrown |jumping velocity. They use this |

|ball to fall varying distances. |down object. Factors such as velocity and |information and the vertical motion |

|Predictions are made prior to this by using|choice of metric units must be considered |equation to determine whether or not they |

|the quadratic formula for a dropped object.|when needed, as well as creating the |would be able to dunk a basketball |

| |scenario itself. Illustrations are required |according to the minimum height |

|They then compare this time prediction to |to enhance the realism of the scenario as |stipulation in the application problem. |

|the actual time gauged in the experiment. |well as to heighten the understandability of| |

|Students record their data along with |the problem. Students are to write |Questions to consider: Was your answer |

|predictions and solutions on a table. |explanations of their answer as it relates |reasonable or unreasonable? Given your |

| |to the problem, as well as explain any |present height, how high do you think the |

|Question: What factors might have affected |answers that didn’t make sense to keep. |rim would have to be in order for you to |

|the outcome of your experiment? What | |be able to dunk if your answer says you |

|difference in the data do you think there | |should or shouldn’t? Explain. Comparing |

|might have been if the ball had been thrown| |the initial velocity of the person in the |

|instead of being dropped? What would have | |applications problem to your own, might |

|caused this difference? Would it make | |there be factors that are not being |

|sense to have a negative value for time? | |considered in deciding whether or not you |

|Why or why not? | |would be able to dunk? Explain. |

| | | |

|Activity (continued): Students work | |Homework: Practice problems w/getting the |

|backwards to derive the initial height of a| |discriminant. Students continue working on|

|tennis ball before it was thrown vertically| |the creation of their word problem. |

|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. | | |

| | | |

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