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Forming Quadratics
During the
Lesson
  • Whole-Class Interactive Introduction: Key Features of Quadratics (10 minutes)
    • If you have a short lesson, or you find the lesson is progressing at a slower pace than anticipated, then we suggest you end the lesson after the first collaborative activity and continue in a second lesson.

      • Give each student either a mini-whiteboard, pen and eraser, or graph paper.
      • Introduce the lesson with:
        • Today, we are going to look at the key features of a quadratic curve.
        • On your mini-whiteboards, draw the x- and y- axis, and sketch two quadratic curves that look quite different from each other.
      • Allow students to work for a few minutes, and then ask them to show you their whiteboards. Be selective as to which student you ask to explain his or her graphs. Look for two sets of curves in particular:
        • one of which has a maximum point, the other a minimum;
        • one of which has two roots, the other one or none;
        • that are not parabolas.
        • What makes your two graphs different?
        • What are the common features of your graphs?
      • Elicit responses from the class, and try to keep your own interventions to a minimum.

        Encourage students to use mathematical terms, such as roots, y-intercepts, turning points, maximum, minimum.

      • As students suggest key features, write them as a list on the board under the heading "Key Features of a Graph of a Quadratic."
      • Ask about turning points:
        • How many turning points does each of your graphs have? Is this turning point a maximum or minimum?
        • Can the curve of a quadratic function have more than one turning point/no turning points?
      • If all students have drawn graphs with minimums, ask students to draw one with a maximum.
      • Ask about roots:
        • How many roots does each of your graphs have?
        • Where are these roots on your curve?
        • Does anyone have a graph with a different number of roots?
        • How many roots can a quadratic have?
      • If all students have drawn graphs with two roots, ask a student to draw one with one or no roots.
      • Ask about y-intercepts:
        • Has anyone drawn a graph with different y-intercepts?
        • Do all quadratic curves have a y-intercept?
        • Can a quadratic have more than one y-intercept?
      • Write on the board these three equations of quadratic functions:
        Standard Form: Factored Form: Completed Square Form:
        1. y = x2 − 10x + 24 2. y = (x − 4) (x − 6) 3. y = (x − 5)2 − 1
        • Here are the equations of three quadratic functions.
        • Without performing any algebraic manipulations, write the coordinates of a key feature of each of their graphs.
        • For each equation, select a different key feature.
      • Explain to students they should use key features from the list on the board.

      For example, students may answer:

      Equation 1: The y-intercept is at the point (0, 24). The graph has a minimum because the coefficient of x is positive.

      Equation 2: The graph has a minimum and has roots at (4, 0) and (6, 0).

      Equation 3: The graph has a minimum turning point at (5, −1).

      • If students struggle to write anything about Equation 3, ask:
        • How can we obtain the coordinates of the minimum from Equation 3?
        • To obtain the minimum value for y, what must be the value of x? How do you know?

      Equation 3 shows that the graph has a minimum when x = 5. This is because (x − 5)2 is always greater than or equal to zero, and it takes a minimum value of 0 when x = 5.

        • What do the equations have in common? [They are different representations of the same function.]

      Completed square form can also be referred to as vertex form.

      • Now write these two equations on the board:
        4. y = −(x + 4) (x − 5) 5. y = −2(x + 4) (x − 5)
        • What is the same and what is different about the graphs of these two equations? How do you know?

      For example, students may answer:

      • Both parabolas have roots at (−4, 0) and (5, 0).
      • Both parabolas have a maximum turning point.
      • Equation 2 will be steeper than Equation 1 (for the same x value Equation 2's y value will be double that of Equation 1).
  • Whole-Class Introduction to Dominoes (10 minutes)
      • Organize the class into pairs.
      • Give each pair of students cut-up "dominoes" A, E, and H from Domino Cards 1 and Domino Cards.
      • Explain to the class that they are about to match graphs of quadratics with their equations in the same way that two dominoes are matched.
      • If students are unsure how to play dominoes, spend a couple of minutes explaining the game.
        • The graph on one "domino" is linked to its equations, which is on another "domino."
        • Place Card H on your desk. Figure out which of the two remaining cards should be placed to the right of Card H and which should be placed to its left.
      A

      y = x2 + 2x − 35

      -----------------------

      -----------------------


      H

      y = x2 − 8x + 15

      y = (x − 3)(x − 5)

      y = (x − 4)2 − 1


      E

      y = −x2 − 6x + 16

      y = −(x − 8)(x − 2)

      y = −(x − 3)2 + 25


      • Encourage students to explain why each form of the equation matches the curve:
        • Dwaine, explain to me how you matched the cards.
        • Alex, please repeat Dwaine's explanation in your own words.
        • Which form of the function makes it easy to determine the coordinates of the roots/y-intercept/turning point of the parabola?
        • Are the three different forms of the function equivalent? How can you tell?

      The parabola on Domino A is missing the coordinates of its minimum.

      The parabola on Domino H is missing the coordinates of its y-intercept.

      • Ask students to use the information in the equations to add these coordinates.
        • What are the coordinates of the minimum of the parabola on Card A? What equation did you use to work it out? [(4, −1)]
        • What are the coordinates of the y-intercept of the parabola on Card H? What equation did you use to work it out? [(0, 16)]
      • At this stage, students may find it helpful to write what each form of the function reveals about the key features of its graph.
      • If you think students need further work on understanding the relationship between a graph and its equations, then ask students to make up three different algebraic functions, the first in standard form, the second in factored form, and the third in completed square form.
      • Students are then to take these equations to a neighboring pair and ask them to explain to each other what each equation reveals about its curve.
  • Collaborative Work: Matching the Dominoes (15 minutes)
      • Give each pair of students all the remaining cut-up Domino Cards.
      • Explain to students that the aim is to produce a closed loop of dominoes with the last graph connecting to the equations on "domino" A.

        Students may find it easier to begin by laying the dominoes out in a long column or row rather than in a loop.

      • You may want to use Slide P-1 of the projector resource to display the following instructions.
        • Take turns at matching pairs of dominoes that you think belong together.
        • Each time you do this, explain your thinking clearly and carefully to your partner.
        • It is important that you both understand the matches. If you don't agree or understand, ask your partner to explain their reasoning. You are both responsible for each other's learning.
        • On some cards an equation or part of an equation is missing. Do not worry about this as you can carry out this task without this information.
      • You have two tasks during small-group work: to make a note of student approaches to the task, and to support student problem solving.

      Make a note of student approaches to the task.

      Notice how students make a start on the task, where they get stuck, and how they respond if they do come to a halt.

      You can use this information to focus a whole-class discussion towards the end of the lesson.

      Support student problem solving.

      Try not to make suggestions that move students towards a particular approach to this task.

      Instead, ask questions to help students clarify their thinking.

      • If several students in the class are struggling with the same issue, write a relevant question on the board.
      • You might also ask a student who has performed well on a particular part of the task to help a struggling student.

      The following questions and prompts may be helpful.

      • Which form of the function makes it easy to determine the coordinates of the roots/y-intercept/turning point of the parabola?
      • How many roots does this function have? How do you know? How are these shown on the graph?
      • Will this function be shaped like a hill or a valley? How do you know?
  • Sharing Work (5 minutes)
      • As students finish matching the cards, ask one student from each group to visit another group's desk.
        • If you are staying at your desk, be ready to explain the reasons for your group's matches.
        • If you are visiting another group, write your card matches on a piece of paper. Go to another group's desk and check to see which matches are different from your own.
        • If there are differences, ask for an explanation. If you still don't agree, explain your own thinking.
        • When you return to your own desk, you need to consider as a pair whether to make any changes to your own work.
      • You may want to use Slide P-2 of the projector resource to display these instructions.
  • Collaborative Work: Completing the Equations (15 minutes)
        • Now you have matched all the domino cards, I would like you to use the information on the graphs to fill in the missing equations and parts of equations.
        • You shouldn't need to do any algebraic manipulation!
      • Support the students as in the first collaborative activity.
      • For students who are struggling, ask:

        • This equation is in standard form, but the final number is missing. Looking at its graph, what is the value for y when x is zero? How can you use this to complete the standard form equation?
        • You need to add the factored form equation. Looking at its graph, what is the value for x when y is zero? How can you use this to complete the factored form equation?
  • Sharing Work (5 minutes)
      • When students have completed the task, ask the student who has not already visited another pair to check their answers with those of another pair of students.

      Students are to share their reasoning as they did earlier in the lesson unit.

      Extension Work:

      • If a pair of students successfully completes the task, then they could create their own dominoes using the reverse side of the existing ones.

      To do this, students will need to use algebraic manipulation to figure out all three forms of the function.

      Once students have written on all the dominoes, they should give them to another pair to match up.

      This is a demanding task, so you may want to limit the number of dominoes students use.

  • Whole-Class Discussion: Overcoming Misconceptions (10 minutes)
      • Organize a discussion about what has been learned.
      • The intention is to focus on the relationships between the different representations of quadratic functions not checking that everyone gets the right answers.

        • Ella, where did you place this card? How did you decide?
        • Ben, can you put that into your own words?
        • What are the missing equations for this graph? How did you work them out?
        • Did anyone use a different method?
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