Planning to Create a Math-Talk Community
general outline of lesson activities including homework assignment and formative assessment
Lesson Title: _Quadratic Functions______Grade Level/Course: Grade 11 Alg2____
● Source Credit (if applicable):
Sub-unit:
Finding zeros, graphing from vertex form
Connection to Content Standards (include prior grade level standards if applicable) :
● Primary:
F-IF
● Secondary:
F-LE
Connection to Mathematical Practice Standards:
● Primary:
4
● Secondary:
1, 2, 3, 7
What prior knowledge is important for students to understand before starting this lesson?
x-intercepts, y-intercepts, symmetry
Materials Needed by Students and by Teachers (including worksheets, solution keys, power points, etc):
● Dice
● Dice Game worksheet
● KWL worksheet
● Quadratics Vocabulary Chart
● Rich task (2 pages)
PART 1: SELECTING AND SETTING UP A MATHEMATICAL TASK
(i) What are your mathematical goals for the lesson (i.e., what do you want students to know and understand about mathematics as a result of this lesson)?
Students will recognize the x- and y-intercepts.
Students will be able to locate the vertex given an equation.
Students will be able to locate the vertex given a graph
Students will be able to locate the axis of symmetry.
Students will know how the values of a, h, and k in the equation affect the graph.
(ii) What is the rich task that students will explore?
Students will be given a picture of a realistic parabolic path to evaluate. They will come up with an equation to fit the situation. They will be able to identify the vertex and determine if the path will be a successful attempt at a goal.
(iii) In what ways does the task build on students’ previous knowledge, life experiences, and culture? What definitions, concepts, or ideas do students need to know to begin to work on the task? What questions will you ask to help students access their prior knowledge and relevant life and cultural experiences?
Students will build on their previous knowledge of x-intercepts and the ideas of parabolas. Students need to know the definition of maximum and minimum. Where else have you seen parabolic shapes? Can you think of any sports that consist of these shapes? Where and why?
(iv) What are all the ways the task can be solved?
Students will estimate the completion of the path to determine if the path is successful to see if they are in the direct path.
Students will estimate the x-intercepts to determine an equation and then determine if the goal lies on a point in the equation.
(v) Which of these methods do you think your students will use? What misconceptions might students have? What errors might students make?
Students might not be able to determine where the vertex is making the graph go further. Students might make a mistake in writing the equations with the zeros of the function. Students may not understand how the a value affects the graph.Students will estimate the end of the path and use that to determine if the path is a success.
(vi) What particular challenges might the task present to struggling students? to students who are English Language Learners (ELL)? How will you address these challenges?
Kids struggling with reading and vocabulary will need extra time to grasp the vocabulary. Students will have pictures to go along with their vocabulary.
(vi) What are your expectations for students as they work on and complete this task?
● What resources or tools will students have to use in their work that will give them entry into, and help them reason through, the task?
● How will the students work—independently, in small groups, or in pairs—to explore this task? How long will they work individually or in small groups or pairs? Will students be partnered in a specific way? If so, in what way?
● How will students record and report their work?
Students will use their notes from class activities to help come up with the equation for their picture. They will work in small groups in class but more independently at home. Students will be in groups on day 2 to discuss their conclusions and then present to the class with their explanation of why.
(vii) How will you introduce students to the exploration task so as to provide access to all students while maintaining the cognitive demands of the task? How will you ensure that students understand the context of the problem? What will you hear that lets you know students understand what the task is asking them to do?
Can you guys think of anywhere outside of our classroom where these types of parabolas take place? Why is the shape of the parabola important or helpful? Why would it be helpful for someone to know the equation?
PART 2: SUPPORTING STUDENTS’ EXPLORATION OF THE TASK
(i) As students work independently or in small groups, what questions will you ask to—
● help a group get started or make progress on the task?
Can you sketch the path of the object?
What do you know about the shape?
Is there a coordinate that you know to help you write an equation?
● focus students’ thinking on the key mathematical ideas in the task?
What information are you given that will help you solve the problem?
What patterns do you see?
● assess students’ understanding of key mathematical ideas, problem-solving strategies, or the representations?
What can you tell me about the situation?
How does the shape of the graph help you to figure out an equation to model the situation?
If you know the line of symmetry and vertex, what other information should you investigate to find an
equation that models the situation?
● advance students’ understanding of the mathematical ideas?
How does the shape of the graph help you to understand the key points?
What tools could you use to analyze the graphs to understand the situation further?
● encourage all students to share their thinking with others or to assess their understanding of their peers’ ideas?
Does anyone have any ways they think there’s might be different? Is it wrong if it’s different? I am looking for the most abstract way of understanding how this is all connected.
(ii) How will you ensure that students remain engaged in the task?
● What assistance will you give or what questions will you ask a student (or group) who becomes quickly frustrated and requests more direction and guidance in solving the task?
Do you see where the vocabulary fits into the pictures? Can you possibly label anything on your picture?
● What will you do if a student (or group) finishes the task almost immediately? How will you extend the task so as to provide additional challenge?
What if we didn’t graph height vs displacement and we graphed height vs time? Would there be a difference in the parabola? Why?
● What will you do if a student (or group) focuses on non-mathematical aspects of the activity
(e.g., spends most of his or her (or their) time making a poster of their work)?
Ask questions to redirect thoughts. Ask questions to confirm their understanding of what they are doing such as, where is the vertex in the picture? Look at the vocabulary words and help me determine where this all fits in.
PART 3: SHARING AND DISCUSSING THE TASK
(i) How will you orchestrate the class discussion so that you accomplish your mathematical goals?
● Which solution paths do you want to have shared during the class discussion? In what order will the solutions be presented? Why? In what ways will the order in which solutions are presented help develop students’ understanding of the mathematical ideas that are the focus of your lesson?
Students will notice how the maximum or minimum points help us find the equation of the parabola. Also, students should identify why some parabolas open up and some open down, where does this show up in our equation? Students will be able to use the vertex point to predict where the objects will fall given their flight paths. This will build on the idea of horizontal distance or displacement.
● What specific questions will you ask so that students will—
○ make sense of the mathematical ideas that you want them to learn?
○ expand on, debate, and question the solutions being shared?
○ make connections among the different strategies that are presented?
○ look for patterns? begin to form generalizations?
What do you notice about the speed of the parabolic flight path?
Why do you think that some parabolas open up and some open down?
At what point does the object or person return to the ground?
How do your answers to the previous questions help us find the a h and k in the vertex form of the
parabola?
Is the equation in vertex form alone enough to tell us where the objects take off from and where they land?
(ii) How will you ensure that, over time, each student has the opportunity to share his or her thinking and reasoning with their peers?
Give students the opportunity to think pair share some of their ideas. The rich task allows for students to form generalizations and share with peers about the vertex form of the quadratic.
(iii) What will you see or hear that lets you know that all students in the class understand the mathematical ideas that you intended for them to learn?
I am looking for students to use the correct vocabulary from the dice game activity. Students should also be able to identify maximums and minimums or the graphs and what we call that point. Students will also understand how this point will help us write the equation of the parabola in vertex form. Finally going backwards how the equation will help us draw the graph of the parabola.
(iv) What closure will you bring to the lesson? If the lesson is a multi-day lesson, what are some possible stopping points? What closure will you bring at these stopping points?
Everything we have seen so far has been put in vertex form? Is that the only way? Is it the easiest way? Why or why not? What kind of job would actually deal with the equation vs just adjusting to be successful?
(v) What assignment will you give students to do before the next class?
Try to take a couple of these equations and put them in standard form. Do you see any connections or patterns with the equation in standard form that will help determine any of our other points? Given the equation in vertex form, I can easily determine the vertex and line of symmetry but what if I wanted to “easily” recognize the x-intercepts? What form do you think that would be in?
(vi) What will you do tomorrow that will build on this lesson?
Introduce factoring and solving to find roots. Make sure to tie in why I might need to know the roots (solutions) to fit into a realistic situation.
Adapted from Smith, Bill, and Hughes, Thinking Through a Lesson Protocol: A Key for Successfully Implementing High-Level Tasks, Mathematics Teaching in the Middle School, NCTM, Oct., 2008, p.34; also found in Smith and Stein, 5 Practices for Orchestrating Productive Mathematics Discussions, Corwin Press, 2011, p.79.)