##
* *Reflection: Diverse Entry Points
Zeroing In - Section 3: Wrap it Up

I selected these two students to show the range of thinking about x-intercepts and zeros. Student A clearly "gets" it, making a simple but accurate explanation of the connection. Student B still has a way to go and it's hard to tell from her explanation if she really understands the full relationship because she says that x-intercepts "help" to find zeros but doesn't make the clear connection that the intercepts are zeros. She also puts in the comment about intercepts being whole numbers or "easy" fractions so I think she does recognize the limitations of reading zeros directly from a graph. I will speak with student B one-on-one to determine just what her misconceptions are and work from there. Possible questions to help clarify her understanding might be "How do the x-intercepts help you find the zeros" and "what kind of fractions don't work?"

*Diverse Entry Points: Identifying student levels of understanding*

# Zeroing In

Lesson 5 of 11

## Objective: SWBAT identify the zeros of given polynomials. SWBAT use identified zeros to construct a rough graph of the polynomial.

## Big Idea: Help your students "zero in" on what the graph of a polynomial looks like using zeros to plot key points.

*60 minutes*

#### Set the Stage

*10 min*

I begin this lesson with a polynomial graph projected on my front board. As my students come in I expect them to discuss what they see because they know from previous experience that I'll be asking questions! After the bell I ask them to identify points of interest, reminding them to use appropriate vocabulary! Because I project directly onto my whiteboard I can mark the points as my students identify them. When they've identified several "points of interest" I ask for volunteers to explain what each point represents on the graph. **(MP2)** For example I expect someone to mention that the function doesn't go through the origin, that it crosses the x-axis three times and the y-axis once. If one of my volunteers mentions that the points where the function crosses the x-axis are called "zeros", fantastic! If not, I use leading questions like, "Do these points of intersection have any other names? and simpler examples (perhaps quadratic) to help my students make that connection.

#### Resources

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#### Put it into Action

*45 min*

**Individual work** *15 minutes:* I tell my students that today they get to work with several quadratic and cubic functions to solve for the zeros. I distribute the ZEROS handout and assure them that they can use whichever factoring method works best for them since some students prefer traditional while others like synthetic division or rectangular factoring. **(MP1, MP2, MP5)** As they're working I walk around giving encouragement and assistance as needed paying particular attention to those students who are still struggling with factoring.

**Graphing** *15 minutes:* When everyone has finished finding all the zeros, I tell my students that now they get to use graphing calculators to graph **only** the first five functions and compare the zeros they've calculated to the points of interest on each graph. I ask them to sketch the graphs next to their calculations (or on a separate piece of paper if they need more room) and mark the zeros they've calculated directly on their sketches. I ask for a volunteer to summarize the relationship between zeros, factors and points on a graph then ask for fist-to-five to check for understanding before moving on. After answering any questions I tell my students to put their calculators away and use what they've just discussed to sketch graphs of the remaining five polynomials. **(MP1, MP2) **This is a challenge for some because they haven't really made the connection between zeros and x-intercepts, but I can usually help those students individually as I walk around the room.

**Real-world connection** *10 minutes:* When everyone is done I tell my students that I have one more challenge for them today. I post the Real-World Polynomial example on the board and ask if they see any connection between the example and the graph at the beginning of class. I tell my students that while some of the math they've done has been simply to gain skills, like running drills for sports, the purpose of all those drills is so that they can make sense of real-world problems like the one I've just posted. I ask them to use their skills to explain what the zeros of the function are **AND** what they represent in terms of the problem. I tell them the challenge will be to communicate their thinking to their right-shoulder partner so that he/she can understand it.** (MP1, MP4, MP6)**

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#### Wrap it Up

*5 min*

To wrap up this lesson I have my students write in their own words the relationship between the x-intercepts of a graph and the zeros/factors of a polynomial. I tell them that I expect complete sentences and that while they can include a sketch in their explanation it must be supplementary to the writing rather than the primary description. Sometimes my students complain that they're in math class not English to which I reply that even Einstein would be an unknown today if he didn't write down his ideas well enough for other people to understand them! ask if they see any connection between the example and the graph at the beginning of class. As I discuss in my video, a shift new to most in the common core is the importance of mathematical written expression and discourse.

#### Resources

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- UNIT 1: First Week!
- UNIT 2: Algebraic Arithmetic
- UNIT 3: Algebraic Structure
- UNIT 4: Complex Numbers
- UNIT 5: Creating Algebraically
- UNIT 6: Algebraic Reasoning
- UNIT 7: Building Functions
- UNIT 8: Interpreting Functions
- UNIT 9: Intro to Trig
- UNIT 10: Trigonometric Functions
- UNIT 11: Statistics
- UNIT 12: Probability
- UNIT 13: Semester 2 Review
- UNIT 14: Games
- UNIT 15: Semester 1 Review