This blog post contains two lesson plans intended for Algebra I classrooms. The lessons can be adapted for similar topics in pre-algebra or Algebra II with moderate alterations. The two thinking strategies that I chose to highlight are the See-Think-Wonder routine and the Circle of Viewpoints routine.

**Lesson One: **Introduction to Quadratic Functions: See – Think – Wonder

**Lesson Objective: Students will use prior knowledge of quadratic functions from pre-algebra as well as linear equations to identify mathematical properties of the images presented. Students will begin to explore properties of quadratic functions as well as any other mathematical ideas and concepts as they appear in the image. Students will make connections to the mathematics they already know, what they want to know and the phenomena they observe through a mathematical lens. This activity is designed to be an open-ended activity. There is not one specific property of a quadratic function that should be discovered through the activity, but rather a starting point for students forming meaningful connections to quadratic functions, their properties and graphs.**

**Lesson Outline: **This lesson is intended to get Algebra I students thinking about what a parabola looks like outside of the mathematics classroom. Ideally, this lesson would be used to introduce quadratic functions and their properties so students can associate properties of quadratic functions and graphs of quadratic functions with concrete examples that exist outside of the classroom.

**Timing: **This lesson is designed for a 45 minute class period and will only take one day. However, if good conversations are occurring, the conversation and extensions may run over into a second day.

**Prior knowledge****:** Students do not necessarily need prior knowledge of quadratic functions to be successful in this activity. There are other mathematical aspects of the images that students may pick up on and wish to discuss.

**Standards: **

**A.CED.1 – I CAN create quadratic equations equations in one variable.**- A.REI.4.1 – I CAN solve quadratic equations using completing the square.
- A.REI.4.2 – I CAN solve quadratic equations by taking square roots, factoring and the quadratic formula.
- F.IF.1 – I CAN understand domain and range as they relate to a quadratic function.
- F.IF.7a.1: – I CAN graph quadratic functions and show key features.

**Set up****:** Class begins with three images on the board

**See****:** “Are these images mathematical?”

- Students record what they notice so that they don’t forget by the time the discussion begins.
- Students discuss their observations with one peer and record shared and different ideas.

**Think****:** Think about how these images are connected to the mathematics you already know. Think about the mathematics you *want* to know about these images.

- Students record what they think so that they don’t forget by the time the discussion begins.
- Students discuss their observations a different peer and record shared and different ideas.

**Wonder****:** Why are these images mathematical? What other structures or images are mathematical? Where else do you see mathematical images such as these outside of the classroom?

- Students record what they think so that they don’t forget by the time the discussion begins.
- Students discuss their observations a third peer and record shared and different ideas.

**Discussion: **A class discussion follows the activity to explore what mathematics students saw, what they thought about that mathematics and where else they see similar mathematics. * If the discussion strays from these topics, that is okay. Discuss any appropriate mathematics students want to explore in the activity.

**Video**: After the discussion, this video can be used to show students other parabola examples outside of the classroom: Quadratic Functions and Parabolas in the Real World

**Assessment:**

*See*: informal assessment of student’s ability to recognize any mathematical properties in the images. As students participate in similar activities, assessment can look for improvement in student’s ability to notice mathematical details and relate those details to prior knowledge*Think*: Focus on support that students are able to provide for their observations as well as connections to prior mathematical knowledge*Wonder*: listen to students musings for broad questions and connections that are making relations to mathematics. These questions should not have concrete answers. Also when students suggest other mathematical images, those images should relate to quadratic functions in an out of the box matter.

**Materials:**

- Images to project on board (above)
- Paper and pencil for students to record observations and thoughts
- Recording space for student observation and thoughts during discussion

**Teacher Notes: **Be sure to allow enough time for students to examine each image through a mathematical lens. This may be tough at first, but don’t guide them to specific properties of quadratics or parts of the image, such as the vertex. Students will draw on their prior knowledge of parabolas, which will take longer for some students to recall than others. The discussion will be the most insightful part of this activity; allow and encourage students to build on each other’s discoveries and thoughts. Avoid pointing out mathematical features of the images and asking students leading questions that point to various aspects of quadratics functions.

**Lesson Two: **Transformations – Circle of Viewpoints

**Lesson Objective: **Students will be exploring the various representations of quadratic transformations as they relate to one another as well as the parent function. Students will be provided a graph, equation and table for a quadratic function and will be asked to describe the function using mathematical language. Students will then engage in justifying that each of the mathematical representations are representing the same quadratic transformation. They will communicate using mathematical language and problem solving in order to justify their conclusions. This activity is designed to be a connection creating activity. Students have knowledge of quadratic transformations, tables, graphs, equations and are well versed in communicating mathematically. Thinking is made visible in this activity through proving posing and problem solving.

**Lesson Outline: This lesson focuses on examining quadratic transformations of functions from different representational perspectives. Quadratic transformations can be expressed verbally, graphically, by an equation and by a table (this is not an extensive list, but the ones I would focus on for this activity).**

**Timing: This lesson is designed for a 45 minute class period and will take two days. Day one is designed to identify the viewpoints, and for students to explore the different representations with their group. Day two will be finishing up those discussions and exploring the different groups’ perspectives as a whole class.**

**Prior Knowledge: Students are beginning this lesson after having been exposed to linear transformation, properties of quadratic functions and base knowledge of parent functions. **

**Standards: **

- A.CED.1 – I CAN create equations in one variable and use them to solve problems
- 8.F.2 – I CAN compare properties of functions using graphs, tables, algebraically and by verbal descriptions.
- A.SSE.1a – I CAN interpret terms, coefficients and exponents of expressions
- A.SSE.1b – I CAN complex expressions by viewing them as individual pieces.

**Set Up:** Multiple representations of a quadratic function that has been transformed is given to students who will be working in small groups)

**Identify viewpoints:** The representations will be apparent to students as they will physically see a table, a graph and an equation and will be given directions to describe the transformation from the parent function verbally (orally within the group and written on paper)

**Select a Viewpoint to Explore: **Students will be given topics to explore for the various representations such as:

- Are each of these representations depicting the same quadratic transformation?
- Why do you believe this to be true?
- How can you support your claim?
- Would this be true of all verbal descriptions, graphs, tables and equations (assuming they are mathematically accurate)?

**Respond to the “I think….” Prompt: **Students will explore how each individual representation is representing the quadratic transformation and how the representation relates to the parent function. Students will be asked to provide concrete evidence that each of the representations in fact do represent the same quadratic transformation and justify their evidence.

**Respond to the “A question I have from this viewpoint….” Prompt: **This step will happen naturally throughout the activity as students discuss the multiple representations of the quadratic transformation and justify whether or not they all represent the same quadratic transformation. Questions will arise regarding the connections between different representations as well as the representations themselves and groups will address these questions as they arise.

**Share the Thinking****:** Student groups will share their findings and discuss these findings with their classmates in an open-idea environment. This allows for all students to ask questions, clarify confusion, expand upon discoveries, reflect upon the process and revise their findings if necessary/desired. In this discussion, common misconceptions regarding transformations (that were most likely noticed during the activity) should be addressed.

**Assessment****:** Groups will be assessed based upon their justifications provided for whether or not each mathematical representation describes the same quadratic function. Students will be provided a rubric for their justifications. Participation in the discussion will also be included as part of the assessment.

**Materials**:

- Prepared tables, graphs and equations of quadratic functions for each group
- Appropriate technology for exploration (graphing calc, access to interactive dynamic technology, etc)
- Recording devices (pencil paper, google doc, etc) for student groups
- Recording space for student observations and thoughts during discussion

**Teacher Notes: The purpose of this activity is to let students create connections and justify their thinking while connecting knowledge. Allow students to participate in this activity by not telling them if they are on the right or wrong track with their justifications. Students seek approval and will want to know if they are on the right track. Instead of saying yes or no, try to pose an open question that keeps students thinking in terms of their task and where they are at in justifying their conclusions. **

**Extension:** After students have been successful thought the graphing transformations unit, Marble slides, a Dan Meyers Desmos activity, can be used to formally or informally student’s knowledge and application of domain, range and transformations.

Check out @HollyClarkEdu for some great ideas on making thinking visible in classrooms with technology! Here’s a link to her blog too!

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“listen to students musings for broad questions and connections that are making relations to mathematics” – LOVE THIS STATEMENT! We need more ‘student musings’ in ALL classes because that is where long term memory connections are made….

The video on parabolas in the read world was fascinating…. I never thought of the parabolas in nature and in sports. My experience was left off at the St. Louis Arch… sad, but true. So thank you for enlightening me the vast world of parabolas.

Your blog is filled with such good ideas… Yes, the SEE-THINK-WONDER is truly an open-ended experience, especially as you have developed here.

Thank you for the link to the making thinking visible blog…. it is terrific but ours is even better (but then I may be biased!)….

Thank you, Dr. Mackenzie!

I like your lesson on parabolas. I really enjoy the part when they turn to their neighbors and have to write down something that is in common and something that was different. I feel sometimes when students turn to talk they never write anything more down so this is a good way for them to be intentional on writing more down on their lists.

Also, I just remembered that when I student taught many years ago we did a parabola activity involving pop tarts because when you bite down it forms a parabola. It was pretty fun activity and students definitely saw some real world connections with that. Here is a link to a similar activity http://300math.weebly.com/uploads/5/2/5/1/52513515/day_2_-_the_parabola_of_best_fit_-_a_dental_dilemma_final.pdf

Thanks for the feedback! I agree that sometimes students have great conversations and often forget to write down their thoughts. I’ve also found the exact opposite where students see another “good idea” on another student’s page and write it down on their paper too. By writing something in common and something different, students are still engaging in those conversations and hopefully asking each other where they saw those similarities and differences.

The pop tart activity sounds fun! I’ll definitely look into it 🙂

I love that you are taking images and having students wonder about what they would like to know about them. I think that in every lesson there should be some aspect of wonder, and I like the way you have connected that into your lesson. I might even have a set of linear images for students to compare and contrast against the quadratic images.

I really like this lesson to as well. I just might think of what the “Catch” or “hook” for the lesson is.

These look like really great lessons.

Thanks so much Cory! I am definitely struggling with the hook for the second lesson as well. I think in developing the graphs, tables and equations I would link them to a real world transformation scenario, but haven’t quite figured out how I would want to do that yet. I definitely agree that each lesson should have an aspect of wonder – it keeps students so much more engaged. And great suggestion to compare images with linear properties. Students will definitely enjoy the link to prior knowledge to build new knowledge. Thanks for all of your input!