I originally used this assessment in an 8th grade Honors Algebra 1 class; however, it can be used in any Algebra 1 classroom. This project involves a lot of physics connections, and could also be used as a joint project between math and science content areas. 

Before starting this learning module, students should be able to:

• Describe and identify inear and exponential functions

• Identify key characteristics of functions (maximums, minimums, domain, range, intercepts)

• Be able to graph points and create appropriate axes on a coordinate plane

• Solve multi-step equations

Time needed to complete this module:

• The first three lessons should each take one 60 minute class period
• Depending on student progress, the performance assessment (lessons four through six) should take between three to five 60 minute periods

Common Core Standards Addressed:

• CCSS.MATH.CONTENT.HSF.IF.B.4
  For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.*
• CCSS.MATH.CONTENT.HSF.IF.B.5
  Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.
• CCSS.MATH.CONTENT.HSF.IF.C.7
  Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.*
• CCSS.MATH.CONTENT.HSF.IF.C.7.A
  Graph linear and quadratic functions and show intercepts, maxima, and minima.
• CCSS.MATH.CONTENT.HSF.IF.C.8
  Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
• CCSS.MATH.CONTENT.HSF.IF.C.8.A
  Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.
  (Common Core State Standards Initiative, 2010)

The first three lessons in this learning module are meant to review the key topics addressed in this performance assessment: key characteristics of quadratic functions, being able to find the vertex, and domain and range. The last three lessons are the performance assessment: the experiment, the calculations and graph, and the presentation and peer review. Because of the experiment part of this performance assessment, this module is best taught in-person. 

Learning Target: Students will be able to identify the x-intercepts, y-intercept, and axis of symmetry of a quadratic function from a graph. 

The first lesson of this learning module helps students review the most important features of quadratic functions. Students also begin to contextualize these features, which is an important part of the performance assessment. 

Students will start with a notice/ wonder warm up. Students previously learned about exponential and linear functions, and should be able to identify them from the tables. They may notice that the third table does not look like the other two, as two of the outputs are repeated.

The primary part of the lesson in on Desmos. The teacher-facing lesson is here (Simon, n.d) and the teacher can assign individual session codes or assign the lesson to Google Classroom. Desmos is similar to Nearpod, but it is made specifically for math lessons. This Desmos lesson covers how to find the x-intercepts, y-intercept, vertex, axis of symmetry, and maximums/minimums. You can either teach this lesson live or make it self-paced. There are plenty of opportunities for formative assessment. Teachers can see student progress on the teacher dashboard. The following video shows how to use the Desmos dashboard to gather formative assessment data. 

Video: Desmos teacher dashboard running an activity (Best, 2018)

For the update, students need to choose a problem from the activity "Domain, Vertex, and Zeros" from Illustrative Mathematics. Students will not only identify key characteristics, but explain them in a given context. Students will need to complete similar tasks for their own experiment later in the unit. 

Learning Target: Students will be able to find the x and y-coordinates of the vertex of a parabola in standard form. 

The purpose of the warm up of this lesson is to get students thinking about the connections between equations and their graphs. Many students will likely grasp that 4 is the y-coordinate of the y-intercept, but struggle with seeing that 2 is the slope. Encourage students to determine whether the linear function is increasing or decreasing. Further scaffolding may include plotting points or making a table. Students may also grasp how to find the x-intercept from a graph, but not from an equation. Encourage them to think about what the y-coordinate of the x-intercept is.

This lesson is on Desmos. The teacher-facing activity is linked here (Desmos, n.d.). You can create individual codes for each class or assign the lesson on Google Classroom. The lesson "Where is the vertex?" details a non-traditional approach to finding the vertex of a parabola in standard form. Typically, students need to complete the square to find the vertex of the parabola, but due to the nature of certain curriculums, some students may not have learned that method yet. This method is a useful alternative that promotes conceptual understanding of the structure of a parabola. If students have learned how to complete the square, you can substitute this lesson with one reviewing completing the square. 

Finding the vertex in this lesson involves more critical thinking about symmetry and midpoints. Students will identify the y-intercept, and then use the graph of the parabola and eventually the equation to find the other point with the same y-value. Since parabolas are symmetrical, the vertex should be located at the x-value that is in the middle of the two points. Students can then plug this x-value back into the original equation to find the y-coordinate of the vertex. This process can be used to conceptualize the traditional equation of finding the vertex: \(x =-b/2a\)

The update for this lesson encourages students to find real-world examples of parabolas and determine how they could find the vertex of that parabola using methods that make sense in the context that they chose. The methods for measuring a parabola created by a baseball would be measured differently than the St. Louis arch. 

Learning Target: Students will be able to find the domain and range of a parabola and interpret it in the context of a real-world problem.

The purpose of the warm-up for this lesson is to remind students how they found the domain and range of functions that they have learned about in the past: linear and exponential functions. The linear function has a range of (-∞, ∞) and a domain of (-∞, ∞). The exponential function has a range of (0, ∞) and a domain of (-∞, ∞). As students are determining these values, ask conceptual questions like "Is 0 included in the range of the exponential function?" and "Why might the outputs (range) be different for these two functions?".

The main lesson is on Desmos. The link to the teacher-facing dashboard is here (McPherson, n.d.). This is a longer lesson with many real-world examples. Consider splitting students up into small groups after the Fireworks activity. Each group can complete one real-world situation (rocket launch, area of a rectangle, electric power, high-rise profit, operating costs, and business cards). Students can then present their findings to the class. This lesson uses the Desmos graphing calculator to graph each function. Students may need to adjust the viewing windows. For more on how to use the Desmos graphing calculator, see the video below.

Video: Introduction to the desmos graphing calculator (Desmos, 2020)

The update for this lesson requires students to build on what they learned in the lesson. They must find another real-world example using parabolas. While it can be similar to the ones they did in the lesson, it should not be the same. Students will graph the function, then find the domain and range and explain what they mean in the context of that problem. 

Learning Task: Students will be able to design an experiment in a small group to find the maximum height and initial velocity of a tennis ball.

This is the first of three days spent on the performance assessment for this unit. Today, students will design an experiment to find the initial velocity and maximum height of a thrown tennis ball. Tomorrow, students will model this function with a graph and an equation. The rubrics for the performance assessment are included below:

"How High Can You Throw?" rubrics (Woodward, n.d.)

Students should be working in groups of four. Depending on your class, you can either assign heterogenous groups or allow students to pick their own groups. Students will be reviewing their group members at the end of the project, so there is a level of accountability for either grouping option. This part of the performance task should be supervised, rather than led, by the teacher. If students are struggling, some questions you could ask include:

• What roles are each of you taking in this experiment?
• How can you make sure that your measurements are consistent?
• What are some factors that you might not be able to control?

This part of the perfomance assessment includes an outdoor component for the experiment. If you do not have access to an outdoor space, a gymnasium with high ceilings will also suffice. If weather or time becomes an issue, this part of the performance assessment can be done over two days instead of one. 

Learning Target: Students will be able to calculate the initial velocity and maximum height given the data from their experiment. They will be able to write an equation, graph that equation, and identify key features of the graph in context.

This lesson is day two of the three day performance assessment. Today, students will be making most of their mathematical calculations, graphing their function, and making a written report of their findings. Like the previous day, this learning should be primarily student-driven, with teacher support. Since this is a perfomance assessment, productive struggle should be expected and encouraged; students have the materials they need to complete this task, and it is up to each group on how to best use them. 

While the calculations can be made by each group, the report should be done individually. Having students use mathematical language and contextualize their equation and its features is one of the important parts of this performance assessment. This is the part of the project that will be peer reviewed tomorrow. For students with disabilites or EL students, you can provide sentence stems like the ones below to help scaffold the report writing. 

Math Journal Sentence Starters (Panicked Teacher, 2015)

Learning Target: Students will be able to give positive and critical feedback and use others' feedback to improve their own work

Today, students will be peer reviewing each others' final reports. While this project is set up for two peer reviews per student, that number can be adjusted based on time and the number of students. The peer reviews can be done either digitally or on paper. If you want to keep the peer review anonymous, having students submit their work using Google Docs would allow the papers to be distributed and commented on anonymously. See the video below for how to use this resource.

Video: Google forms for peer evaluation (Teacher Tech Cafe, 2017)

Students may use the sentence starters given in the student section to help them write feedback. You may also use student examples from the update in lesson 5 to provide good examples of feedback. Students will then use this feedback to edit the draft of their reports. They will then turn in their report, worksheet, and rubrics. 

After grading using the rubrics in update 4, you can tally all the scores in the Assessment List, which is included below. Students will already have filled out the self-grading column. 

"How High Can We Throw?" Assessment List

Students will also fill out a short survey about the performance assessment. The purpose of this survey is to determine what went well in the performance assessment and gain feedback on what could be improved the next time this project or similar projects are done. 

The final update has students reflect on an aspect of the project of their choosing. They then must explain what they learned and how it might apply in other courses or in their lives outside of school. It is important that this performance assessment is not viewed as an isolated task, but as a larger part of learning and knowledge.