Learning Progression for Trigonometric Functions in Precalculus: CCSS.MATH.HSF.TF.A.1-4

This learning progression was designed primarily for a high school Precalculus course. The three Common Core State Standards that this learning progression will be satisfying are from the cluster titled “Extend the domain of trigonometric functions using the unit circle,” these are HSF.TF.A.1, HSF.TF.A.3, and HSF.TF.A.4. In this course, students are focusing on mastering the Common Core State standards for Functions. Throughout this learning progression, students will focus on three mathematical practices which are MP5, MP7, and MP8.

One of the instructional tasks that is included in the Learning Progression is creating a unit circle.
In order to construct the unit circle students first have to form the two special right triangles. Through this activity they will from connections between the unit circle and the special right triangles which will strengthen their understanding of the concepts of the lesson.

The assessment used in this learning progression is the green sheet quiz. On this quiz students are given three angles and then they have to draw the angle and evaluate the 6 trigonometric functions for it. The tasks in the learning progression prepare students for this assessment.

Learning Progression

Learning Progression for: Reason quantitatively and use units to solve problems.

This learning progression was designed primarily for a 9th grade algebra course. The three Common Core State Standards that this learning progression will be satisfying are from the cluster titled “Reason quantitatively and use units to solve problems,” these are HSN-Q.A.1, HSN-Q.A.2 and HSN-Q.A.3. In this course, students are focusing on mastering the Common Core State standards for Algebra. Throughout this learning progression, students will focus on three mathematical practices which are MP2, MP4, and MP6.

There are three instructional tasks/activities included in this learning progression. Each one was provided by Illustrative Mathematics.

The following is an excerpt from the learning progression regarding the task “Fuel Efficiency”

Learning Target:

I can use unit conversions and proportions to determine fuel efficiency.

Task: Fuel Efficiency

Sadie has a cousin Nanette in Germany. Both families recently bought new cars and the two girls are comparing how fuel efficient the two cars are. Sadie tells Nanette that her family’s car is getting 42 miles per gallon. Nanette has no idea how that compares to her family’s car because in Germany mileage is measured differently. She tells Sadie that her family’s car uses 6 liters per 100 km. Which car is more fuel efficient?

Guiding questions:

Are the two mileages given in the same form? (no, our task says they aren’t)

When we say 42 miles per gallon this is an example of a (rate)?

How should we set up the problem? (use proportions)

____________________________________________________

At the end of the unit the students will do a project for the summative assessment. The students will be told to look through newspapers and magazines and collect 2 examples of situations to be expressed algebraically.  They are then supposed to come up with the algebraic expression that accompanies each. They must also write up what each variable represents, what quantities are involved and what units are being used.

Combined Learning Progression and Formative Assessment

Mystery (Function) Machine F-IF.1&7

CCSS.MATH.F-IF.1

Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).

CCSS.MATH.F-IF.7.a

Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.★

  1. Graph linear and quadratic functions and show intercepts, maxima, and minima.

The math concept that is the focus of this picture problem is functions. The math thinking that can take place via this picture problem includes simplifying, analyzing patterns, and synthesizing the information followed up by representing the information in a variety of ways. For starters, the student can simplify what is represented in the picture. They can pull out the numbers and then they can realize from the mystery machine that they will be evaluating a function. They can analyze the pattern between the numbers that will give them the equation of the function. Then they will represent their synthesis of the information in a variety of ways including a table and a graph. Their processes will be evaluating the given number and coming up with an equation for it. Then they will have to determine the graph from either the points or the equation.

The picture brings relevance because in this unit students have been learning about functions and the curriculum will frequently use the concept of a function machine to describe how each input has only one output. As students are already familiar with the concept of a function machine, so bringing in a physical model such as the Mystery Machine Function Machine or even the above photo of the Mystery Machine Function Machine will be very relevant to what they are learning. As for engagement, this will also get students engaged because they will have to take what is supplied in the photo and compile it into a table, establish what the function is and then graph it. Teachers can put the picture on the board and then ask the students to do what they think they should do with what is supplied in the picture or the teacher can give them explicit directions.

Crossover Between Exercise and Mathematics F.F-IF.4

Many students are not quiet about their opinions of a typical math class, one where the teacher lectures, the students write notes and there’s time at the end to work on the homework. While there may be a need for this sometimes, this is not the most efficient way to get students engaged in the class and the lesson.

An easy way to mix things up and increase student engagement is by implementing an activity, other than a basic worksheet to complete. With all the technological advances occurring all around us, it’s important that we expose our students to some of these new technologies especially when they relate to math.

Students learn best and engage when the activity and topics can be related to their life or daily activities. This is where the “Crossover between Exercise and Mathematics” activity comes into play. Even if not all students enjoy exercise, they are all familiar with it because they are required to take a certain number of gym credits in high school.

Image result for CBR 2In this activity students are to use a Vernier CBR 2 Motion Detector with a compatible TI graphing calculator, either a TI-83 or TI-84. The students will work in groups of three to complete this math exercise activity. They will have to come up with an exercise were they can measure distance and amount of time while they are performing it.

The students will take turns with each person performing the exercise, operating the equipment, and recording the information on their worksheet.

Once the students have decided on the exercise to perform they will make a prediction of what their graph of distance over time will look like. The distance will be graphed on the y-axis and the time on the x-axis.

Once their prediction graph is drawn they will use the Vernier CBR 2 motion detector and TI graphing calculators to create a graph of the exercise. After using the software and getting the graph, the students will record the graph onto their sheet and explain the differences between their graph and the graph done by the software. They can also explain which aspects of the exercise are visible in each part of the graph. For example, if the exercise is pushups then the graph would go up and down and the peaks would be when the person is in the fully extended position of the push-up. And the down of the pushup would be in the valley of the graph.

This activity relates to the following Common Cores State Standard:

CCSS.MATH.HSF.F-IF.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.

Connection Between Exercise and Math worksheet

ClassKick Keeps the Class Kicking!

Image result for classkick

Abstract:

ClassKick is a browser application that allows teachers to give assignments out digitally, and for students to ask for help anonymously. With ClassKick, teachers can easily create worksheets for their lessons that are accessible via a Class Code.  In acquiring this class 

Image result for classkick

code and putting it to use students are not required to provide their personal information. Students can work on an assignment at home or at school, and the teacher will receive real-time progress of their students. This technology also makes it possible for teachers to  track student work for multiple students at once.  Students are also able to ask their teacher and peers questions while working on their assignments and then receive immediate feedback. This also makes it possible for teachers to act upon student feedback more quickly during class time.

 

Resources:

CCSS.MATH.CONTENT.HSF.BF.A.1

Write a function that describes a relationship between two quantities.

Teacher Guide for Using ClassKick

ClassKick Article