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

Systems of Inequalities Assessment A.REI.D

System of inequalities (CCSS.Math.Content.HSA.REI.D.12)


Alignment to Content Standards

Alignment: CCSS.Math.Content.HSA.REI.D.12

Alignment: CCSS.Math.Content.HSA.REI.D.10



Find the system of inequalities for the graph below. Write the inequalities in slope-intercept form.

Then, determine if the following points are in the solution set of the system of inequalities:

(0,0), (1,1), (-1,1).



The purpose of this assignment is to assess whether the students understand systems of linear inequalities and the solution set. In this task, the students will be working with two major skills. They will be creating equations for two inequalities in two variables for a system of equations from a graph. The students will also be determining if a point satisfies the solution set from the system of inequalities. The students will have to have a full understanding of the concept to be able to complete this assessment since the students are given the solution and must work backward to find the system that satisfies the solution.



The first step to completing this assignment, the students must first create inequality equations for each line. Since the assignment says the equations must be in slope-intercept form, the student can use the plotted points to calculate the slope and find the y-intercept for each then create the formula for the equation. They must then choose a point in the solution set, that is the shaded region, and plug that into their equations and choose the correct inequality symbol to create a true statement.

For the purple line, the slope is -3 and the y-intercept is 1; therefore, the equation for the line is y=-3x+1. Then by putting a point into the equation that is in the solution region and simplifying, we can find the correct inequality symbol. For this one I put in the origin point, (0,0) and simplified.

0 ? -3*0 + 1

0 ? 1, 0 ≤ 1

Therefore, the inequality should read y ≤ -3x + 1. The inequality has the equal to part since the line is a solid line and not a dashed line. If it was dashed then the symbol would only be less then, <.

For the blue line, the slope is 2 and the y-intercept is 4; hence, the equation for the line is y=2x+4. Similarly to the purple line, by putting in the origin, (0,0), which is a point in the solution shaded region, we can find which inequality symbol to use.

0 ? 2*0 + 4

0 ? 4, 0 ≤ 4

Thus, inequality is y ≤ 2x + 4. Again, since this line is solid the symbol contains the equal to part as well as the less then, <.

Finally, the student need to determine if the three points are in the systems solution set or not. The students can do this in two different ways. They can either look on the graph and see if the point is in the shaded region then it is in the solution set. Or the students can put the point into both equations and see if the statement is true for both inequalities. If the point creates a true statement for both, then the point is in the systems solution set. Therefore, (0,0) and (-1,1) are in the systems solution set and (1,1) is not.

Introducing Calculus Through Exploration of Motion CCSS.Math.HS.F-IF.6

Calculus is notorious for frightening students away from higher math and because it is “too difficult” or “too abstract”. This does not have to be the case. Calculus can be made approachable and tangible.

The idea for the lesson below introduces rate-of-change to students with hands-on engagement. Students will be analyzing the motion of a ball to understand how physical objects experience rate-of-change.

For this exercise, students will need to form small groups. Each group will be given a worksheet (how many is at the digression of the teacher), a small ball that should fit comfortably in one hand, a foot-long ruler, and an iPad with the Video Physics app installed on it. Video Physics is an app created by Vernier. The app can only be installed on Apple devices, and will cost $4.99. However, if the app is purchased, it should be easy to install on all of the iPads if they are all linked to the same account. More information can be found here.

Groups will use Video Physics to videotape the ball rolling off a desk and observe its velocities in the x and y-directions. Based off the videotape, and some student input, the app will produce two graphs with equations: rate of change along the x-axis, and rate of change along the y-axis. Based on this information, students will fill in two plot maps with vectors representing the velocity for each axis. Students will notice that the velocity of the ball along the x-direction does not change, but along the y-direction it does. Furthermore, this change can be graphed using the slope formula!

By using tools students already have, calculus can be easily introduce by showing rate-of-change with velocities relationship to acceleration. Students won’t be working out derivatives or integrals of equations in this lesson, but understanding rate-of-change can be measured is a large conceptual hurdle that will be met by this lesson.

Pre-made Material: AnalyzingMovementExercise_Worksheet-1kw8fvm

Common Core associated with this lesson:

  • CCSS.Math.HS.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.
  • CCSS.Math.HS.F-IF.6 – Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

A Beginning Look at Calculus IF.B4

This learning progression is for a high school calculus class. This is the first unit of Calculus, where students will be introduced to the two goals of calculus. The learning progression will start first by students developing concepts of slope and slope functions. Students will show how particular functions change by examining finite differences.

The learning progression aligns the following Common Core State Standards:


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).


Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.


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.


The learning progression aligns with the following Mathematical Practices:


Make sense of problems and persevere in solving them.


Reason abstractly and quantitatively.


Look for and express regularity in repeated reasoning.


The learning progression can viewed at the link below

calculus Continue reading

Going, going, GONE! HSF.TF.A.4

If you’ve ever played golf, when your on the fairway and you have a large distance to cover, you grab your lowest iron in the bag because the 3-iron has the lowest angle which will produce the largest horizontal distance. Similarly, if you want to get the largest vertical distance, you’ll grab a wedge with the higher angle. Using the Vernier projectile launcher, this activity will task students to use knowledge of the unit circle and cos/sin functions to verify their calculations of which angles will produce the largest horizontal/vertical distances for projectile motion. Teacher has flexibility of changing concept of cannon in activity to any projectile motion that comes to mind. A few are golf, baseball, soccer, a cannon, throwing rocks into the river, etc.



CO.B.8 We will find YOU

Throughout the years airplanes and air transportation have suffering from laser attacks. In this activity, students will solve a right triangle using trigonometry functions. Students will determine how far the laser point of origin is and the distance the authorities have to travel to catch the perpetrator. Watch the following CNN video before starting the assignment.


Click here for Picture Problem Worksheet:  Picture Problem

Common Core Standards:

Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

Explain and use the relationship between the sine and cosine of complementary angles.

Good Visualization Project for Linear and Quadratic Function Writing

I found a great activity using Logger Pro to teach student the difference between linear and quadratic functions. Logger Pro is a program from Vernier and is free for one month. In this activity student use an interactive video to write linear and quadratic equations for the movement of a basketball. The project and a picture of the video are attached.