Formative Assessment Learning Progression

Learning Progression

Grade level: 12th

Book: Thomas’ Calculus

CCSS math domain: Functions

Cluster: Interpreting functions that arise in applications in terms of the context

Interpret functions that arise in applications in terms of the context.


            Students in this class have analyzed functions in their previous math classes in terms of algebraic, graphic, and even trigonometric terms. They will now apply these previously learned skills to analyze functions that are both theoretical and that model real world situations in terms of calculus. They will do this by exploring the properties of functions in relation to their derivatives. They will also be asked to find appropriate approximations for the derivative so that they will become comfortable with quickly estimating the derivative of a function. This ability will assist students later in the course when they are calculating derivatives more quickly. If they can quickly estimate derivates then they will be able to quickly identity and egregious errors they make. At the end of this learning progression there will be a quiz that is complied from questions similar to those used in the activities. This will be the benchmark assessment for this learning progression.



To ease students into the process of taking the derivative of a function simply by using the limit definition or applying the shortcut rule of calculus the students will be asked to estimate the derivative of a function by looking at the equation of a function and finding its average slope over smaller and smaller intervals. An example of this type of problem can be found in the column on the right. At the same time students will be asked to follow the same process for tables of values for a given function. Once they students have an accurate approximation for the derivatives of these functions at certain points, they will interpret these findings in terms of the original problem. This process will give students the opportunity to become comfortable with the units associated with the derivate of a function as well as the meaning of the derivate of a function in specific circumstances.



CCSS.Math.Content.HSF-IF.B.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.


Example Problems:

1. For the function  answer each of the following questions.

(a) Evaluate the function the following values of x compute (accurate to at least 8 decimal places).

(i) 2.5              (ii) 2.1             (iii) 2.01            (iv) 2.001          (v) 2.0001

(vi) 1.5            (vii) 1.9           (viii) 1.99          (ix) 1.999          (x) 1.9999

Estimate the derivative of the function from the graph for the following values:

a)    x=2

b)   x=-3

c)    x=0













The activities that are designed to help students meet this standard should help student better articulate the important points on the graph of a derivative. The student will practice graphing derivatives by identifying the points of inflection. They will also be asked to describe the graph of a derivative in words by looking at a graph of the original function. They will also perform this task based on a description of a graph of an original function. And example of this type of problem is provided in the right hand column. This gives the student the opportunity to understand derivatives in a new way by describing it in words.


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.


Example Problems:

Find the inflection points for the following graph by sketching a picture of its derivative.


Describe the graph of the derivative of this function. Be sure to describe any holes in the graph.


After spending some time estimating the derivative of functions from their graphs and tables of their values the student will begin predicting what the derivates of functions will look like when graphed. The students will do this by observing how the domains of continuous and discontinuous functions affect the graph of their derivatives. While working on this standard the students will continue to work with derivatives in terms of their units. This will help the students make connections from the derivative to real world functions.


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.


Example Problem

A cup of hot tea is placed on a table in a room that is 72 degrees. Sketch a graph of the temperature of the tea as a function of time and determine if the derivative of this graph is entirely positive, negative or zero.



Lesson Title: Slope over small intervals

Unit Title: Introduction to the derivative

Teacher Candidate: Albany Thompson

Subject, Grade Level, and Date: Calculus, 12th grade, 2/4/14


Placement of Lesson in Sequence

This lesson will come after a unit on limits and before the unit on the shortcuts to taking the derivatives. This lesson will be the first in a short sequence that is designed to teach students to be comfortable with the concept of a derivative.

Central Focus and Essential Questions

Central focus: Relating the concept of a derivative with preexisting knowledge of functions. Once this conceptual connection had been made the students will be able to build upon their knowledge with the procedures of calculating the derivative and using this procedure to solve real world problems.

Essential Question: How is the derivative related to the average slope over an interval of an equation?

Content Standards

CCSS.Math.Content.HSF-IF.B.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.

Learning Outcomes Assessment
The students should be able to approximate the derivative of a function after the completion of this lesson. After the next lesson in the sequence the students will be assess through a quiz. For this lesson the students will receive feedback on their homework but will only receive a grade for competition of the assignment.


Learning Targets Student Voice
I will be able to estimate a derivative by taking the slope of a function over very small intervals.

I will be able to find the units associated with the derivative of a function.

The students will have an opportunity to express their understanding through



Prior Content Knowledge and Pre-Assessment

The students will have to know the algebra associated with functions from Algebra 2. This should be evidenced on the students’ grades from this course. I will partner with the teachers of this class to gather information of the knowledge of my students.

Academic Language Demands
Vocabulary & Symbols Language Functions Precision, Syntax & Discourse
  • Continuous
  • Differentiable
  • Limit
  • Interval notation



The terminology in the lesson is used to categorize functions based on their properties that relate to their derivatives. Mathematical Precision: be able to find the units of a derivative.

Syntax: explain in words the meaning of the value of the derivative at a point in terms of the original function.

Discourse: Be able to talk about the problem in context of the derivative of the function associated with it.


Language Target Language Support Assessment of Language Target
The students should be able to talk about the derivative in terms of the original context of the problem. This includes knowing the units of a derivative. The students will be able to reference their note from prior lectures to determine the meaning of words that have been used previously in class. Also, when students are reading their answers out loud I will have them practice interpreting the symbols of their answers. I will also model appropriate use of this language and symbols during my lecture and my discussions with the students. I will be able to assess the students’ comfort with the language through conversations with them and with the exit slip provided below.


Lesson Rationale (Connection to previous instruction and Objective Standards)

This lesson is designed to give students a conceptual understanding of the derivative before they are asked to work with it procedurally or reason with it. It should also help segue from talking about taking a limit to taking a derivative and give the students a good idea of how the two are related.

Differentiation, Cultural Responsiveness and/or Accommodation for Individual Differences

For students with visual or auditory impairments I will place them near the front of the classroom. I will also make the note available to those students that have a difficult time following a lecture format, such as English language learners.

Materials – Instructional and Technological Needs (attach worksheets used)

  • Doc camera
  • Textbooks
Teaching & Instructional Activities
Time Teacher Activity Student Activity Purpose
10 minutes Have the students complete the following warm up activity. Once they finish go through the answers as a class. Ask students questions such as:

Is the graph continuous everywhere?

Is this graph differentiable everywhere?

How would you answer the question if the function had a gap in it?

Find the limit of the graph of a function below for each of the following values.

a)    x=-1

b)   x=0

c)    x=2

d)   x=5

e)    x=-4



Find the limit of a function at certain points. Answer questions for the class discussion. Review the material covered in the previous unit and prepare students for more interpretation of graphs of functions.
20 minutes Give lecture on estimating the derivative of a function at a point by finding the average rate of change over smaller and smaller intervals. Take notes and listen to lecture Direct instruction
15 minutes Give students homework assignment and be available to answer students’ questions. Work on homework Guided practice
5 minutes Hand out exit slips (found below) to students. Be available to answer student questions. Collect slips at the end of the period. Fill out exit slips. Formatively assess students understanding of content and vocabulary.




a) My formative assessment technique will be an exit slip. An example of this exit slip is provided below.

Exit slip:

List 3 concepts that we covered today and indicate whether you would like to cover any of them again.







List two new terms from the lesson today and if possible define them






List one thing you found interesting from the lesson today.





b) The type of formative assessment will help me know if the students not only know what the standards are but also how well they feel they know the material. The information provided in from these slips will let me know what should be covered in any reengagement lessons.

c) I will give the students five minutes at the end of class to fill out these sheets. The first time that I hand them out I will give the students extra time to ask me any questions they have about the requirements for and purpose of these slips.

d) For students that have trouble articulating themselves in writing I will spent the last five minutes of class discussing their answers to these questions. Any English language learners many write one word answers to these questions. If students are unable to think of the full number of responses for these questions they may leave parts of it blank.

e) I will keep a list of the topics covered in this learning progression and mark the number of times that students indicate that they need to cover the material again. I will also keep track of the exit slips from specific students in order to track their progress throughout the progression.

How can tutors be used effectively to improve student performance?

How can we use tutors to effectively enhance student performance?

While I have not had much experience teaching in the classroom yet, I do have a great opportunity to work with remedial math students and their tutors next year.  Right now Central Washington University’s remedial mathematics program seems to be producing unsatisfactory results for the CWU mathematics department.  It is the job of the remedial mathematics program to get students who have not succeeded in traditional math courses offered in high schools to succeed in the remedial mathematics program at CWU.  Next year I will not have any control over the way that remedial mathematics professors will teach their classes, but I will be able to closely work with and monitor the tutors that are working with these remedial students. 

In order to make these tutors even more effective I will work with the teachers of CWU’s higher level mathematics education students.  These teachers will require their students to tutor remedial mathematics students for at least two hours a week.  My job in this is to be sure that the tutors are fully trained and capable of working with remedial mathematics students.  They will be expected to support and encourage the students, while building on their content knowledge through constructivist tutoring methods.  While I cannot change how the teachers teach their classes, I can change how the tutors work with the students and how the tutoring center is ran.  It is my hope that by properly training tutors with specific methods, and matching them up correctly with remedial students, that I will be able to see improvement in the performance of remedial students.  

While it may be a process to measure this improvement, I plan to do use qualitative analysis.  First I will evaluate studens performance by comparing students that have been tutored (experimental group) and students who have not been tutored (control group).  I will put end of course standardized test scores as well as SAT/ACT scores of all students in each group into ANCOVA.  If ANCOVA outputs a p-value less than .05 I will assume that tutoring was effective in enhancing performance of remedial mathematics students. 

While I do have a partial plan on how to use these tutors and measure success, I could still use any help or suggestions possible.  If anyone has any idea of how to use these tutors more effectively please share!  Another isssue I am having a difficult time decided is how I will pair the tutor’s up with students from the remedial math center.  It would be greatly appreciative and helpful if anyone has ideas on how to do this as well.