# Modeling with Equations and Functions

In today’s Algebra I session, participants explored the lessons in Module 5 of Algebra I, “A Synthesis of Modeling with Equations and Functions.” The module is packed with experiences that pull together the cohesiveness of the topics covered throughout the year and is loaded with application problems that develop fluency, but not computational fluency alone. This module drives home the fact that students need to be fluent in pulling their prior knowledge to the forefront in a variety of settings. This can be a challenging and interesting task to incorporate into the design of a lesson. Some key problems/exercises that participants looked at were the following:

Lesson 1, Exercise 2:
Students examine a graph of a function and recognize the function type and state the parent function. Students then need to be able to identify what transformation took place to the parent function to produce the graph, a more challenging task and perhaps one that students will struggle with. Lastly, students need to write the equation of the function. Lesson 2, exercise 2 had some concrete examples of the same nature.

Lesson 2, Exercise 4:
This exercise was an excellent example of allowing students the opportunity to communicate their conceptual understanding and critique the reasoning of others.  This problem is highly recommended and generated great discussion amongst the crowd.

Lessons progressed through problems that had students analyzing data sets, verbal descriptions and graphs.

Lesson 4, Exercise 2:
This exercise provided an opportunity for students to determine what type of function best models the data displayed in a graph. The graph appears to be a quadratic, but as participants learned at the last NTI, looks can be deceiving. As it turned out, the graph was quadratic and it provided an opportunity for the sharing of great techniques of solution. These strategies included solving a system of equations, using the second differences (common theme of the day) to find the leading coefficient for the quadratic, and estimating the other root and working backwards to find the quadratic. This problem was well received because of the opportunity it presented for students to be successful.

Another good example of a modeling problem was the opening exercise discussed for Lesson 5 that involved exercise time and rest time for interval training. We quickly learned as a group that part of the modeling process is learning how to handle any assumptions that are made and determining how those assumptions will affect the desired outcome.

Finally, participants looked at problems that involved modeling exercises from sequences and investigated the question: should we believe in patterns? Participants examined an interesting example that involved the appearance of a pattern from points on a circle that crashes after the 6th term. The example reinforced that a pattern can disappear.

One of the biggest takeaways of the session is that students need to be able to recognize whether they have enough information to be sure that the function they have created is an accurate representation of the data being described.

The presenters touched briefly on how to support learning throughout this module and any other module. They shared three key points:

1. Be attentive to language. Teachers need to be clear with their mathematical vocabulary. They need to accurate and precise with the mathematical language being used in the classroom, so this can transfer to the students.
2. Teachers need to remember that conceptual knowledge precedes fluency.
3. Conceptual understanding is achieved through strong questioning techniques, progressing from the concrete-pictorial-abstract, and knowing and showing the progression of the content.

The materials for this session are available here: