School Improvement in Maryland

Gr. HS Unit: Linear and Exponential Relationships

Unit Overview

In earlier grades, students define, evaluate, and compare functions, and use them to model relationships between quantities. In this unit, students will learn function notation and develop the concepts of domain and range. Students move beyond viewing functions as processes that take inputs and yield outputs and start viewing functions as objects in their own right. Students explore many examples of functions, including sequences; they interpret functions given graphically, numerically, symbolically, and verbally, translate between representations, and understand the limitations of various representations. Students work with functions given by graphs and tables, keeping in mind that, depending upon the context, these representations are likely to be approximate and incomplete. Their work includes functions that can be described or approximated by formulas as well as those that cannot. When functions describe relationships between quantities arising from a context, students reason with the units in which those quantities are measured. Students explore systems of equations and inequalities, and they find and interpret their solutions. Students build on and informally extend their understanding of integer exponents to consider exponential functions. Students compare and contrast linear and exponential functions with domains in the integers, distinguishing between additive and multiplicative change. Students interpret arithmetic sequences as linear functions and geometric sequences as exponential functions.

Essential Questions:

  • When and how is mathematics used in solving real world problems?
    • What characteristics of a real world problem indicate that the situation could be modeled by a functional relationship?
    • How can systems of equations and inequalities model and be used to solve real-world problems?
  • What characteristics of problems would determine how to model the situation and develop a problem solving strategy?
    • How can multiple representations of functions be used to reason and make sense of relationships and model change?
    • What characteristics of a problem help determine if a linear or exponential model could serve as an appropriate function to represent the situation?
  • What is the most appropriate structure to represent mathematical situations?
    • How are the symbolic, numeric, graphic, and verbal representations of functions and equations related?
    • What are the similarities and differences between linear and exponential functions?
    • How are recursive and explicit formulas related?

A question is essential when it stimulates multi-layered inquiry, provokes deep thought and lively discussion, requires students to consider alternatives and justify their reasoning, encourges re-thinking of big ideas, makes meaningful connections with prior learning, and provides students with opportunities to apply problem-solving skills to authentic situations.

Unit Lesson

Additional information such as Teachers Notes, Enduring Understandings,Content Emphasis by Cluster, Focus Standards, Possible Student Outcomes, Essential Skills and Knowledge Statements and Clarifications, and Interdisciplinary Connections can be found in this Lesson Unit.

Available Model Lesson Plans

The lesson plan(s) have been written with specific standards in mind. Each model lesson plan is only a MODEL - one way the lesson could be developed. We have NOT included any references to the timing associated with delivering this model. Each teacher will need to make decisions related ot the timing of the lesson plan based on the learning needs of students in the class. The model lesson plans are designed to generate evidence of student understanding.

This chart indicates one or more lesson plans which have been developed for this unit. Lesson plans are being written and posted on the Curriculum Management System as they are completed. Please check back periodically for additional postings.

CCSC Alignment: A.RE1.6

This lesson plan uses art to motivate students for a lesson on solving systems of linear equations. In this lesson students will:

  • understand that the point of intersection of a system of equations satisfies both equations.
  • translate a real world problem to a system of equations.
  • solve systems of linear equations algebraically, graphically and numerically.
  • select the most efficient method to solve a system of equations.
  • verify that the coordinates of the point of two graphs makes both equations in the system true.

Available Model Lesson Seeds

The lesson seed(s) have been written with specific standards in mind. These suggested activity/activities are not intended to be prescriptive, exhaustive, or sequential; they simply demonstrate how specific content can be used to help students learn the skills described in the standards. Seeds are designed to give teachers ideas for developing their own activities in order to generate evidence of student understanding.

This chart indicates one or more lesson seeds which have been developed for this unit. Lesson seeds are being written and posted on the Curriculum Management System as they are completed. Please check back periodically for additional postings.

CCSC Alignnment: A.REI.6

Reinforcement
This activity will help students better understand how systems of equations are created in order to demystify this topic. This lesson seed describes a non traditional approach that helps to build conceptual understanding of the fact that the solution to a system of equations is an ordered pair whose coordinates are solutions of all equations in the system. (addition/ linear combination).

CCSC Alignnment: A.REI.10, A.REI.11

Introduction
This seed provides an introductory activity that could be used for a t lesson on solving systems of equations graphically.