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School District:Hill City
Unit of Study Title:Introduction to Limits
Grade Level:9-12
Time Frame:4 days
Key Words:Math, Calculus
Unit Designer:John Hoag
Unit Designer
Date Added:6/6/2002 12:40:22 PM
Date Last Revised:6/24/2002 3:44:47 PM
Topic Area:Math
Peer Reviewer:Cory Ginsbach
Peer Reviewer

Brief Summary of Unit: This unit will cover what the definition of a limit is and how it can be associated to the real world. There will be extensive use of calculators to find the limit graphically and by using tables.
Students will Understand: Students will understand the definition of a limit. Students will be able to find the limit of an equation, a graph or from numerical data that is presented to them. Students will be able to find the limit of an equation, a graph or from numerical data as a variable approaches + or - infinity. Students will be able to use their ti-83+ calculators to determine the limit graphically or by using a table.

Link to Standards #1

Students will use the language of algebra to explore, describe, represent, and analyze number expressions and relations that represent variable quantities.
Students will develop and use number sense to investigate the characteristics of numbers in a variety of forms and modes of operation.
Students will discover, analyze, extend, and create patterns, relations, or functions to model mathematical ideas in a variety of forms.

Indicator:GOAL 1: ALGEBRA
Indicator 1: Analyze procedures to transform algebraic expressions.
Indicator 2: Use a variety of algebraic concepts and methods to solve problems.

Indicator 2: Apply number operations with real numbers and other number systems.
Indicator 4: Analyze the concept of value, magnitude, and relative magnitude of real numbers.

Indicator 1: Analyze and describe the properties and behaviors of relations, functions, and related inverses.
Indicator 3: Analyze the applications of the concept of mathematical limit.

Benchmark(s):GOAL 1: ALGEBRA
Indicator 1: 9 – 12 Benchmarks: c. transform algebraic expressions and analyze the changes in graphs.
Indicator 2: 9 – 12 Benchmarks: c. use the graphs of functions to solve problems.

Indicator 2: 9 – 12 Benchmarks: analyze algebraic expressions
Indicator 4: 9 – 12 Benchmarks: c. create generalizations discovered through investigations of finite and infinite sets of numbers.

Indicator 1: 9 – 12 Benchmarks: a. analyze the relationship between dependent and independent variables.
Indicator 3: 9 – 12 Benchmarks: c. demonstrate the concept of limit using various geometric, numeric, and algebraic models.
Other Content Standards:California Standards Calculus--
1.0 Students demonstrate knowledge of both the formal definition and the graphical interpretation of limit of values of functions. This knowledge includes one-sided limits, infinite limits, and limits at infinity. Students know the definition of convergence and divergence of a function as the domain variable approaches either a number or infinity:
1.2 Students use graphical calculators to verify and estimate limits.

Other Components
Essential Questions to Guide this Unit and Focus Teaching and Learning: What are the 3 conditions of a limit?
In your own words, describe a limit using anything that is not related to math. (Example: an engine with a governor.)
How can you use your calculators to find a limit by only using the graph mode? by the table mode?
Give a real world example of a limit as some variable approaches infinity.
Key Knowledge and Skills Students will Acquire: After working through the material, the student should be able:

(a)to obtain numerical evidence for the calculation of limits;
(b)to determine what appears to be the limit from the numerical evidence; and
(c)to become aware of some of the problems in using numerical evidence for the calculation of limits.
Assessment Plan: Worksheets will be used daily to check for comprehension.
A quiz will be given at the end of the week.
Tests will be given throughout the year with questions from this unit.
Learning Activities: Students will be presented with many different mediums of instruction. These will include flash presentations on the internet, white board lecture, cooperative learning and instruction on the use of the ti-83+ calculator with an overhead projector. Students will be presented worksheets that will have riddles answered when they find the appropriate answers to questions.
Internet Resource Link:

Adapted from the the work of Jay McTighe and Grant Wiggins. Understanding by
Design. Association for Supervision and Curriculum Development. Alexandria, VA. 1998.

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