Algebra I Unit Development

A Function-Based Approach

Equation-based Algebra

Building and solving equations is the dominant theme

The major use of letters is for unknown quantities in equations

The study of functions is at the end of the curricula, mostly isolated from the equations

Emphasis is on grasping and applying established routines to symbolically manipulate expressions and equations

Many exercises are skill-driven and content free

Function-based Algebra Understanding the relationship between variables is

the dominant theme Letters are used to denote variables that are

involved in change In modeling real-world situations, students are

expected to determine variables and their interdependencies, formulate function rules, find solutions, and make predictions.

Problems and solution techniques may utilize multiple representations (verbal, symbolic, tabular, graphical)

Distinction in posing the question between the two

approaches Snow begins falling steadily at time t = 0. The ground already has some snow on it, and at time t = 2, it is covered to a depth of 8 inches. Four hours later (at time t = 6), the depth is 12 inches.

(a) (equations-based question) When will there be 17 inches of snow on the ground?

(b) (function-based question) Find a possible formula for d, the depth of snow coverage (in

inches), as a function of t, the amount of time elapsed (in hours). What does your formula tell you about the snow cover?

Distinguishing among families

What family would you use to represent the following quantities? Explain.

a. The price of gas if it grows by $0.02 a week.b. The speed of personal computers if it doubles every 3

years.c. The area of a rectangle that is 6 inches narrower than it

is long.d. The value of an investment that drops by a third n times

in a row.e. The height of a ball thrown straight up in the air.f. The population of a town that is growing at 2% a year.

Issues to Consider

Research Shows Students are more likely to remember the

mathematics taught because we capitalize on associations made through using a function approach.

Learning is made simpler, faster, and more understandable by using pattern building as a teaching tool.

Taken from Why Use a Function Approach When Teaching Algebra? by Ed Laughbaum

Issues to Consider (continued)

Research Shows Students cannot learn if they are not paying

attention. The graphing calculator is used to draw attention to the mathematics through its basic functionality including various applications software.

Without visualizations, students do not understand or remember the mathematics as well. In the function approach visualizations are used first before any symbolic development.

Taken from Why Use a Function Approach When Teaching Algebra? by Ed Laughbaum

Issues to Consider (continued)

The enriched teaching/learning environment promotes correct memory of math content. The wide variety of teaching activities facilitated by the function approach provides the enriched environment.

Contextual situations (represented as functions) provide meaning to the algebra learned. Algebra taught without meaning creates memories without meaning that are quickly forgotten.

Taken from Why Use a Function Approach When Teaching Algebra? by Ed Laughbaum

Linking Formal Mathematical Understanding to Informal Reasoning

Which of these problems is most difficult for a beginning algebra student?

STORY PROBLEM – When Ted got home from his waiter job, he multiplied his hourly wage by the 6 hours he worked that day. Then he added the $66 he made in tips and found he had earned $81.90. How much does Ted make per hour?

WORD PROBLEM – Starting with some number, if I multiply it by 6 and then add 66, I get 81.9. What number did I start with?

EQUATION – Solve for x: x*6 + 66 =81.90

Urban High School Students’ Performance

STORY PROBLEM – WORD PROBLEM – EQUATION –

Urban High School Students’ Performance

STORY PROBLEM – 66%

WORD PROBLEM – 62%

EQUATION – 43%

Key Point to Remember

Investigating students’ written work helps explain why. Students often solved the verbal problems without using the equation.

A2.2.1 Combine functions by addition, subtraction, multiplication, and division.

Unit 1 - Add and subtract linear functions to get other linear functions.

Unit 2 - Multiply two linear functions to get a quadratic function.

Unit 3 - Multiply three or more linear functions to get polynomial functions.

Unit 4 - Polynomial functions can also be thought of as the addition/subtraction of power functions

A2.2.2 Apply given transformations to basic functions and represent symbolically.

Unit 1 – Use vertical shifts when solving linear equations graphically.

Unit 2 – Use horizontal shifts, stretching and shrinking with quadratic functions.

Unit 4 – Use reflections and rotations about the y-axis in power functions.

A2.1.1 Recognize whether a relationship is a function and identify its domain and range.

Unit 1 – Linear Functions Unit 2 – Quadratic Functions Unit 3 – Polynomial Functions Unit 4 – Exponential and Power Functions Unit 5 – Mathematical Modeling