Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum

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Insights Into Algebra 1 - Teaching For Learning
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Workshop 2 Linear Functions and Inequalities Topic Overview
Topic Overview:

Part 1: Linear Functions

Part 2: Linear Equations and Inequalities
Download the Workshop 2 Guide

Tool Box
Graphing Calculator
NCTM Standards

Part 1: Linear Functions

Linear functions are those that exhibit a constant rate of change, and their graphs form a straight line. They are also described as polynomial functions of degree one.

Mathematical Definition
Role in the Curriculum


Linear functions model a wide variety of real-world situations, including predicting the cost of a telephone call that lasts a given amount of time, the profit of a hot dog stand, and the amount of tax paid for a given income. Linear functions arise when there is a constant rate of change. Students should solve problems in which they use tables, graphs, words, and symbolic expressions to represent and examine linear functions and linear patterns of change.

Linear functions can be written in three different forms, as shown in the table below. Each form provides different information about the function.

Form: Standard Form Point-Slope Form Slope-Intercept Form
Equation: Ax + By = C,
where A 0, B 0
y - y1 =
m(x - x1),
where m 0
y = mx + b, where m 0
Information: Slope: m

Point on the line: (x1, y1)
Slope: m

y-intercept: (0, b)

Students should be able to recognize a linear function in a table of values, a graph, or in algebraic form. When studying linear functions graphically, students should understand that the slope of the line represents a constant rate of change for the function, and that the y-intercept is the point where the graph crosses the y-axis and often represents the initial condition or starting point for the function. Through practical experience solving linear function problems in context, students will develop an understanding of the concepts and real-world meanings of the slopes and y-intercepts of lines.

Other examples:

  • The linear function F = 1.8C + 32 can be used to convert temperatures between Celsius and Fahrenheit.

  • If a utility company charges a fixed monthly rate plus a constant rate for each unit of power consumed, a linear function will show the monthly cost of power. If the fixed rate is $25, and the cost for each unit of power is $0.02, the linear function is C = 0.02P + 25.

  • The linear function I = 400C + 1,500 yields the total monthly income of a car salesman who makes a monthly base salary of $1,500 and receives $400 dollars for each car sold.
Mathematical Definition

A linear function, whose graph is a line, can be written in the form y = mx + b, where m and b are constants and m 0. As with any function, students can represent a linear function as a table, an equation, or a graph.

Table Equation Graph
Minutes Cost

0 0.85 y = 0.24x + 0.85
1 1.09
2 1.33
3 1.57
4 1.81
5 2.05

Role in the Curriculum

Linear functions are fundamental to the study of mathematics. Students can transfer many of the important concepts learned through the study of linear functions to the understanding of other functions. According to the National Council of Teachers of Mathematics (NCTM):
It is essential that [students] become comfortable in relating symbolic expressions containing variables to verbal, tabular, and graphical representations of numerical and quantitative relationships. Students should develop an initial understanding of several different meanings and uses of variables through representing quantities in a variety of problem situations.
(NCTM, Principles and Standards for School Mathematics,
2000, p. 223)
Students should be able to move fluently between the different representations of linear functions and, given a description of a situation, should be able to produce a table, equation, and graph. Likewise, when given one representation of a linear function, students should be able to produce the others. To do this, they need ample opportunity to explore situations involving linear functions in all representations. For example, in the video for Workshop 2, Part I, Tom Reardon started with a verbal description and a table of values. The students then produced an equation and a graph to further describe the situation. They discussed the meaning of the constants in the equation and how those constants affected both the table and the graph.

See what Diane Briars has to say about the important aspects of linear functions that Tom Reardon applied in his lesson:

Read transcript from teacher educator Diane Briars
When the students were talking about the slope and they said that 24 cents was the slope, he didn't stop there. Read More

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Next: Part 2: Linear Equations and Inequalities
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