Title
Suggested Time Frame
Algebra I
2016-17 ~ Unit 3
2nd Six Weeks
Suggested Duration – 15 days
Guiding Questions
Linear Functions and Equations
*CISD Safety Net Standards: A.2A
Big Ideas/Enduring Understandings
Linear functions can be represented in a variety of ways and translated
among various representations.
Algebraic skills are important and necessary in order to simplify and solve
equations and inequalities.
Through a variety of ways equations and inequalities can be solved that
were formulated from problem situations.
Slope and intercepts have specific meanings when applied to linear
functions.
Changing the parameters (slope and intercepts) changes the linear function
but allows generalization to be used in order to interpret the meaning.
How can you use a linear function to solve real-world problems?
How can you use different forms of linear equations to solve real-world
problems?
How can you use linear equations and inequalities to solve real-world
problems?
How can you use linear modeling and regression to solve real-world
problems?
Vertical Alignment Expectations
TEA Vertical Alignment Grades 5-8, Algebra 1
Sample Assessment Question
COMING SOON………………………………..
The resources included here provide teaching examples and/or meaningful learning experiences to address the District Curriculum. In order to address the TEKS to the proper
depth and complexity, teachers are encouraged to use resources to the degree that they are congruent with the TEKS and research-based best practices. Teaching using only the
suggested resources does not guarantee student mastery of all standards. Teachers must use professional judgment to select among these and/or other resources to teach the
district curriculum. Some resources are protected by copyright. A username and password is required to view the copyrighted material. District Specificity/Examples TEKS
clarifying examples are a product of the Austin Area Math Supervisors TEKS Clarifying Documents.
Algebra I ~ Unit 3
Updated September 14, 2016
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Algebra I
2016-17 ~ Unit 3
Ongoing TEKS
Math Processing Skills
A.1 Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding.
The student is expected to:
(A) apply mathematics to problems arising in everyday life, society, and the
workplace;
(B) use a problem-solving model that incorporates analyzing given information,
formulating a plan or strategy, determining a solution, justifying the solution,
and evaluating the problem-solving process and the reasonableness of the
solution;
(C) select tools, including real objects, manipulatives, paper and pencil, and
technology as appropriate, and techniques, including mental math, estimation,
and number sense as appropriate, to solve problems;
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Focus is on application
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Students should assess which tool to apply rather than trying only one or all
(E) create and use representations to organize, record, and communicate
mathematical ideas;
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Students should evaluate the effectiveness of representations to ensure they are
communicating mathematical ideas clearly
Students are expected to use appropriate mathematical vocabulary and
phrasing when communicating ideas
(F) analyze mathematical relationships to connect and communicate
mathematical ideas; and
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Students are expected to form conjectures based on patterns or sets of
examples and non-examples
(G) display, explain, and justify mathematical ideas and arguments using
precise mathematical language in written or oral communication
•
Precise mathematical language is expected.
(D) communicate mathematical ideas, reasoning, and their implications using
multiple representations, including symbols, diagrams, graphs, and language
as appropriate;
Algebra I ~ Unit 3
Updated September 14, 2016
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Algebra I
2016-17 ~ Unit 3
Knowledge and
Skills with Student
Expectations
District Specificity/ Examples
A.2 Linear functions,
equations, and
inequalities. The
student applies the
mathematical
process standards
when using
properties of linear
functions to write
and represent in
multiple ways, with
and without
technology, linear
equations,
inequalities, and
systems of
equations. The
student is expected
to:
*CISD Safety Net*
(A) determine the
domain and range of
a linear function in
mathematical
problems; determine
reasonable domain
and range values for
Vocabulary
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A.2(A)
Previously introduced in Unit 2
Misconceptions
The student may confuse x and y values.
The student may confuse which inequality symbol to use (< or >, > or
≥, etc.)
Algebra I ~ Unit 3
Updated September 14, 2016
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Between
Between, inclusive
Closed circle
Constant
Constant of
proportionality
Continuous
Direct variation
Discrete
Domain
Function
Greater than (>)
Greater than or equal
to (≥)
Horizontal
Inequality
Less than (<)
Less than or equal to
(≤)
Linear equation
Open circle
Opposite reciprocal
Parallel
Perpendicular
Point
Point-slope form (of a
linear equation)
Suggested Resources
Resources listed and
categorized to indicate
suggested uses. Any additional
resources must be aligned with
the TEKS.
Textbook Resources
HMH Algebra I
Unit 3
Web Resources:
Region XI: Livebinder
NCTM: Illuminations
Domain and Range
(video)
Page 3 of 22
real-world
situations, both
continuous and
discrete; and
represent domain
and range using
inequalities.
The student may have trouble recognizing whether a real-world
situation should be represented with discrete or continuous
variables.
The student may confuse domain and range on the graph and see
domain as the “height” of the graph and the range as the “width” of
the graph.
Including, but not limited to:
1. Use the definition of a function to determine whether a
relationship is a function given a table, graph or words.
2. Given the function f(x), identify x as an element of the domain, the
input, and f(x) is an element in the range, the output.
3. Know that the graph of the function, f, is the graph of the equation
y=f(x).
4. When a relation is determined to be a function, use f(x) notation.
5. Evaluate functions for inputs in their domain.
6. Interpret statements that use function notation in terms of the
context in which they are used.
7. Given the graph of a function, determine the practical domain of
the function as it relates to the numerical relationship it describes.
(B) write linear
equations in two
variables in various
forms, including y =
mx + b, Ax + By = C,
and y – y1 = m(x –
x1), given one point
and the slope and
given two points.
A.2(B)
Write equations in slope-intercept form.
Write equations in standard form.
Write equations in point-slope form.
Misconceptions
Students may forget that in standard form A must be positive.
Students may confuse the signs in point-slope form.
1. Students identify the rate of change (slope) and initial value (y-
Algebra I ~ Unit 3
Updated September 14, 2016
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Algebra I
2016-17 ~ Unit 3
Range
Slope
Slope-intercept form
(of a linear equation)
Standard form (of a
linear equation)
Strictly between
Undefined
Vertical
X-axis
X-values
Y-axis
Y-intercept
Y-values
Writing Equations in
Slope-Intercept Form
From Various
Representations
(Videos)
Forms of Equations
(videos)
Page 4 of 22
intercept) from tables, graphs, equations or verbal descriptions to
write a function (linear equation).
2. Students understand that the equation represents the relationship
between the x-value and the y-value; what math operations are
performed with the x-­‐value to give the y-value. Slopes could be
undefined slopes or zero slopes.
Algebra I
2016-17 ~ Unit 3
Tables:
Students recognize that in a table the y-intercept is the y-value when
x is equal to 0. The slope can be determined by finding the ratio y/x
between the change in two y-values and the change between the two
corresponding x-values.
Example 1:
Write an equation that models the linear relationship in the table
below.
Solution: The y-intercept in the table below would be (0, 2). The
distance between 8 and -1 is 9 in a negative direction -9; the distance
between -2 and 1 is 3 in a positive direction. The slope is the ratio of
rise to run or y/x or -9/3 = -3. The equation would be y = -3x + 2
Graphs:
Using graphs, students identify the y-intercept as the point where the
line crosses the y-axis and the slope as the rise/run.
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Algebra I
2016-17 ~ Unit 3
Example 2: Write an equation that models the linear relationship in
the graph below.
Equations:
In a linear equation the coefficient of x is the slope and the constant
is the y-intercept. Students need to be given the equations in formats
other than y = mx + b, such as y = ax + b (format from graphing
calculator), y = b + mx (often the format from contextual situations),
etc.
Point and Slope:
Students write equations to model lines that pass through a given
point with the given slope.
Example 1:
A line has a zero slope and passes through the point (-5, 4). What is
the equation of the line?
Solution: y = 4
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Algebra I
2016-17 ~ Unit 3
Example 2:
Write an equation for the line that has a slope of ½ and passes
through the point (-2, 5)
Solution: y = ½ x + 6
Students could multiply the slope . by the x-coordinate -2 to get -1.
Six (6) would need to be added to get to 5, which gives the linear
equation.
Students also write equations given two ordered pairs. Note that
point slope form is not an expectation at this level. Students use the
slope and y-intercepts to write a linear function in the form y = mx
+b.
Contextual Situations: In contextual situations, the y-intercept is
generally the starting value or the value in the situation when the
independent variable is 0. The slope is the rate of change that occurs
in the problem. Rates of change can often occur over years. In these
situations it is helpful for the years to be “converted” to 0, 1, 2, etc.
For example, the years of 1960, 1970, and 1980 could be represented
as 0 (for 1960), 10 (for 1970) and 20 (for 1980).
Example 3:
The company charges $45 a day for the car as well as charging a onetime $25 fee for the car’s navigation system (GPS). Write an
expression for the cost in dollars, c, as a function of the number of
days, d, the car was rented.
Solution: C = 45d + 25
Algebra I ~ Unit 3
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Students interpret the rate of change and the y-intercept in the
context of the problem. In Example 3, the rate of change is 45 (the
cost of renting the car) and that initial cost (the first day charge) also
includes paying for the navigation system. Classroom discussion
about onetime fees vs. recurrent fees will help students model
contextual situations.
Algebra I
2016-17 ~ Unit 3
Example 4:
Write an equation to represent the graph below.
Solution: y= -3/2x
Example 5:
Students write equations in the form y = mx + b for lines not passing
through the origin, recognizing that m represents the slope and b
represents the y-intercept.
Solution: y= 2/3x - 2
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Algebra I
2016-17 ~ Unit 3
(D) write and solve
equations involving
direct variation.
A.2(D)
Write and solve equations for direct variation problems.
1. Students build on their work with unit rates from 6th grade
and proportional relationships in 7th grade to compare
graphs, tables and equations of proportional relationships.
2. Students identify the unit rate (or slope) in graphs, tables and
equations to compare two proportional relationships
represented in different ways.
Example:
Compare the scenarios to determine which represents a greater
speed. Explain your choice including a written description of each
scenario. Be sure to include the unit rates in your explanation.
Direct Variation
Notes
Direct Variation
(video)
Solution: Scenario 1 has the greater speed since the unit rate is 60
miles per hour. The graph shows this rate since 60 is the distance
Algebra I ~ Unit 3
Updated September 14, 2016
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traveled in one hour. Scenario 2 has a unit rate of 55 miles per hour
shown as the coefficient in the equation.
Algebra I
2016-17 ~ Unit 3
Given an equation of a proportional relationship, students draw a
graph of the relationship. Students recognize that the unit rate is the
coefficient of x and that this value is also the slope of the line.
(E) write the
equation of a line
that contains a given
point and is parallel
to a given line.
A.2(E)
Parallel lines have the same slope.
Triangles are similar when there is a constant rate of proportionality
between them. Using a graph, students construct triangles between
two points on a line and compare the sides to understand that the
slope (ratio of rise to run) is the same between any two points on a
line.
Parallel Lines (Video)
Example:
The triangle between A and B has a vertical height of 2 and a
horizontal length of 3.
The triangle between B and C has a vertical height of 4 and a
horizontal length of 6.
The simplified ratio of the vertical height to the horizontal length of
both triangles is 2 to 3, which also represents a slope of ⅔ for the
line, indicating that the triangles are similar.
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Algebra I
2016-17 ~ Unit 3
Given an equation in slope-­‐intercept form, students graph the line
represented.
Students write equations in the form y = mx for lines going through
the origin, recognizing that m represents the slope of the line.
(F) write the
equation of a line
that contains a given
point and is
perpendicular to a
given line.
A.2(F)
The product of the slopes of perpendicular lines is -1.
The slopes of perpendicular lines are opposite reciprocals.
Perpendicular Lines
(Video)
Example:
Given the following graph, write an equation that is perpendicular to
the line and goes through the point (4,-5).
Solution: y = -x – 1
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Updated September 14, 2016
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Students must find the slope from the given graph, find the slope that
is perpendicular to it, and use the point given to write a linear
equation.
(G) write an
equation of a line
that is parallel or
perpendicular to the
x- or y-axis and
determine whether
the slope of the line
is zero or undefined.
A.2(G)
Emphasize that an undefined slope is not equal to zero slope.
The equation of a vertical line is x = #. The equation of a horizontal
line is y = #.
Algebra I
2016-17 ~ Unit 3
Horizontal and
Vertical Lines
(Videos)
Misconceptions
Students will confuse the slopes of horizontal and vertical lines.
Students think that x=# runs parallel to the x-axis.
Students use their knowledge of parallel and perpendicular slopes
and their knowledge of linear equations to write an equation of a line
that is either parallel or perpendicular to the X or Y axis.
Example 1:
1.
Algebra I ~ Unit 3
Updated September 14, 2016
Write a linear equation of the line that is parallel to the y-axis.
Page 12 of 22
2.
Algebra I
2016-17 ~ Unit 3
Write a linear equation of the line that is parallel to the x-axis.
Example 2
What would be the slope of a line that is perpendicular to the x-axis?
Solution: undefined or no slope
A.3 Linear functions,
equations, and
inequalities. The
student applies the
mathematical
process standards
when using graphs
of linear functions,
key features, and
related
transformations to
represent in
multiple ways and
solve, with and
without technology,
equations,
inequalities, and
systems of
equations. The
student is expected
to:
(A) determine the
slope of a line given
a table of values, a
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A.3(A)
Slope should be identified as –A/B.
Algebra I ~ Unit 3
Updated September 14, 2016
Algebraic
representation
(equation)
Change in x (“run”)
Change in y (“rise”)
Graph
Linear
Parent function
Rate of change
Shaded region
Solution
Solution set
Table
Transformation
Translation (up, down,
right, left)
X-intercept
Zero
Finding Slopes
(Videos)
Page 13 of 22
graph, two points on
the line, and an
equation written in
various forms,
including y = mx + b,
Ax + By = C, and y –
y1 = m(x – x1).
For struggling students, use the calculator to support finding the
equation of a line between two points.
Algebra I
2016-17 ~ Unit 3
Misconceptions
Students put x over y instead of y over x.
Make sure students have a clear understanding of which formula to
use when.
1. Students will need to be able to move from one form to another,
Understanding how to manipulate the various forms.
2. They will need to be familiar with the slope formula:
Example 1
Determine the slope of the linear equation 3x + 2y = 4.
Solution:
Rearrange the standard form given to slope-intercept form:
y = -3/2 + 4 , so slope is -3/2.
Example 2
Given the following points, find their slope.
(2,-5) and (3,4).
Solution:
Using slope formula, m = (3+5)/(4-2) = 8/2 = 4.
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(B) calculate the rate
of change of a linear
function
represented
tabularly,
graphically, or
algebraically in
context of
mathematical and
real-world problems.
A.3(B)
Determine the rate of change from multiple representations in realworld problems.
Algebra I
2016-17 ~ Unit 3
Rate of Change
Between Two Points
(video)
Misconceptions
The student may switch values for x and y in the slope formula.
The student may make sign errors when computing rate of change
(positive or negative).
Students may neglect to note the scale when determining rate of
change from a graphical representation.
Example 1
Compare the following functions to determine which has the greater
rate of change.
Function 1: y = 2x + 4
Function 2:
Solution:
The rate of change for function 1 is 2; the rate of change for function
2 is 3. Function 2 has the greater rate of change.
Example 2
Compare the two linear functions listed below and determine which
has a negative slope.
Function 1: Gift Card
Samantha starts with $20 on a gift card for the bookstore. She spends
$3.50 per week to buy a magazine. Let y be the amount remaining as
a function of the number of weeks, x.
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Algebra I
2016-17 ~ Unit 3
Function 2: Calculator rental
The school bookstore rents graphing calculators for $5 per month. It
also collects a non-refundable fee of $10.00 for the school year.
Write the rule for the total cost (c) of renting a calculator as a
function of the number of months (m). c = 10 + 5m
Solution:
Function 1 is an example of a function whose graph has a negative
slope. Both functions have a positive starting amount; however, in
function 1, the amount decreases 3.50 each week, while in function
2, the amount increases 5.00 each month.
(C) graph linear
functions on the
coordinate plane
and identify key
features, including xintercept, yintercept, zeros, and
slope, in
mathematical and
real-world problems.
A.3(C)
Graph linear functions.
Identify x-intercepts and y-intercepts, zeros, and slope of a linear
function in mathematical and real-world problems.
Misconceptions
The student may switch values for the change in x and the change in
y when identifying slope.
The student may make sign errors (positive or negative) when
identifying slope.
Students may confuse x-intercepts and y-intercepts.
Identifying Key
Features x and yintercepts (Video)
Students will need to not only graph linear functions, but also identify
the linear function when given the graph of a line.
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Algebra I
2016-17 ~ Unit 3
Example
Write an equation that models the linear relationship in the graph
below.
Solution:
The y-intercept is 4. The slope is 1/4 , found by moving up 1 and right
4 going from (0, 4) to (4, 5). The linear equation would be
y = 1/4x + 4.
The x-intercept can be found by plugging in 0 for the y-value in
y = 1/4x + 4. So,
0 = 1/4x + 4
-4
-4
-4 = 1/4x
-4(4) = ¼(4)x
-16 = x
So, the x-intercept is -16.
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(E) determine the
effects on the graph
of the parent
function f(x) = x
when f(x) is replaced
by a∙f(x), f(x) + d, f(x
– c), f(bx) for specific
values of a, b, c, and
d.
A.3(E)
*Consider presenting parameter changes to linear and quadratic
parent functions simultaneously as introduction only.
Analyzing parameter changes on the linear parent function.
Changes in slope, changes in y-intercepts, and horizontal change are
all foundational elements for parameter changes in future functions.
Misconceptions
Horizontal changes and f(bx) are new to linear functions and will not
come naturally to students.
Algebra I
2016-17 ~ Unit 3
Transformations of
Linear Functions
(Videos)
Experiment with cases and illustrate an explanation of the effects on
the graph using technology. Include recognizing even and odd
functions from their graphs and algebraic expressions for them.
Example
The function of this graph is represented by f(x)=x2
How would the parent function change if the y-intercept was 3
instead of 0?
Solution:
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Algebra I
2016-17 ~ Unit 3
If 3 is the y-intercept, then f(x)=x2 + 3.
A.4 Linear functions,
equations, and
inequalities. The
student applies the
mathematical
process standards to
formulate statistical
relationships and
evaluate their
reasonableness
based on real-world
data. The student is
expected to:
(A) calculate, using
technology, the
correlation
coefficient between
two quantitative
variables and
interpret this
A.4(A)
Linear regression using technology including regression coefficient.
Use a calculator or computer to find the correlation coefficient for a
linear association. Interpret the meaning of the value in the context
of the data.
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Linear function
Association/correlation
Linear regression
Correlation coefficient
Strength
Causation
Correlation and
Regression Using the
TI-84
Example:
The correlation coefficient measures the “tightness” of the data
points about a line fitted to data, with a limiting value of 1 (or -1) if all
points lie precisely on a line of positive (or negative) slope. For the
line fitted to cricket chirps and temperature (figure 1), the correlation
is 0.84, and for the line fitted to boys’ height (figure 2), it is about 1.0.
However, the quadratic model for tree growth (figure 3) is non-linear,
so the value of its correlation coefficient has no direct
interpretation
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Algebra I
2016-17 ~ Unit 3
quantity as a
measure of the
strength of the
linear association.
Figure 1
Figure 2
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Algebra I
2016-17 ~ Unit 3
Figure 3
Scatterplots and
Correlation
(B) compare and
A.4(B)
contrast association
Explain the difference between correlation and causation in real
and causation in
world problems.
real-world problems.
Example:
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Updated September 14, 2016
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Algebra I
2016-17 ~ Unit 3
In situations where the correlation coefficient of a line fitted to data
is close to or 1, the two variables in the situation are said to have a
high correlation. Students must see that one of the most common
misinterpretations of correlation is to think of it as a synonym
for causation. A high correlation between two variables (suggesting a
statistical association between the two) does not imply that one
causes the other. It is not a cost increase that causes calories to
increase in pizza, and it is not a calorie increase per slice that causes
cost to increase; the addition of other expensive ingredients cause
both to increase simultaneously. Students should look for examples
of correlation being interpreted as cause and sort out why that
reasoning is incorrect. Examples may include medications versus
disease symptoms and teacher pay or class size versus high school
graduation rates. One good way of establishing cause is through the
design and analysis of randomized experiments.
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