Download Algebra I Unit 8 - Cleburne Independent School District

Title
Suggested Time Frame
Algebra I
Unit 8
4th & 5th Six Weeks
Suggested Duration – 22 days
Guiding Questions
Quadratic Equations and Modeling
Big Ideas/Enduring Understandings
Factoring a quadratic equation can be used to solve real-world
problems.
Quadratic Equations can be used to solve real-world problems.
How can you use factoring a quadratic equation to solve real-world
problems?
How can you use quadratic equations 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. A portion of the District Specificity and
Examples are a product of the Austin Area Math Supervisors TEKS Clarifying Documents available on the Region XI Math website.
Algebra I Unit 8
Updated November 19, 2015
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Algebra I
Unit 8
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;
•
Focus is on application
•
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;
•
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
•
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 8
Updated November 19, 2015
•
Page 2 of 13
Algebra I
Unit 8
Knowledge and Skills
with Student
Expectations
District Specificity/ Examples
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:
(C) write, with and
without technology,
linear functions that
provide a reasonable
fit to data to estimate
solutions and make
predictions from realworld problems.
•
•
•
•
•
Linear function
Association/correlation
Linear regression
Correlation coefficient
Strength
Resources listed and
categorized to indicate
suggested uses. Any
additional resources must be
aligned with the TEKS.
HMH Algebra I
Unit 8
A.4(C)
Collect data and model with a linear equation with and without
technology.
•
•
•
•
Algebra I Unit 8
Updated November 19, 2015
Vocabulary
Suggested Resources
Create a scatter plot from two quantitative variables.
Describe the form, strength and direction of the
relationship.
Categorize data as linear or not. Use algebraic methods and
technology to fit a linear function to the data. Use the
function to predict values.
Explain the meaning of the slope and y-­‐intercept in
context.
Page 3 of 13
Algebra I
Unit 8
Example:
1. If you have a keen ear and some crickets, can the cricket
chirps help you predict the temperature? What does 20
cricket chirps tell you?
2. The model is used to draw conclusions:
The line estimates that, on average, each added chirp
predicts an increase of about 3.29 degrees
Fahrenheit. What does this represent?
A.6 Quadratic
functions and
equations. The student
applies the
mathematical process
standards when using
properties of quadratic
functions to write and
represent in multiple
Algebra I Unit 8
Updated November 19, 2015
•
•
•
•
•
•
•
Domain
Range
X-values
Y-values
Function
Inequality
Greater than (>)
Page 4 of 13
ways, with and
without technology,
quadratic equation.
The student is
expected to:
(A) determine the
domain and range of
quadratic functions
and represent the
domain and range
using inequalities.
•
A.6(A)
•
•
Determine domain and range from a graph, table,
equation, or verbal situation.
Represent in the form of an inequality.
Misconceptions
The student may confuse x and y values.
The student may confuse which inequality symbol to use ( < or
>, ≤ or ≥, etc.)
The student may have trouble interpreting the independent
and dependent variables in a real-world situation.
The student may have difficulty determining how a problem
situation can limit the domain or the range.
•
•
•
•
•
•
•
Algebra I
Unit 8
Greater than or equal
to (≥)
Less than (<)
Less than or equal to (≤)
Open circle
Closed circle
Between
Strictly between
Between, inclusive
Example:
Algebra I Unit 8
Updated November 19, 2015
Page 5 of 13
Algebra I
Unit 8
Solution:
Domain: All real numbers or all real solutions
Range: y ≥ 6.2 or {y| y ≥ 6.2}
A.7 Quadratic
functions and
equations. The student
applies the
mathematical process
standards when using
graphs of quadraic
functions and their
realted
transformations to
represent in multiple
ways and determine,
with and without
A.7(B)
technology, the
solutions to equations. Linear factors are the zeros of the quadratic function, such as
The student is
(x+3)(x+2), where ‐3 and ‐2 are your x‐intercepts (or zeros).
expected to:
Example:
(B) describe the
relationship between
y=(x+4)(x-5)
the linear factors of
Set both linear factors equal to zero.
quadratic expressions
x+4=0
and the zeros of their
x=-4
x-5=0
associated quadratic
x=5
functions.
Algebra I Unit 8
Updated November 19, 2015
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•
•
•
•
Quadratic function
x-intercept(s)
Zero(s)
Solutions(s)
Factors
Page 6 of 13
A.8 Quadratic
functions and
equations. The student
applies the
mathematical process
standards to solve,
with and without
technology, quadratic
equations and
evaluate the
reasonableness of
their solutions. The
student formulates
statistical relationships
and evaluates their
reasonableness based
on real-world data.
The student is
expected to:
(A) solve quadratic
equations having real
solutions by factoring,
taking square roots,
completing the square,
and applying the
quadratic formula.
Algebra I Unit 8
Updated November 19, 2015
• Quadratic equations
• Solutions
• Real solutions
• Factor
• Square root
• Coefficients
• Quadratic function
• Regression
Algebra I
Unit 8
A.8(A)
Students need to be familiar with all forms of solving quadratic
equations, and be able to recognize when one form is more
appropriate over another.
Misconceptions
• When factoring, the student may confuse whether pairs
of numbers have a common product or common sum.
• The student may make sign errors when determining
solutions from factors.
Page 7 of 13
•
(B) write, using
technology, quadratic
functions that provide
a resonable fit to data
to estimate solutions
and make predictions
for real-world
problems.
•
Algebra I
Unit 8
When evaluating the quadratic formula, the student
may make arithmetic mistakes involving integers,
squares, square roots, fractions, and/or the order of
operations.
Students may have difficulty remembering how to
complete the square due to lack of understanding of the
process of factoring perfect square trinomials.
Example 1
Find the solutions to the following quadratic equation:
x2 + 5x + 6 = 0
Solution:
x2
x2 + 5x + 6 = 0
+ 5x + 6 = (x + 3)(x + 2)
0 = (x + 3)(x + 2)
0 = x + 3 or 0 = x + 2
x = −3 or x = −2
Example 2
Find the solutions to the following quadratic equation:
(x − 4)2 = 5
Solution:
Algebra I Unit 8
Updated November 19, 2015
(x − 4)2 = 5
�(x − 4)2 = 5
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Algebra I
Unit 8
x – 4 = √5
x = ±√5
x = 4 + √5 and x = 4 - √5
Example 3
Find the solutions to the following quadratic equation:
Solution:
+7 +7
+9 +9
x2 + 6x − 7 = 0
x2 + 6x − 7 = 0
x2 + 6x = 7
x2 + 6x + 9 = 16
(x + 3)2 = 16
�(x + 3)2 = √16
x + 3 = ±4
x = −3 ± 4
x = −7 and x = 1
Example 4
Find the solutions to x! + 3x − 4 = 0 using the quadratic formula.
Solution below:
Algebra I Unit 8
Updated November 19, 2015
Page 9 of 13
Algebra I
Unit 8
So, our solutions are x=-4 and x=1.
A.8(B)
Students need to be familiar with how to use a graphing
calculator to write quadratic functions.
Example:
• Create a scatter plot from two quantitative variables.
• Describe the form, strength and direction of the
relationship.
• Categorize data as quadratic or not. Use algebraic
methods and technology to fit a quadratic function to
the data. Use the function to predict values.
• Explain the meaning of the x-­‐intercepts in concept.
Algebra I Unit 8
Updated November 19, 2015
Page 10 of 13
A.10 Number and
algebraic methods.
The student applies
the mathematical
process standards and
algebraic methods to
rewrite in equivalent
forms and perform
operations on
polynomial
expressions. The
student is expected to:
(E) factor, if possible,
trinomials with real
factors in the form
ax2 + bx + c, including
perfect square
trinomials of degree
two.
A.10(E)
Factor
Misconceptions
The student may make errors in distribution or combining like
terms when checking the factors of a trinomial (especially with
minus signs and/or negative numbers).
The student may think you can “distribute” the exponent to
both terms in a binomial that is being squared. For example,
students may incorrectly rewrite (2x+3)2 as (2x)2 + (3)2 = 4x2 + 9.
Or, students may incorrectly factor 16y2 – 25 as (4y – 5)2.
The students may “factor” polynomials that are prime.
The student may factor a polynomial, but not its simplest form.
For example, students may factor 8x2 + 4x -12 as (4x + 6)(2x – 2)
rather than 4(2x + 3)(x – 1).
• Factor
• Polynomial
• Trinomial
• Terms
• Leading coefficient
• Product
• Sum
• Binomial
• Difference
• Square (or, perfect
square)
Algebra I
Unit 8
Example 1 (perfect square trinomial):
Factor x2 + 6x + 9
Solution:
(x + 3)(x + 3) = (x + 3)2
Example 2:
(F) decide if a binomial
can be written as the
Solution:
diffence of two squares
and, if possible, use the
Algebra I Unit 8
Updated November 19, 2015
Factor x2+ 2x − 15
(x − 3)(x + 5)
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structure of a
difference of two
squares to rewrite the
binomial.
Algebra I
Unit 8
A.10(F)
To factor a difference of squares, students must be able to
recognize if a term is a square or not.
Example 1:
x2 − 64
Solution:
(x − 8)(x + 8)
Example 2:
9y2 − 121
Solution:
(3y − 11x)(3y + 11x)
Example 3:
Solution:
x4 − 81
(x2 − 9)( x2 + 9)
(x − 3)(x + 3)( x2 + 9)
If students have a hard time understanding why this pattern
works each time when factoring a difference of squares, have
Algebra I Unit 8
Updated November 19, 2015
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them FOIL out the factored solution. They will end up with the
original binomial.
Algebra I Unit 8
Updated November 19, 2015
Algebra I
Unit 8
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