Grade 9 Math (Integrated Math 1)
Big Ideas:
The following Big Ideas frame the instruction:
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Equivalence: Two expressions are equivalent if they have the same value; we look for helpful equivalence to turn the complex
and unfamiliar into the simpler and familiar in order to solve problems.
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Variables can mean many things, while constants always mean the same thing *
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Solution: A solution of an equation or inequality is any value that creates a true statement when replacing the variable.*
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Representations: Relationships can be expressed verbally, numerically, algebraically, or graphically. One may be more useful
than another, depending upon the context.
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Problem Solving: Algebra is a tool for problem solving in situations containing variables relating to a constant or other variables.
Algebra allows us to isolate the most important relationships in a situation, in order to solve problems more efficiently.*
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Communication: We follow conventions, such as the Order of Operations, and identification of variables, in order to be well
understood.*
Learning bridges (according to Common Core Standards implemented 2012-2013):
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Operations on positive and negative rational numbers, extension to the real number systems with the inclusion of irrationals and
negative exponents.
Equivalence of algebraic expressions.
Solutions of linear equations and inequalities, algebraically and graphically.
Function notation for linear functions y=mx+b
Transformations in the coordinate plane
Similarity and congruence as they relate to transformations and proportions
Geometry of parallel and perpendicular lines, relating to slope, alternate interior angles, etc.
Long-term Transfer Goals:
Students will independently use their learning to:
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Make sense of problems and persevere in solving them.
Use mathematics to make more efficient and effective decisions.
Move fluently among multiple representations of data.
Create and carry out a plan to solve a non-routine problem.
Model the quantifiable or spatial relationships in the real world using expressions, equations, numerical tables, and/or graphs.
The course is designed to spiral through the big ideas of equivalence and expression, solution, and multiple representations.
Problem solving experiences are intended to occur throughout the course.
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Unit 1
Data Analysis and Representation – One-Variable Statistics
Common Core State Standards & Unpacking
* Standards marked with an asterisk are unifying standards and are revisited throughout the year, increasing in depth as the content
differs
Reason quantitatively and use units to solve problems.
N-Q.1* Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units
consistently in formulas; choose and interpret the scale and origin in graphs and data displays.
Ex. What scale would be appropriate for making a histogram of the following data that describes the level of lead
in the blood of children (in micrograms per deciliter) who were exposed to lead from their parents’ workplace?
10, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 23, 24, 25, 27, 31, 34, 34, 35, 35, 36, 37, 38, 39, 39, 41, 43, 44, 45, 48, 49,62, 73
N-Q.2 Define appropriate quantities for the purpose of descriptive modeling.
N-Q.3* Choose a level of accuracy appropriate to the limitations on measurement when reporting quantities.
Summarize, represent, and interpret data on a single count or measurement variable.
S-ID.1 Represent data with plots on the real number line (dot plots, histograms, and box plots).
Students will know and be able to:
In grades 6-8, students describe center and spread of a data distribution. Here they choose a
summary statistic appropriate to the characteristics of the data distribution, such as the shape (skewed vs.
normal) or the presence of outliers.
Sample Performance Tasks:
Construct appropriate graphical displays (dot plots, histogram, and box plot) to describe sets of data values.
a.
Make a dot plot of the number of siblings that members of your class have.
b.
Create a frequency distribution table and histogram for the following set of data:
c.
Construct a box plot of the number of buttons each of your classmates has on their clothing today.
S-ID.2 Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range,
standard deviation) of two or more different data sets.
Students will know and be able to:
S-ID.2 Understand which measure of center and which measure of spread is most appropriate to describe a given
data set. The mean and standard deviation are most commonly used to describe sets of data. However, if the distribution is extremely
skewed and/or has outliers, it is best to use the median and the interquartile range to describe the distribution since these measures are
not sensitive to outliers.
Sample Performance Tasks:
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Ex. You are planning to take on a part time job as a waiter at a local restaurant. During your interview, the boss told you that their best
waitress, Jenni, made an average of $70 a night in tips last week. However, when you asked Jenni about this, she said that she made an
average of only $50 per night last week. She provides you with a copy of her nightly tip amounts from last week (see below). Calculate the
mean and the median tip amount.
a. Which value is Jenni’s boss using to describe the average tip? Why do you think he chose this value?
b. Which value is Jenni using? Why do you think she chose this value?
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c. Which value best describes the typical amount of tips per night? Explain why
Day Tip Amount
Sunday $50
Monday $45
Wednesday $48
Friday $125
Saturday $85
S-ID.2 Select the appropriate measures to describe and compare the center and spread of two or more data sets in context.
Sample Performance Tasks:
Ex. Delia wanted to find the best type of fertilizer for her tomato plants. She purchased three types of fertilizer and
used each on a set of seedlings. After 10 days, she measured the heights (in cm) of each set of seedlings. The data
she collected is shown below. Construct box plots to analyze the data. Write a brief description comparing the
three types of fertilizer. Which fertilizer do you recommend that Delia use?
Fertilizer A
7.1 6.3 1.0
5.0 4.5 5.2
3.2 4.6 2.4
5.5 3.8 1.5
6.2 6.9 2.6
Fertilizer B
11.0 9.2 5.6
8.4 7.2 12.1
10.5 14.0 15.3
6.3 8.7 11.3
17.0 13.5 14.2
Fertilizer C
10.5 11.8 15.5
14.7 11.0 10.8
13.9 12.7 9.9
10.3 10.1 15.8
9.5 13.2 9.7
S-ID3 Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data
points (outliers).
Understand and be able to use the context of the data to explain why its distribution takes on a particular shape (e.g. are there real-life
limits to the values of the data that force skewness? are there outliers?)
Sample Performance Tasks:
Ex. Why does the shape of the distribution of incomes for professional athletes tend to be skewed to the right?
Ex. Why does the shape of the distribution of test scores on a really easy test tend to be skewed to the left?
Ex. Why does the shape of the distribution of heights of the students at your school tend to be symmetrical?
Understand that the higher the value of a measure of variability, the more spread out the data set is.
Sample Performance Tasks:
Ex. On last week’s math test, Mrs. Smith’s class had an average of 83 points with a standard deviation of 8 points. Mr. Tucker’s class had
an average of 78 points with a standard deviation of 4 points. Which class was more consistent with their test scores? How do you know?
Explain the effect of any outliers on the shape, center, and spread of the data sets.
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Sample Performance Tasks:
Ex. Explain the relationship between the mean and the median for a data set that has a few high outliers. What would most likely be the
shape of its distribution?
Ex. The heights of Washington High School’s basketball players are: 5 ft 9in, 5 ft 4in, 5 ft 7 in, 5ft 6 in, 5 ft. 5 in, 5ft. 3 in, and 5 ft 7 in. A
student transfers to Washington High and joins the basketball team. Her height is 6 ft 10in.
a. What is the mean height of the team before the new player transfers in? What is the median height?
b. What is the mean height after the new player transfers? What is the median height?
c. What affect does her height have on the team’s height distribution and stats (center and spread)?
d. How many players are taller than the new mean team height?
e.Which measure of center most accurately describes the team’s average height? Explain.
Unit 1: Data and Representation (One-Variable Statistics)
Pacing: 10 days MS/ 5 days HS
Lesson 1: Quantities, Estimates, Precision and Appropriate Measures – (2 days MS; 1 day HS)
Lesson 2: Representing Data: Boxplots, Histograms, Line Plots, Frequency Tables – (3 days MS, 1.5 days HS)
Lesson 3: Interpreting Data: Measures of Center, Measures of Spread, Effect of Outliers on data, Compare and Contrast Data
Sets, Choosing Appropriate measures of Center and Spread – (5 days MS; 2.5 days HS)
Overview:
In this unit, students will study descriptive statistics for one-variable data. Using data they will create box plots and histograms.
They will calculate measures of center and measures of spread using technology. Students will describe the shape, center, and
spread of a data distribution in context. They will learn to decide which measure of center most accurately describes a particular
data set. They will identify outliers by inspecting graphs and using the 1.5IQR Rule. They will look at the effect of outliers on
measures of center and spread. Students will compare and contrast two data sets.
Rationale:
Students have always seen multiple representations of data thru graphs, formulas, and charts; however, they haven’t always
moved between them for the same data. Throughout this year, students will learn to move between the various representations
in order to interpret data, graphs, charts, lists, etc. In this unit, the data is presented categorically, quantitatively and statistically.
Critical thinking abilities are fostered as students learn that the same data can lead to different, even opposing, conclusions
according to the representation.
Essential Questions
The following questions should drive students’ thinking throughout this unit.
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Do we really need different representations of data?
Which representation is the best?
Essential Understandings
By the end of this unit, students will understand that….
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Questioning the appropriateness of how data is presented is important in the real world.
Conclusions drawn from analysis of data displays should be realistic. They shouldn’t conflict with physical laws and
basic common sense.
Graphical representation helps to uncover patterns in data sets (shape of the distribution, unusual features such as
outliers, gaps, and clusters).
There are specific measures of center and spread that are more appropriate for a given data set or comparison of
multiple data sets depending on the shape of the distribution(s).
Unit Skills
By the end of this unit, students will be able to…
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Interpret data displays in terms of units and scales.
Understand how to model data and report quantities with an appropriate level of accuracy.
Represent data set(s) using an appropriate plot (dotplot, histogram, or boxplot) and to compare two or more data sets.
Summarize a data set using the appropriate plot, and measures of center (mean and median) and spread (range,
interquartile range, standard deviation).
Interpret the summary statistics for center and spread in the context of the problem.
Use the 1.5IQR rule to find outliers in a data set.
Explain the effects of an outlier on the measures of center and spread in the context of the problem (for the given data
set).
Compare and contrast the distribution of two or more data sets in context using appropriate summary statistics and
displays.
Interpret the similarities and differences of the distribution of two or more data sets in context using appropriate
summary statistics and displays.
Use technology for calculating summary statistics and visual representation of data.
Unit Facts & Knowledge
By the end of the unit, students will know…
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Units are based on the context of the problem.
Units and scales have the ability to change the accuracy of displays of data.
What limitations exist and why these limitations exist in relation to data.
Information given in the problem is not always needed to answer the question.
A histogram, dotplot, and boxplot is a graphical representation of data.
The shape of the graphical representation is the determinant of the appropriate summary statistics.
The measures of center are mean and median.
Center is the measure of “average” value within the data set.
The measures of spread are range, interquartile range, and standard deviation.
Spread is the measure of variability within the data set.
Outliers are extreme data points that that do not fit the pattern of the data set.
An outlier is a data point that is at least 1.5*IQR below the first quartile or above the third quartile.
Resources
Textbook Option 1:
Core Plus Course 1
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Unit 2, Lesson 2: Measuring Variability
Investigation 1: Measuring Position (pp. 104-108)
Investigation 2: Measuring and Displaying Variability: The Five-Number Summary and Box Plots (p. 108-112)
Investigation 3: Identifying Outliers (p. 113 – 116)
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Investigation 4: Measuring Variability: The Standard Deviation (p. 116-123)
Textbook Option 2:
Prentice Hall Pre-Algebra
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12-2: Box-and-Whisker Plots
Prentice Hall Algebra 1
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Sections 2-6 and 2-7 Formulas and Measures of Central Tendency
Line Plots and Frequency tables p. 735
Histograms p. 737
Box and Whisker Plots: Skills Handbook p. 740
Choosing an Appropriate Graph p. 741
Misleading Graphs p. 742
Prentice Hall Algebra 2
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12-3 Analyzing Data
12-4 Standard Deviation
Core Plus Student Study Guide:
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Measures of Center: pp. 78-81
Project-based Assessment:
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Data Collection and Graphical Display Project
Technology Applications:
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http://www.nctm.org/standards/content.aspx?id=32704 - statistics
Estimate Center
Estimate Center and Spread
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http://www.khanacademy.org/math/statistics/#math/statistics?k
Additional Activities:
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Mean, Median, Mode Introduction Activity: This is a review activity, to warm students up for the week. Students
examine the concepts of mean, median, and mode. They define key vocabulary terms, observe the teacher modeling
examples of central tendency, mean, median, and mode, and independently complete a worksheet. Students then
discuss the solutions to the independent worksheet.
Mean and Median: In this applet from Illuminations, students investigate the mean, median, and box-and-whisker plot
for a set of data that they create. The data set may contain up to 15 integers, each with a value from 0 to 100.
Are You Watching Your Computer Time?: This is an interactive group activity provided by Texas Instruments. Students
will record the amount of time that they spend on the computer (collect data), and determine the mean, median, mode
and standard deviation of collected data. They will make a frequency histogram of the grouped data, both on graph
paper, and on the TI 83+.
The Bell Curve: This is an Interactivate extension activity, but a good one for practical application. The goal of this
lesson is to introduce the concepts of the normal curve, skewness and the standard deviation. The controversy over
the 1994 book is also examined.
Using NBA Statistics for Box and Whisker Plots: This is an introduction activity from NCTM Illuminations for identifying
outliers in gathered data. Students use information from NBA statistics to make and compare box and whisker plots.
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The data provided in the lesson come from the NBA, but you could apply the lesson to data from the WNBA or any
other sports teams or leagues for which player statistics are available.
What are the Odds?: Students predict the outcome of a basketball game. In this what are the odds lesson, students
convert from standard units of measure to metric. They organize data and create stem-and-leaf plots, box-and-whisker
plots, and scatter plots. Students identify the mean, median, mode, and outliers of given data.
Graphs and More: In this lesson, students create and interpret various types of graphs from data gathered in class. In
the analysis of the graphs, students classify data as univariate or bivariate and as quantitative or qualitative. Students
also interpret frequency distributions based on spread, symmetry, skewness, clusters and outliers.
Comparison of Univariate and Bivariate Data: This is an optional activity, as it provides extension by exposing students
to the differentiation between univariate and bivariate data. Students will gain experience determining what types of
graphs and measures are appropriate for each type of data. Have students reflect on their learning process as they
interact with each graph type in their math journal/reflective blogs.
Unit 2
One-Variable Equations and Inequalities
Common Core State Standard
* Standards marked with an asterisk are unifying standards and are revisited throughout the year, increasing in depth as the content
differs.
Extend the properties of exponents to rational exponents.
N-RN.1 Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those
values, allowing for a notation for radicals in terms of rational exponents. For example, we define 51/3 to be the cube root of 5 because we
want (51/3)3 = 5(1/3)3 to hold, so (51/3)3 must equal 5.
In order to understand the meaning of rational exponents, students can initially investigate them by considering a pattern such as:
24 = 16; 22 = 4; 21 = 2; 21/2 = ?
What is the pattern for the exponents? They are reduced by a factor of ½ each time. What is the pattern of the simplified values? Each
successive value is the square root of the previous value. If we continue this pattern, then 21/2 = √2.
Once the meaning of a rational exponent (with a numerator of 1) is established, students can verify that the properties of integer exponents
hold for rational exponents as well.
Ex. Use an example to show why x m / x n = x m-n holds true for expressions involving rational exponents like
½ or 1/5.
N-RN.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents.
Students should be able to use the properties of exponents to rewrite expressions involving radicals as expressions using rational
exponents. At this level, focus on fractional exponents with a numerator of 1. Students should be able to use the properties of exponents to
rewrite expressions involving rational exponents as expressions using radicals. At this level, focus on fractional exponents with a numerator
of 1.
Reason quantitatively and use units to solve problems.
N-Q.1* Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units
consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.
N-Q.3* Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.
Interpret the structure of expressions.
A-SSE.1* Interpret expressions that represent a quantity in terms of its context.
a. Interpret parts of an expression, such as terms, factors, and coefficients.
Ex. The expression 20(4x) + 500 represents the cost in dollars of the materials and labor needed to build a square fence with side length x
feet around a playground. Interpret the constants and coefficients of the expression in context.
b. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r)n as the
product of P and a factor not depending on P.
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Create equations that describe numbers or relationships.
A-CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and
quadratic functions, and simple rational and exponential functions.
(Note: At this level, focus on linear and exponential functions.)
Ex. The Tindell household contains three people of different generations. The total of the ages of the three family members is 85.
a. Find reasonable ages for the three Tindells.
b. Find another reasonable set of ages for them.
c. One student, in solving this problem, wrote C + (C+20)+ (C+56) = 85
1. What does C represent in this equation?
2. What do you think the student had in mind when using the numbers 20 and 56?
3. What set of ages do you think the student came up with?
Ex. A salesperson earns $700 per month plus 20% of sales. Write an equation to find the minimum amount of sales needed to receive a
salary of at least $2500 per month.
A-CED.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange
Ohm’s law V = IR to highlight resistance R.
(Note: At this level, limit to formulas that are linear in the variable of interest, or to formulas involving squared or cubed variables.)
Ex. If
, solve for
Understand solving equations as a process of reasoning and explain the reasoning.
A-REI.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting
from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.
Ex. Solve 5(x+3)-3x=55 for x. Use mathematical properties to justify each step in the process.
Solve equations and inequalities in one variable.
A-REI.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
Ex. Solve, Ax +B =C for x. What are the specific restrictions on A?
Ex. What is the difference between solving an equation and simplifying an expression?
Ex. Grandma’s house is 20 miles away and Johnny wants to know how long it will take to get there using various modes of transportation.
a. Model this situation with an equation where time is a function of rate in miles per hour.
b. For each mode of transportation listed below, determine the time it would take to get to Grandma’s.
Mode of Transportation
Bike
Car
Walking
Rate of Travel (miles per hour)
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55
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Time of Travel (hours)
Ex. A parking garage charges $1 for the first half-hour and $0.60 for each additional half-hour or portion thereof. If you have only $6.00 in
cash, write an inequality and solve it to find how long you can park.
Ex. Compare solving an inequality in one variable to solving an equation in one variable. Compare solving a linear inequality to solving a
linear equation.
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Represent and solve equations and inequalities graphically.
A-REI.11* Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions
of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or
find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and
logarithmic functions. (Note: At this level, focus on linear and exponential functions.)
Ex. How do you find the solution to an equation graphically?
A-REI.11 Solve graphically, finding approximate solutions using technology. At this level, focus on linear and
exponential functions.
Ex. Solve the following equations by graphing. Give your answer to the nearest tenth.
Ex. Road Trip
Given three destinations with varying miles to travel and one rental car company to choose from, have students decide on a destination
given the distance, then decide on the destinations given a specified cost. Have students demonstrate their rationales algebraically and
graphically.
Ex. Going Over
Students will investigate the maximum amount of overage minutes they can have in a month given a specified monthly budget. Only focus
on voice calls and overlook data and texting charges. Project or investigate real cellular plans from provides such as; AT&T, Verizon, or
US Cellular
A-REI.11 Solve by making tables for each side of the equation. Use the results from substituting previous values of x to decide whether
to try a larger or smaller value of x to find where the two sides are equal. The x-value that makes the two sides equal is the solution to the
equation. At this level, focus on linear and exponential functions.
Ex. Solve the following equations by using a table. Give your answer to the nearest tenth.
Geometric Measurement and Dimension
Explain volume formulas and use them to solve problems.
G-GMD.1 Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and
cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments. (Note: Informal limit arguments are not the intent at
this level. )
Understanding the formula for the circumference of a circle can be taught using the diameter of the circle or the radius of the circle.
Measure either the radius or diameter with a string or pipe cleaner. As you measure the distance around the circle using the measure of
the diameter, you will find that it’s a little over three, which is pi. Therefore the circumference can be written as
. When
measuring the circle using the radius, you get a little over 6, which is
. Therefore, the circumference of the circle can also be
expressed using
.
Understanding the formula for the area of a circle can be shown using dissection arguments. First dissect portions of the circle like pieces
of a pie. Arrange the pieces into a curvy parallelogram as indicated below.
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Understanding the volume of a cylinder is based on the area of a circle, realizing that the volume is the area of the circle over and over
again until you’ve reached the given height, which is a simplified version of Cavalieri’sprinciple. In Cavalieri’s principle, the cross-sections
of the cylinder are circles of equal area, which stack to a specific height. Therefore the formula for the volume of a cylinder is
. Informal limit arguments are not the intent at this level
G-GMD.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. (Note: At this level, formulas for pyramids,
cones and spheres will be given.)
Ex. Given the formula
, for the volume of a cone, where B is the area of the base and h is the height of the figure.
If a cone is inside a cylinder with a diameter of 12in. and a height of 16 in., find the volume of the cone.
Unit 2 Overview
Pacing
Essential Questions
Essential Understandings
Unit Skills
Unit Facts & Knowledge
Unit 2
Overview
Essential Questions
Solutions
One-Variable
Equations and
Inequalities
By the end of this
unit students will be
able to answer the
following questions:
Students will
develop their
ability to solve
problems
by defining
variables and
relating those
variables to one
another.
They will
develop
their ability
to express
situations using
equations or
inequalities.
They will learn
to interpret
solutions
in problem
contexts.
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How do you apply
the properties of
exponents to solve
various problems
involving rational
exponents?
Why is it important
to manipulate and
explain the individual
parts of expressions?
When is it appropriate
to create an equation
vs. create an
inequality for problem
solving?
How does a formula
compare to a rule?
Essential
Understandings
Unit Skills
Unit Facts &
Knowledge
By the end of this
unit, students will
understand that….
By the end of this
unit, students will be
able to…
By the end of the
unit, students will
know . . .
15 days
Middle
School
-The properties of
integer exponents are
the same for rational
exponents (ex: fractional
exponents)
-certain situations
require a more precise
unit than others
-data can represent
real-world problems
-units are specific for
formulas and real-world
problems
- the graph scale affects
the shape of a graph
Understand the
meaning of rational
exponents
Each successive
value is the square
root of the previous
value.
8 days High
School
Expressions can be
broken into individual
parts (terms, factors,
coefficients)
How can a formula
be solved like an
equation/inequality?
Equations and
inequalities must
appropriately represent
the given information
Why is it important to
rearrange a formula?
Equations and
inequalities are the basis
Investigate rational
exponents by
considering a pattern
such a
= 16,
=2,
Use the properties
of exponents to
rewrite expressions
involving radicals as
expressions using
rational exponents and
vice versa
Choose the correct
unit to represent a
solution
Interpret the meaning
of units in formulas
Choose the
Rational numbers
are numbers that
can be written as
fractions.
A radical undoes
exponentiation (ex:
square root/cube
root)
The structure of
an expression is
made up terms,
factors, and
coefficients that
have meaning
within a context.
Complicated
expressions can
be manipulated by
looking at its parts
as single entities.
Pacing
Lesson
1: Using
Variables.
Exponents
& Order of
Operations.
(2 day MS, 1
day HS)
Lesson 2: Real
Number
System
(operations &
properties) (2
days MS, 1 day
HS)
Lesson
3: Solving
equations and
inequalities.1,
2 & multi- step;
variables on
both sides. (6
days MS, 3
days HS)
12
13
Unit 3
Two Variable Equations and Functions
Common Core State Standards
* Standards marked with an asterisk are unifying standards and are revisited throughout the year, increasing in depth as the content
differs
Reason quantitatively and use units to solve problems.
N-Q.1 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently
in formulas; choose and interpret the scale and the origin in graphs and data displays.
Represent and solve equations and inequalities graphically.
A-REI.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often
forming a curve (which could be a line).
(Notes: At this level, focus on linear and exponential equations)
Ex. Which of the following points are on the graph of the equation
graph? Explain.
a. (4, 0)
b. (0, 10)
c. (-1, 7.5)
d. (2.3, 5)
How many points are on this
Understand the concept of a function and use function notation.
F-IF.1 Understand that a function from one set (called the domain) to another set (called the range) assigns to then f(x) denotes the output
of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).
Ex. When is an equation a function? Explain the notation that defines a function.
Ex. Describe the domain and range of a function and compare the concept of domain and range as it relates to a
function.
F-IF.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a
context.
(Note: At this level, the focus is linear and exponential functions.)
Ex. Evaluate
for the function
Ex. The function
seconds after it is shot
straight up into the air from a pitching machine. Evaluate
problem.
.
describes the height h in feet of a tennis ball x
and interpret the meaning of the point in the context of the
F-IF.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the
Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1.
Ex. You just got a pair of baby rabbits for your birthday – one male and one female. You decide that you will breed the rabbits, but need to
plan a budget for the upcoming year. To help prepare your budget, you need an estimate of how many rabbits you will have by the end of
the year. In order to build a mathematical model of this situation, you make the following assumptions:
14
•
A rabbit will reach sexual maturity after one month.
•
The gestation period of a rabbit is one month.
• Once a female rabbit reaches sexual maturity, she will give birth every month.
•
A female rabbit will always give birth to one male rabbit and one female rabbit.
•
Rabbits never die.
So how many male/female rabbit pairs are there after one year (12 months)?
Ex. In August 2011, the population in the United States was approximately 312 million. Suppose that in recent trends the birth rate was
1.7% of the total population. Use the word NOW to represent the population of the United States in any given year and the word NEXT to
represent the population the following year. Using NOW-NEXT, write a rule that shows how to calculate next year’s population from the
current population. What is the initial value of this sequence?
Interpret functions that arise in applications in terms of the context.
F-IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the
quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals
where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and
periodicity. ★
(Note: At this level, focus on linear functions.)
Ex. Below is a table that represents the relationship between daily profit, P for an amusement park and the number of paying visitors in
thousands, n.
n
0
1
2
3
4
5
6
P
0
5
8
9
8
5
0
a. What are the x-intercepts and y-intercepts and explain them in the context of the problem.
b. Identify any maximums or minimums and explain their meaning in the context of the problem.
c. Determine if the graph is symmetrical and identify which shape this pattern of change develops.
d. Describe the intervals of increase and decrease and explain them in the context of the problem.
F-IF.4 When given a verbal description of the relationship between two quantities, sketch a graph of the
relationship, showing key features.
Ex. Elizabeth and Joshua tried to get a monthly allowance from their mother. If their mother initially paid them a penny and 2 pennies for
the first day of the month, 4 pennies for the second day, and so on. How much would their mother have to pay on the 10th, 20th, and 30th
day of the month? Sketch the graph of the relationship between the two quantities and explain what the point (0, 1) represents.
F-IF.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if
the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an
appropriate domain for the function. ★
(Note: At this level, focus on linear and exponential functions).
Ex. If Jennifer buys a cell phone and the plan she decided upon charged her $50 for the phone and $0.10 for each minute she is on the
phone. What would be the appropriate domain that describes this relationship? Describe what is meant by the point (10, 51).
Ex. Graph the function
exist.
15
and determine the domain and range, identifying any restrictions on that
F-IF.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval.
Estimate the rate of change from a graph.
(Note: At this level, focus on linear functions whose domain is a subset of the integers.)
Ex. What is the average rate at which this bicycle rider traveled from four to ten minutes of her ride?
Build a function that models a relationship between two quantities
F-BF.1 Write a function that describes a relationship between two quantities. ★
a. Determine an explicit expression, a recursive process, or steps for calculation from a context.
Ex. A single bacterium is placed in a test tube and splits in two after one minute. After two minutes, the resulting two bacteria
split in two, creating four bacteria. This process continues for one hour until test tube is filled up. How many bacteria are in the
test tube after 5 minutes? 15 minutes? Write a recursive rule to find the number of bacteria in the test tube after n minutes.
Convert this rule into explicit form. How many bacteria are in the test tube after one hour?
b. Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a
cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.
Ex. Suppose Kevin had $10,000 to invest in a CD account paying 8% interest compounded yearly. The function representing this
situation is
. When the constant function y=50 is added to the function, what effect
does it have on the exponential function? What does that mean in the context of the problem?
(Note: At this level, limit to addition or subtraction of constant to linear, exponential or quadratic functions or addition of linear functions to
linear or quadratic functions.)
F-BF.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and
negative); find the value of k given the graphs. 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.
16
(Note: At this level, limit to vertical and horizontal translations of linear and exponential functions. Even and odd functions are not
addressed.)
Ex. Given
describe the changes in the graph of g(x), that occurred to create
.
Ex. If f(x) represents a diver’s position from the edge of a pool as he dives from a 5ft. long board 25ft. above the water. If his second dive
was from a 10ft. long board that is 10ft above the water, what happens to my equation of f(x) to model the second dive?
Unit 3
Overview
Students
should develop
the ability to
recognize
and describe
important
patterns that
relate quantitative
variables.
They will use
data tables,
graphs, words,
and symbols
to represent
relationships.
They will be able
to use reasoning
and calculating
tools to answer
questions and
solve problems.
Essential Questions
The following
questions should
drive students’
thinking throughout
this unit.
Why would the scale
and origin of a graph
affect the way data is
displayed?
How can I represent
an equation/inequality
graphically?
How can solutions be
represented? (Table,
Graph, Shading)
What is a function?
Is every relationship
between the domain
and range a function?
What is the relationship
between the graphs
of a function and the
change in the domain
and/or range?
Does the sequence
of objects/numbers
represent a function?
What is the meaning of
the key features of the
graph as it relates to a
real world situation?
17
How can I build a
function using a NOW
= NEXT (Recursive)
process?
How can I describe
a relationship
between sets of
Essential
Understandings
Unit Skills
Unit Facts &
Knowledge
Pacing
By the end of this
unit, students will
understand that….
By the end of this
unit, students will
be able to…
By the end of the
unit, students will
know…
13 days MS
Appropriate scales
should be used when
displaying data.
Display a graph
using an appropriate
scale and units.
Where the x and y
axis are located.
Solutions fit different
models.
Generate solutions
from a function rule.
Lesson 1:
Graphing data
on x-y plane;
relations &
functions;
(3 days MS/1.5
days HS)
The direct correlation
between the set of all
solutions represented
algebraically,
graphically, or in a
table.
Represent the
ordered pairs in a
table or graph
Functions are used
to compare the
relationship between
in an input and an
output
How does the real
life context of the
problem relate to the
function notation?
Recognize
differences between
models (linear,
quadratic, absolute
value, step function,
exponential)
Recognize
functions in multiple
representations.
The difference
between the axes
and what they can be
used for.
The values in the
table are derived
from the function rule
(equation)
The points on a
graph (including
the shading from
an inequality)
are solutions to
the function rule
(equation)
Describe the
relationship between
a domain and range.
The pattern of the
graph is dictated
by the appropriate
model.
Use function
notation.
That a set is a
collection of elements
Recognize the rate of
change, y-intercept
Evaluate functions
given the domain.
Interpret the key
features (rate of
change, y-intercept)
of the graph
Interpret statements
that use function
notation.
The domain is the
input values of a
relation/function
Functions can be
used to solve real
world situations
Sketch the graph
of a function given
description and using
key features (rate of
change, y-intercept)
Recognize that
sequences are
functions.
Determine on which
intervals a function
The range is the
output values of a
relation/function
A function is a
relationship in which
each output (range
element) is paired
6.5 days HS
Lesson 2:
relating graphs
to events/cause
& effect
(2 days MS/1
day HS)
Lesson 3:
Connect rules,
tables & graphs
(linear, quad,
exp, absolute)
(5 days MS/2.5
days HS)
Lesson 4:
Change over
time-connect
rate of change
& slope
(3 days MS/1.5
days HS)
18
Unit 4
Linear Functions
Common Core State Standard
* Standards marked with an asterisk are unifying standards and are revisited throughout the year, increasing in depth as the content
differs
Reason quantitatively and use units to solve problems.
N-Q.1 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently
in formulas; choose and interpret the scale and the origin in graphs and data displays.
Interpret the structure of expressions.
Perform arithmetic operations on polynomials.
A-APR.1 Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition,
subtraction, and multiplication; add, subtract, and multiply polynomials.
(Note: At this level, limit to addition and subtraction of quadratics and multiplication of linear expressions.)
A-SSE.1 Interpret expressions that represent a quantity in terms of its context.«
a. Interpret parts of an expression, such as terms, factors, and coefficients.
Ex. The expression 20(4x) + 500 represents the cost in dollars of the materials and labor needed to build a square fence with side length x
feet around a playground. Interpret the constants and coefficients of the expression in context.
b. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r)n as the
product of P and a factor not depending on P.
(Note: At this level, limit to linear expressions.)
Ex. Without expanding, explain how the expression
having the structure of a quadratic expression. (Note: At this level, limit to linear expressions.)
+ 1 can be viewed as
A-SSE.2 Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a
difference of squares that can be factored as (x2 – y2)(x2 + y2).
A.SSE.2 Students rewrite algebraic expressions by combining like terms or factoring to reveal equivalent forms of the same expression.
Ex. Expand the expression
.
to show that it is a quadratic expression of the form
Create equations that describe numbers or relationships.
A-CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with
19and scales. (Note: At this level, focus on linear.)
labels
A-CED.2 Given a contextual situation, write equations in two variables that represent the relationship that exists between the quantities.
Also graph the equation with appropriate labels and scales. Make sure students are exposed to a variety of equations arising from the
functions they have studied. At this level, focus on linear equations.
Ex. If Jennifer buys a cell phone and the plan she decided upon charged her $50 for the phone and $0.10 for each minute she is on the
phone. What would be the appropriate domain that describes this relationship? Describe what is meant by the point (10, 51). Determine the
domain and range, identifying any restrictions on that exist.
F- IF.5 From a graph students will identify the domain. In context, students will identify the domain, stating any restrictions and why they
are restrictions. At this level, focus on linear functions.
F-IF.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval.
Estimate the rate of change from a graph.★
20
Note: At this level, focus on linear functions and exponential functions whose domain is a subset of the integers.
Ex. What is the average rate at which this bike rider travels from 4 to 10 minutes?
F-IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more
complicated cases.
Graph linear and quadratic functions and show intercepts, maxima, and minima.
F-IF.7 Students should graph functions given by an equation and show characteristics such as but not limited to intercepts, maximums,
minimums, and intervals of increase or decrease. Students may use calculators. At this level, focus on linear and exponential functions
only.
Ex. Graph f(x) =
identifying it’s intercepts and maximum or minimum.
F-IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by
verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the
larger maximum. (Note: At this level, focus on linear, exponential, and quadratic functions.)
F-IF.9 Students should compare the properties of two functions represented by verbal descriptions, tables, graphs, and equations. For
example, compare the growth of two linear functions, two exponential functions, or one of each. At this level, limit to linear, exponential,
and quadratic functions.
Ex. Compare the functions represented below. Which has the lowest minimum?
a. f(x) = 3x2 +13x +4
b.
Ex. Compare the patterns of (x, y) values when produced by these functions: f(x) =
by completing these tasks.
a. Write a NOW-NEXT equation that would provide the same pattern of (x, y) values for each function.
b. How would you describe the similarities and differences in the relationships of x and y in terms of their graphs, tables, and equations.
Build a function that models a relationship between two quantities.
F-BF.1 Write a function that describes a relationship between two quantities. ★
a. Determine an explicit expression, a recursive process, or steps for calculation from a context.
F-BF.1a Recognize when a relationship exists between two quantities and write a function to describe them. Use steps, the recursive
process, to make the calculations from context in order to write the explicit expression that represents the relationship.
Ex. A single bacterium is placed in a test tube and splits in two after one minute. After two minutes, the resulting two bacteria split in
two, creating four bacteria. This process continues for one hour until test tube is filled up. How many bacteria are in the test tube after
5 minutes? 15 minutes? Write a recursive rule to find the number of bacteria in the test tube after n minutes. Convert this rule into
explicit form. How many bacteria are in the test tube after one hour?
21
b. Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a
cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.
F-BF.1b Students should take standard function types such as constant, linear and exponential functions and add, subtract, multiply
and divide them. Also explain how the function is effected and how it relates to the model. At this level, limit to addition or subtraction
of a constant function to linear, exponential, or quadratic functions or addition of linear functions to linear or quadratic functions.
Ex. Suppose Kevin had $10,000 to invest in a CD account paying 8% interest compounded yearly. The function representing this
situation is y = 10000
When the constant function y=50 is added to the function, what effect does it have on
the exponential function? What does that mean in the context of the problem?
F-BF.2 Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate
between the two forms.
Note: At this level, formal recursive notation is not used. Instead, use of informal recursive notation (such as NEXT = NOW + 5 starting at
3) is intended.
F-BF.2 Write the recursive and explicit forms of the arithmetic and geometric sequences. Translate between the recursive and explicit
forms. Use the recursive and explicit forms of arithmetic and geometric sequences to model real-world situations. In an arithmetic
sequence, each term is obtained from the previous term by adding the same number each time. This number is called the common
difference. In a geometric sequence, each term is obtained from the previous term by multiplying by a constant amount, called the common
ratio. Connect arithmetic sequences to linear functions and geometric sequences to exponential functions. At this level, formal recursive
notation is not used. Instead, use of informal recursive notation (such as NEXT = NOW + 5, starting at 3) is intended.
Ex. A concert hall has 58 seats in Row 1, 62 seats in Row 2, 66 seats in Row 3, and so on. The concert hall has 34 rows of seats. Write a
recursive formula to find the number of seats in each row. How many seats are in a row 5? Write the explicit formula to determine which
row has 94 seats?
F-BF.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and
negative); find the value of k given the graphs. 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. (Note: At this level, limit to
vertical and horizontal translations of linear and exponential functions. Even and odd functions are not addressed.)
F-BF.3 Know that when adding a constant, k, to a function, it moves the graph of the function vertically. If k is positive, it translates the
graph up, and if k is negative, it translates the graph down.
If k is either added or subtracted from the x-value, it translates the graph of the function horizontally. If we add k, the graph shifts left and
if we subtract k, the graph shifts right. The expression (x + k) shifts the graphs k units to the left because when x + k = 0, x = -k. Use the
calculator to explore the effects these values have when applied to a function and explain why the values affect the function the way
it does. The calculator visually displays the function and its translation making it simple for every student to describe and understand
translations. At this level, limit to vertical and horizontal translations of linear and exponential functions. Even and odd functions are not
addressed. Relate the vertical translation of a linear function to its y-intercept.
Ex. Given g(x) = x2 describe the changes in the graph of g(x) that occurred to create f(x) = 2(x – 5)2 + 7.
Ex. If f(x) represents a diver’s position from the edge of a pool as he dives from a 5ft. long board 25ft. above the water. If his second dive
was from a 10ft. long board that is 10ft above the water, what happens to my equation of f(x) to model the second dive?
Construct and compare linear and exponential models and solve problems.
F-LE.1 Distinguish between situations that can be modeled with linear functions and with exponential functions
•
Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal
factors over equal intervals.
F-LE.1 Decide whether a situation can be represented using a linear model or an exponential model.
22
Recursive forms of functions will show that linear models grow by a constant rate over equal intervals, and exponential models grow by
equal factors over equal intervals. These can also be seen through NOW-NEXT equations. NOW-NEXT equations are equations that show
how to calculate the value of the next term in a sequence from the value of the current term. The arithmetic sequence NEXT = NOW + c is
the recursive form of a linear function. The common difference C corresponds to the slope m in the explicit form of a linear function, y = mx
+ b. the initial value of the sequence corresponds to the y intercept, b. The geometric sequence NEXT = B·NOW is the recursive form of an
exponential function. The common ratio B corresponds to the base b in the explicit form of an exponential function, y = a
. The initial
value corresponds to the y intercept, a.
Ex. If one person does good deeds for three new people, then the three new people each do good deeds for three more new people. Next,
nine people each do good deeds for three more new people, and so on. Does this situation represent a linear or exponential model? Why
or why not?
Ex. Describe the similarities and differences of a linear and an exponential NOW- NEXT equation.
F-LE.1a Given a linear function, use a table or graph to locate points that are at equal intervals of x. Calculate the difference between
values of f(x), showing that these differences are the same. Hence linear functions grow by equal differences over equal intervals.
F-LE.1a Given an exponential function, use a table or graph to locate points that are at equal intervals of x. Calculate the ratio of the
sequential values of f(x) for these points, showing that these ratios are the same. This ratio represents the constant factor that is used as
the base of the function. In NOW-NEXT form, the base is the factor that is multiplied by NOW to get to NEXT value. Hence exponential
functions grow by equal factors over equal intervals.
Ex. Describe how you decide whether a problem situation is represented by a linear or an exponential model?
b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.
F-LE.1b Given a real-life relationship between two quantities, determine whether one quantity changes at a constant rate per unit interval
of the other quantity. When working with symbolic form of the relationship, if the equation can be rewritten in the form y = mx + b, then
the relationship is linear and the constant rate per unit interval is m. When working with a table or graph, either write the corresponding
equation and see if it is linear or locate at least two pairs of points and calculate the rate of change for each set of points. If these rates are
the same, the function is linear. If the rates are not all the same, the function is not linear.
Ex. Describe and compare the rates of change of a linear function to that of an exponential function.
c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
F-LE.1c Given a real-life relationship between two quantities, determine whether one quantity changes at a constant percent rate per unit
interval of the other quantity. When working with symbolic form of the relationship, if the equation can be rewritten in the form y = a (1 + r)x,
then the relationship is exponential and the constant percent rate per unit interval is r. When working with a table or graph, either write the
corresponding equation and see if it is exponential or locate at least two pairs of points and calculate the percent rate of change for each
set of points. If these percent rates are the same, the function is exponential. If the percent rates are not all the same, the function is not
exponential.
Ex. Town A adds 10 people per year to its population, and town B grows by 10% each year. In 2006, each town has 145 residents. For
each town, determine whether the population growth is linear or exponential. Explain. Report the constant rate per unit interval (linear) or
the constant percent rate per unit interval (exponential).
F-LE.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a
relationship, or two input-output pairs (include reading these from a table).
F-LE.2 Students use graphs, a verbal description, two-points(in the linear case), and reading from a table of values to write the linear or
exponential function that each of these representations describes. Also include arithmetic and geometric sequences because an arithmetic
sequence is linear in pattern and a geometric sequence has an exponential pattern.
Ex. Suppose a single bacterium lands in a cut on your hand. It begins spreading an infection by growing and splitting into two bacteria
every 10 minutes. The table below represents the number of bacteria in the cut after several 10-minute intervals.
Number of
23
1
2
3
4
5
6
7
10-minute
periods
Bacteria
Count
2
4
8
16
32
64
128
a. Use NOW-NEXT to write a rule relating the number of bacteria at one time to the number 10 minutes later.
b. Write an equation showing how the number of bacteria can be calculated from the number of stages in the growth and division process.
F-LE.5 Interpret the parameters in a linear or exponential function in terms of a context.
F-LE.5 Understand the difference between the practical and the non-practical domain in linear and exponential situations and explain their
meaning in terms of their context. Identify any values for which the exponential function may approach but does not reach and interpret
its meaning in terms of the context it’s in. You may want to talk about asymptotes here, which will be a connection with simple rational
functions.
Ex. A function of the form f(n) = P(1 + r)n is used to model the amount of money in a savings account that earns 8% interest, compounded
annually, where n is the number of years since the initial deposit. What is the value of r? What does it mean in terms of the savings
account? What is the meaning of the constant P in terms of the savings account? Explain your reasoning. Will n or f(n) ever take on the
value 0? Why or why not?
24
25
Unit 4
Overview
Students should
develop their
understanding of
the characteristics
of linear functions
in 2 variables and
the connections
between their
numeric,
graphic, verbal,
and symbolic
representations.
(rule, table, graph)
They will use these
representations
to solve linear
equations and
inequalities.
Students will
become more
proficient in
recognizing
patterns
in function
representations
and writing
function rules
describing them.
Students will
explore ways
to rewrite linear
expressions in
equivalent forms.
Essential
Questions
Essential
Understandings
Unit Skills
The following
questions
should drive
students’
thinking
throughout this
unit.
By the end of this unit,
students will understand
that….
By the end of this unit,
students will be able
to…
By the end of the
unit, students will
know…
37 days
MS/18.5
days HS
Units must be consistent
throughout the problem
solving process.
Use units to understand
problems.
Real-world problems
require units.
Lesson
1: Rate of
Change &
Slope
Guide the solutions of
multi-step problems.
Some situations only
require Quadrant 1
when graphing.
(4 days
MS/2 day
HS)
What factors
should be
considers when
choosing units
and scales?
What is the
importance of
labeling and
scaling a graph?
What does the
origin represent
in a real world
situation?
How do you
interpret the
structure of
the linear
expression?
What is the
meaning of the
variables in the
expression y= mx
+ b?
The consistence in
formulas relates to its
units.
m represents the rate
of change and b is the
initial amount of a linear
function.
26
How does writing
an equation show
the effect one
variable will have
on the other if
changed?
How do we graph
equations and
Choose and interpret
units consistently in
formulas.
M represents the rate
of change (slope)
Choose an appropriate
window for a graphical
representation.
B represents the
initial amount
Rewrite linear expressions
to get to y=mx+b
Linear relationships exist.
Correct scales and axes
are important to prevent
misconceptions
Relationships can be
modeled (through graphs,
equations, and tables)
There are more solutions
to an inequality than an
equation.
The graph of a linear
equation is a line.
How do you
create a linear
equation based
on a “real life”
relationship?
Unit Facts &
Knowledge
The benefits of
technology integrated into
mathematics.
The benefits of relating
linear functions to real live
situations.
Interpret a linear
function through multiple
representations.
A function can be a
composition of two simpler
Identify and interpret
the rate of change, or
coefficient, and an
initial amount in a given
context.
Create a linear equation
to model a relationship.
Ways to rewrite linear
expressions as mx+ b
A rate of change
should be rewritten in
its simplest form ( 10
burgers for 5 dollars
should be expressed
as 1 burger for 2
dollars)
Correctly label and
graph a linear equation.
Plot a linear equation/
inequality using the x
and y intercepts.
Plot a linear equation/
inequality using slope
intercept form.
Determine if a given
Linear equations
with two or more
variables are based
on relationships
between quantities.
Pacing
Lesson
2: Given
a table,
graph,
or data
pairs,
develop
rule in
y= mx
+ b and
convert it
to other
forms;
interpr
etation
of linear
function
rule in
real world
context.
(10 days
MS/5 day
HS)
Lesson
3:Modeli
ng data &
predicting
(6 days
MS/3
days HS)
Lesson
4:
Equations can
be graphed on a
coordinate plane.
The difference
Model
real world
situations
using
linear
inequali
ties and
27
Resources:
Textbook: Core Plus Course 1
• Unit 3 Lesson 1 Investigations 1, 2. (Postpone Investigation 3 until Unit 8. Focus on patterns & function rule development.
Students can also begin estimating lines of best fit for a given data set)
• Unit 3 Lesson 2 Investigations 1-3. (postpone Investigation 4 until Unit 7)
• Unit 3 Lesson 3 Investigations 1-2
Textbook: Prentice Hall Algebra 1
• Sections 6-1 thru 6-4, 6-6. (6-5 done in Unit 7)
• Sections 7-5
Project-based Assessment:
Technology Applications:
•
•
•
•
•
•
•
Prentice Hall Algebra 1 pg. 290 Investigation y = mx + b
http://mathbits.com/mathbits/TISection/Openpage.htm
Determining Functions using regression: http://illuminations.nctm.org/LessonDetail.aspx?id=U180;
Linear functions (many to choose from): http://www.thinkfinity.org/partner-search?
start=0&partner=6&partner_value=no&from_links=&txtKeyWord=linear+functions&txtKeyWord2=&narrow=1&chkPartner%5B%5
D=Illuminations
Slopes and Equations of Lines: http://www.regentsprep.org/Regents/math/ALGEBRA/AC1/indexAC1.htm
http://www.khanacademy.org/math/algebra/#linear-equations-and-inequalitie
http://www.khanacademy.org/math/algebra/#solving-linear-inequalities
Additional Activities:
NCDPI Indicators for Algebra I (Notebook)- Making Sense of Slope (B44-46); How Do They Fit (A11/B94); Picture Tells a Linear Story
(B39-42); Toothpick Triangles (A23/B1, 2, 105); What Shape are You (A17/B2, 102); Scoring and Winning (A7, B87); I Have, Who Has?
(B81-82); How Do They Fit (A11/B97); Polynomial Four in a Row;
Algebra Uno (B63); How Do They Fit (B91-B92); Algebra tiles (A29, B110, B53); Factoring with Algebra Tiles (A31, B111-112); Digits 0-9
Puzzle (B66); Polynomials Four In a Row (B22); How Do They Fit (B93)
Sample Performance Task(s):
Unit 5
Exponential Functions
Common Core State Standards
* Standards marked with an asterisk are unifying standards and are revisited throughout the year, increasing in depth as the content
differs
Reason quantitatively and use units to solve problems.
28
N-Q.1 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently
in formulas; choose and interpret the scale and the origin in graphs and data displays.
Interpret the structure of expressions.
A-SSE.1 Interpret expressions that represent a quantity in terms of its context.«
a. Interpret parts of an expression, such as terms, factors, and coefficients.
b. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r)n as the
product of P and a factor not depending on P.
(Note: At this level, limit to linear expressions, exponential expressions with integer exponents and quadratic expressions.
A-SSE.2 Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a
difference of squares that can be factored as (x2 – y2)(x2 + y2).)
Linear, Quadratic, and Exponential Models
Construct and compare linear and exponential models and solve problems.
F-LE.1 Distinguish between situations that can be modeled with linear functions and with exponential functions
a. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
F-LE.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a
relationship, or two input-output pairs (include reading these from a table).
F-LE.3 Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly,
quadratically, or (more generally) as a polynomial function.
(Note: At this level, limit to linear, exponential, and quadratic functions; general polynomial functions are not addressed.)
Interpret expressions for functions in terms of the situation they model.
F-LE.5 Interpret the parameters in a linear or exponential function in terms of a context.
Create equations that describe numbers or relationships.
A-CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with
labels and scales.
(Note: At this level, focus on linear, exponential and quadratic. Limit to situations that involve evaluating exponential functions for integer
inputs. )
Represent and solve equations and inequalities graphically.
A-REI.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often
forming a curve (which could be a line).
(Notes: At this level, focus on linear and exponential equations.)
Understand the concept of a function and use function notation.
F-IF.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a
context.
(Note: At this level, the focus is linear and exponential functions.)
Interpret functions that arise in applications in terms of the context.
F-IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the
quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals
where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and
periodicity. ★
(Note: At this level, focus on linear, exponential and quadratic functions; no end behavior or periodicity.)
F-IF.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if
the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an
appropriate domain for the function.
(Note: At this level, focus on linear and exponential functions.)
29
F-IF.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval.
Estimate the rate of change from a graph.
Note: At this level, focus on linear functions and exponential functions whose domain is a subset of the integers.
F-IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more
complicated cases.
Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline,
and amplitude.
(Note: At this level, for part e, focus on exponential functions only.)
F-IF.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions
such as y = (1.02)t, y = (0.97)t, y = (1.01)12t, y = (1.2)t/10, and classify them as representing exponential growth or decay.
F-IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by
verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the
larger maximum.
(Note: At this level, focus on linear, exponential, and quadratic functions.)
Build a function that models a relationship between two quantities.
F-BF.1 Write a function that describes a relationship between two quantities.
a. Determine an explicit expression, a recursive process, or steps for calculation from a context.
b. Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a
cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.
(Note: At this level, limit to addition or subtraction of constant to linear, exponential or quadratic functions or addition of linear functions to
linear or quadratic functions.)
F-BF.2 Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate
between the two forms.
(Note: At this level, formal recursive notation is not used. Instead, use of informal recursive notation (such as NEXT = NOW + 5 starting at
3) is intended.)
Build new functions from existing functions.
F-BF.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and
negative); find the value of k given the graphs. 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.
(Note: At this level, limit to vertical and horizontal translations of linear and exponential functions. Even and odd functions are not
addressed.)
Unit 5
Overview
This unit develops
student ability
to recognize
and represent
exponential growth
and decay patterns,
to express those
patterns in symbolic
forms, to solve
problems that involve
exponential change,
and to use properties
of exponents to
write
30expressions in
equivalent forms.
Topics Include:
Exponential growth
and decay functions,
Essential
Questions
The following
questions
should drive
students’
thinking
throughout
this unit.
Essential
Understandings
Unit Skills
Unit Facts &
Knowledge
By the end of this
unit, students will
understand that….
By the end of this
unit, students will be
able to…
By the end of the
unit, students
will know…
The equation represents
the initial value and the
rate of change for a
given situation.
Interpret what the
parts of an expression
(coefficients, constants)
tell you about a given
situation.
The purpose of
parameters, when
and how they are
used
How are the
basic patterns
of exponential
growth/decay
expressed with
symbolic rules?
The context of a
problem determines the
domain and range as
well as an appropriate
scale.
How can a
given situation
determine the
Difference between
linear and exponential
functions
Determine the
boundaries
Choose an appropriate
scale.
Construct an
exponential function
given a graph, situation
or table
That constant
percent rate
per unit interval
refers to the
common ratio or
the percent rate of
change
A constant
percent rate of
Pacing
18 days MS/
9 days HS
Lesson 1:
Exponential
Functions-form,
rules, graphs,
evaluation,
percent rate of
change.
(2 days MS/1
day HS)
Lesson 2:
Exponential
Growth
(patterns,
Resources:
Textbook: Core Plus Course 1
•
•
Unit 5 Lesson 1: Investigations 1-5
Unit 5 Lesson 2: Investigations 1-5
Textbook: Prentice Hall Algebra 1
•
Lessons 8-6 thru 8-8
Project-based Assessment:
Technology Applications:
•
•
Prentice Hall Algebra 1: Pg 436 Fitting Exponential Curves to Data
http://www.khanacademy.org/search?page_search_query=exponential+functions
Additional Activities:
Sample Performance Task(s):
31
Unit 6
Quadratic Functions
Common Core State Standards
* Standards marked with an asterisk are unifying standards and are revisited throughout the year, increasing in depth as the content
differs
Reason quantitatively and use units to solve problems.
N-Q.1 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently
in formulas; choose and interpret the scale and the origin in graphs and data displays
Interpret the structure of expressions.
A-SSE.1 Interpret expressions that represent a quantity in terms of its context.«
a. Interpret parts of an expression, such as terms, factors, and coefficients.
b. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r)n as the
product of P and a factor not depending on P.
(Note: At this level, limit to linear expressions, exponential expressions with integer exponents and quadratic expressions.)
A-SSE.2 Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a
difference of squares that can be factored as (x2 – y2)(x2 + y2).
Write expressions in equivalent forms to solve problems.
A-SSE.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the
expression.«
a.
Factor a quadratic expression to reveal the zeros of the function it defines.
(Note: At this level, the limit is quadratic expressions of the form ax2 + bx + c.)
Create equations that describe numbers or relationships.
A-CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with
labels and scales.
(Note: At this level, focus on linear, exponential and quadratic. Limit to situations that involve evaluating exponential functions for integer
inputs.)
Interpret functions that arise in applications in terms of the context.
F-IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the
quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals
where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and
periodicity. ★
(Note: At this level, focus on linear, exponential and quadratic functions; no end behavior or periodicity.)
F-IF.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if
the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an
appropriate domain for the function.
(Note: At this level, focus on linear and exponential functions.)
F-IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more
complicated cases.
32
Graph linear and quadratic functions and show intercepts, maxima, and minima.
F-IF.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph,
and interpret these in terms of a context.
33
Resources:
Textbook: Core Plus Course 1
• Unit 7 Lesson 1 Investigations 1-3
• Unit 7 Lesson 2 Investigations 1-2
• Unit 7 Lesson 3 Investigations 1-2
Textbook: Prentice Hall Algebra 1
• Sections 10-1, 10-2
• Sections 10-4, 10-5
• Section 10-7
• Section 10-9
Project-based Assessment:
Technology Applications:
•
•
•
•
•
Prentice Hall Algebra 1 pg. 535 Investigation roots (zeros)
http://mathbits.com/mathbits/TISection/Openpage.htm
http://www.khanacademy.org/math/algebra/#quadtratics
http://www.khanacademy.org/math/algebra/#algebra-functions
http://www.khanacademy.org/math/algebra/ck12-algebra-1/v/linear--quadratic--and-exponential-models
Additional Activities:
NCDPI Indicators for Algebra I (Notebook)- Picture Tells a Quadratic Story(B70-71); Maximum Gardens (A41); Getting to the Root of a
Number (A35/B116-118); How Do They Fit (A11/B98); Shuttle Launch (A43/B129); Quadratic Functions (A37/B119-121); Review Games
(E19-21); Song “Pop Goes the Weasel”; Patterns with Exponential Equations(A45/B130-131)
Goals and Objectives (G41-43);
Sample Performance Task(s):
Unit 7
34
Systems of Equations and Inequalities
Common Core State Standards
Resources:
Textbook: Core Plus Course 1
Unit 3 Lesson 2 Investigation 1 & 4
Textbook: Prentice Hall Algebra 1
•
•
•
•
Sections 7-1 thru 7-6
Section 11-2
Section 11-3
Pg. 360-361 Prentice Hall Algebra 1 Matrices and Solving Systems.
Project-based Assessment:
•
•
Car buying project (car vs. car)
Bumper Sticker Project
Technology Applications:
•
•
•
•
Pg. 385 Graphing Linear Inequalities
http://mathbits.com/mathbits/TISection/Openpage.htm
http://www.mathsteacher.com.au/year9/ch05_simult/01_sub/method.htm
http://www.khanacademy.org/math/algebra/#systems-of-eq-and-ineq?k
Additional Activities:
•
•
http://regentsprep.org/Regents/math/ALGEBRA/AE3/PracAlg.htm
http://www.oercommons.org/courses/hs-algebra-the-cycle-shop/view
NCDPI Indicators for Algebra I (Notebook)- Applications (G37-40); Calculator Wars (G39);
Sample Performance Task(s):
Cell Phone Challenge: (http://www.leadered.com/GSLs/new/Cell%20Phone%20Challenge%20Math%209-11.pdf)
The Cycle Shop: (http://www.oercommons.org/courses/hs-algebra-the-cycle-shop/view)
35
36
37
Unit 8
Two Variable Statistics
Common Core State Standards
* Standards marked with an asterisk are unifying standards and are revisited throughout the year, increasing in depth as the content
differs
Construct and compare linear and exponential models and solve problems.
F-LE.1 Distinguish between situations that can be modeled with linear functions and with exponential functions
a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors
over equal intervals.
b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.
c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
Summarize, represent, and interpret data on two categorical and quantitative variables.
S-ID.5 Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data
(including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.
S-ID.6 Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.
a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or
choose a function suggested by the context. Emphasize linear and exponential models.
b.
c.
Informally assess the fit of a function by plotting and analyzing residuals. (Note: At this level focus on linear models.)
Fit a linear function for a scatter plot that suggests a linear association.
Interpret linear models.
S-ID.7 Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.
S-ID.8 Compute (using technology) and interpret the correlation coefficient of a linear fit.
S-ID.9 Distinguish between correlation and causation
38
Unit 8
Overview
Essential
Questions
Essential
Understanding
s
Unit Skills
Unit
Facts &
Knowledg
e
By the end
By the
By the
Pacing
Friel, O'Connor, Mamer (2006) p. 119
The following
17 days MS/
39
Resources:
Textbook: Core Plus Course 1
•
•
Unit 8 Lesson 1 Investigations 1,2
Unit 3 Lesson 1 investigation 3
Textbook: Core Plus Course 2
•
•
Unit 4 Lesson 1 Investigations 1-2
Unit 4 Lesson 2 Investigations 1-4
Textbook: Prentice Hall Algebra 1
•
•
•
•
•
Lessons 1-9 (Graphing Data on Coordinate Plane)
Lesson 6-1 (Rate of Change, Slope)
Lesson 6-2 (Slope-Intercept Form)
Lesson 6-6 (Scatterplots, correlation, line of best fit)
P336-337 (Mathematically Inclined-rate of change, slope, graphing, line of best fit)
Textbook: Prentice Hall Algebra 2
•
Lesson 12-2 (Conditional Probability)
Textbook: Prentice Hall Pre-Algebra
•
Lesson 8-5 (Scatterplots)
Project-based Assessment:
Technology Applications:
http://www.khanacademy.org/math/statistics/#math/statistics?k
http://www.khanacademy.org/math/probability/#math/probability?k
Vocabulary:
•
Bivariate Quantitative: Scatterplot, Independent/Explanatory/Predictor variable, Dependent/Response variable, Linear, YIntercept, Slope, Rate of change, Linear model, Linear Regression, Residuals, Correlation Coefficient, Exponential model
•
Bivariate Categorical:Two Way Table, Relative frequency, Joint frequency, Marginal frequency/distribution, Conditional relative
frequencies/distribution.
Additional Activities:
•
•
40
State Names – This NCTM Illumination activity is an INTRODUCTION activity, to remind students of how to interpret a
frequency table. Using multiple representations, students analyze the frequency of letters that occur in the names of all 50
states.
Two-Way Frequency Table Discussion - In this short activity with the teacher, students will analyze categorical data that
involve variables such as pet preference and gender.
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
Two-Way Frequency Table Tutorial Video – This VCE tutorial video provides guidance in how to fill in a two-way
frequency table, but provides extension (not required) by showing students how to gather information using percentage
segmented bar charts. For an activity, have students create a survey to replicate this activity, and create a two-way frequency
table to illustrate the data.
Linear Functions: Are You Ready For The Real World?: Review of data on a coordinate grid and graphing a linear
function. This is a refresher activity.
Barbie Bungee - This Illuminations lesson, students model a bungee jump using a Barbie doll and rubber bands. The
distance to which the doll will fall is directly proportional to the number of rubber bands, so this context is used to examine linear
functions.
Murder In The First Degree – The Death of Mr. Spud: In this Texas Instruments activity, Students investigate
exponential functions in a real-world scenario. In this exponential functions in a real-world scenario lesson, students collect data
on the temperature of a cooling baked potato. Students enter the data into lists and graph the data using a scatter plot. Students
find an exponential function that models the data.
Shedding Light on the Subject: In this lesson, one of a multi-part unit from Illuminations, students explore the development
of a mathematical model for the decay of light passing through water, in a rich exploration of exponential models in context. They
use an interactive Java applet to explore related algebraic functions.
Have students reflect on their learning process as they interact with each graph type in their math journal/reflective blogs.
Slope: One Step At A Time: In this activity, the students will explore the concept of slope by exploring the steepness of
different staircases and graphing these on the TI-73/TI-84 calculator. (This is a review of slope)
Understanding and Applying Slope: Students develop the concept of slope with teacher aid. In this slope lesson,
students receive a detailed lesson about slope. Students find slope by graphing and given two ordered pairs. Students graph
linear equations and find slope given a word problem.
Personal Trainers: Working with Slope: A LearnNC activity highlighting slope. Students investigate slopes as they apply
it to the real world. In this lesson, students calculate the slope using the slope formula. They analyze the rate of change as it
applies to a work out regime, and graph their findings.
Slope and Popular Trends: Students review slope and use the Internet to research true data. They creatively display their
found research by a graph and explain their results to their classmates. They study a graph showing Manhattan's population from
1790 - 1980.
Cell Phenomena: Have students compare the rates of local cell phone plans.
How Tall?: In this NCTM Illuminiations lesson, by graphing different body measurements versus height and comparing their
correlation coefficient, students decide which body measurement is the best predictor.
Who’s Got The Fastest Legs?: Students use a stopwatch to collect information for a scatterplot. In this fastest legs
lessons, students collect data through the taking of measurements, create and find a median number. Students develop an
equation and answer questions based on their data.
The Demographics of Immigration: Using United States Census Data: PBS Lesson that has students work
together to analyze United States Census data on immigration. They compare and contrast the data and determine how
immigration numbers have changed over time. They calculate percentages and make their own conclusions about the data.
Effects of Urban Growth: Students explore the population growth in the United States and the impact it has had on our
society. In this social studies "population growth" lesson, students brainstorm in small groups to determine reasons for and
results of population growth. Surveys are designed, results are recorded, and related research takes place. Groups present a
final project.
Have students reflect on their learning process as they interact with each graph type in their math journal/reflective blogs
Sample Performance Task(s):
Authentic-Learning tasks (weekly practice):
•
Students propose 2 categorical variables that they believe may have a relationship, such as favorite subject and favorite sport.
Students create categories, collect data from at least 20 individuals, and display the data in a two-way frequency table.
Skill-Based tasks (daily practice):
•
41
What is the joint frequency of students who have chores and a curfew? Which marginal frequency is the largest?
•
•
•
•
The correlation coefficient of a given data set is 0.97. List three specific things this tells you about the data.
Give an example of a data set that has strong correlation but no causation and describe why this is so. Give an example of a
data set that has both strong correlation and causation and write a description of why this is so.
The following data shows the age and average daily energy requirements for male children and teens (1, 1110), (2, 1300), (5,
1800), (11, 2500), (14, 2800), (17, 3000). Create a graph and find a linear function to fit the data. Using your function, what is the
daily energy requirement for a male 15 years old? Would your model apply to an adult male? Explain.
Collect power bills and graph the cost of electricity compared to the number of kilowatt hours used. Find a function that models
the data and tell what the intercept and slope mean in the context of the problem.
Problem-Based tasks (daily practice):
•
•
•
•
Collect data that compares populations of countries with square miles. What trends emerge when we compare living in
geographically large countries with those that are highly populated?
Hypothesize the correlation between two sets of seemingly related data. Gather data to support or refute your hypothesis.
Find media artifacts that make claims of causation and evaluate them.
Collect data on forearm length and height in a class. Plot the data and estimate a linear function for the data. Compare and
discuss different student representations of the data and equations they discover. Could the equation(s) be used to estimate the
height for any person with a known forearm length? Why or why not? Create a poster of bivariate data with a linear relationship.
Describe for the class the meaning of the data, including the meaning of the slope and intercept in the context of the data.
Transdisciplinary Connections:
•
•
•
Science – video of The Golden Ratio within nature and the human body.
English Language Arts – Writing a summary report for final project.
Potential science or social science connections – data that are collected can pertain to almost any subject and should raise
questions relevant to that subject area
S.ID.5-6 Instructional Strategies:
•
•
•
In this cluster, the focus is that two categorical or two quantitative variables are being measured on the same subject.
In the categorical case, begin with two categories for each variable and represent them in a two-way table with the two values of
one variable defining the rows and the two values of the other variable defining the columns. (Extending the number of rows and
columns is easily done once students become comfortable with the 2 x 2 case.) The table entries are the joint frequencies of how
many subjects displayed the respective cross-classified values. Row totals and column totals constitute the marginal frequencies.
Dividing joint or marginal frequencies by the total number of subjects define relative frequencies (and percentages), respectively.
Conditional relative frequencies are determined by focusing on a specific row or column of the table. They are particularly useful
in determining any associations between the two variables.
In the numerical or quantitative case, display the paired data in a scatterplot. Note that although the two variables in general will
not have the same scale, e.g., total SAT versus grade-point average, it is best to begin with variables with the same scale such
as SAT Verbal and SAT Math. Fitting functions to such data will avoid difficulties such as interpretation of slope in the linear case
in which scales differ. Once students are comfortable with the same scale case, introducing different scales situations will be less
problematic.
S.ID.5 – 6: Common Misconceptions:
• Students may believe: that a 45 degree line in the scatterplot of two numerical variables always indicates a slope of 1 which is
the case only when the two variables have the same scaling. that residual plots in the quantitative case should show a pattern of
some sort. Just the opposite is the case.
S.ID.7-9 Instructional Strategies:
•
42
In this cluster, the key is that two quantitative variables are being measured on the same subject. The paired data should be
listed and then displayed in a scatterplot. If time is one of the variables, it usually goes on the horizontal axis. That which is being
predicted goes on the vertical; the predictor variable is on the horizontal axis.
•
Note that unlike a two-dimensional graph in mathematics, the scales of a scatterplot need not be the same. Even if they are
similar, such as SAT Math and SAT Verbal, they still need not have the same spacing. Thus, visual rendering of slope makes no
sense in most scatterplots. For example, a 45 degree line on a scatterplot need not mean a slope of 1.
•
Often the interpretation of the intercept (constant term) is not meaningful in the context of the data. For example, when the zero
point on the horizontal is of considerable distance from the values of the horizontal variable, or in some case has no meaning
such as for SAT variables.
•
Noting that a correlated relationship between two quantitative variables is not causal (unless the variables are in an experiment)
is a very important topic.
S.ID.7-9 Common Misconceptions:
43
•
Students may believe that a 45 degree line in the scatterplot of two numerical variables always indicates a slope of 1 which is
the case only when the two variables have the same scaling. Because the scaling for many real-world situation varies greatly,
students need to be give opportunity to compare graphs of differing scale.
•
Asking students questions such as “What would this graph look like with a different scale or using this scale?” is essential in
addressing this misconception.
•
That when two quantitative variables are related, i.e., correlated, that one causes the other to occur. Causation is not necessarily
the case. For example, at a theme park, the daily temperature and number of bottles of water sold are demonstrably correlated,
but an increase in the number of bottles of water sold does not cause the day’s temperature to rise or fall.
Teacher Common Core Curriculum Web Resource References
http://maccss.ncdpi.wikispaces.net/home (NC DPI COMMON CORE SITE FOR MATHEMATICS)
http://www.smarterbalanced.org/ (ASSESSMENTS FOR COMMON CORE)
http://www.buncombe.k12.nc.us/PAGE/22066 (Stefanie Buckner’s Core Plus Mathematics Curriculum Resources)
http://www.livebinders.com/play/play/187117 (Common Core lessons, activities, links)
http://www.uen.org/core/core.do?courseNum=5600 (Utah Common Core High School Math Resources)
http://www.uen.org/core/core.do?courseNum=5180 (Utah Common Core High School 8th Grade Resources)
www.classscape.org (assessment resources)
http://www.khanacademy.org/ (video instruction)
http://www.acps.k12.va.us/curriculum/design/sample-algebra-course.pdf (Alexandria City Algebra 1 curriculum
resources)
Cumulative Daily Pacing Summary
Unit 1: Data and Representation (One-Variable Statistics)……………………….10 days MS/ 5 days HS
MS DAYS 1-10
HS DAYS 1-5
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Lesson 1: Quantities, Estimates, Precision and Appropriate Measures – (2 days MS; 1 day HS)
Lesson 2: Representing Data: Boxplots, Histograms, Line Plots, Frequency Tables – (3 days MS, 1.5 days HS)
Lesson 3: Interpreting Data: Measures of Center, Measures of Spread, Effect of Outliers on data, Compare and Contrast Data Sets,
Choosing Appropriate measures of Center and Spread – (5 days MS; 2.5 days HS)
Unit 2: One-Variable Equations and Inequalities……………………….15 days MS/8 days HS
MS DAYS 11-25
HS DAYS 6-13
Lesson 1: Using Variables.
Exponents & Order of Operations.
(2 day MS, 1 day HS)
Lesson 2: Real Number
System (operations & properties) (2 days MS, 1 day HS)
Lesson 3: Solving equations and inequalities.1, 2 & multi- step; variables on both sides. (6 days MS, 3 days HS)
Lesson 4:
Properties of Exponents and radicals. (5 days MS, 2.5 days HS)
Unit 3: Two Variable Equations and Functions…………….13 days MS/6.5 days HS
MS DAYS 26-38
HS DAYS 14-20
Lesson 1:
Graphing data on x-y plane; relations & functions;
(3 days MS/1.5 days HS)
Lesson 2:
relating graphs to events/cause & effect
(2 days MS/1 day HS)
Lesson 3:
Connect rules, tables & graphs (linear, quad, exp, absolute)
(5 days MS/2.5 days HS)
Lesson 4:
Change over time-connect rate of change & slope
(3 days MS/1.5 days HS)
Unit 4: Linear Functions……………………………………………………………. 37 days MS/18.5 days HS
MS DAYS 39-75
HS DAYS 21-39
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Lesson 1: Rate of Change & Slope
(4 days MS/2 day HS)
Lesson 2: Given a table, graph, or data pairs, develop rule in y= mx + b and convert it to other forms; interpretation of linear function rule in
real world context.
(10 days MS/5 day HS)
Lesson 3: Modeling data & predicting
(6 days MS/3 days HS)
Lesson 4:
Model real world situations using linear inequalities and equations. Interpret the inequality or equation and its solution in real world
context.
(4 days MS/2 days HS)
Lesson 5:
Adding & Subtracting polynomials (2 days MS/1 day HS)
Lesson 6:
Multiply polynomials (include monomial ) & factor out GCF.
(2 days MS/1 day HS)
Lesson 7:
Factor by grouping.
(3 days MS/1.5 days HS)
Lesson 8:
Factor Trinomials including special cases.
(3 days MS/1.5 days HS
Assessment:
(3 days MS/1.5 days HS)
Notes:
Substantial time given to pacing of unit to allow for multiple investigations.
Unit 5: Exponential Functions……………………………..18 days MS/9 days HS
MS DAYS 76-93
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HS DAYS 22-30
Lesson 1:
Exponential Functions-form, rules, graphs, evaluation, percent rate of change.
(2 days MS/1 day HS)
Lesson 2:
Exponential Growth (patterns, NOW-NEXT, compound interest, exponent properties)
(6 days MS/3 days HS)
Lesson 3:
Exponential Decay (bouncing balls, half-life, exponent properties, NOW-NEXT, etc.)
(7 days MS/3 days HS)
Assessment:
(3 days MS/2 days HS)
Unit 6 Quadratic Functions……………………………18 Days MS/9 days HS
MS DAYS 94-111
HS DAYS 31-39
Lesson 1:
Exploring forms of quadratic graphs & functions. (min, max, symmetry, intercepts, domain restrictions)
(3 days MS/1.5 days HS)
Lesson 2:
Understanding zeros-solving quadratics by graphing, factoring & square roots.
(4 days MS/ 2 ys HS)
Lesson 3:
Modeling and solving problems using quadratic expressions. (quadratic formula, factoring)
(4 days MS/2 days HS)
Lesson 4:
Choosing the best function (linear, quadratic, exponential) to model data presented in various forms. Comparing rate of increase for linear,
quadratic, exponential functions.
(4 days MS/2 days HS)
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Assessment:
(3 days MS/1.5 days HS)
Unit 7: Systems of Equations & Inequalities……………………………22 days M; 11 days HS
MS DAYS 112-133
HS DAYS 40-50
Lesson 1:
Solving systems of linear equations & inequalities by graphing.
(4 days MS/ 2 days HS)
Lesson 2:
Solving systems of linear equations using substitution.
(3 days MS/ 2 days HS)
Lesson 3:
Solving systems of linear equations using elimination.
(2 days MS/ 1 day HS)
Lesson 4:
Applying systems of linear equations & inequalities.
(4 days MS/ 2 day HS)
Lesson 5:
Parallel & Perpendicular lines.
(2 days MS/ 1 days HS)
Lesson 6:
Distance, midpoint, and slope used in proving that a figure in a coordinate system is a special geometric shape.
(3 days MS/ 1.5 days HS)
Lesson 7 (Middle School only for 8th grade Math):
Pythagorean Theorem (1 days)
Assessment:
(3 days MS; 1.5 days HS)
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Unit 8: Two Variable Statistics………………………………………..17 days MS/8.5 days HS
MS DAYS 134-150
HS DAYS 51-60
Lesson 1:
How Often Does That Happen? (CCSS: S-ID.5)
2 Way Frequency Tables; Joint, conditional and marginal relative frequency; Conditional Probability.
Begin lesson by presenting data in a frequency table and assessing prior knowledge. Using contextual situations have students create
a two-way frequency table showing the relationship between two categorical variables such as height and weight or blood pressure and
incidence of heart disease. If possible, use digital technology to create two-way tables. Have students compare various tables and discuss
frequencies that are evident.
(3 days MS/1.5 days HS)
Lesson 2:
Making it Fit (CCSS: S-ID.6, S-ID.7)
Students may need review in plotting data on a coordinate grid and graphing a linear function. Students should be able to recognize
characteristics of linear and exponential functions and write an equation of a line given two points. Students will need to review how to
determine the slope and y-intercept of lines. Have students find and graph data sets from the internet and discuss the meaning of the
slopes and intercepts in context. Have students create scatter plots for data gathered, find a trend line and evaluate the fit by analyzing
residuals. Build on students' work with linear relationships in eighth grade and introduce the correlation coefficient. The focus here is on the
computation and interpretation of the correlation coefficient as a measure of how well the data fit the relationship.
(5 days MS/2.5 days HS)
Lesson 3:
Correlation Does Not Necessarily Mean Causation (S-ID.8, S-ID.9)
Students use graphing technology to enter data, calculate the regression equation, and interpret what the correlation coefficient is telling
about the data. Students must have prior knowledge of how to compute the correlation coefficient of a set of linearly related data using
technology and how to determine whether the correlation coefficient shows a weak positive, strong positive, weak negative, strong
negative, or no correlation. To provide scaffolding this unit includes lessons that review these concepts. Next, students will analyze data
sets and their graphs to discuss data that has correlation but no causation (height vs. foot length) or data that has correlation and causation
(number of M&Ms in a cup vs. the weight of the cup).
(5 days MS/2.5 days HS)
Assessment:
(3 days MS/1.5 days HS)
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