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Math 1 Toolkit*
Canyon Crest Academy
Student Name:___________________________________________
Year:________________
Teacher Name:___________________________________________
Room:_______________ * Use the toolkit during class work, homework,
and in studying for assessments
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1
Table of Contents Module 1:
Getting Ready Equations (solving
using balance analogy)
...........................................................................4
Distributive Property
............................................................................................................4
Expressions (algebraic)
.........................................................................................................5
Slope
.....................................................................................................................................5
Slope-‐Intercept
Form............................................................................................................6
Equations (meaning of
solution)...........................................................................................6
Equations (solving literal equations)
....................................................................................7
Inequalities (meaning of solution and
solving).....................................................................7
System of Equations (solving by
graphing)
...........................................................................8
Equations (writing one or a system
for a
situation)..............................................................8
Inequalities (writing for a situation)
.....................................................................................9
Module 2: Systems of Equations and
Inequalities Constraint
.............................................................................................................................9
Linear Inequality (graphing on a
half-‐plane).........................................................................10
Standard Form of a Linear Equation
(finding intercepts)
.....................................................10
System of Linear
Inequalities................................................................................................11
Feasible
Region.....................................................................................................................11
System of Equations (solve by
substitution).........................................................................12
System of Equations (solve by
elimination)..........................................................................12
System of Equations (determining the
number of solutions: 0, 1, or
infinite) .....................13
Module 3: Arithmetic and Geometric
Sequences Function
Notation.................................................................................................................13
Arithmetic
Sequences...........................................................................................................14
Recursive and Explicit Rules for
Arithmetic Sequences
........................................................14
Geometric
Sequences...........................................................................................................15
Graphing a Table of Data (inputs,
outputs, and axes)
.........................................................15
Recursive and Explicit Rules for
Geometric Sequences
........................................................16
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Module 4: Linear and Exponential
Functions Discrete and Continuous
Relationships (in context and
graphically) ...................................16
Linear and Exponential Relationships (in
context and graphically)
......................................17
Point-‐Slope Form (given slope and
a
point)..........................................................................17
Simple
Interest......................................................................................................................18
Compound
Interest...............................................................................................................18
Module 5: Features of Functions
Function Basics
.....................................................................................................................19
Increasing and Decreasing Intervals
.....................................................................................19
Domain and
Range................................................................................................................20
Maximum and Minimum
Values...........................................................................................20
Module 6: Congruence, Construction, and
Proof
Translation............................................................................................................................21
Reflection..............................................................................................................................21
Rotation
................................................................................................................................22
Pythagorean Theorem
..........................................................................................................22
Parallel and Perpendicular lines
...........................................................................................23
Quadrilaterals
.......................................................................................................................23
Symmetry (line and
rotational).............................................................................................24
Congruence...........................................................................................................................24
Similar Figures
......................................................................................................................25
Notation................................................................................................................................25
Triangle congruence
.............................................................................................................26
Bisector.................................................................................................................................26
Module 7: Connecting Algebra and
Geometry Distance
Formula..................................................................................................................27
Vertical Translation (of a function)
.......................................................................................27
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3
Module 8: Modeling Data
Histogram
.............................................................................................................................28
Measures of Central Tendency (mean,
median, and mode)
................................................28
Box and Whiskers Plot
..........................................................................................................29
Data Distribution (shape, center, and
spread)
.....................................................................29
Scatterplot
............................................................................................................................30
Two-‐Way Frequency Table
...................................................................................................30
Relative Frequency
Table......................................................................................................31
Line of Best Fit
......................................................................................................................31
Correlation Coefficient
.........................................................................................................32
Residual Plot
.........................................................................................................................32
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4
Definition An equals sign can
be thought of as a balancing
point between the two sides of
an equation. You may add
or remove equal amounts to both
sides of the equation to keep
it in balance.
Properties
• Perform the same math to both
sides of the equation to get
the variable alone on one side
• Solved equations look like this:
m = 5
or x = -‐3
or
2 = N
Example (diagram to equation)
Example (equation to solution)
Definition The distributive
property takes a multiplier on
the outside of parentheses and
multiplies each term inside the
parentheses.
Properties
• Terms in parentheses are separated
by addition or subtraction
• Each term in parentheses gets
multiplied by the outside multiplier
Example
Example
Equations: Balance Analogy
Distributive Property
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5
Definition An algebraic expression
has at least one variable and
usually numbers and math operations.
Properties
• Most expressions use addition,
subtraction, multiplication, and division
• Some formats are more typical
e.g. use “n + 3”
instead of “3 + n”
e.g. use “6y” instead
of “y6”
Examples
Nonexamples
Definition Slope can be
defined as:
• The steepness of a line
•
rise
run or
ΔyΔx
or vertical change
horizontal change
• two sides of a slope triangle
on a grid
•
y2− y
1
x2− x
1
or y1− y
2
x1− x
2
Properties
• Positive slope looks like this:
• Negative slope looks like this:
Example (two points on grid)
Example (two points – no grid)
Expressions: Algebraic
Slope
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Definition The slope-‐intercept
form of a linear equation
presents the slope and y-‐intercept
as values in the equation.
In the form, the variable “y”
must be alone on one side
of the equation.
Properties
• In y =mx + b the value of
“m” is the slope and the
value of “b” is the
y-‐intercept
• In y = a+ bx the value of
“b” is the slope and the
value of “a” is the
y-‐intercept
Example
Example
Definition To solve an
equation means to determine the
value(s) of the variable(s) that
make the equation true.
Properties
• To verify a solution substitute
the variable(s) with value(s) to
see if a true statement
results.
Example (one variable) Is
x = 7 a solution to 5+
x = 11?
Example (two variables) Is the
ordered pair (−3, 1) a
solution to 2x + y = −5 ?
Slope-‐Intercept Form
Equations: Solution Meaning
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Definition An equation is
solved for a variable when that
variable is isolated on one
side of the equation.
Properties
• Some equations contain several
variables but are solved for
only one of them
• The solution is one variable
written in terms of the other
variables and numbers
Example Solve 3x + 5y = −4
for y.
Example Solve A(B+C)= 5
for B.
Definition To solve an
inequality means to rewrite it
so that the variable is alone
on one side of the inequality
sign. Unlike most equations,
inequalities have many solutions.
Properties
• When solving an inequality, only
switch the inequality sign if
you multiply or divide both
sides with a negative number
• Solved inequalities look like this:
Example
Example
Equations (literal): Solving
Inequalities: Meaning, Solving
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Definition A system of
equations usually involves two
equations using the same two
variables. A solution to a
system of equations is usually
one point (x, y)
Properties
• Graph both equations on the same
grid to find their point of
intersection
• The point of intersection is the
solution
• Verify that this point makes
both equations true
Example Solve y = 3x −1,
y = −x + 3 by graphing.
Definition A situation
requires you to define variable(s)
and write equation(s) that, when
solved, answer the question.
Properties
• Clearly define what each variable
represents
• Verify the accuracy of your
solution(s).
• Be sure to answer the question
in a complete sentence.
Example (write and solve) Erik
paid $13.20 for two pounds of
dog food and a three dollar
dog toy. How much was
each pound of dog food?
Example (write*, do not solve)
A teacher has eight siblings.
The number of sisters is two
more than the number of
bothers. How many sisters and
brothers does the teacher have?
*use two variables
System of Equations: Solve by
Graphing
Equations: Writing for a situation
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Definition Some words imply an
inequality sign while other words
imply a math operation.
• greater than is > • greater
than or equal to is ≥ •
less than is < • less than
or equal to is ≤
Properties
• Addition can be implied by: more
than, increased, sum, etc.
• Subtraction can be implied by:
less than, decrease, difference, etc.
• Multiplication can be implied by:
product, times, etc.
• Division can be implied by:
quotient
Example (write an inequality) Six
more than the number of girls
is greater than 28.
Example (write an inequality) The
product of 5 and three less
than the cost of a shirt
is less than 47.
Definition A constraint is
a fact or statement that places
a limit on something.
Properties
• A constraint can be an equation
or an inequality
• A solution to a situation must
make all constraints true
Example A bakery makes dozens
of iced cookies and dozens of
plain cookies. Each dozen of
plain cookies uses one pound of
dough and each dozen of iced
cookies uses 0.7 pounds of
dough. The bakery has 100
pounds of dough available. The
ovens at the bakery can handle
140 dozens of cookies. Write
constraints for the dough and
the oven space.
Inequalities: Writing for a situation
Constraint
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Definition The graph of a
linear inequality is a shaded
section of a grid called a
half-‐plane. A half-‐plane (i.e. half
of the grid) is the part
of the grid cut in half
by the boundary line. A
boundary line separates the points
that satisfy a linear inequality
from the points that do not.
Properties
• The points on the boundary line
are included and the graph uses
a solid line if we have ≥
or ≤
• The points on the boundary line
are not included and the graph
uses a dashed line if we
have > or <
Example Graph y ≥ 3x −1 .
Example Graph y < −x + 3 .
Definition The standard
form of a linear equation looks
like this:
Ax +By = C
Properties
• This form it convenient for
finding the x-‐intercept and
y-‐intercept
• Using intercepts to
graph the
equation can be a quick method
Example Graph 2x + 3y = 12
using intercepts
Example
Graph 5x −2y = 10 using
intercepts
Standard Form
Linear Inequality (graphing)
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Definitions A system of
linear inequalities is two or
more inequalities using the same
variables. The solution
to the system is the shaded
area that makes both inequalities
true.
Properties
• To graph the system first graph
the boundary line for each
inequality
• Then decide which side of each
boundary line contains the solution
for that inequality
• Where the shaded areas overlap
is the solution
Example Solve y ≥ −3x
−1, 4x −2y
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Definition Solving a system
using substitution requires solving
one equation for a variable and
then substituting this amount into
the other equation. This new
equation should only have one
type of variable and can be
solved for this variable.
Properties
• Sometimes one equation is already
solved for a variable and this
makes a good equation to start
with
• Remember to solve for all
variables to create a complete
solution
Example Solve using the
Substitution Method:
2x − y = −8, y = 4x
Definition Solving a
system using elimination requires
combining the standard form equations
using addition, subtraction, and
possibly multiplication to create a
new equation. This new
equation should only have one
type of variable and can be
solved for this variable.
Properties
• Make sure equations are in
standard form
• The goal of combining equations
is to eliminate a variable type
• Remember to solve for all
variables to create a complete
solution
Example Solve using the
Elimination Method:
2x − y = −8, y + x = 14
System of Equations (solve by
elimination)
System of Equations (solve by
substitution)
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Definition A system of
equations may have no solution,
one solution, or an infinite
number of solutions.
Properties
• The no solution option has a
graph of two parallel lines
sharing no points
• The one solution option has two
lines that intersect at one
point
• The infinite solutions option has
two lines that graph as the
same line and share all points
Example (graph each option)
Definition Function notation
indicates the function name and
the variable used. As an
example: the notation f(x) indicates
a function named “f” using the
variable “x”.
Properties
• f(x) is often used in place
of y in an equation
• f(7) means use x = 7 in
the function f
Example
Example
System of Equations (number of
solutions)
Function Notation
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Definition An arithmetic
sequence has terms created by
choosing a starting amount and
adding the same value each time
to create the next term.
Properties
• Adjacent terms have a common
difference, commonly called “d”
• Some rules generate arithmetic
sequences
Example
Example
Definition A recursive
rule states a formula that
calculates the next term based
on the previous term. An
explicit rule states a formula
that calculates the term based
on the term number you want
(e.g. first term, fourth term,
etc.)
Properties
• Recall that consecutive terms of
an arithmetic sequence have a
common difference
Example Write both a recursive
rule and an explicit rule for
the following sequence: 11,
16, 21, 26, 31, 36, …
Recursive and Explicit Rules
(arithmetic)
Arithmetic Sequences
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Definition A geometric sequence
has terms created by choosing a
starting amount and multiplying the
same value each time to create
the next term.
Properties
• Adjacent terms have a common
ratio, commonly called “r”
• Some rules generate geometric
sequences
Example
Example
Definition A table of
data contains values on the
left side and right side.
The left side is the input
(often the variable “x”) and is
called the independent variable.
The right side is called the
output (often the variable “y”)
and is called the dependent
variable.
Properties
• The independent variable (the input
-‐ often representing time) graphs
on the horizontal axis
• The dependent variable (the output)
graphs on the vertical axis
• Scale the axes so the data
fits
Example (table)
Example (graph)
Geometric Sequences
Graphing a Table of Data
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Definition A recursive rule
states a formula that calculates
the next term based on the
previous term. An explicit
rule states a formula that
calculates the term based on
the term number you want (e.g.
first term, fourth term, etc.)
Properties
• Recall that consecutive terms of
a geometric sequence have a
common ratio
Example Write both a recursive
rule and an explicit rule for
the following sequence: 3, 6,
12, 24, 48, 96, …
Definition A discrete
relationship creates data that can
only take on certain values
(e.g. the number of cars can
only be a positive whole
number). A continuous relationship
creates data that can take on
any value, within reason (e.g.
the length of a snake)
Properties
• The graph of a discrete
relationship is a set of
unconnected points
• The graph of a continuous
relationship is a smooth line or
curve connecting the points
Example (discrete data)
Example (continuous data)
Discrete and Continuous Relationships
Recursive and Explicit Rules (geometric)
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Definition A linear relationship
shows equal differences over equal
intervals. An exponential
relationship shows equal factors over
equal intervals.
Properties
• A linear relationship creates values
with an adding pattern like an
arithmetic sequence
• An exponential relationship creates
values with a multiplying pattern
like a geometric sequence
Example (linear) Start = 5
d = 3 table: rule: graph:
Example (exponential) Start = 5
r = 3 table: rule: graph:
Definition The point-‐slope
form of a linear equation is
useful if you want to write
a linear equation and are given
the slope and one point.
Properties
• The point-‐slope form is:
y − y
1=m(x − x
1)
• The slope is the value “m”
and the
given point is (x
1, y1)
Example slope = point =
Example
slope = point =
Point-‐Slope Form
Linear and Exponential Relationships
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Definition Simple interest is
a calculation that only pays
interest on the original amount
of money (called the principal).
The formula is: i = Prt
Properties
• In the formula: o i = the
interest o P = the principal
(money) o r = the interest rate
as a
decimal o t = the time in
years
Example P = r =
t =
Example
P = r = t =
Definition Compound interest
is a calculation that pays
interest on the original amount
of money (called the principal)
and any additional interest already
earned. The formula is:
A = P(1+ r)
t
Properties • In the formula:
o A = the amount of money in
the account after t years
o P = the principal (original
amount of money)
o r = the annual interest rate
as a decimal
o t = the time in years
Example P = r =
t =
Example P = r =
t =
Compound Interest
Simple Interest
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Definition A function creates an
In-‐Out table with only one Out
for every In. This means
that any vertical line intersects
the graph of a function at
exactly one point (i.e. the
vertical line test).
Properties
• Functions can be represented as
a graph, table, set of ordered
pairs, a rule, or a situation
• A function cannot have the same
input paired with two different
outputs
Examples
Nonexamples
Definition A graph, as
read from left to right using
horizontal axis values, can have
sections of increasing intervals
and/or decreasing intervals.
Interval notation uses parentheses
when a value is approached but
not reached and brackets when a
value is included.
Properties
• In interval notation: o (2, ∞)
means from two up to
infinity o [−3, 4] means from
-‐3 to 4
including both values o (−∞, 6]
means from very
small up to and including 6
Example
Example
Function Basics
Increasing and Decreasing Intervals
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Definition The domain is
the set of numbers used as
inputs on the left side of
an In-‐Out table. The
range is the set of numbers
used as outputs on the right
side of an In-‐Out table.
Properties
• The domain is: o the horizontal
axis values o related to the
independent
variable • The range is:
o the vertical axis values o related
to the dependent
variable
Example (graph)
Example (rule and table)
Definition A graph, when
read from left to right, can
have high points and/or low
points.
Properties
• Maximum values (i.e. maxima) are
points that sit on a “hilltop”
of a graph
• Minimum values (i.e. minima) are
points that sit on a “valley
floor” of a graph
Example (graph)
Example (table)
Maximum and Minimum Values
Domain and Range
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Definition A translation is
a transformation that slides an
object a certain distance in a
given direction.
Properties
• The original object and its
translation have the same size
and shape
• Notation on the coordinate grid:
o e.g. (x, y)→ (x + 5, y − 3)
makes every x value 5 larger
and every y value 3 smaller
Example Graph a triangle and
translate it using
(x, y)→ (x −2, y + 3)
Definition A reflection is
a transformation that flips an
object over some line (called
the line of reflection).
Properties
• Each point of the original
figure (pre-‐image) has an image
that is the same distance from
the line of reflection but on
the opposite side
Example: Diagram (triangle)
pre-‐image
Reflection
Translation
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Definition A rotation is a
transformation that turns a figure
about a fixed point (called the
center of rotation).
Properties
• The figure and its rotation are
the same size and shape
• The angle of rotation measures
the amount of turn from the
figure and its rotation
• The turn can be clockwise or
counterclockwise
Example: Graph a triangle
Rotate the triangle 90!
counterclockwise about the origin
Definition In any right
triangle, the sum of the
squares of the lengths of the
legs is equal to the square
of the length of the
hypotenuse.
Properties
• This is only true for right
triangles • Diagram:
Example: Graph a right triangle
Determine the lengths of each
side:
Example: Graph two points
Determine the distance between the
points:
Pythagorean Theorem
Rotation
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Definitio PARALLEL LINES: Lines
whose graphs never intersect (i.e.
they never cross)
PERPENDICULAR LINES: Lines whose
graphs intersect and form 900
right angles
Properties
• Parallel lines have the same
slope
e.g. 6
and 6
• Perpendicular lines have slopes that
are negative reciprocals of each
other
e.g.
3
4 and −
4
3
Example of parallel Graph
line b : y = 2x − 3, line c : y
= 2x +1
Example of perpendicular
Graph line e : y = −3x + 4,
line f : y =
1
3x +1
Definition A quadrilateral
is any four-‐sided figure.
Diagrams for the common types:
• Trapezoid (one set of parallel
sides)
• Parallelogram (two sets of parallel
sides)
• Rectangle (a parallelogram with
four
right angles)
• Rhombus (a parallelogram with four
congruent sides)
• Square (a rhombus with four
right angles)
Parallel and Perpendicular Lines
Quadrilaterals
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24
Definition Line symmetry occurs
when two halves of a figure
mirror each other across a line
(called the line of symmetry).
Rotational symmetry occurs when a
figure has a center point
around which the figure is
rotated a number of degrees and
the figure looks the same.
Properties
• A line that reflects a figure
onto itself is called a line
of symmetry
• A figure that can be
carried onto
itself by a rotation is said
to have rotational symmetry
Example: Line symmetry The
capital letter H:
Example: Rotational symmetry The
capital letter H:
Definition Figures are
congruent if they have exactly
the same shape and the same
size.
Properties
• Congruent figures are duplicates of
one another
• Congruent polygons have corresponding
sides equal in length and
corresponding angles equal in degree
Example: Congruent triangles
ΔABC ≅ ΔDEF
Congruence
Symmetry
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25
Definition Figures are similar
if they have the same shape
but not the same size.
Properties
• Similar polygons have corresponding
angles that are congruent
• Similar polygons have the ratio
of their corresponding sides in
proportion
Example: Similar triangles
ΔABC ∼ ΔDEF
Definition Many figures in
geometry use special notation.
Notation
• Segment (a fixed length)
• Ray (continues in one direction)
Notation
• Line (continues in opposite
directions)
• Angle (the joining of two rays)
Notation
• Triangles (equal markings implies
congruence)
Notation
Similar Figures
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26
Definition Triangles are congruent
if they have the exact same
shape and size. With limited
information, two triangles can be
shown to be congruent.
Methods to show two triangles
congruent
• ASA (angle – side – angle)
• SAS (side – angle – side)
• SSS (side – side – side)
• AAS (angle – angle – side)
Definition A bisector
divides an object into two
equal parts.
Properties
• An angle bisector divides the
angle into two equal halves
• A perpendicular bisector of a
line segment intersects the segment
at a right angle and divides
the segment into two equal
halves
Example
Example
Bisector
Triangle Congruence
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27
Definition The distance formula
calculates the length of a
segment using the coordinates of
the endpoints. Here is the
formula:
d = (x
2− x
1)2 + (y
2− y
1)2
Properties
• The formula uses the endpoints
(x
1,y
1) and (x
2,y
2)
• The formula is a coordinate
geometry way of using the
Pythagorean Theorem
Example
Example
Definition A vertical
translation slides the graph of
a function vertically upward or
downward.
Properties
• A vertical translation of f(x)
is written as f(x) + k
• If k > 0, the graph
slides upward • If k < 0,
the graph slides downward • If
g(x) is a vertical translation
of f(x),
then the translation form of g(x)
is written as g(x) = f(x)
+k
Example
Example
Vertical Translation
Distance Formula
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28
Definition A histogram displays
data as bars along a horizontal
axis. The vertical axis is
scaled to show the frequencies
of each interval.
Properties
• The scale on the horizontal axis
shows the intervals of the data
• The bars are rectangular and
touch from interval to interval
Example: Ask 20 students for the
season of their birthday
Spring: 3/21-‐6/20 Summer:
6/21-‐9/20 Fall: 9/21-‐12/20
Winter: 12/21-‐3/20
Definition Measures of
central tendency give information
about a set of data. The
measures are mean, median, and
mode. If a set of data
is labeled X, the mean of
this set
is labeled X and is called
“X bar”.
The measures
• Mean (average): the sum of the
data set divided by the number
of data
• Median: the middle value when
the data are in numerical order
• Mode: the value that appears
most in the data set
Example (test scores) Data: 80,
75, 90, 95, 65, 65, 80,
85, 70, 100
Histogram
Measures of Central Tendency
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29
Definition A box and whiskers
plot uses five numbers to
display information about the data:
minimum, maximum, median (second
quartile), first quartile, and third
quartile. Using a number line
containing these numbers, place a
dot above the line for each
number and box all quartiles
and extend whiskers to the
minimum and maximum vales.
Properties
• The median separates the data
into two parts and gives the
second quartile
• The first quartile is the median
of the lower half of the
data
• The third quartile is the median
of the upper half of the
data
Example (test scores) Data: 80,
75, 90, 95, 65, 65, 80,
85, 70, 100
Example
Definition Data distribution
refers to the shape, center,
and spread of data. Data
modes can be uniform (no
obvious mode), unimodal (one main
peak), bimodal (two main peaks),
or multimodal (many peaks)
Data that are close together
have low variability. Sometimes
a center value best describes
the data set.
Properties
• Data skewed to one side leaves
a tail on that side (e.g.
skewed left has a tail on
the left)
• Outliers are values that stand
away from the set
• A normal distribution is unimodal
and symmetric
Diagrams Normal
Distribution
Skewed Right
Bimodal
Skewed Left
Data Distribution
Box and Whiskers Plot
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30
Definition A scatterplot is a
graph comparing two sets of
data.
Properties
• Do not connect the points in
a scatterplot
• The data can show a positive
correlation, negative correlation, or
no correlation
Examples of correlation
Positive
Negative
No Correlation
Definition A two-‐way
frequency table is a visual
representation of the possible
relationships between two sets of
categorical data. The categories
are labeled on the top and
left side with data in the
middle cells and totals in the
bottom row and right column.
Properties
• Entries in the body (middle) of
the table are called joint
frequencies
• Entries in the “totals” cells
for any row or column (except
the grand total cell) are
called marginal frequencies
Example Of 60 males, 21
wanted an SUV while the rest
wanted a sports car. Of
180 females, 45 wanted a sports
car.
Two-‐Way Frequency Table
Scatterplot
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31
Definition When a two-‐way
frequency table displays data as
percentages instead of frequency
counts, the table is called a
relative frequency table. The
inner values as a percent are
called conditional frequencies.
Properties
• A Relative Frequency of Row
Table uses row totals to
calculate percentages
• A Relative Frequency of Column
Table uses column totals to
calculate percentages
• A Relative Frequency Table uses
whole table totals to calculate
percentages
Example Of 60 males, 21
wanted an SUV while the rest
wanted a sports car. Of
180 females, 45 wanted a sports
car.
Definition A line of
best fit is a straight line
that best represents the data
of a scatterplot. This line
may pass through some or none
of the points.
Properties
• Lines of best fit are also
called trend lines
• Technology uses regression techniques
to determine a line of best
fit called a regression line
Example
Example
Line of Best Fit
Relative Frequency Table
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32
Definition The correlation coefficient
is a value that indicates how
well a model fits a particular
set of data. Positive values
describe positive association (similar
to slope) while negative values
describe negative association.
Properties
• The coefficient is designated by
the letter r
• The coefficient falls into the
range −1≤ r ≤1
• If r is close to 1 or
-‐1 the model is a good
linear fit while if r is
close to zero the model is
a weak linear fit
Examples
r = 0.832
r
= -‐ 0.0121
r = -‐ 0.9998
r =
0.54
Definition A residual plot
shows how far the actual data
are from the regression line.
Residual values can be positive,
negative, or zero.
Properties
• A point above the regression
line gives a positive residual
value
• A point below the regression
line gives a negative residual
value
Example of plot
Example of plot
Residual Plot
Correlation Coefficient
Math 1 Toolkit CoverMath 1 Toolkit (with text-no index)