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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
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
2
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
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
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
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
6
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
7
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
8
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
9
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
10
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)
11
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
12
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)
13
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
14
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
15
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
16
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)
17
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
18
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
19
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
20
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
21
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
22
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
23
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
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
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
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
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
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
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
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
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
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)