unit 2 homework 7 algebra 2

Free Printable Math Worksheets for Algebra 2

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• Order of operations
• Evaluating expressions
• Simplifying algebraic expressions
• Multi-step equations
• Work word problems
• Distance-rate-time word problems
• Mixture word problems
• Absolute value equations
• Multi-step inequalities
• Compound inequalities
• Absolute value inequalities
• Discrete relations
• Continuous relations
• Evaluating and graphing functions
• Review of linear equations
• Graphing absolute value functions
• Graphing linear inequalities
• Direct and inverse variation
• Systems of two linear inequalities
• Systems of two equations
• Systems of two equations, word problems
• Points in three dimensions
• Systems of three equations, elimination
• Systems of three equations, substitution
• Basic matrix operations
• Matrix multiplication
• All matrix operations combined
• Matrix inverses
• Geometric transformations with matrices
• Operations with complex numbers
• Properties of complex numbers
• Rationalizing imaginary denominators
• Properties of parabolas
• Vertex form
• Graphing quadratic inequalities
• Factoring quadratic expressions
• Solving quadratic equations w/ square roots
• Solving quadratic equations by factoring
• Completing the square
• Solving equations by completing the square
• Solving equations with the quadratic formula
• The discriminant
• Naming and simple operations
• Factoring a sum/difference of cubes
• Factoring by grouping
• Factoring quadratic form
• Factoring using all techniques
• Factors and Zeros
• The Remainder Theorem
• Irrational and Imaginary Root Theorems
• Descartes' Rule of Signs
• More on factors, zeros, and dividing
• The Rational Root Theorem
• Polynomial equations
• Basic shape of graphs of polynomials
• Graphing polynomial functions
• The Binomial Theorem
• Evaluating functions
• Function operations
• Inverse functions
• Simplifying radicals
• Operations with radical expressions
• Dividing radical expressions
• Radicals and rational exponents
• Simplifying rational exponents
• Square root equations
• Rational exponent equations
• Graphing radicals
• Graphing & properties of parabolas
• Equations of parabolas
• Graphing & properties of circles
• Equations of circles
• Graphing & properties of ellipses
• Equations of ellipses
• Graphing & properties of hyperbolas
• Equations of hyperbolas
• Classifying conic sections
• Eccentricity
• Systems of quadratic equations
• Graphing simple rational functions
• Graphing general rational functions
• Simplifying rational expressions
• Multiplying / dividing rational expressions
• Adding / subtracting rational expressions
• Complex fractions
• Solving rational equations
• The meaning of logarithms
• Properties of logarithms
• The change of base formula
• Writing logs in terms of others
• Logarithmic equations
• Inverse functions and logarithms
• Exponential equations not requiring logarithms
• Exponential equations requiring logarithms
• Graphing logarithms
• Graphing exponential functions
• Discrete exponential growth and decay word problems
• Continuous exponential growth and decay word problems
• General sequences
• Arithmetic sequences
• Geometric sequences
• Comparing Arithmetic/Geometric Sequences
• General series
• Arithmetic series
• Arithmetic/Geometric Means w/ Sequences
• Finite geometric series
• Infinite geometric series
• Right triangle trig: Evaluating ratios
• Right triangle trig: Missing sides/angles
• Angles and angle measure
• Co-terminal angles and reference angles
• Arc length and sector area
• Trig ratios of general angles
• Exact trig ratios of important angles
• The Law of Sines
• The Law of Cosines
• Graphing trig functions
• Translating trig functions
• Angle Sum/Difference Identities
• Double-/Half-Angle Identities
• Sample spaces and The Counting Principle
• Independent and dependent events
• Mutualy exclusive events
• Permutations
• Combinations
• Permutations vs combinations
• Probability using permutations and combinations

Algebra 2 Worksheets with answer keys

Enjoy these free printable math worksheets . Each one has model problems worked out step by step, practice problems, as well as challenge questions at the sheets end. Plus each one comes with an answer key.

• Absolute Value Equations
• Simplify Imaginary Numbers
• Adding and Subtracting Complex Numbers
• Multiplying Complex Numbers
• Dividing Complex Numbers
• Dividing Complex Number (advanced)
• End of Unit, Review Sheet
• Exponential Growth (no answer key on this one, sorry)
• Compound Interest Worksheet #1 (no logs)
• Compound Interest Worksheet (logarithms required)
• Simplify Rational Exponents
• Solve Equations with Rational Exponents
• Solve Equations with variables in Exponents
• Factor by Grouping
• 1 to 1 functions
• Evaluating Functions
• Composition of Functions
• Inverse Functions
• Operations with Functions
• Functions Review Worksheet
• Product Rule of Logarithms
• Power Rule of Logarithms
• Quotient Rule of Logarithms
• Logarithmic Equations Worksheet
• Dividing Polynomials Worksheet
• Solve Quadratic Equations by Factoring
• Solve Quadratic Equations by Completing the Square
• Quadratic Formula Worksheets
• Quadratic formula Worksheet (real solutions)
• Quadratic Formula Worksheet (complex solutions)
• Quadratic Formula Worksheet (both real and complex solutions)
• Discriminant Worksheet
• Sum and Product of Roots
• Radical Equations
• Rationalizing the Denominator
• Simplify Rational Expressions Worksheet
• Dividing Rational Expressions
• Multiplying Rational Expressions
• Adding and Subtracting Rational Expressions (with like denominators)
• Adding and Subtracting Ratioal Expressions with Unlike Denominators
• Mixed Review on Rational Expressions

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Common Core Algebra II

The full experience and value of eMATHinstruction courses are achieved when units and lessons are followed in order.  Students learn skills in earlier units that they will then build upon later in the course.  Lessons can be used in isolation but are most effective when used in conjunction with the other lessons in this course. All Lesson/Homework files, and videos are available for free.  Other resources such as answer keys and more, are accessible with a paid membership.

Each month August through May we release new resources for this course that are accessible with a Teacher Plus membership. We release new resources in unit order throughout the school year.  You can see a list of our new releases by visiting our blog and selecting the most recent newsletter.

If you have an Algebra II membership, please note that you also have access to the paid resources for Algebra 2 with Trigonometry.

Standards Alignment – Powered by EdGate

• Table of Contents and Standards Alignment for Common Core Algebra II
• Unit 1 - Algebraic Essentials Review
• Unit 2 - Functions as the Cornerstones of Algebra II
• Unit 3 - Linear Functions, Equations, and Their Algebra
• Unit 4 - Exponential and Logarithmic Functions
• Unit 5 - Sequences and Series
• Unit 6 - Quadratic Functions and Their Algebra
• Unit 7 - Transformations of Functions
• Unit 8 - Radicals and the Quadratic Formula
• Unit 9 - Complex Numbers
• Unit 10 - Polynomial and Rational Functions
• Unit 11 - The Circular Functions
• Unit 12 - Probability
• Unit 13 - Statistics
• Statistical Simulators

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Using Normal Distributions for Experiment Analysis

14.1: Notice and Wonder: Some Distributions (5 minutes)

Routines and Materials

Instructional Routines

• Notice and Wonder

The purpose of this warm-up is to elicit the idea that these distributions are symmetric and approximately normal, which will be useful when students study the origin of the distributions in a later activity. While students may notice and wonder many things about these images, the symmetry, shape, and center are the important discussion points.

Display the images for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice and wonder with their partner, followed by a whole-class discussion.

Student Facing

What do you notice? What do you wonder?

Description: <p>Dot plot from negative 5 to 5 by point 6’s. difference in means. Beginning at negative 5, number of dots above each increment is 1, 1, 2, 3, 5, 7, 9, 11, 14, 16, 18, 19, 20, 20, 19, 18, 16, 14, 11, 9, 7, 5, 3,  2, 1, 1, 0.</p>

Description: <p>Dot plot from negative 6 to 6 by point 3’s, difference in means. Beginning at negative 6, number of dots above each increment is 1, 1, 2, 3, 5, 7, 9, 11, 13, 19, 22, 28, 32, 39, 42, 48, 51, 55, 55, 58, 55, 55, 51,  48, 42, 39, 32, 28, 22, 19, 13, 11, 7, 5, 3, 2, 1, 1.</p>

Student Response

For access, consult one of our IM Certified Partners .

Activity Synthesis

Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the images. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information.

If the symmetry, shape, and center of the distributions does not come up during the conversation, ask students to discuss this idea.

14.2: A Theoretical Experiment (15 minutes)

CCSS Standards

Building Towards

• MLR3: Clarify, Critique, Correct

In this lesson students begin to examine how their understanding of normal distributions can help to determine whether the results from an experiment are significant. For this activity, students recognize that the results from randomizing the groupings for data tend to be approximately normally distributed centered around 0. In the next activity, students will use the properties of normal distribution to determine the likelihood that the observed difference in means is due to the treatment rather than the groupings.

Students must reason abstractly and quantitatively (MP2) when the analysis leads them to look at the differences in means from possible groups created from the original experiment data.

To see what might be happening when we regroup data, consider an experiment that takes 12 subjects and divides them into 2 groups at random. The control group contains 6 subjects and the treatment group contains 6 subjects. To explore what's possible, assume the control group results in the data: 1, 3, 4, 6, 8, and 10. The treatment group results in the data: 2, 5, 7, 9, 11, and 12.

• Find the difference in means for the original groups by subtracting the control group mean from the treatment group mean.

With a smaller data set like this, we can actually consider all of the different arrangements of the data. There are 924 distinct ways to separate the 12 values into 2 groups of 6. The frequency table shows all the possible differences in means and how often they occur. Notice that a difference in means of 4.33 occurs 7 times and a difference of -4.33 also occurs 7 times. The dot plot shows the same information.

What proportion of possible groupings have a difference at least as great as the difference in means for the original groups? Explain or show your reasoning.

The proportion you calculate represents the probability that the original difference in means could be due to the groupings themselves. Based on the proportion you calculated for this situation, which description is most accurate? Explain your reasoning.

Because the proportion is so low, it is unlikely that the difference in means is due to the randomized groupings. This means that the difference in means is most likely caused by the treatment.

Because the proportion is not that low, it is still rather possible that the original difference in means is due to the random groupings. This means that there is not enough evidence to determine that the difference in means is likely caused by the treatment.

Anticipated Misconceptions

Students may be confused about how to interpret the frequency table with the \(\pm\) included. Remind students that the same data is displayed in the dot plot and to look particularly at the dots representing a difference of 6 and -6. The frequency of 1 in the table for the difference labeled \(\pm 6\) represents each of those differences.

The purpose of the discussion is to help students understand that redistributing the data from an experiment into groups at random can help determine whether the original results are significant or due to the way the subjects were grouped.

Select students to share their solutions and reasoning. 

Tell students that the use of statistics cannot  prove  that the results of an experiment are due to the treatment or groups. An analysis of this type can merely provide strong evidence that the treatment is the reason the original means are different or the analysis can tell us that there is not enough evidence to make that conclusion.

Consider asking:

• "Why do you think the distribution of differences in means is approximately symmetric? Why do you think the center of the distribution is 0?" (For every redistribution of data like 1, 2, 3 and 4, 5, 6 there is another that switches the two groups like 4, 5, 6 and 1, 2, 3. By a similar reasoning, for every difference of means like 3 there is another for the opposite like -3. So, the center of the distribution is 0.)
• "Suppose the original data from the control group was 1, 2, 3, 4, 5, 8 and the original data from the treatment group was 6, 7, 9, 10, 11, and 12. Based on the data alone, do you think there is a significant difference in the means from the group? The means are different by 5.33. Using the table, does the proportion of possible groupings with a difference of means that are at least 5.33 support your claim?" (I think there is a significant difference in the means. This is supported by using the table, since only 8 of the groupings area that extreme, so there is less than 1% chance ( \(\frac{8}{924} < 0.01\) ) that this grouping could have happened by chance.)

14.3: Simulating to Decide (15 minutes)

Building On

• MLR6: Three Reads

Required Materials

• Statistical technology

In this activity, students work through a case in which looking at all the possible combinations of regrouping experimental data is too much to consider. In many cases, even the simulations are too many to use to find exact proportions of mean differences to compare with the original difference in means. In these cases, we can use the approximately normal distribution of the simulations to estimate a proportion of redistributed data that would be at least as extreme as the original difference in means.

Students model with mathematics (MP4) when they use a normal distribution to model the simulated difference in means based on redistributing the data. Students must also select appropriate tools (MP5) to find the area under the normal curve for the analysis. 

Provide access to devices that can run GeoGebra or other statistical technology.

Researchers want to know the effect of captively raising birds on the weight of the birds. The researchers begin with 100 birds divided into 2 groups of 50 each. One group of 50 will be raised in captivity and the other 50 are tagged and released into the wild. After 5 years, all 100 birds are collected and weighed.

There are more than \(10^{29}\) different ways to regroup the 100 birds into groups of 50 again, so looking at all the combinations would be too time consuming to reproduce. In this case, we can run simulations to determine how the original difference in means compares to those from regrouping the data.

Attribution: tagged red knot birds calidris canutus rufa, by pixnio. Public Domain. pixnio. Source .

The original groups have a difference of means of 0.27 grams. Researchers run 1,000 simulations regrouping the data into 2 groups at random and record the differences in means for the groups in each simulation. The histogram shows the differences in means from the simulations.

Description: <p>Histogram, difference of means, from negative 0 point 4 to 0 point 4 by 0 point 1. Intervals have width 0 point 0 1. Beginning at 0 point 4 5, bars have height 1 and then get generally taller until 0 point 1 with height 45, and then generally get lower until 0 point 4.</p>

They determine that the mean of the differences of means from the simulations is 0.0021 grams and the standard deviation for the differences of means from the simulations is 0.112 grams.

What features of the distribution in the histogram let you know that modeling with a normal distribution is reasonable?

Model the simulations using a normal distribution with a mean of 0.0021 and a standard deviation of 0.112. What is the area under this normal curve that is more extreme than 0.27?

How can this area be used to compare the difference of means from the simulations to the difference of means from the original groups?

Based on the area under the normal curve, is there evidence that the original difference in means is likely due to where the birds spent the 5 years? Explain your reasoning.

Are you ready for more?

Suppose we decide that if the probability of observing our difference or a more extreme difference happens less than 5% of the time in simulations we will conclude that it was captivity that caused the difference, and if the probability is greater than 5% we will not draw any conclusions. In what ways could the conclusion we make (or decide not to make) be wrong?

The purpose of this discussion is to show that areas under normal curves can be used to help decide whether the difference in means from an experiment are significant.

Select students to share their solutions and reasoning. To promote discussion, consider asking: "In this problem, a 1.5% chance of the difference in means from the original groups happening by random chance seems low enough to conclude that the difference in means is likely due to the treatment rather than the way the subjects were grouped. What percentage might make you change your opinion on whether the groupings were the cause of the original difference in means?"

Tell students that standard cutoffs used in statistics are 1%, 5%, or 10% before the results are called into question.

Ask students, "Why do you think it would be important to set your expectations for a cutoff  before  doing the experiment?" (Setting the expectations can help decide how many subjects to include in the study and does not unfairly bias the results. Some experimenters may continue to run new simulations until their results become significant.)

Lesson Synthesis

The purpose of the discussion is to help students understand what they can do to analyze the significance of results from an experiment. To promote discussion, consider asking:

• "What can you do to determine if the original difference in means is significant if the number of subjects in the experiment are large and finding all the different combinations of data is difficult?" (Use simulations to determine differences in mean for many possible groupings, then find the mean and standard deviation of those values to create a normal distribution. The normal distribution can then be used to estimate the likelihood of the original difference in means happening by chance.)
• "Why is a normal distribution a good model for differences of means from simulated regrouping of data?" (The differences in means should produce a distribution that is approximately symmetric and bell-shaped which is a type of distribution that is usually represented fairly well by a normal distribution.)
• "What does the total area on either end of the normal distribution that is more extreme than the original difference in means represent?" (The area represents the approximate proportion of the regroupings that have a difference of means at least as large as the one from the original groups.)
• "If you want the analysis to show that there is evidence that the difference in means from the experiment is due to the treatment, would you want the areas on either end of the normal distribution to be large or small? Explain your reasoning." (Small, because the area represents the likelihood that the original difference in means is due to the way the subjects were grouped. If that likelihood is small, then the difference is likely due to another effect like the treatment.)

14.4: Cool-down - The Science Experiment (5 minutes)

Student lesson summary.

To analyze the significance of the data collected from an experiment, a randomization distribution can be used. In some cases, where the number of subjects is small, all of the possible ways to regroup the data can be used to compare the original difference in means. When the difference in means is more extreme than most of the differences seen from the randomized regroupings (usually more than 90%, 95%, or 99% depending on the situation), we can say that we have evidence that the difference in means is due to the treatment rather than the way the subjects were originally grouped.

The more subjects included in the experiment, the greater number of possible regroupings. For example, 14 subjects divided into 2 groups of 7 can have their data redistributed into groups 3,432 different ways. When there are 60 subjects divided into 2 groups of 30, there are more than 118 quadrillion ( \(1.18 \times 10^{17}\) ) different ways to redistribute the data into groups of 30. This large number of ways to regroup the data makes looking at the distribution of every possible regrouping difficult.

In these cases, we often do a simulation and redistribute the data many times to get a sense of the true distribution of all possibilities. For example, this histogram shows the difference of means for 1,000 simulations of redistributing 60 data values into 2 groups of 30 each.

The simulations should produce approximately normal distributions with a center near 0. This allows us to use our understanding of normal distributions to estimate the proportion of regroupings that are at least as extreme as the original difference in means from the experiment. When the proportion is small enough, we should conclude that there is enough evidence to say that the difference in means from the original groups is most likely due to the treatment.

For example, using the values from the histogram, the mean is 0.04 and the standard deviation is 9.07. That provides enough information to create a normal distribution that models the data. In the image, we see the normal distribution and the regions for which the difference of means might be significant since there is only a 5% chance of the original difference in means being in the shaded region (less than -17.74 or greater than 17.82).

Description: <p>Normal distribution graph, origin O. Horizontal axis from negative 25 to 25 by 5’s, difference of means. Vertical axis from 0 to 0 point 0 3 by 0 point 0 1’s. The normal distribution shows a mean of 0 point 0 4 with a standard deviation of 9 point 0 7. The height of the graph is approximately 0 point 0 2 5. There are shaded regions of the graph less than negative 17 point 7 4 and greater than 17 point 8 2.</p>

If the original difference in means is something like 20, then we can conclude that there is evidence to show that the difference in means is due to the treatment. On the other hand, if the original difference in means is something like 10, then we should say that there is not enough evidence to conclude that the difference in means is due to the treatment, since there is still a good chance that the difference in means is due to the way the subjects were originally grouped.

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Expressions (Math 7 Curriculum - Unit 2) | All Things Algebra®

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Description

This Expressions Unit Bundle includes guided notes, homework assignments, three quizzes, a study guide, and a unit test that cover the following topics:

• Order of Operations

• Translating Expressions

• Evaluating Expressions

• Combining Like Terms

• The Distributive Property

• Simplifying Expressions (Distribute and Combine Like Terms)

• Factoring Expressions/Prime Expressions

• Adding and Subtracting Expressions

• Multiply Monomials (Product Rule)

• Divide Monomials (Quotient Rule)

• Powers of Monomials (Power Rule)

• Properties

ADDITIONAL COMPONENTS INCLUDED:

(1) Links to Instructional Videos: Links to videos of each lesson in the unit are included. Videos were created by fellow teachers for their students using the guided notes and shared in March 2020 when schools closed with no notice.  Please watch through first before sharing with your students. Many teachers still use these in emergency substitute situations. (2) Editable Assessments: Editable versions of each quiz and the unit test are included. PowerPoint is required to edit these files. Individual problems can be changed to create multiple versions of the assessment. The layout of the assessment itself is not editable. If your Equation Editor is incompatible with mine (I use MathType), simply delete my equation and insert your own.

(3) Google Slides Version of the PDF: The second page of the Video links document contains a link to a Google Slides version of the PDF. Each page is set to the background in Google Slides. There are no text boxes;  this is the PDF in Google Slides.  I am unable to do text boxes at this time but hope this saves you a step if you wish to use it in Slides instead! 

Note: This unit is similar to my Pre-Algebra Curriculum in that it contains similar topics. However, all material has been rewritten to ensure there is no duplication. The biggest differences include no square roots, cube roots, or absolute value in the order of operations and evaluating expressions, and no negative exponents in the monomials. This unit is written for Math 7 or advanced Math 6 students, whereas Pre-Algebra is written for Math 8 or advanced Math 7 students.

This resource is included in the following bundle(s):

Math 7 Curriculum

More Math 7 Units:

Unit 1 – Number Sense

Unit 3 – Equations and Inequalities

Unit 4 – Ratios, Proportions, and Percents

Unit 5 – Functions and Graphing

Unit 6 – Geometry

Unit 7 – Measurement: Area and Volume

Unit 8 – Probability and Statistics

LICENSING TERMS: This purchase includes a license for one teacher only for personal use in their classroom. Licenses are non-transferable , meaning they can not be passed from one teacher to another. No part of this resource is to be shared with colleagues or used by an entire grade level, school, or district without purchasing the proper number of licenses. If you are a coach, principal, or district interested in transferable licenses to accommodate yearly staff changes, please contact me for a quote at [email protected].

COPYRIGHT TERMS: This resource may not be uploaded to the internet in any form, including classroom/personal websites or network drives, unless the site is password protected and can only be accessed by students.

© All Things Algebra (Gina Wilson), 2012-present

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All Things Algebra®

Math 7 Unit 2: Expressions

This unit includes 56 pages of guided notes, homework assignments, three quizzes, a study guide, and a unit test that cover the topics listed in the description below.

• Description
• Additional Information
• What Educators Are Saying

This unit contains the following topics:

• Order of Operations • Translating Expressions • Evaluating Expressions • Combining Like Terms • The Distributive Property • Simplifying Expressions (Distribute and Combine Like Terms) • Factoring Expressions/Prime Expressions • Adding and Subtracting Expressions • Multiply Monomials (Product Rule) • Divide Monomials (Quotient Rule) • Powers of Monomials (Power Rule) • Properties

This unit does not contain activities.

This is the guided notes, homework assignments, quizzes, study guide, and unit test only.  For suggested activities to go with this unit, check out the ATA Activity Alignment Guides .

This resource is included in the following bundle(s):

Math 7 Curriculum

License Terms:

This purchase includes a single non-transferable license, meaning it is for one teacher only for personal use in their classroom and can not be passed from one teacher to another.  No part of this resource is to be shared with colleagues or used by an entire grade level, school, or district without purchasing the proper number of licenses.  A t ransferable license is not available for this resource.

Copyright Terms:

No part of this resource may be uploaded to the internet in any form, including classroom/personal websites or network drives, unless the site is password protected and can only be accessed by students.

What standards is this curriculum aligned to?

What format are the files in, will i have access to materials if they are updated, are answer keys included, are videos included.

Thank goodness this curriculum is sold by the unit! I was still in the beginning stages of 7th/8th grade math blended course when I purchased this, but the curriculum I use in alignment with ATA doesn't require my classes to go in as much depth as ATA's pre-algebra equivalent of this unit. While I used the pre-algebra expressions unit with success last year, my students this year aren't quite as developed mathematically. This unit was perfect to use in place of ATA's pre-algebra expressions to meet my students where they are at.

-JESSICA M.

LOVE every resource from ATA! The notes pages are super kid-friendly. I also really like how there's everything you need for the Unit! Not just worksheets, but fun little activities, etc. Thank you for your attention to detail and thoroughness of your resources!

-JENNIFER B.

Guided notes have made my class so much smoother! Students are not rushing to copy everything down and those that struggled to write notes are now filling out the packets. Activities are a great way to practice in a fun way. Love the riddles and mazes.

-MELISSA R.

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