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1: Algebra - Mathematics

1: Algebra - Mathematics


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  • 1.1: Dividing Polynomials
    We are familiar with the long division algorithm for ordinary arithmetic. We begin by dividing into the digits of the dividend that have the greatest place value. We divide, multiply, subtract, include the digit in the next place value position,. Division of polynomials that contain more than one term has similarities to long division of whole numbers. We can write a polynomial dividend as the product of the divisor and the quotient added to the remainder.
  • 1.2: Zeros of Polynomial Functions
    In the last section, we learned how to divide polynomials. We can now use polynomial division to evaluate polynomials using the Remainder Theorem. If the polynomial is divided by (x–k), the remainder may be found quickly by evaluating the polynomial function at (k), that is, (f(k)).
  • 1.3: Rational Functions
    In the last few sections, we have worked with polynomial functions, which are functions with non-negative integers for exponents. In this section, we explore rational functions, which have variables in the denominator.
  • 1.4: Logarithmic Properties
    Recall that the logarithmic and exponential functions “undo” each other. This means that logarithms have similar properties to exponents. Some important properties of logarithms are given here.

Algebra Practice Test 1

The following proportion may be written: 1/p=x/5. Solving for the variable, x, gives xp = 5, where x=5/p. So, Lynn can type 5/p pages, in 5 minutes.

Sally can paint 1/4 of the house in 1 hour. John can paint 1/6 of the same house in 1 hour. In order to determine how long it will take them to paint the house, when working together, the following equation may be written: 1/4 x+1/6 x=1. Solving for x gives 5/12 x=1, where x= 2.4 hours, or 2 hours, 24 minutes.

Sale Price = $450 – 0.15($450) = $382.50, Employee Price = $382.50 – 0.2($382.50) = $306

$12,590 = Original Price – 0.2(Original Price) = 0.8(Original Price), Original Price = $12,590/0.8 = $15,737.50

In order to solve for A, both sides of the equation may first be multiplied by 3. This is written as 3( 2A /3)=3(8+4A) or 2A=24+12A. Subtraction of 12A from both sides of the equation gives -10A=24. Division by -10 gives A = -2.4.

Three equations may initially be written to represent the given information. Since the sum of the three ages is 41, we may write, l + s + j = 41, where l represents Leah’s age, s represents Sue’s age, and j represents John’s age. We also know that Leah is 6 years older than Sue, so we may write the equation, l = s + 6. Since John is 5 years older than Leah, we may also write the equation, j = l + 5. The expression for l, or s + 6, may be substituted into the equation, j = l + 5, giving j = s + 6 + 5, or j = s + 11. Now, the expressions for l and j may be substituted into the equation, representing the sum of their ages. Doing so gives: s + 6 + s + s + 11 = 41, or 3s = 24, where s = 8. Thus, Sue is 8 years old.

Simple interest is represented by the formula, I = Prt, where P represents the principal amount, r represents the interest rate, and t represents the time. Substituting $4,000 for P, 0.06 for r, and 5 for t gives I = (4000)(0.06)(5), or I = 1,200. So, he will receive $1,200 in interest.

$670 = Cost + 0.35(Cost) = 1.35(Cost), Cost = $670/1.35 = $496.30

The amount of taxes is equal to $55*0.003, or .165. Rounding to the nearest cent gives 17 cents.

The GPA may be calculated by writing the expression, ((3*2)+(4*3)+(2*4)+(3*3)+(4*1))/13, which equals 3, or 3.0.

From 8:15 A.M. to 4:15 P.M., he gets paid $10 per hour, with the total amount paid represented by the equation, $10*8=$80. From 4:15 P.M. to 10:30 P.M., he gets paid $15 per hour, with the total amount paid represented by the equation, $15*6.25=$93.75. The sum of $80 and $93.75 is $173.75, so he was paid $173.75 for 14.25 hours of work.

If she removes 13 jellybeans from her pocket, she will have 3 jellybeans left, with each color represented. If she removes only 12 jellybeans, green or blue may not be represented.

The value of z may be determined by dividing both sides of the equation, r=5z, by 5. Doing so gives r/5=z. Substituting r/5 for the variable, z, in the equation, 15z=3y, gives 15(r/5)=3y. Solving for y gives r = y.

50 cents is half of one dollar, thus the ratio is written as half of 300, or 150, to x. The equation representing this situation is 300/x*1/2=150/x.

The following proportion may be used to determine how much Lee will make next week: 22/132=15/x. Solving for x gives x = 90. Thus, she will make $90 next week, if she works 15 hours.

The given equation should be solved for x. Doing so gives x = 6. Substituting the x-value of 6 into the expression, 5x + 3, gives 5(6) + 3, or 33.

The amount you will pay for the book may be represented by the expression, 80+(80*0.0825). Thus, you will pay $86.60 for the book. The change you will receive is equal to the difference of $100 and $86.60, or $13.40.

The amount you have paid for the car may be written as $3,000 + 6($225), which equals $4,350.

You will need 40 packs of pens and 3 sets of staplers. Thus, the total cost may be represented by the expression, 40(2.35) + 3(12.95). The total cost is $132.85.

Substituting 3 for y gives 3 3 (3 3 -3), which equals 27(27 – 3), or 27(24). Thus, the expression equals 648.


1: Algebra - Mathematics

Driven by student discourse, IM Certified™ curricula are rich, engaging core programs built around focus, coherence, and rigor. The curricula are trusted, expert-authored materials developed to equip all students to thrive in mathematics.

About the curriculum

Spark discussion, perseverance, and enjoyment of mathematics.

IM Algebra 1, Geometry, and Algebra 2 are problem-based core curricula rooted in content and practice standards to foster learning and achievement for all. Students learn by doing math, solving problems in mathematical and real-world contexts, and constructing arguments using precise language. Teachers can shift their instruction and facilitate student learning with high-leverage routines that guide them in understanding and making connections between concepts and procedures.

Intentional lesson design that promotes mathematical growth.

IM 9-12 Math, authored by Illustrative Mathematics, is highly rated by EdReports for meeting all expectations across all three review gateways. Read the report.

The purpose and intended use of the Algebra 1 Supports Course.

Students who struggle in Algebra 1 are more likely to struggle in subsequent math courses and experience more adverse outcomes. The Algebra 1 Extra Support Materials are designed to help students who need additional support in their Algebra 1 course. Each Algebra 1 Extra Support Materials lesson is associated with a lesson in the Algebra 1 course. The intention is that students experience each Algebra 1 Extra Support Materials lesson before its associated Algebra 1 lesson. The Algebra 1 Extra Support Materials lesson helps students learn or remember a skill or concept that is needed to access and find success with the associated Algebra 1 lesson.


After STAAR Fun

I’m Texas teacher and your page is amazing. I’m going to implement it in part this year and in full next year. Thanks for sharing.

Thanks so much, Kiara! I’m glad it’s been useful so far. Let me know when you find areas for improvement!

My name is Andy and I’m an Algebra/French Teacher in Alaska. Your website and course structure is well-organized and most importantly ENGAGING. Would you mind if used some of the pieces in creating my own school webpages and specific pieces as templates?

Thank you for the kind words! I really appreciate it. You’re more than welcome to use anything on the site. I’m happy to share! My only ask is that everything remains free. Thanks for asking!

Thank you for sharing all of your resources! Sometimes I feel like I’m searching all over the place for different resources, so I love that you have warm-ups and concepts all in one place. Keep up the awesome work!

Thank you for the encouragement, Rachel! I’m glad the site has been useful. Feel free to send questions or feedback anytime!

First of all, EXCELLENT website and curriculum. I greatly appreciate your dedication and hardwork you have put into all your lesson plans. I stumbled across it the end of last year while researching Dan Meyer’s 3 Act Tasks. I have been using your geometry curriculum to great effect. My kids have been greatly engaged and seem to be truly learning. I have also applaud your school for grading based on knowledge, not just test taking skills. I have applied the same kind of grading in my own classroom! I really want to thank you for helping me teach better lesson plans, especially as a newer teacher.

I do wonder, do you guys not teach any kind of piecewise functions in Algebra 1? Or go over absolute value functions/equations/inequalities? I am also teaching a remedial Algebra 1 class and have been looking for good lesson plans to help them have a better understanding.

Thanks a bunch,
Kevin Grogaard

Thank you so much for the kind words! I really appreciate it. I’m happy to hear the site has been useful. Feel free to send feedback for improvement anytime as well!

In Texas, Algebra 1 does not include piecewise functions or absolute value functions. I believe we cover those things in Algebra 2. As for remedial Algebra 1, I’ve taught that many times, and I use most of the lessons on this site. The only difference is that I tend to spend a little more time on each concept to help solidify. This may lead to some concepts not getting enough time, so it’s important to spend the most time on the most heavily tested concepts in your state.

Hope that helps. Thanks again!

Dane, your site is absolutely terrific. The TEKS unfortunately encourage wide and shallow learning but you’ve done an masterful job in turning your curriculum into meaningful learning that encourage deep understandings. Thank you for your willingness to share. My colleague is teaching Algebra 2 next year and is interested in this style of teaching. Do you have any recommendations for places to look for Algebra 2 activities that incorporate the same style as you?

Thank you for the kind words! I really appreciate it!

For Algebra 2, New Visions has great content and a lot of curriculum already built out. Some blogs that have Algebra 2 content are Julie Reulbach, Dylan Kane, Sam Shah, and Jonathan Claydon.

Hello,
Thank you for this great resource. I noticed your answer key folder is empty except for day 126-138. Is there a place I can find them?
Thanks !

Thank you for the kind words! Hmm that’s strange. It may be a glitch. I just checked, and all the answer keys are showing up now. Let me know if you check again and still can’t see the rest.

Dane,
I see it now. It was a glitch on my end. Thanks again!

I’m teaching Algebra 1 this year and love your resources and concept quizzes. Out of curiosity, I noticed there is a large jump from the day 10 quiz to day 23 quiz. Is there any particular reason why the some of the content in between those days is not assessed?

Thanks for the question and kind words! Here’s a detailed breakdown about why some concepts are quizzed and some aren’t. For the un-quizzed concepts between day 11 and 17, I decided that they were important, but not crucial enough to be included in the top 20ish concepts for Algebra 1. Solving One-Variable Inequalities was the toughest one to keep out. My reasoning for that one is that I’ve quizzed it in the past, and I had trouble finding ways to assess the inequality aspect at a high level. When I was grading, the solving equations portion of it ended up being the main focus. Therefore, I kept solving equations as a quizzed concept, but just decided to only spend a decent chunk of days teaching inequalities without formally assessing it.

Let me know if that helps!

First off…I LOVE your stuff. What you do with your concepts and instructional strategies are fantastic. I love the pacing and the outline of all the activities that you do. Last year my geometry PLC and I followed your timelines and addressed the same topics that you have listed. We pretty much followed it full-blown, with our twist on certain thing of course, and I must say it was a fantastic year. The kids learned so much….on a conceptual level, and retained it! We want to do the same thing this coming school year with Algebra. Couple questions….

Do you have other math teachers at your school that teach the same classes? Are you on a PLC? If so, do they do things the same way as you? Do you have to convince other teachers to do it the way you are? I’m curious what other math teachers in your building think about all this….We will have 7 different teachers teaching Algebra 1 next year and we are all supposed to use the same timelines, same assessments, same essential standards, etc. You can imagine this is difficult…

Do you feel like how you have algebra 1 right now is pretty solid? With the concepts and the pacing?

Also, do I noticed at the end of the year in Geometry you had probability lessons but no links.
We are starting the year next year in Algebra with a probability/statistics unit. Wondering if you had anything to share as far as probability goes and/or statistics? Keep up the great work! You inspire me.

Thank you so much for the kind words! I’m really happy to hear that the curriculum has been useful.

I just sent you an email to answer your questions. Let me know if you received it. Looking forward to continuing the conversation.

Hi Dane,
I started the summer with a general search of SBG and came across your blog. I was overwhelmed at first, but have printed off each topic on SBG and feel like I’ve gotten a good understanding of what you are doing.

I teach Algebra 1 and am excited to use this new idea in the fall.
I am curious what your recent reply was to Jake concerning your feeling on how solid you felt the Algebra concepts and pacing were from last year.
Also, would there be any changes you would implement moving forward(retakes, faux quizzes, student analysis, etc)?

I’m excited to hear that you’re diving into SBG! Feel free to ask questions any time.

As for the Algebra 1 curriculum, I feel very confident about the pacing and sequencing of the curriculum. Based on what I currently know about curriculum design, it’s as complete as I can make it.

Moving forward, I’m thinking about requiring all kids to retake their quiz the first time (if they don’t make a 100) and not necessarily require an analysis handout to be completed. However, I will basically force them to complete the analysis without technically requiring it because it’s so crucial to their learning. I do want kids to know that I want all of them to retake so they can witness their own growth. I don’t want to hinder the growth opportunity. It’ll be an interesting line to balance, but that’s what I’m reflecting on right now.

As for any retake after the 1st, those will definitely require an analysis handout and tutoring before offering the retake opportunity. The 2nd retake (or higher) is where students tend to try to work the system and do hail mary retakes.

Also, always go with your gut when you see things you would do differently in the curriculum. If problems need to be done in a different order, or there need to be more challenging problems, make those changes. None of this is fool proof!

Hi Dane,
I’m probably not seeing it but are there answer keys for the concept quizzes?


Algebra I

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Study math for free!

Math planet is an online resource where one can study math for free. Take our high school math courses in Pre-algebra, Algebra 1, Algebra 2 and Geometry. We have also prepared practice tests for the SAT and ACT.

The educational material is focused on US high school maths. However, since maths is the same all over the world, we welcome everybody to study math with us, for free.

Mattecentrum is a Swedish non for profit member organization founded in 2008 in Sweden. Since then, the center has been providing free help in mathematics to all who study math. The purpose of Mattecentrum is to promote equal knowledge acquisition and to increase knowledge and interest in mathematics and other STEM related subjects. The members age varies between 6-26 years old.

The center runs FREE math labs in Sweden at schools, libraries and other premises across 34 cities.
More than 5000 students across the country monthly receive individual help with their studies from the Mattecentrum's 500 active volunteers. In 2019, approx. 30 000 students participated in our Math labs, summer camps and Maths Convention.

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By supporting Mattecentrum you help us to increase the level of knowledge and interest in mathematics among children and young people. Click on the donation button and support Mattecentrum!


Algebra 1 Worksheets

Looking for free printable Algebra 1 worksheets and exercises to help you prepare for the Algebra I test?

Want to measure your knowledge of Algebra 1 concepts and assess your exam readiness? Need the best Algebra 1 worksheets to help your students learn basic math concepts? If so, then look no further. Here is a perfect and comprehensive collection of FREE Algebra 1 worksheets that would help you or your students in Algebra 1 preparation and practice.

Download our free Mathematics worksheets for Algebra 1.

IMPORTANT: COPYRIGHT TERMS: Worksheets may not be uploaded to the internet in any form, including classroom/personal websites or network drives. You can download the worksheets and print them as many as you need. You have permission to distribute the printed copies to your students, teachers, tutors, and friends.

You Do NOT have permission to send these worksheets to anyone in any way (via email, text messages, or other ways). They MUST download the worksheets themselves. You can send the address of this page to your students, tutors, friends, etc.


This 4-week curriculum unit we have developed is one that can be used to introduce algebraic concepts at any grade level. It draws upon algebraic research showing that it is more helpful for students to learn algebra through studying pattern growth where a variable represents a case number, and can vary, before learning about “solving for x.” When students start learning algebra by solving for x they come to believe that a variable stands for a single number and does not vary. Later when they need to understand that variables can vary, they meet a conceptual barrier, and many do not ever get past that barrier. We recommend that students learn first about pattern growth and see that algebra can be useful for describing growth. Later, when they encounter situations when the variable stands for one missing number, they see this as a subset of their broader learning about variables and there is no confusion.

This 4-week curriculum unit we have developed is one that can be used to introduce algebraic concepts at any grade level. It draws upon algebraic research showing that it is more helpful for students to learn algebra through studying pattern growth where a variable represents a case number, and can vary, before learning about “solving for x.” When students start learning algebra by solving for x they come to believe that a variable stands for a single number and does not vary. Later when they need to understand that variables can vary, they meet a conceptual barrier, and many do not ever get past that barrier. We recommend that students learn first about pattern growth and see that algebra can be useful for describing growth. Later, when they encounter situations when the variable stands for one missing number, they see this as a subset of their broader learning about variables and there is no confusion. For a detailed review of research on algebra learning, see Kieran, (2013).

A second goal of our curriculum is for students to learn that algebra is a problem-solving tool. Students will learn to examine different functions that they explore visually, numerically, graphically, physically constructed, and algebraically. Students will be generalizing, representing, modeling, describing, and interpreting the relationships between two quantities. They will also be distinguishing between linear, quadratic, cubic, and exponential growth within multiple representations.

A third goal of our curriculum is to help students develop stronger number sense, as many students fail algebra, not because the algebra is difficult but because they lack a strong foundation in number sense (Gray & Tall, 1994 Boaler, 2016). In our number sense activities students will learn ways to adapt numbers and to use grouping symbols that will help them understand and use algebraic expressions.

Throughout the 4-week unit students will receive opportunities to make important brain connections, as they experience algebra in different ways, forms and representations.

References & Further Reading.

Boaler, J (2016) Mathematical Mindsets: Unleashing Students’ Potential through Creative Math, Inspiring Messages and Innovative Teaching. Jossey-Bass/Wiley: Chappaqua, NY.

Gray, E. and D. Tall (1994). “Duality, Ambiguity, and Flexibility: A “Proceptual” View of Simple Arithmetic.” Journal for Research in Mathematics Education 25(2): 116-140.

Kieran, C. (1992). The learning and teaching of school algebra. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning: A project of the National Council of Teachers of Mathematics (pp. 390-419). New York, NY, England: Macmillan Publishing Co, Inc.

Mason J. (1996) Expressing Generality and Roots of Algebra. In: Bernarz N., Kieran C., Lee L. (eds) Approaches to Algebra. Mathematics Education Library, vol 18. Springer, Dordrecht.

Schoenfeld, A. & Arcavi, A. (1988) On the Meaning of Variable. The Mathematics Teacher, 420-427.

Thompson, P. W., McCallum, W., Harel, G., Blaire, R., Dance, R., & Nolan, E. (2007). Intermediate algebra. In Algebra: Gateway to a technological future.


Free Algebra 1 Lesson Plans

10 Outstanding Sites for Downloading Lesson PlansScholastic Teaching Resources and Student Activities. An immense lesson database offering "timely lessons and units for all your teaching needs". . NCTM Illuminations. . HotChalk's Lesson Plans Page. . SqoolTech. . Teach-nology. . A to Z Teacher Stuff. . Teachers.net. . EdHelper. . Lesson Planet. .

What is the curriculum of Algebra 1?

The Algebra I curriculum is one of five Math courses offered at the high school level. Algebra I is taught using a combination of multimedia lessons, instructional videos, worksheets, quizzes, tests and both online and offline projects. The Algebra I course is designed to prepare students for the Algebra II course.

What is a Math Tutorial?

The mathematics tutorial seeks to give students an insight into the fundamental nature and intention of mathematics and into the kind of reasoning that proceeds systematically from definitions and principles to necessary conclusions.


1: Algebra - Mathematics

Linear Equations, Inequalities, and Systems

  • Writing and Modeling with Equations
  • Manipulating Equations and Understanding Their Structure
  • Systems of Linear Equations in Two Variables
  • Linear Inequalities in One Variable
  • Linear Inequalities in Two Variables
  • Systems of Linear Inequalities in Two Variables

Unit 3

  • Two-way Tables
  • Scatter Plots
  • Correlation Coefficients
  • Estimating Lengths

Unit 4

  • Functions and Their Representations
  • Analyzing and Creating Graphs of Functions
  • A Closer Look at Inputs and Outputs
  • Inverse Functions
  • Putting it All Together

Unit 5

Introduction to Exponential Functions

  • Looking at Growth
  • A New Kind of Relationship
  • Exponential Functions
  • Percent Growth and Decay
  • Comparing Linear and Exponential Functions
  • Putting It All Together

Unit 6

Introduction to Quadratic Functions

  • A Different Kind of Change
  • Quadratic Functions
  • Working with Quadratic Expressions
  • Features of Graphs of Quadratic Functions

Unit 7

  • Finding Unknown Inputs
  • Solving Quadratic Equations
  • Completing the Square
  • The Quadratic Formula
  • Vertex Form Revisited
  • Putting It All Together

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