Analyze and Solve Linear Equations Math B Honors
Module #1 Student Edition 2017-2018
Created in collaboration with Utah Middle School Math Project A University of Utah Partnership Project
San Dieguito Union High School District
Table of Contents MODULE #1: ANALYZE AND SOLVE LINEAR EQUATIONS* STANDARDS FOR MATHEMATICAL PRACTICE: A GUIDE FOR STUDENTS AND PARENTS......................................... 4 MODULE #1: PROPERTIES LIST ........................................................................................................................ 5
SECTION 1.1: CREATING AND SOLVING MULTI-STEP LINEAR EQUATIONS AND INEQUALITIES .............................. 7 1.1A LESSON: SIMPLIFYING EXPRESSIONS AND ALGEBRA TILES*....................................................................... 8 1.1A EXTENSION: FILL IN THE BOX ................................................................................................................. 13 1.1B LESSON: LINEAR EXPRESSIONS AND REAL-W ORLD CONTEXTS* .............................................................. 14 1.1C LESSON: SOLVING LINEAR EQUATIONS* ................................................................................................. 16 1.1C EXTENSION: MOBILES ........................................................................................................................... 21 1.1D LESSON: LINEAR EQUATIONS AND REAL-W ORLD CONTEXTS* .................................................................. 22 1.1E LESSON: SOLVING LINEAR INEQUALITIES ................................................................................................ 26 1.1F LESSON: SOLVING EQUATIONS WITH ABSOLUTE VALUE ........................................................................... 30 1.1F EXTENSION: ABSOLUTE VALUE EQUATIONS ............................................................................................ 34 1.1G LESSON: ABSTRACTING THE SOLVING PROCESS (PART 1)* ..................................................................... 36 SECTION 1.2: CREATING AND SOLVING MULTI-STEP LINEAR EQUATIONS WITH VARIABLES ON BOTH SIDES* .... 38 1.2A LESSON: SOLVING MULTI-STEP LINEAR EQUATIONS (WITH VARIABLES ON BOTH SIDES)* ........................... 39 1.2A EXTENSION: DOG KENNELS ................................................................................................................... 43 1.2B LESSON: SOLVING LINEAR EQUATIONS WITH DIFFERENT SOLVING OUTCOMES* ....................................... 44 1.2B EXTENSION: NUMBER TOWERS .............................................................................................................. 48 1.2C LESSON: SOLVING LINEAR EQUATIONS TO MODEL REAL-W ORLD CONTEXTS* .......................................... 50 1.2D LESSON: EXAMINING REAL-W ORLD CONTEXTS WITHOUT SOLUTIONS ...................................................... 54 1.2E LESSON: ABSTRACTING THE SOLVING PROCESS (PART 2)* ..................................................................... 56 1.2F ACTIVITY: EXAMINING REAL-W ORLD CONTEXTS ..................................................................................... 58
* Denotes a lesson that was adapted from Utah Middle School Math Project © Utah Middle School Math Project & University of Utah http://utahmiddleschoolmath.org/ This work is licensed under a Creative Commons Attribution-NonCommercial 2.5 Generic License http://creativecommons.org/licenses/by-nc/2.5/ This work is licensed under a Creative Commons Attribution-NonCommercial 3.0 Unported License http://creativecommons.org/licenses/by-nc/3.0/legalcode
SDUHSD Math B Honors Module #1 - STUDENT EDITION 2017-2018
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Module #1: Analyze and Solve Linear Equations* Online support for this module can be found at http://goo.gl/s2vDtM (case sensitive) or using the QR code below. This website includes copies of student lessons, homework, and instructional support videos.
Common Core Standard(s): Solve linear equations in one variable. (8.EE.7) -
-
Give examples of linear equations in one variable with one Curriculum Support Website solution, infinitely many solutions, or no solution. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form results (where a and b are different numbers). Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.
Academic Vocabulary: linear expression, evaluate, variable, constant term, simplify, linear equation, solve, solution, like terms, distributive property, no solution, infinitely many solutions Module Overview: This module begins with a review of simplifying and writing expressions and then progresses to solving multi-step linear equations in one variable. The module includes equations with one solution, no solution, and infinitely many solutions. Students use algebra tiles to model, simplify, and solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. While working with the concrete representation of an equation, students are simultaneously manipulating the symbolic representation of the equation. By the end of the module, students should be fluently solving multi-step equations represented symbolically. They should feel comfortable with the laws of algebra that allow them to simplify expressions and the properties of equality that allow them to transform a linear equation into its simplest form, thus revealing the solution if there is one. While solving, students will become comfortable with the inverse operations that allow them to transform a linear equation into its simplest form. An important feature of allowable operations on equations is that they can be reversed. This module utilizes error analysis to highlight common mistakes that are made when solving equations. In the last section of the module, students look at equations with infinitely many or no solutions. They analyze what it is about the structure of the equation and the solving outcome that results in one solution, infinitely many solutions, or no solution. Applications are interwoven throughout the module so that students learn to write and solve linear equations pertaining to real-world situations. The ability to solve real world problems by writing and solving linear equations gives purpose to the skills students are learning in this module. Connections to Content: Prior Knowledge In previous coursework, students used properties of operations to generate equivalent expressions, including those that require the use of the distributive property. Students solved one- and two-step equations. Students have solved real-life mathematical problems using numerical and algebraic expressions and equations.
Future Knowledge As students continue in this course, they will begin to study linear functions. They should understand that when working with linear equations in one variable, the variable takes on a specific value as studied here. As they proceed to the study of linear functions, they will see that functions can take on an infinite number of values. They will analyze and solve pairs of simultaneous linear equations, including systems with one solution, no solution, and infinitely many solutions. They will also write and solve linear equations in two variables to solve real-world problems. SDUHSD Math B Honors Module #1 - STUDENT EDITION 2017-2018
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Standards for Mathematical Practice: A Guide for Students and Parents The Standards for Mathematical Practices are central to the Common Core. These practices build fluency and help students become better decision-makers and problem solvers. The practices reflect the most advanced and innovative thinking on how students should interact with math content. Students and parents will develop skill with these standards by asking some of these questions:
Make Sense of Problems and Persevere in Solving Them. What is the problem that you are solving for? Can you think of a problem that you recently solved that is similar to this one? H w will y u g ab ut s lving the p blem?(i.e. What’s y u plan?) Are you progressing towards a solution? How do you know? Should you try a different solution plan? How can you check your solution using a different method? Construct Viable Arguments and Critique the Reasoning of Others. Can you write or recall an expression or equation to match the problem situation? What do the numbers or variables in the equation refer to? What’s the c nnecti n am ng the numbe s and va iables in the equati n? Reason Abstractly and Quantitatively. Tell me what your answers(s) mean(s) How do you know that your answer is correct? If I t ld y u I think the answe sh uld be (a w ng answe ), h w w uld y u explain t me why I’m wrong? Model with Mathematics. Do you know a formula or relationship that fits this problem situation? What’s the c nnecti n am ng the numbe s in the p blem? Is your answer reasonable? How do you know? What do(es) the number(s) in your solution refer to? Use Appropriate Tools Strategically. What tools could you use to solve this problem? How can each one help you? Which tool is most useful for this problem? Explain your choice. Why is this tool (the one selected) better to use than (another tool mentioned)? Before you solve the problem, can you estimate the solution? Attend to Precision. What do the symbols that you used mean? What units of measure are you using (for measurement problems) Explain to me what (term from the lesson) means. Look For and Make Use of Structure. What do you notice ab ut the answe s t the exe cises y u’ve just c mpleted? What do different parts of the expression or equation you are using tell you about possible correct answers? Look for and Express Regularity in Repeated Reasoning. What shortcut can you think of that will always work for these kinds of problems? What pattern(s) do you see? Can you make a generalization?
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Module #1: Properties List Name: PROPERTY
Period: ALGEBRAIC DEFINTION
Commutative Property of Addition
If a and b are real numbers, then a + b = b + a
Commutative Property of Multiplication
If a and b are real numbers, then ab = ba
Associative Property of Addition
If a, b, and c are real numbers, then (a + b) + c = a + (b + c)
Associative Property of Multiplication
If a, b, and c are real numbers, then (ab)c = a(bc)
Distributive Property
If a, b, and c are real numbers, then a(b + c) = ab + ac and (b + c)a = ba + ca
Identity Property of Addition
There is a unique real number 0 such that, for every real number a, a + 0 = a and 0 + a = a
Identity Property of Multiplication
There is a unique real number 1 such that, for every real number a, a 1= a and 1 a = a
EXPLAIN PROPERTY IN YOUR OWN WORDS
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NUMERICAL EXAMPLE
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Definition of Subtraction Property
For all real numbers a and b, a – b = a + (-b)
Additive Inverse Property
For every real number a, there is a unique real number -a such that a + (-a) = 0 and –a + a = 0
Multiplicative Property of Zero
For every real number a, a 0 = 0 and 0 a = 0
Cancellation property
For all real numbers a, -(-a) = a
Substitution Property of Equality
For all real numbers a and b, If a = b, then a can be substituted in for b in any equation, and b can be substituted for a in any equation For every nonzero real number a,
Multiplicative Inverse Property
there is a unique real number such that a
1 a
1 1 1 and a 1 a a
Addition Property of Equality
For all real numbers a, b and c, if a = b then a + c = b + c and c+a=c+b
Multiplication Property of Equality
For all real numbers a,b and c, if a = b, then ac = bc and ca = cb
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Section 1.1: Creating and Solving Multi-Step Linear Equations and Inequalities Section Overview: This section begins with a review of writing and simplifying algebraic expressions. Students evaluate algebraic expressions and create expressions to represent real-world situations. They apply their understanding of algebraic expressions to solving linear equations. Students solve linear equations by collecting like terms, applying the distributive property, and clearing fractions. They write contexts to match equations and write equations to match contexts. Students solve linear inequalities and equations involving absolute value.
Concepts and Skills to Master: By the end of this section, students should be able to: Understand the meaning of linear expression and linear equation. Simplify linear expressions, including those requiring expanding using the distributive property and collecting like terms. Write and simplify linear expressions that model real-world contexts. Translate between the concrete and symbolic representations of an expression and an equation. Solve multi-step linear equations with rational coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Write and solve multi-step linear equations that model real-world contexts. Solve multi-step linear inequalities. Solve equations involving absolute value.
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1.1A Lesson: Simplifying Expressions and Algebra Tiles* Name:
Period:
1. Define the sets of rational numbers. a. Rational numbers:
b. Whole Numbers:
c. Natural Numbers:
d. Integers:
2. Complete the Venn Diagram by filling in the correct number set and placing the following numbers in the correct position: ̅
√
3. Explain in words how natural and whole numbers differ.
4. Explain in words how integers and rational numbers differ.
Rational numbers are often used to create linear expressions. A linear expression is a mathematical phrase consisting of
. and
are two different examples of linear expressions.
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5. Oliver explains that the expression equals 20 and his work is shown to the right. a. Did Oliver simplify the expression correctly? Explain why or why not.
b. Sa a disag ees with Olive ’s process and states that multiplication and division must always be simplified first. Is Sara correct in her thinking? Explain why or why not.
6. Place parentheses in the expression
, so that the value of the expression is
7. Emma is playing a popular video game and is determined to beat the high score. The game saves her place so that each time she plays it again, she picks up in the same place with the same number of points. Emma downloads the video game on Monday night and starts playing, scoring a bunch of points. On Tuesday, she scores an additional 500 points. On Wednesday, she doubles her total score from the previous day. On Thursday, she scores the same number of points that she scored on Monday. a. Sam writes an expression that represents Emma’s t tal sc e afte she is finished playing on ) Thursday. Sam writes the expression ( . Write in w ds what each piece f Sam’s expression represents in the story.
b. Beth writes the expression t ep esent Emma’s sc e n Thu sday. H w did Beth represent the problem differently from Sam?
c. Write another expression to represent Emma’s sc e n Thu sday.
Substituting a specific number for the unknown in an expression and calculating the resulting value is called evaluating the expression. 8. If Emma scored 700 points on Monday, evaluate each of the three expressions above to determine how many points Emma has on Thursday. Which expression was most efficient to evaluate? Explain why. Expression from part a:
Expression from part b:
SDUHSD Math B Honors Module #1 - STUDENT EDITION 2017-2018
Expression from part c:
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) F Emma’s situati n, the exp essi ns ( and and are equivalent linear expressions. Expressions are considered equivalent when a substitution of any number for the unknown in each expression produces the same numerical result. Algebra tiles can be used to model and simplify linear expressions. Key for Tiles:
=1
= –1
= x
= –x
= x2
= -x2
A positive tile and a negative tile, when added together, become zero. This is called a zero pair.
9. To the right is a model of the expression . a. Circle and eliminate the zero pairs and write the simplified form of this expression.
b. Evaluate this expression when
10. Use Algebra tiles to model the expression
Circle and eliminate the zero pairs.
a. Simplified expression:
b. Evaluate when
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11. Does simplify to
simplify to .
? Explain why or why not. If not, rewrite the expression so that it does
The distributive property is often used to simplify expressions. The distributive property is applied when there is a number or variable that needs to be multiplied by an addition or subtraction problem in ) ( parenthesis. For example, the linear expression ( ) simplifies to ( ) which equals .
12. Use Algebra tiles to model the expression ( ) . Circle and eliminate the zero pairs. a. Simplified expression:
b. Evaluate when
Directions: Simplify the expression without using Algebra tiles. Evaluate for the given value. Show all of your work. (
13.
15.
(
)
)
(
)
14.
(
)
16.
[ (
)]
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Directions: Simplify the expression without using Algebra tiles. Show all of your work 17.
(
)
(
)
Directions: Evaluate the expression for
18.
and
Show all of your work.
19.
20.
21.
22.
(
)
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1.1A Extension: Fill in the Box Name:
Period:
1. Fill in the boxes with whole numbers 1 through 9, using each number at most once, so that the sum is as close to as possible. Show all of your attempts.
2.
Fill in the boxes with whole numbers 1 through 9, using each number at most once, find two different mixed numbers that will make the equation true. Show all of your work.
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1.1B Lesson: Linear Expressions and Real-World Contexts* Name:
Period:
Linear expressions can be used to model many real-world contexts. For example, if you buy 3 pieces of chicken that cost dollars each, a slice of pie for $3, and a drink for $2, we can write an expression that represents the total amount you spent: . The simplified form of this expression would be . 1.
A salesperson gets a base salary of $300 per week plus $20 for each item he sells. He sells in one week. The salesperson also spends $40 a week on travel expenses.
items
a. Identify the unknown quantity. Write an expression that represents the total amount of money that the salesperson had at the end of the week. Simplify the expression.
b. If the salesperson sold 50 items, how much money did he earn at the end of the week?
2.
Tim took his friends to the movies. He started with $40 and bought 3 movie tickets that each cost dollars. He also bought one tub of popcorn that cost $5.75. a. Identify the unknown quantity. Write an expression that represents the amount of money remaining. Simplify the expression.
b. If one movie ticket costs $8, how much money did Tim have left?
3.
Sara bought 3 baby outfits and one bottle of baby lotion. The baby lotion costs Each outfit costs $2 more than a bottle of baby lotion.
dollars per bottle.
a. Identify the unknown quantities. Write an expression that represents the total amount spent. Simplify the expression.
b. How much money was spent if one bottle of baby lotion costs $3.50?
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4.
In the NBA, the length of the basketball court is 44 feet longer than the width. The width of the court is . a. Identify the unknown quantities. Write an expression that represents the perimeter of the basketball court. Simplify the expression.
b. If the width of a basketball court is 50 feet, what is the perimeter of the entire court?
5. Peter is paid dollars per hour. For every hour he works over 40 hours per week, he is paid an additional $10 per hour. Peter worked 46 hours last week. a. Identify the unknown quantities. Write an expression for the amount of money Peter earned last week. Simplify the expression.
b. If Peter is paid $12 per hour, how much money did he earn?
6. Molly has nickels, dimes and quarters in her piggybank. She has twice as many nickels as quarters and 15 more dimes than quarters. Molly has amount of quarters. a. Identify the unknown quantities. Write an expression that represents the total number of coins that Molly has in her piggybank. Simplify the expression.
b. If Molly has 22 quarters in her piggybank, how many nickels and dimes does she have? .
c. How much money does Molly have in her piggybank?
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1.1C Lesson: Solving Linear Equations* Name:
Period:
In the previous lessons, we studied linear expressions. When we set two linear expressions equal to each other, we create a linear equation. Let’s l k at a p evi us example from lesson 1.1B: 1. A salesperson earns a base salary of $300 per week plus $20 for each item he sells. He sells one week. The salesperson also spends $40 a week on travel expenses.
items in
a. Write an expression that represents the total amount made last week.
b. If the salesperson made $860 last week, how many items did the salesperson sell? Create a linear equation to solve. Show all of your work.
c. A solution to an equation is a number that makes the equation true when substituted for the variable. Verify that your solution to the example above is correct.
2. Algebra tiles can be used to solve linear equations. Use the model to complete the following. a. What linear equation is represented by the model?
Model
b. Model and solve using Algebra tiles. Circle and eliminate all zero pairs.
c. Solve algebraically.
d. Verify your solution.
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3. Use the model to complete the following. Model
a. What linear equation is represented by the model?
b. Model your solving process using Algebra tiles. c. Solve algebraically.
d. Verify your solution.
4. Justify each step when solving the equation Step
. Property/Justification
1.
2.
3.
4.
5.
6.
7.
8.
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5. Show algebraically two different methods to solve the equation ( different in each method.
Method #1
6. To solve the equation ( reasoning.
)
Method #2
)
, which method above would be more efficient? Explain your
7. Show algebraically two different methods to solve the equation different in each method.
Method #1
The first step must be
. The first step must be
Method #2
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8. Show algebraically two different methods to solve the equation different in each method. Method #1
The first step must be
Method #2
Problems #7 and #8 can be solved by clearing decimals or fractions first to make the solving process more efficient. This method is justified by the properties of equality. Directions: Solve the following equations without using Algebra tiles. Show all of your work. 9.
10.
(
11.
13.
(
)
)
(
12.
14.
(
SDUHSD Math B Honors Module #1 - STUDENT EDITION 2017-2018
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)
( )
19
(
15.
)
21.
)
18.
17.
19.
(
16.
(
)
(
(
)
)
20.
(
)
(
)
22.
23. Build an equation that contains parentheses and has a solution of
SDUHSD Math B Honors Module #1 - STUDENT EDITION 2017-2018
.
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1.1C Extension: Mobiles Name:
Period:
Directions: Every beam in these mobiles is balanced. The strings and beams weigh nothing. Find the weight of each shape. 1.
=
=3
2.
=6
=
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1.1D Lesson: Linear Equations and Real-World Contexts* Name:
Period:
1. Part of the context that matches the expressions and equation shown on the left has been written for you. Complete the context, solve the equation, and determine how much time Theo spends training for each sport. Triathalon Training Schedule
Context:
Minutes spent swimming:
Theo is training for a triathalon. He runs twice as long as he swims. He bikes
Minutes spent running: Minutes spent biking: (
)
(
)
He swims three times a week, runs four times a week, and bikes If he spends a total of 510 minutes per week training, how many minutes does he spend on each exercise at a time?
Time spent swimming: Time spent running: Time spent biking:
Directions: Write a context that represents the given information and equation. Solve and write your solution in a complete sentence. Show all of your work. 2. The Cost of Lunch
Context:
Cost of a pear: Cost of juice: Cost of a sandwich:
x 1.5x ( x 1.40) 3.50
Solution:
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Context:
3. Ages Talen’s age n w: Pete ’s age n w: (
)
(
)
Solution:
4. Angles
Context:
: :
a (4a 5) 90
Solution:
5.
Rectangles
Context:
Width of a rectangle: Length of a rectangle: ( [
(
)
)]
Solution:
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6.
Coins
Context:
Number of quarters: Number of dimes: (
)
Solution:
Directions: Write an expression for each unknown quantity and an equation that represents the word problem. Solve your equation and write your solution in a complete sentence. Show all of your work. 7. The length of a certain rectangle is 4 meters greater than five times its width. Find the dimensions of the rectangle if its perimeter is 20 meters.
8. There are eight fewer students in the Debating Club than on the Math Team, and there are only half as many students on the Ecology Force as in the Debating Club. How many students are in each group if there are 68 students in all and each student participates in only one of the activities?
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9. The measure of the first angle of a triangle is 20 degrees more than the measure of the second angle. The measure of the third angle is 30 degrees more than twice the sum of the measures of the first two angles. Find the measures of each angle.
10. Sam got a 90% on his first math exam, a 76% on his second math exam, and a 92% on his third math exam. What must he score on his fourth exam to have an average of 88% in the class?
11. Kelly works 40 hours a week as a nurse practitioner. She earns time and a half for every hour she works over 40 hours per week. If she worked 60 hours in one week and made $3360, what is her regular hourly wage and her overtime hourly wage?
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1.1E Lesson: Solving Linear Inequalities Name:
Period:
1. Solve and graph its solution set on a number line.
2. Solve ( a number line.
)
and graph its solution set on
3. Examine your graphs for problems #1 and #2. What is different about the graphs?
4. Explain in words how the symbols > and < affect the graph of an inequality.
5. Explain in words how the symbols ≥ and ≤ affect the g aph f an inequality.
( ) 6. Savannah solved the inequality , and her steps are shown below. Savannah made a mistake in her solving process. Explain the mistake or mistakes she made and solve the inequality correctly. Explain mistake(s) in words
Inequality solved correctly
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Directions: Solve each inequality. Show all of your work. State four values that satisfy the inequality and four values that do not. 7.
8.
Values that satisfy the inequality
Solution
(
Values that do not satisfy the inequality
Whole #:
Whole #:
Natural #:
Natural #:
Integer #:
Integer #:
Rational #:
Rational #:
)
Values that satisfy the inequality
Solution
Values that do not satisfy the inequality
Whole #:
Whole #:
Natural #:
Natural #:
Integer #:
Integer #:
Rational #:
Rational #:
Directions: Solve each inequality and graph its solution set on a number line. Show all of your work. 9.
10.
(
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11.
13.
(
)
12.
14.
(
)
(
)
Directions: Write an expression for each unknown quantity and an inequality that represents the word problem. Solve your inequality and write your solution in a complete sentence. Show all of your work. 15. You and your family want to rent a limousine for your graduation party. The limousine costs $650 for the evening and $1.30 per mile. If you want to spend at most $700, how many miles can you travel in the limousine? Round your answer to the nearest mile.
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16. Andy earns $75,000 per year and receives 10% commission on his sales. How much money in sales must Andy make in order to earn at least $80,000 this year?
17. The sum of three consecutive integers is less than 739. Find three integers with the greatest sum.
18. The sum of two consecutive odd integers is at most 123. Find the pair with the greatest sum.
19. The longest side of a triangle is 2 times the shortest side. The third side is 4 inches longer than the shortest side. How long is the shortest side if the perimeter is more than 52 inches?
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1.1F Lesson: Solving Equations with Absolute Value Name:
Period:
1. What is the value of |
|? Use the number line to support your explanation.
| |. He explains, “The expression equals 2. Ryan wants to simplify the expression beacuse times equals 16.” Ma issa disag ees with Ryan, and explains that the absolute value of must be found fi st bef e multiplying. Ma issa says, “The expression equals . The absolute value of is and times equals .” Who is correct in their reasoning? Explain.
3. Can the absolute value of an expression equal a negative value? Explain why or why not.
4. For what values of
is
| | negative?
5. For what values of
is
| | positive?
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Directions: Simplify each expression. Show all of your work. 6.
|
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8.
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10. |
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12.
14.
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9.
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16. When solving the equation | |
, how many solutions occur? Explain your reasoning.
Directions: Solve the following equations. Show all of your work. | |
17.
| |
19.
25. |
20.
| |
21.
23.
18. | |
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| |
22.
(
24. |
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26.
(| |
| |)
| |
)
27. For problem #26 above, explain why this type of solution occurs.
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Directions: A mistake was made when solving the equations below. Circle and explain the mistake made. Solve the equations correctly. | |
28. Equation: | | | |
Equation solved correctly:
| |
29. Equation: |
|
|
|
Mistake made: .
Mistake made:
Equation solved correctly:
30. For which value(s) of
is the statement | |
31. For which value(s) of
is the statement | |
32. For which value(s) of x is the statement | |
| | true? Explain your reasoning.
true? Explain your reasoning.
true? Explain your reasoning
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1.1F Extension: Absolute Value Equations Name:
Period:
1. What values of
2. Is
make the equation |
the only solution to the equation |
|
true? Justify your answer using the number line below.
|
? Is there another solution? Explain why or why not.
Directions: Solve the following equations. Show all of your work. 3. |
|
4. |
5. |
|
6. |
|
|
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7. |
9. |
11. |
|
|
|
8. |
|
10. |
|
12. |
|
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1.1G Lesson: Abstracting the Solving Process (Part 1)* Name:
Period:
1. Oliver and Sara both solved the equation (
)
Oliver’s Method: (
for . Describe their solving process. Sara’s Method:
)
(
)
Examine the solutions. Did both Oliver and Sara solve the equation correctly? Explain why or why not.
2. The equation below has the same structure as the equation above; however the numbers 2, 3, and 12 in the original equation have been replaced with the variables p, q, and r which represent any real number. Solve the equation below for , using both Olive ’s and Sa a’s meth ds. Explain y u s lving p cess. Oliver’s Method:
(
)
Sara’s Method:
(
)
Are the resulting exp essi ns in Olive ’s and Sa a’s meth ds equivalent? Explain why your reasoning.
3. Explain in words how to solve for
in the equation (
)
why n t. Justify
. Then solve for .
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4. Explain the mistake made when solving for Then solve for correctly.
Directions: Solve the following equations for
in the equation.
if
and
5.
6.
7.
8.
9.
10.
11.
12.
represent real numbers not equal to 0.
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Section 1.2: Creating and Solving Multi-Step Linear Equations with Variables on Both Sides* Section Overview: In this section, students solve equations with unknowns on both sides of the equal sign. Students extend their understanding of linear equations with unique solutions to linear equations with either no solution or infinitely many solutions. Students use models and skills developed and solidified in section 1.1 to analyze what it is about the structure of an equation and the solving outcome ( , or where and are different numbers) that results in one solution, no solution, or infinitely many solutions. Students transform equations to solve for a particular unknown.
Concepts and Skills to Master: By the end of this section, students should be able to: Solve multi-step linear equations that have one solution, infinitely many solutions, or no solution. Understand what it is about the structure of a linear equation that results in equations with one solution, infinitely many solutions, or no solutions. Identify and provide examples of equations that have one solution, infinitely many solutions, or no solutions. Transform equations to solve for a particular unknown.
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1.2A Lesson: Solving Multi-Step Linear Equations (with variables on both sides)* Name:
Period:
1. Two students, Theo and Lance, each have some chocolates. They know that they have the same number of chocolates. Theo has ten full bags of chocolates and forty-seven loose chocolates. Lance has one and a half as many full bags of chocolates as Theo, but thirty-five fewer loose chocolates than Theo.
a. Define a variable, write an equation, and solve to determine how many chocolates are in one full bag.
b. How many chocolates does Theo and Lance have? Show all of your work.
Directions: Solve the following equations. Verify your solution and show all of your work. 3.
2.
Verify:
1 4 x 8 12 x 3 3
Verify:
SDUHSD Math B Honors Module #1 - STUDENT EDITION 2017-2018
4.
Verify:
39
Directions: Solve the following equations. Show all of your work. 5.
(
7.
9.
(
6.
(
)
)
8.
10.
(
)
)
(
SDUHSD Math B Honors Module #1 - STUDENT EDITION 2017-2018
(
)
)
(
)
40
11.
12.
(
)
13. Build an equation that has parentheses on at least one side of the equation, variables on both sides of the equation, and a solution of .
14. Show algebraically two different methods to solve the equation ( must be different in each method.
Method #1
)
(
). Your first step
Method #2
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15. Show algebraically two different methods to solve the equation the step must be different in each method. Method #1
(
)
(
). Your first
Method #3
Directions: Solve each inequality and graph the solution on a number line. Show all of your work. 16.
18.
17.
(
)
(
)
19.
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1.2A Extension: Dog Kennels Name:
Period:
Directions: Use the prices for three different kennels to complete the following. Dynomite Kennels: Charges a flat fee of $40 for manners lesson, $16.25 per day to board a dog, and $8.50 per day for play time with a dog
Lucky Kennels: Charges a flat fee of $40 for manners lesson, $15.25 per day to board a dog, and $9.50 per day for play time with a dog Super Kennels: Charges a flat fee of $45.99 for manners lesson, $18 per day to board a dog, and $6.75 per day for play time with a dog 1. Allie boards her dog, Delta, at Dynomite Kennels and pays $163.75. How many days did her dog stay at the kennel? Assume her dog received a manners lesson and play time each day. Show all of your work.
2. Laura boards her dog, Lilly, at Lucky Kennels and pays $213.25. How many days did her dog stay at the kennel? Assume her dog received a manners lesson and play time each day. Show all of your work.
3. Joy boards her dog, Sophie, at Super Kennels and pays $144.99. How many days did her dog stay at the kennel? Assume her dog got a manners lesson and play time each day. Show all of your work.
4. After how many days would Allie and Laura pay the same fee to Dynomite Kennels and Lucky Kennels? Assume they both pay all the fees described above. Show all of your work.
5. After how many days would Laura and Joy pay the same fee to Lucky Kennels and Super Kennels? Assume they both pay all the fees described above. Show all of your work.
SDUHSD Math B Honors Module #1 - STUDENT EDITION 2017-2018
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1.2B Lesson: Solving Linear Equations with Different Solving Outcomes* Name:
Period:
1. Use the model to complete the following. a. What observations can you make?
b. Write the algebraic representation for this model and solve. Explain in words what happened when you solved the equation.
c. What is it about the structure of the equation that led to the solution?
d. Create a model of an equation with different grouping and variables on both sides that would result in the same solution as in part b. Write the algebraic representation for your model. Model
Algebraic Representation:
2. Use the model to complete the following. a. What observations can you make?
b. Write the algebraic representation for this model and solve. Explain in words what happened when you solved the equation.
c. What is it about the structure of the equation that led to the solution?
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d. Create a model of an equation with different grouping and variables on both sides that would result in the same solution as in part b. Write the algebraic representation for your model. Model
Algebraic Representation:
3. Use the model to complete the following. a. What observations can you make?
b. Write the algebraic representation for this model and solve. Explain in words what happened when you solved the equation.
c. What is it about the structure of the equation that led to the solution?
Directions: Without solving, determine the number of solutions by examining the structure of the equation. 4.
5.
6.
7.
(
)
SDUHSD Math B Honors Module #1 - STUDENT EDITION 2017-2018
(
)
45
Directions: Examine the structure of each equation. State if there are infinitely many solutions, no solution, or one solution. If there is one solution, solve the equation and state the solution. If there are infinitely many solutions or no solution, justify why using the structure of the equation. 8.
10.
12.
(
)
(
)
9.
(
)
(
11.
)
(
)
(
(
)
(
)
)
13.
14. Replace the ? in the equation with a numerical or a variable expression so that the following conditions are satisfied. Justify your reasoning. a. The solution set is 2.
b. The solution set is infinitely many solutions.
c. The solution set is no solution.
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15. What is it about the structure of an equation that leads to one solution, infinitely many solutions, or no solution? Complete the table below and provide examples to support your claim. One Solution
No Solution
Infinitely Many Solutions
Structure of Equation
Model
Algebraic Equation of Model
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1.2B Extension: Number Towers Name:
Period:
1. Describe what is happening in the tower below.
29 14 8
15 6
9
2. Using the rule you developed for the tower above, complete the number towers by writing the algebraic expressions in the empty boxes.
-6 4y + 3 -7y
3
-7
2x + 5 x
5
4x
3. Find the value of the variables for each of the number towers above. x=
y=
4. Identify the equations that have the same x and y values as the number towers. a.
b.
c.
d.
e.
f.
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5. Describe what is happening in the tower below.
896 56 7
16 8
2
6. Using the rule you developed for the tower above, complete the number towers by writing the algebraic expressions in the empty boxes. 240 20 5
y+6
300
3
x
4
7. Find the value of the variables for each of the number towers above. x=
y=
8. Identify the equations that have the same
and
values as the number towers.
a.
b.
c.
d.
e.
f.
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1.2C Lesson: Solving Linear Equations to Model Real-World Contexts* Name:
Period:
Directions: Write a context that represents the given expressions and equation. Solve and write your solution in a complete sentence. Show all of your work. 1.
Birthday Parties
Context:
Number of guests at birthday party: Cost of party at Boondocks: Cost of party at Raging Waters:
Solution:
a. If you are inviting 2 friends to your party, which option is cheaper, Boondocks or Raging Waters? How much cheaper is it?
b. If you are inviting 7 friends to your party, which option is cheaper, Boondocks or Raging Waters? How much cheaper is it?
2.
Coins
Context:
number of quarters: number of nickels: number of dimes: (
)
( )
Solution:
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3.
Savings
Context:
Number of weeks: S phie’s money: 300 40w Raphael’s money: 180 20w
300 40w 180 20w Solution:
a. If Sophie continues to spend at this rate, when will she run out of money?
b. If Raphael continues to save at this rate, how long will it take for him to have $1,000?
Directions: Define your variables and write an expression for each unknown quantity and an equation that represents the word problem. Solve your equation and write your solution in a complete sentence. Show all of your work. 4. Horizon Phone Company charges $15 a month plus 10 cents per text. G-Mobile charges a flat rate of $55 per month with unlimited texting. At how many texts would the two plans cost the same? Which plan is the better deal if you send 200 texts per month?
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5. The enrollment in dance class is currently 80 students and is increasing at a rate of 4 students per term. The enrollment in choir is 120 students and is decreasing at a rate of 6 students per term. After how many terms will the number of students in dance equal the number of students in choir? How many students will be in each class?
6. The Panama Canal is 2 km shorter than twice the length of the Suez Canal. The sum of the lengths of the two canals is 121 km greater than three fourths the length of the Panama Canal. Find the length of each Canal.
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7. Jackson is eight years older than Sam. Three years ago, Jackson was twice as old as Sam was then. How old is Jackson now?
8. A box contains 30 coins. Some of the coins are nickels and the rest are dimes. The total value of coins in the box is $2.40. How many nickels and dimes are in the box?
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1.2D Lesson: Examining Real-World Contexts without Solutions Name:
Period:
1. Not every word problem has a solution. For example, can you find four consecutive even integers whose sum is 224? Explain why or why not.
2. When might a word problem involving coins have no solution?
3. When might a word problem involving perimeter have no solution?
Directions: Write an expression for each unknown quantity and an equation or inequality to represent the word problem. Solve your equation or inequality and write your solution in a complete sentence. If the word problem has no solution, explain your reasoning. 4. The width and length of a rectangle are consecutive multiples of 5. If the perimeter of the rectangle is 125 m, what is the length and width of the rectangle?
5. In a triangle, the first angle is 20 degrees more than the measure of the second angle. The third angle is three times the measure of the second angle. Find the measure of each angle.
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6. One-half f M lly’s age tw yea s f m n w, added t twenty years. How old is Molly now?
ne-third of her age three years ago, is equal to
7. A total of $458 was collected for admission from 132 people who attended a show. If adults paid $4 each and children paid $2.50 each, how many children attended show?
8.
The math club can buy ready-made shirts for $15 each. Alternately, it can buy plain T-shirts for $6.25 each, fabric paint for $35.70, and a pack of stencils for $8.50. For how many shirts is stenciling plain Tshirts cheaper than buying ready-made shirts?
9. Colby is 5 years older than his brother Joe. Ten years from now, Colby will be three times as old as Joe will be. How old are Colby and Joe now?
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1.2E Lesson: Abstracting the Solving Process (Part 2)* Name:
Period:
1. Solve the following equations for . Explain your solving process. Assume no variables equal 0. a.
b.
Directions: Solve the following equations for
if
and
2.
3.
4.
5.
6.
(
)
represent real numbers not equal to 0.
(
)
7.
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8.
9.
( 10. The equation degrees Fahrenheit.
) gives the temperature
in degrees Celsius in terms of the temperature
in
a. Solve the equation for
b. What Fahrenheit temperature would be equivalent to a temperature of 30 degrees Celsius?
11. The equation height .
can be used to find the surface area
of a cylinder given radius
and
a. Solve the equation for height .
b. If the surface area of a cylinder was Round your answer to the nearest tenth.
with a radius of
SDUHSD Math B Honors Module #1 - STUDENT EDITION 2017-2018
, what is the height of the cylinder?
57
1.2F Activity: Examining Real-World Contexts Name:
Period:
Directions: Working in collaborative groups, write an expression for each unknown quantity and an equation for each word problem. Solve your equation and write your solution in a complete sentence. Show all of your work in the appropriate box. #1
#2
#3
#4
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#5
#6
#7
#8
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