math 6 expressions and equations ee.d1 model and solve problems involving whole number exponents....

Math 6Expressions and Equations

EE.D1 Model and solve problems involving whole number exponents.

EE.D3 Write expressions using variables from contextual problems.

EE.D2 Use mathematical properties to generate and identify equivalent expressions.

Write inequalities from contextual problems and graph or describe the solution.

EE.D4 Solve real-world problems by writing and solving equations.

Use variables to represent two quantities in a contextual problem.

Analyze the relationship between variables using graphs and tables.

EE Learningtargets

Vocabulary of this unit

Term, Expression, Equation, Variable, Variable Structure, Coefficient, Exponent, Factors, Simplify, Solve

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Algebra Vocabulary

TermParts of an expression or series separated by + or -

Expression:

Terms:

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More Vocabulary

exponentCoefficient

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What’s a factor?

What are the factors of:

80 w3125

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Expressions vs Equations

An Expression -

An Equation - 5x + 3x = 10

What’s the difference between an expression and an equation?

A group of terms added or subtracted

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Solving vs Simplifying

Solve – Make a true statement, find a value that makes a true statement.Example: Solve for n when 3 + n = 5

Simplify - Rewrite an expression so it is more simply put. Example: 3 + 5 => 8 or 2n + 3n => 5n

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You try…. EE.D1

Exponents

How many ways can you write thesenumbers using exponents?

125 27 81

48

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Simplify: Write out factors then recombineEE.D1

Simplify: Write out factors then recombine

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Simplify: Write out factors then recombine

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Independent Practice

ICP GM U3D1

U3D1

In your notebook on the vocabulary page, write a definition and give an example of the vocabulary we discussed so far.

Term, Expression, Equation, Variable, Coefficient, Exponent, Factors, Simplify, Solve

Must Do: *Please set yesterday’s assignment on your desk so I can check off that you did it!

Use mathematical properties to generate and identify equivalent expressions.

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What does an equal sign mean?

Let’s discussEE.D2

What is an equivalent Expression?

Write an expression equivalent to:

8

6 + 4

n + 2

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Let’s review and discuss

Order of Operations

1. Calculations must be done from left to right.2. Calculations in brackets (parenthesis) are done first. When you have more than one set of brackets, do the inner brackets first.3. Exponents (or radicals) must be done next.4. Multiply and divide in the order the operations occur.5. Add and subtract in the order the operations occur.

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Let’s Create some Equivalent ExpressionsPractice

Expression Equivalent Expression Simplest Expression

(y + y + y)

9(3y + n)

3(3f + g)

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What’s the difference between

n + n + n

n x n x n

and

How would you simplify each? Are they equivalent?

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What terms can you simplify?

n + 4 4n + 4 4n + n

How do you know if you can combine terms?

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Simplify expressions by combining like termsEE.D2

ChallengingEE.D2

Independent PracticeEE.D2

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Review and Continue:

Use mathematical properties to generate and identify equivalent expressions.

EE.D3 Write expressions using variables from contextual problems.

Write an equivalent equation

3 + x = 5

2n + 5 = 15

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What does it mean to solve an equation?

3n + 4 = 16

Let’s solve it…. Using a bar model, and algebra

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Solving simple equations using a bar model and algebraic manipulation

4n + 6 = 14

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How can we write equivalent equations to help us solve?

Let’s do bar model first.

Discuss: Is this the same as 10/4 + 6/4? Talk about misconceptions in reducing…..

Solve this one

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Writing Expressions and equations from contextual problems EE.D3

Write an expression

Tom is 12 years old. Find his age after

a) 4 years b) 7 years c) x years

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Write an expression

A worker’s salary is $200 less than one-third of a manager’s salary. Find the worker’s salary when the manager’s

salary is

a) $7,200 b) $m

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Write an expression

The price of a papaya is $2 and the price of a mango is $1. Find the total price of

a) 4 papayas and 5 mangosb) x papayas and y mangos

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Write an expression

The price of a cup is $5 and the price of a plate is $12. Find the total price of

a) 5 cups and 6 platesb) N cups and M plates

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Report on conclusions and the reasoning behind them

Do you agree? Prove and explain why you agree or disagree

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Independent Practice EE.D3

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Learning Targets

I can solve equations arising from a context by first drawing a model and then writing an equation to represent the context.

Mathematical Practices

Make sense of problems and persevere in solving themModel with mathematics

Resource: Singapore 3.3-7a

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Review

The price of a cup is $5.50 and the price of a plate is $6.75. Find the total price of

a) 5 cups and 6 platesb) N cups and M plates

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Concept Review

Order of operations

Solve when x = 3 and n = 2

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The average score a student earned on 11

tests was 90. He averaged a score of 88 on the

first 5 tests he took, and the average of his

final 5 test scores was 94. What score did the

student earn on his 6th test?

Problem solving.. Use any method to solveEE.D4

Draw a model, write an equationSolve

At birth, baby George weighs 2 pounds lighter than his twin brother Harry.

a) Suppose Harry’s weight is x pounds, what is George’s weight in terms of x?

b) If the sum of their weight is 15 pounds, how much does each weigh?

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Draw a model, write an equationSolve

A store is selling a notebook computer at $90 more than 4 times the price of a mobile

phone.

a) Suppose the price of the mobile phone is $m. Express the price of the notebook in terms of

m.b) Given that the notebook computer costs $n,

write a formula connecting m and m

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Stella has $11 more than Roland. Together they have $159. How much money does Stella have? How much money does Roland have?

Draw a model, write an equationSolve

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Jared has twice as many DVDs as Eric. Together they have 51 DVDs. How many DVDs does Jared have? How many DVDs does Eric have?

Draw a model, write an equationSolve

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Draw a model, write an equationSolve

The sum of three consecutive integers is 405. What are the three numbers?

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Write an equationSolve

YOU TRY!Write an equation and solve

A corn plant was 4 inches tall. It grew 1.5 inches a day and is now 22 inches tall. How many days did it take?

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Learning Targets

I can differentiate between like and unlike termsI can use the distributive law and add and subtract linear equations

Mathematical Practices

Make sense of problems and persevere in solving themModel with mathematics

Resources: Singapore 4.1 & 4.2 – 7a

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Bell Work

Let x = 4, y = -2, z = 0

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Warm Up EE.D5

Draw a model, write an equation

Kylie is addressing invitations to her party. She has already addressed 17 of them. She finishes the rest of them in 4

hours. If she addressed a total of 81 invitations, how many invitations did she address per hour?

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Like and unlike terms

Review vocab: Terms, Coefficient

Like terms have the same variable structure: same variables with same exponents, but the coefficient need not be the same.

Constants are always “Like Terms”: 5, 24, -73

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Simplifying by adding or subtracting like terms

Examples

If two terms are “like” (have the same variable structure), you can add or subtract them, but add the coefficient in front of the variables. Any numbers without a variable (a constant), can just be added or subtracted.

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You Try EE.D5

The Distributive law of multiplication over addition

Solve 3(4 + 5) using an area model

Demonstrate using area models

Solve 2(4x + 5y) using an area model

Solve -x(4y + 5y) using an area model

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The Distributive Law in general

a(x + y) = ax + ay

a(x - y) = ax - ay

a(x + y + z) = ax + ay + az

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You Try; use the distributive property to remove parenthesis

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4x – x(3 + 2y) + xy

Harder… use distributive property and collect like terms

(2x + 3y) – (3x + 2y)

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Independent PracticeICP EE.D5

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Learning Targets

I can solve equations in one variable

Mathematical Practices

Make sense of problems and persevere in solving themModel with mathematics

Resources: Singapore 5.1 – 7a

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15 minute Skill Check EE.SC1

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Model and write an equationSolve

Sam is 4 less than twice Bill’s age. Jack is 3 years older than Bill. If the

sum of their ages is 47, how old are Sam, Bill & Jack?

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Solving Equations Using Algebra

When we solve an equation using algebra, we want to undo the operations on the unknown in order to find out what it is.

Remember! Keep the equation balanced by doing the same thing on both sides!!!

Example:

Solve 483 x

Explain the opposite of multiplication is division, etc…

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Solving Equations Using Algebra

Solve each equation

1.

2.

3.

682 y

2039 w

232

p

Do first problem with class, have students do 2 & 3.

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Solving Two-Step Equations

What’s the first step in solving this equation? Why?

How about this one? Why?

102

8

p

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Solving Two-Step Equations

Solve each equation

1.

2.

3.

33096 y

863

p

Do first problem with class, have students do 2 & 3.

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A word about notation

Are these equivalent?

What about these?

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Independent Practice

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Learning Targets

I can solve equations involving parenthesis in one variableI can rearrange equations to solve.

Mathematical Practices

Make sense of problems and persevere in solving themModel with mathematics

Resources: Singapore 5.2 – 7a

Bell work

a) The sum of 3 consecutive numbers is 234. Find the numbers

b) The sum of 3 consecutive even numbers is 192. Find the numbers

c) The sum of 3 consecutive odd numbers is 111. Find the numbers

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Review

Let x = 5, y = -3, z = 0

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Problem Solving

Kylie is addressing invitations to her graduation party. She has already addressed 17 of them. She finishes the rest of them in 4 hours. If she addressed a total of 81 invitations, how many invitations did she address per hour?

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Problem Solving

Can us solve with algebra, a model, or some other method?

104 chickens and goats in a farm have 246 legs altogether. How many of each type of animal are there?

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Solving Multi-Step Equations

Let’s review the Distributive Property for a moment…

The Distributive Property says

Simplify

)3(2 x

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Solving Equations with the Distributive Property

What should we do first to solve this equation? Why?

15)12(5 x

Explain we need both sides of equation as simplified as possible to make solving easier…

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Solving Equations with the Distributive Property

Solve each equation

1.

2.

3.

21)4(7 x

2115)4(32 x

10)42(54 xx

Do first problem with class, have students do 2 & 3.

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A little more challenging examples…. How does the distributive property apply here?

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Independent Practice

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Math 6EE.D8

Standards Assessed:

I can solve equations using algebraic manipulationI can create and solve equations arising from a context

Mathematical Practices:

Make sense of problems and persevere in solving themModel with mathematics

U4D8EE.D8

EE.D8Practice Solving proportions

Keep practicing!

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Write an equation and solve. Draw a model if necessary.

The length of a rectangle is 50 meters. This is 6 meters more than twice the width. Find the

width of the rectangle.

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You Try!

Write an equation and solve. Draw a model if necessary.

Twelve decreased by 8 times a number is 36. Find the number.

You Try!!

Write an equation and solve. Draw a model if necessary.

JoGrandpa is 75 years old. This is nine years less than seven times the age of Jo. How old is Jo?

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You Try!!

Write an equation and solve. Draw a model if necessary.

JoGrandpa is 75 years old. This is nine years less than seven times the age of Jo. How old is Jo?

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Write an equation and solve

A-1 Plumbing company pays Sam $45 per day, plus 10.50 for each hour he works. If he earned 566.25 last week, how many hours a day did he work.

(Sam works a 5 day week, and each day he works the same number of hours)

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Write an equation and solve

Lisa is bored, riding in the backseat of her parent’s SUV on their family vacation. She notices the mile markers on the interstate. She adds up three consecutive mile markers and discovers the sum is 312. Which three mile marker signs did her family just pass?

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Independent Practice

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Continue developing algebraic manipulation skills by solving inequalities

Expressing a solution to an inequality

Solutions to equations vs solutions to inequalities

How do we express a solution to the equations:

2n = 10

2n + 1 = 2n + 10

2n = 2n

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Inequalities!

• Vocab: An Inequality is a comparison of two values that may or may

not be equal.

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The Symbols and Their MeaningsSymbol Meaning

a > b a is greater than b

a ≥ b a is greater than or equal to b

a < b a is less than b

a ≤ b a is less than or equal to b

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What does it all mean!?Examples! Graph each inequality on a number line using or

1.

2.

3.

5x

2x

0x

21x

properties ws

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Solutions to an Inequality

• Remember that the solution to an equation gave us a single, unique result… but for inequalities…

• The solution to an inequality is a range of possibilities.

– Example: The solution of allows every real number that is bigger than 2. So we get to use any possible number we can think of as long as it is greater than 2.

2x

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Let’s try some….Examples! Solve each inequality and graph the solution on a

number line.1.

2.

3.

4.

12 x

x312

64 x

32

x

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You Try!

• Solve each inequality and graph the solution on a number line.

1.

2.

3.

86 x

1211 x

43

x

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Standards Addressed Today: Attend to Precision Reason Abstractly and Quantitatively Solve Inequalities Using Algebraic Manipulation Explain the Multiplication Property by Negative

Integers for Inequalities

Concept Review

• Solve each inequality and graph the solution on a number line.

1.

2.

3.

4.

113x

128 x

16

x

357 x

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Now for the tricky part…

• Consider the inequality 4 < 12

1. Divide both sides by 2. Is the inequality still true?

2. Now divide both sides by –2. Is the inequality still true?

3. Divide by 4.

4. Divide by –4.

5. Complete questions 1-4 on your half sheet.

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Solve each inequality

1.

2.

3.

4.

162 x

y48

13

x

3

212

w

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Challenge!

1.

2.

3.

4.

5.

6.

)74(22 w )76(226 ww

2

4816

w

xx 921153

3

16212

w

12

8

3

2

w

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Describe a situation

… that could be modeled by the inequality

103 x

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Work with a partner

• You are building a patio. You want to cover the patio with Spanish Tile that costs $5 per square foot. You budget for the tile is $1700. How wide can you make the patio without going over budget?

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Independent Practice

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EE.D11

Standards: Attend to Precision

Reason Abstractly and Quantitatively Solve Inequalities Using Algebraic

Manipulation Solve Inequalities Arising From a

Context

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Concept Review• Solve each equation

or inequality.

1.

2.

3.

2097

2x

Graph the solution to the inequality.

4.

5.

6.

)38(2)23(8 xx

7817 x

9)12(3 x

3

43

9

7 xx

46

13

x

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Concept Review

Write an inequality and solve.

• Four times a number is at least –48. What is the number?

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Problem Solving

Write an inequality and solve.

• Ilona is saving up to buy a digital camera that costs $495.75. So far, she has $175 saved. She would like to buy the camera 5 weeks from now. At least how much must she save every week to have enough money to purchase the camera?

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Use the definition of mean to write an inequality and solve

• You need a mean score of at least 90 to advance to the next round of the trivia game. What score do you need on the fifth game to advance?

Problem Solving

• For what values of x will the area of the blue region be greater than 12 square units?

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Try this on your own

a) For what values of y will the area of the trapezoid be less than or equal to 10 square units?

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In Class Practice

EE.D11 – ICP

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Unit Review

Solving equations and inequalities