Equations with the Same Equations with the Same How could you start - - PDF document

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Algebra I

Equations

2015-08-21 www.njctl.org

Slide 3 / 79 Table of Contents

Click on a topic to go to that section.

· Equations with the Same Variable on Both Sides · Solving Literal Equations · Glossary & Standards · Substituting Values into an Equation

Slide 3 (Answer) / 79 Table of Contents

Click on a topic to go to that section.

· Equations with the Same Variable on Both Sides · Solving Literal Equations · Glossary & Standards · Substituting Values into an Equation

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Teacher Notes

Vocabulary Words are bolded in the presentation. The text box the word is in is then linked to the page at the end

• f the presentation with the

word defined on it.

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Equations with the Same Variable on Both Sides

Return to Table

• f Contents

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Equations with the Same Variable on Both Sides

Return to Table

• f Contents

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Math Practice

This lesson addresses MP1, MP6 & MP7. Additional Q's to address MP standards: How could you start this problem? (MP1) What operation is given in the problem? (MP1) What do you know about inverse

• perations that apply to this question?

(MP7)

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Now, we will be given an equation with the same variable on both

• sides. These equations will look similar to the following:

These require one additional step to get all the terms with that variable to one side or the other. It doesn't matter which side you choose to move the variables to, but it’s typically most helpful to choose the side in which the coefficient of the variable will remain positive.

Variables on Both Sides

Previously, you solved equations with variables on one side, similar to the following:

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Now, we will be given an equation with the same variable on both

• sides. These equations will look similar to the following:

These require one additional step to get all the terms with that variable to one side or the other. It doesn't matter which side you choose to move the variables to, but it’s typically most helpful to choose the side in which the coefficient of the variable will remain positive.

Variables on Both Sides

Previously, you solved equations with variables on one side, similar to the following:

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Math Practice

MP6: Attend to precision. Emphasize performing the inverse

• peration to BOTH sides of the equation.

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When you have finished solving, discuss the meaning of your answer with your neighbor.

Meaning of Solutions

Before we encounter the new equations, let's practice how to solve an equation with the variable on only one side. Solve for x:

Slide 6 (Answer) / 79 Slide 7 / 79 Meaning of Solutions

Remember that you always have the ability to check your answers by substituting the value you solved for back in to the original equation. It isn't necessary to show on each problem, but is encouraged if you feel unsure about your answer.

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Which side do you think would be easiest to move the variables to?

Variables on Both Sides

Slide 8 (Answer) / 79 Slide 9 / 79 Variables on Both Sides

Which side do you think would be easiest to move the variables to?

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Example: What do you think about this equation? What is the value of x?

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Example: What do you think about this equation? What is the value of x?

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1 Solve for f:

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2 Solve for h:

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3 Solve for x:

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Sometimes, you get an interesting answer. What do you think about this? What is the value of x? 3x - 1 = 3x + 1

• 3x
• 3x
• 1 = +1

Since the equation is false, there is no solution! No value will make this equation true.

No Solution Slide 16 / 79

How about this one? What do you think about this? What is the value of x? 3(x - 1) = 3x - 3 3x - 3 = 3x - 3

• 3x
• 3x
• 3 = -3

Since the equation is true, there are infinitely many solutions! The equation is called an identity. Any value will make this equation true.

Identity Slide 17 / 79

4 Solve for r: A r = 0 B r = 2 C infinitely many solutions (identity) D no solution

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4 Solve for r: A r = 0 B r = 2 C infinitely many solutions (identity) D no solution

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Answer

C infinitely many solutions (identity)

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5 Solve for w: A w = -8 B w = -1 C infinitely many solutions (identity) D no solution

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6 Solve for x: A x = 0 B x = 24 C infinitely many solutions (identity) D no solution

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6 Solve for x: A x = 0 B x = 24 C infinitely many solutions (identity) D no solution

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Answer

D no solution

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8 In the accompanying diagram, the perimeter of ∆MNO is equal to the perimeter of square ABCD. If the sides of the triangle are represented by 4x + 4, 5x - 3, and 17, and one side of the square is represented by 3x, find the length of a side of the square.

5x – 3 4x + 4 17 O M N 3x A B C D

From the New York State Education Department. Office of Assessment Policy, Development and

• Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.

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8 In the accompanying diagram, the perimeter of ∆MNO is equal to the perimeter of square ABCD. If the sides of the triangle are represented by 4x + 4, 5x - 3, and 17, and one side of the square is represented by 3x, find the length of a side of the square.

5x – 3 4x + 4 17 O M N 3x A B C D

From the New York State Education Department. Office of Assessment Policy, Development and

• Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.

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Answer

18

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· When solving an equation with variables on both sides, choose a side to move all of them to, then continue working to isolate the variable. · When solving an equation where all variables are eliminated and the remaining equation is false, there is No Solution. · When solving an equation where all variables are eliminated and the remaining equation is true, there are Infinite Solutions.

RECAP Slide 23 / 79

Solving Literal Equations

Return to Table

• f Contents

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Solving Literal Equations

Return to Table

• f Contents

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Math Practice

This lesson addresses MP1, MP2, MP6 & MP7. Additional Q's to address MP standards: How could you start this problem? (MP1) What operation is given in the problem? (MP1) What do you know about inverse

• perations that apply to this question?

(MP7)

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A good example is , which you may have seen in your physics course. Another example is which we use when studying geometry. In some cases, it is actually easier to work with literal equations since there are only variables and no numbers.

Literal Equations

A literal equation is an equation in which known quantities are expressed either wholly or in part by using letters.

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The variables in this equation are s, d and t. Solving for a variable means having it alone, or isolated. This equation is currently solved for s.

Literal Equations

Our goal is to be able to solve any equation for any variable that appears in it. Let's look at a simple equation first.

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When solving a literal equation you will be asked to isolate a particular variable in the equation. For example, with the formula: you might be asked to solve for p. This means that p will be on one side of the equation by itself. The new formula will look this: You can transform a formula to describe one quantity in terms of the

• thers by following the same steps as solving an equation.

Literal Equations Slide 26 (Answer) / 79

When solving a literal equation you will be asked to isolate a particular variable in the equation. For example, with the formula: you might be asked to solve for p. This means that p will be on one side of the equation by itself. The new formula will look this: You can transform a formula to describe one quantity in terms of the

• thers by following the same steps as solving an equation.

Literal Equations

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Teacher Notes Work through the steps with students, to show them that the same rules apply when moving variables as numbers.

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• 2. You can do anything you want (except divide by zero) to one

side of an equation, as long as you do the same thing to the

• ther.
• 3. If there is more than one operation going on, you must undo

them in the opposite order in which you would do them, the

• pposite of the "order of operations."
• 4. You can always switch the left and right sides of an equation.

Tips for Solving Equations

• 1. To "undo" a mathematical operation, you must perform the

inverse operation.

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• 1. To "undo" a mathematical operation, you must do the opposite.

We learned earlier that for every mathematics operation, there is an inverse operation which undoes it: when you do both

• perations, you get back to where you started.

When the variable for which we are solving is connected to something else by a mathematical operation, we can eliminate that connection by using the inverse of that operation.

Tips Explained Slide 29 / 79

• 2. You can do anything you want (except divide by zero) to one

side of an equation, as long as you do the same to the other side. If the two expressions on the opposite sides of the equal sign are equal to begin with, they will continue to be equal if you do the same mathematical operation to both of them. This allows you to use an inverse operation on one side, to undo an operation, as long as you also do it on the other side. You can just never divide by zero (or by something which turns out to be zero) since the result of that is always undefined.

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• 3. If there is more than one operation going on, you must undo

them in the opposite order in which you would do them, the

• pposite of the "order of operations."

The operations which are connected to a variable must be "undone" in the reverse order from the Order of Operations. So, when solving for a variable, you: first have to undo addition/subtraction, then multiplication/division, then exponents/roots, finally parentheses. The order of the steps you take to untie a knot are the reverse of the order used to tie it.

Tips Explained

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• 4. You can always switch the left and right sides of an equation.

Once an equation has been solved for a variable, it is typically easier to use if that variable is moved to the left side. Mathematically, this has no effect since the both sides are equal.

Tips Explained Slide 32 / 79

Examples: Solve for : Solve for :

Literal Equations Slide 32 (Answer) / 79

Examples: Solve for : Solve for :

Literal Equations

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Answer

*Some students may need to see the division take place in 2 steps; before seeing you can divide by both variables at the same time.

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Let's solve this equation for d That means that when we're done we'll have d isolated.

Practice Solving for a Variable Slide 34 / 79

9 First, is d already alone? If not, what is with it? A s B d C t D it is already alone

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9 First, is d already alone? If not, what is with it? A s B d C t D it is already alone

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Answer

C

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10 What mathematical operation connects d and t? A d is added to t B d is multiplied by t C d is divided by t D t is subtracted from d

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10 What mathematical operation connects d and t? A d is added to t B d is multiplied by t C d is divided by t D t is subtracted from d

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Answer

C

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11 What is the opposite of dividing d by t? A dividing t by d B dividing by s into t C multiplying d by t D multiplying by t by d

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11 What is the opposite of dividing d by t? A dividing t by d B dividing by s into t C multiplying d by t D multiplying by t by d

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Answer

C

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12 What must we also do if we multiply the right side by t?

A divide the left side by t B multiply the left side by t C divide the left side by d D divide the left side by d

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12 What must we also do if we multiply the right side by t?

A divide the left side by t B multiply the left side by t C divide the left side by d D divide the left side by d

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Answer

B

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13 Is there more than one mathematical operation acting

• n d?

Yes No

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13 Is there more than one mathematical operation acting

• n d?

Yes No

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Answer

No

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14 What is the final equation, solved for d? A B C D

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14 What is the final equation, solved for d? A B C D

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Answer

B

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A common kinematics equation is given below. Although they use the same letter, and are different variables. Let's solve this equation for .

Solving for vo

is pronounced "vee naught" & represents the initial (or starting) velocity.

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16 Is already alone? If not, what is with it? A only a B only t C a and t D it is already alone

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16 Is already alone? If not, what is with it? A only a B only t C a and t D it is already alone

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Answer

C

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17 What mathematical operation connects a and t to ? A at is being divided by vo B at is being added to vo C vo is being multiplied by at D vo is being divided by at

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17 What mathematical operation connects a and t to ? A at is being divided by vo B at is being added to vo C vo is being multiplied by at D vo is being divided by at

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Answer

B

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18 What is the opposite of adding at to ? A dividing by vo by at into t B subtracting vo from at C subtracting at from vo D dividing at by vo

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18 What is the opposite of adding at to ? A dividing by vo by at into t B subtracting vo from at C subtracting at from vo D dividing at by vo

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Answer

C

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19 What must we do, if we subtract at from the right side?

A add at to the left side B multiply the left side by at C subtract at from the left side D divide the left side by vo

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19 What must we do, if we subtract at from the right side?

A add at to the left side B multiply the left side by at C subtract at from the left side D divide the left side by vo

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Answer

C

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20 Is there more than one mathematical operation acting

• n ?

Yes No

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20 Is there more than one mathematical operation acting

• n ?

Yes No

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Answer

No

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21 What is your final equation for ? A B C D

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21 What is your final equation for ? A B C D

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Answer

D

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22 Which of the following correctly shows the equation solved for the variable a? A B C D

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22 Which of the following correctly shows the equation solved for the variable a? A B C D

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Answer

C

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23 To convert Fahrenheit temperature to Celsius you use the formula: A B C D Which of the following shows the equation correctly solved for F?

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23 To convert Fahrenheit temperature to Celsius you use the formula: A B C D Which of the following shows the equation correctly solved for F?

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Answer

B

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24 Solve for h: A B C D

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24 Solve for h: A B C D

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Answer

B Slide 51 / 79

Literal Equation questions may be posed in various ways, while still wanting you to isolate a variable. You may encounter some

• f the following phrases.

· Which equation is equivalent... · Solve for ___ in terms of ____ · Isolate the variable ___ · Transform the formula to find ___ · ___ is given by... Remember, the steps for solving all remain the same!

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25 If A represents the area of a circular horse corral, the following equation correctly shows , solved for r. True False

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25 If A represents the area of a circular horse corral, the following equation correctly shows , solved for r. True False

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Answer TRUE Good question for discussion. While it is true that when you initially solve for r, you must include the ± version, we are given a real life situation in where the radius of the horse corral cannot be negative.

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26 Solve for t in terms of s: A B C D

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26 Solve for t in terms of s: A B C D

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Answer

A

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27 A satellite's speed as it orbits the Earth is found using the formula . In this formula, m stands for the mass of the Earth. Transform this formula to find the mass of the Earth. A B C D

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27 A satellite's speed as it orbits the Earth is found using the formula . In this formula, m stands for the mass of the Earth. Transform this formula to find the mass of the Earth. A B C D

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Answer

D

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28 Which equation is equivalent to ? A B C D

From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.

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28 Which equation is equivalent to ? A B C D

From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011. [This object is a pull tab]

Answer

C

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30 The formula for finding the perimeter, P, of a rectangle with length l and with width w is given. Which formula shows how the length of a rectangle can be determined from the perimeter and the width? A B C D

From PARCC PBA sample test calculator #1

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30 The formula for finding the perimeter, P, of a rectangle with length l and with width w is given. Which formula shows how the length of a rectangle can be determined from the perimeter and the width? A B C D

From PARCC PBA sample test calculator #1

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Answer

B Slide 58 / 79

31 Caroline knows the height and the required volume of a cone-shaped vase she's designing. Which formula can she use to determine the radius of the vase? Recall the formula for volume of a cone: Select the correct answer. A B C D

From PARCC EOY sample test calculator #8

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31 Caroline knows the height and the required volume of a cone-shaped vase she's designing. Which formula can she use to determine the radius of the vase? Recall the formula for volume of a cone: Select the correct answer. A B C D

From PARCC EOY sample test calculator #8

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Answer

B Slide 59 / 79

Substituting Values into an Equation

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Substituting Values into an Equation

Return to Table

• f Contents

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Math Practice

This lesson addresses MP1, MP2, MP3, MP4, MP6 & MP7. Additional Q's to address MP standards: How could you start this problem? (MP1) What operation is given in the problem? (MP1) How does Substitution relate to Evaluating Equations and Literal Equations? What do you know about inverse

• perations that apply to this question?

(MP7)

Slide 60 / 79 Evaluating Expressions

In previous courses you have learned to evaluate expressions given the values for specific variables. Recall - Evaluate given In this section we will extend that knowledge to include literal equations, and use substitution to solve for unknown quantities.

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The value of any variable in an equation depends on the values of the other variables. To find an unknown value:

• 1. Identify an equation, if not given to you, which relates the values of

the variables you know with that of the variable you don't know.

• 2. Solve for the variable of interest.
• 3. Substitute numbers for the known variables (using parentheses

around each number).

• 4. Then do the arithmetic to find the unknown value.
• 5. Assign units to solution, if necessary.

Solving for Unknowns Slide 62 / 79 Slide 62 (Answer) / 79 Slide 63 / 79 Solving for Unknowns

The units to the solution in the last question turned out to be . Discuss with your neighbor why this was the case.

Slide 63 (Answer) / 79 Solving for Unknowns

The units to the solution in the last question turned out to be . Discuss with your neighbor why this was the case.

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Teacher Notes

no units feet feet

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Example: A car travels 800m in 480s. At what speed was it traveling?

• 1. Identify a useful equation:
• 2. Solve for the unknown:
• 3. Substitute known values:
• 4. Calculate:
• 5. Assign units:

(Can you think of any equations to relate distance and time?)

Solving for Unknowns Slide 64 (Answer) / 79

Example: A car travels 800m in 480s. At what speed was it traveling?

• 1. Identify a useful equation:
• 2. Solve for the unknown:
• 3. Substitute known values:
• 4. Calculate:
• 5. Assign units:

(Can you think of any equations to relate distance and time?)

Solving for Unknowns

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Answer The equation is already solved for s.

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Example: A car travels at a speed of 75 miles/hour for 1.5 hours. How far did it travel?

• 1. Identify a useful equation:
• 2. Solve for the unknown:
• 3. Substitute known values for variables:
• 4. Calculate:
• 5. Apply units:

Solving for Unknowns Slide 65 (Answer) / 79

Example: A car travels at a speed of 75 miles/hour for 1.5 hours. How far did it travel?

• 1. Identify a useful equation:
• 2. Solve for the unknown:
• 3. Substitute known values for variables:
• 4. Calculate:
• 5. Apply units:

Solving for Unknowns

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Answer

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Acceleration is found using the following formula, which takes the change in velocity over time. Turn to a partner, where do you hear about acceleration outside of class?

Acceleration

acceleration = change of velocity elapsed time

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Acceleration is found using the following formula, which takes the change in velocity over time. Turn to a partner, where do you hear about acceleration outside of class?

Acceleration

acceleration = change of velocity elapsed time

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Teacher Notes Students in PSI Algebra based Physics are learning about acceleration in their first unit, Kinematics. They may have heard about acceleration when referring to cars, roller coasters, gravity, etc.

Slide 67 / 79 Units for Acceleration

You can derive the units for acceleration by substituting the correct units into the right hand side of the equation. Change in velocity (v - v0) is in meters/second (m/s) Time, t, is in seconds (s) Acceleration, a, is in meters/seconds (m/s2)

Slide 67 (Answer) / 79 Units for Acceleration

You can derive the units for acceleration by substituting the correct units into the right hand side of the equation. Change in velocity (v - v0) is in meters/second (m/s) Time, t, is in seconds (s) Acceleration, a, is in meters/seconds (m/s2)

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Teacher Notes MP6: Attend to Precision On the next 7 slides, the students are only entering the numerical answer with their Responder, so emphasize the correct units of measurement for each question with the class, since not all responders let you put in units.

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32 A particle traveled for 10 seconds at a rate of 32 m/s. How far did the particle travel?

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32 A particle traveled for 10 seconds at a rate of 32 m/s. How far did the particle travel?

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Answer d = rt d = 32(10) d = 320 m

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33 A particle traveled for 2.5 seconds at a rate of 25 m/s. How far did the particle travel?

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33 A particle traveled for 2.5 seconds at a rate of 25 m/s. How far did the particle travel?

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Answer d = rt d = 25(2.5) d = 62.5 m

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34 A particle increased its speed from 18 m/s to 98 m/s in 25

• seconds. What is the acceleration of the particle?

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34 A particle increased its speed from 18 m/s to 98 m/s in 25

• seconds. What is the acceleration of the particle?

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Answer a = (v - vo) / t a = (98 - 18) /(25) a = 3.2 m/s2

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35 A particle increased its speed from 20 m/s to 65 m/s in 40

• seconds. What is the acceleration of the particle?

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35 A particle increased its speed from 20 m/s to 65 m/s in 40

• seconds. What is the acceleration of the particle?

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Answer a = (v - vo) / t a = (65 - 20) /(40) a = 1.125 m/s2

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36 A particle increased its speed from 12 m/s to 87 m/s in 30

• seconds. What is the acceleration of the particle?

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36 A particle increased its speed from 12 m/s to 87 m/s in 30

• seconds. What is the acceleration of the particle?

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Answer a = (v - vo) / t a = (87 - 12) /(30) a = 2.5 m/s2

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Glossary & Standards

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Glossary & Standards

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Teacher Notes

Vocabulary Words are bolded in the presentation. The text box the word is in is then linked to the page at the end

• f the presentation with the

word defined on it.

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Back to Instruction

Equation

A mathematical statement, in symbols, that two things are exactly the same (or equivalent).

4x + 2 = 14 3y + 2 = 11 11 - 1 = 3z + 1 7x = 21 a.k.a. function d = rt

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Back to Instruction

Identity

An equation that has infinitely many solutions.

3(x - 1) = 3x - 3

3x - 3 = 3x - 3

• 3x -3x
• 3 = -3

7(2x + 1) = 14x + 7 14x + 7 = 14x + 7

• 14x -14x

7 = 7

3x - 1 = 3x + 1

• 3x -3x
• 1 = +1

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Back to Instruction

Inverse Operation

The operation that reverses the effect of another operation.

Addition Subtraction Multiplication Division + _ x ÷ 11 = 3y + 2

• 2
• 2

9 = 3y ÷ 3 ÷ 3 3 = y

• 5 + x = 5

x = 10 + 5 + 5

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Back to Instruction

Literal Equation

An equation in which known quantities are expressed either wholly or in part by means

• f letters.

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Back to Instruction

No Solution

An equation that is false.

3x - 1 = 3x + 1

• 3x -3x
• 1 = +1

8x - 4 = 8x + 6

• 8x -8x
• 4 = 6

3(x - 1) = 3x - 3 3x - 3 = 3x - 3

• 3x -3x
• 3 = -3

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Throughout this unit, the Standards for Mathematical Practice are used. MP1: Making sense of problems & persevere in solving them. MP2: Reason abstractly & quantitatively. MP3: Construct viable arguments and critique the reasoning of

• thers.

MP4: Model with mathematics. MP6: Attend to precision. MP7: Look for & make use of structure. MP8: Look for & express regularity in repeated reasoning. Additional questions are included on the slides using the "Math Practice" pull-tabs (e.g. a blank one is shown to the right on this slide) with a reference to the standards used. If questions already exist on a slide, then the specific MPs that the questions address are listed in the pull-tab.

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Throughout this unit, the Standards for Mathematical Practice are used. MP1: Making sense of problems & persevere in solving them. MP2: Reason abstractly & quantitatively. MP3: Construct viable arguments and critique the reasoning of

• thers.

MP4: Model with mathematics. MP6: Attend to precision. MP7: Look for & make use of structure. MP8: Look for & express regularity in repeated reasoning. Additional questions are included on the slides using the "Math Practice" pull-tabs (e.g. a blank one is shown to the right on this slide) with a reference to the standards used. If questions already exist on a slide, then the specific MPs that the questions address are listed in the pull-tab.

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Math Practice