Multiple Pathways To Success Quarter 2 Learning Module

Aligned with Maryland State Standards

Algebra 1 Linear and Exponential Functions

Copyright July 31, 2014— Written December 15, 2015 Prince George's County Public Schools

Board of Education of Prince George's County, Maryland

PGCPS Pfte.a,, e‘,0rece

Dear Scholars,

As you move through the Algebra I curriculum, the level of academic rigor will increase. This could potentially lead to gaps in your understanding. Therefore, this learning module has been designed to assist you in acquiring and strengthening the essential skills needed for successful completion of Algebra I Common Core. Your experiences with this module will also help to remediate misconceptions, confusion, and rebuild areas of weakness.

Sincerely,

Writers of the Multiple Pathways to Success Modules

Part I: Exponential Equations

Student Learning Outcomes • Write an exponential equation from a word problem • Solve an exponential equation from a word problem and interpret its solution

within the context • Write an exponential equation using two or more variables • Rewrite an exponential equation in terms of one variable (y = form

Mathematical Practices • MP1: Make sense of a problem and persevere in solving them • MP2: Reason abstractly and quantitatively • MP3: Construct a viable argument and critique the reasoning of others • MP4: Model with mathematics • MP6: Attend to precision • MP7: Look for and make use of structure

Resources/Websites • The following page contains the graphic organizer Common Vocabulary Associated

With Law of Exponents. This organizer provides the definitions of basic vocabulary terms associated with solving equations and offers visual representations of how the terms look are actually applied in mathematics.

The Youtube Video "Properties of Exponents" will simplify expressions using properties of exponents. (http://www.youtube.com/watch?v=0GAMbuPJGOY)

The Youtube Video "Negative Exponents" will simplify expressions using negative exponents. (https://www.youtube.com/watch?v=R7Yp5TW1 NTs)

The Youtube Video "Solving Exponential Equations f' will solve exponential equations when both sides of the equation can be written in the same base. (https://www.youtube.com/watch?v=aPyE9SKtczs)

Example

6'=ó

70= 1

44 = 1/4

2+3 =

00O3 =

(x/y)2 /y2

= 1/X3

Law of Exponents

ex11 = x111

xnvxn =

(xm)n

(xy)n = xnyn

(x/y)' = xn/yn

x = 1/x

And the law about Fractional Exponents:

Example A

If 2=2, then

'40 6; 1110 .V?

Example B

Solve for x.

Apply the exponential property of equality, when the bases are the same, the exponent can be set equal.

Apply Subtraction Property of Equality. x+ 1

Apply Division Property of Equal Solution.

Example C

Solve for n.

tY•

x+ 1 = 3x —x=3x—x 1 = iT

1 +2=2r+2

Since 27=3 Simplify.

then

= 272 3n = (33)2

311 =36

n = 6

Example D

Solve for x. 21r-1 =

Rewrite in a common base. 221.-1 = (23)' Simplify. 22x-1 = 2' Set exponents equal. 2x— 1 =3x Apply Subtraction Property of Equality. 2x — 1 — 2x = 3x — 2x Solution: —I =x

1. Use laws of exponents and simplify. Write your answers in positive exponents.

3k2 /-8

a (n6)5 b. s4 •

s10 d. 6k71-2

31- a 243 2. Solve each equation. =

3. Jeanette can invest $2000 at 3% interest compounded annually or she can

invest $1500 at 3.2% interest compounded annually. Which is the better

investment and why?

4. The value of an antique has increased exponentially, as shown in this graph.

Based on the graph, estimate to the nearest $50 the average rate of change in value of the antique for the following time intervals: a. From 0 to 20 years: $

b. From 20 to 40 years: $

Stock A Stock

(yowl)

St

5. Stock Prices The graphs show the estimated prices y (in dollars) of three different stocks x years after the stocks are offered to the public to buy.

a. Complete the table for each function.

Time Interval 0 to 1 yr 1 to 2 yr 2 to 3 yr 3 to 4 yr 4 to 5 yr 5 to 6 yr

Average Rate of Change (dollar/year)

b. Which stock has an average rate of change that is constant? Explain.

c. Describe the average rate of change of the two stocks whose average rate of change is not constant.

d. Which stock would you want to own? Explain your reasoning.

Part II: Linear Functions

Student Learning Outcomes:

• Identify, through experimenting with technology, the effect on the graph of a

function by replacing f(x) + k, kf(x), f(kx), and f(x + k) for specific values of k

(both positive and negative).

• Identify the parent function and shift of a graph of a linear function

• Create the equation for a graph after its translation

• Given the graphs of the original function and a transformation, students will

determine the value of (k).

• Describe the variables in linear functions in terms of a context (y-intercept is

the initial value and the slope is the rate of change

Mathematical Practices: • MP1: Make Sense of a Problem and Persevere • MP2: Reason Abstractly and Quantitatively • MP3: Construct a viable argument and critique the reasoning of others • MP4: Model with Mathematics • MP6: Attend to Precision • MP7: Make Use of Structure

Resources/Websites: • The next page contains the graphic organizer Common Vocabulary Associated With

Linear Functions. This organizer provides the definitions of basic vocabulary terms associated with graphing linear equations and offers visual representations of how the terms look are actually applied in mathematics.

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Graphing a linear function using a table (use this method when the equation is solved for y)

y 2x + 1

-a

0

rxrimai r goverpuy:- .ttrImpw 1ntorry01 F es linear I

Find the intercepts: Ax + By = C

po

1. Summarize the characteristics in common of the graphs below.

y = 2x + 3 y = 4x +3 y=-x+3 y= -10x+3

2. You sold a total of $48 worth of cheese. You forgot how many pounds of each type of cheese you sold. Swiss cheese costs $8 per pound. Cheddar cheese costs $6 per pound.

Pounds of Pounds of • Swiss pound cheddar

a. Let x represent the number of pounds of Swiss cheese_ Let y represent the number of pounds of cheddar cheese, Use the verbal model to write an equation that relates x and y.

b. Solve the equation fory. Then use a graphing calculator to graph the equation. Given the real-life context of the problem, find the domain and range of the function_

c. The x-intercept of a graph is the x-coordinate of a point where the graph crosses the x-axis. The pintercept of a graph is the y-coordinate of a point where the graph crosses the y-axis_ Use the graph to determine the and y-intercepts.

d. How could you use the equation you found in part (a) to determine the x- and y-intercepts? Explain your reasoning

e. Explain the meaning of the intercepts in the context of the problem_

Te

400 0

3 2 0 2.1-0 160

1 2 3 4 5 6

Number of weeks

3.

You conduct a biology experiment with 3 male rats and 3 female rats_ Each male rat weighs 200 grams at the beginning of the experiment. You feed the rats different amounts of food and supplements and weigh them each week You feed one of the mole rats as recommended_ This rat is called the male control rat The graph shows his weighty (in ounces) after x weeks_

a.

Does the graph represent a linear or nonlinear function? Explain_

b.

The female control rat gained weight at the same rate as the male control rat, but her weight was 40 grams lighter at the beginning of the experiment_ How does the graph of her weight compare to the graph of his weight? Justify your answer by writing and graphing an equation that represents her weight over time_

C.

One male rat received more food than recommended and also received supplements in his water. This rat's weight each week was 110% of the male control rat's weight. How does the graph of his weight compare to the graph of the male control rat's weight? Justify your answer by writing and graphing an equation that represents his weight over time.

d.

The other male rat received less food than reconmiended_ If f (x) represents the weight of the male control rat after x weeks and lc f (x)

represents the weight of the male receiving less food, would you expect k to be greater than 1, less than 1, or equal to 1? Explain your reasoning.

Part III: Solving Linear Systems of Equations and Inequalities

Student Learning Objectives:

Students will solve systems of equations using the elimination method sometimes called linear combinations). Students will solve a system of equations by substitution (solving for one variable in the first equation and substitution it into the second equation Solve a system of equations with one solution, no solution, or infinitely many solutions Justify the method used to solve the system of equations Check the solution of the system of equations in the original system of equation Show that multiplying an equation by a coefficient does not affect the solution of the system of equations Solve systems of equations using graphs. Solve a system of equations and check the solutions graphically (one solution, no solution, or infinitely many solutions) Explain that a solution to the system of equations is the intersection point on the graph Match the type of solution (one, no, many) to its corresponding graph (e.g. no solution would be a graph of parallel lines). Use technology to graph the equations and determine their point of intersection, using tables of values, orsuccessive approximations that become closer and closer to the actual value. Calculate the y value given an x value of an equation Identify the associated graph for an algebraic equation (linear and exponential) Substitute the coordinate (x, y) into y= f(x) and y = g(x) to verify that the point is a solution to the system Graph functions and find solutions to systems of equations using technology Prove a point is the solution to a linear/exponential system of equations algebraically Construct a table of values for any equation (linear and exponential) using technology Represent systems of linear inequalities as regions on the coordinate plane. Identify the bounded region for a system of inequalities. Determine if a given point is a solution of a system of inequalities.

Mathematical Practices: • MP1: Make Sense of a Problem and Persevere • MP2: Reason Abstractly and Quantitatively • MP3: Construct a viable argument and critique the reasoning of others • MP4: Model with Mathematics • MP6: Attend to Precision • MP7: Make Use of Structure

Resources/Websites: • The Khan Academy Video, "Converting Linear Equations to Slope-Intercept Form"

provides remediation about transforming equations for standard form to slope-intercept form.

httos://www.khanacademy.org/math/cc-eighth-grade-math/cc-8th-linear-equations-functions/8t h-slope/v/converting-to-slope-intercept-form

• The Khan Academy Video, "Solving Linear Systems by Graphing" will demonstrate how to solve linear systems through the use of the graphing. (https://www.khanacademy.org/math/algebra-basics/core-algebra-systems/core-algebr a-systems-tutorial/v/solving-linear-systems-by-graohinq)

• The Khan Academy Video," Solving Linear Systems by Substitution" will demonstrate how to solve linear systems through the use of substitution.

https://www.khanacademy.org/math/algebra-basics/core-algebra-systems/core-algebra-syste ms-tutorial/v/solving-linear-systems-by-substitution

• The Khan Academy Video, "How to Graph a System of Inequalities and Check the Solution" will demonstrate how to solve linear inequalities through graphing.

https://www.khanacademy.org/math/algebra/two-variable-linear-inequalities/graphing-ineaualiti

es/v/graphing-systems-of-inequalities-2

• For additional practice problems with system of equations and inequalities: http://agmath.com/media//D IR 11806/07 SystemsEquations2.pdf

• Powerpoint for Solving Systems of Equations and Inequalities https://drive.google.com/a/pgcps.org/folderview?id=0BwV4RDNmsGITnRRaXlacFpZM WM&usp=sharinq

Methods to Solve Linear Systems of Equations

GRAPHING

Parallel lines have

no solution because

there is no point of

intersection

NNN, Coinciding lines have an infinite

number of solutions (all the points on the line are solutions of

the system)

(x, )IY=Trix+b}

1.41..11M_

1. Graph Line I. 2. Graph Line 2. 3. Visually identify the

point of intersection.

Intersecting lines have

exactly one solution

which is the point of

intersection

(x, y)

1 Methods to Solve Systems of Linear Systems

SUBSTITUTION

I. Isolate one variable in one of the equations. 2. Substitute that expression into the second equation and solve for the variable. 3. Substitute the value found in step 2 into either of the original equations and solve for the remaininR variable. 4. Write the ordered pair.

2x + 3y = 5 4x - y = 17

COMBINATIONS

1. R.ev,Tiw each equation in standard form. 2. Choose a variable to eliminate and multiply by appropriate number to eliminate it. 3. Solve for remaining variable either by substituting into one of the original equations or by repeating step 2 for the other variable,. 4. Write the ordered pair.

How do you find the solution to a system of linear equations?

What if both of the variables cancel out? Look at the resulting arithmetic equation. *False statement indicates the lines are parallel so there is no solution.

4True statement indicates the lines coincide so there are infinite solutions.

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Graphing Linear Systems (A) Graph each system and identify its solution.

1. 3x — y 4

2. x±y=5

71: ± 2y = 18

2x + 7y = 0

Solution: , Solution: )

3. For each example below, use the window x: [-9.4,9.4], y:[-6.2,6.2], then complete the following steps. Write your answers in the space provided with each example.

a) Graph each system accurately using a graphing calculator and then graph paper. If you change an equation to slope-intercept form, show your steps neatly and clearly. b) Describe the lines using one of the following words: parallel, intersecting, or coinciding c) Tell how many solutions the system has, and then name the solution. d) If the lines are intersecting, verify that the given point satisfies both equations by substitution.

2x + 3y = 11 4x+ y=-3

Use substitution to solve each system.

4) 2x + 8y = 20

5) x = 5

y = 2

2x + y = 10

Solve using elimination.

6. -3x - y = -4

3x + 5y = 8

7. You sold two different types of wrapping paper for your band fund-raiser. One type

sold for $6 a roll and the other for $8 a roll. You collected a total of $92 for the 14 rolls you

sold. How many of each type were sold?

[mark all correct answers]

a. 10 of the $6 rolls

b. 8 of the $6 rolls

c. 4 of the $8 rolls

d. 10 of the $4 rolls

8. The drawer of a cash register contains some quarters and some

dollar bills. x = the number of quarter coins in the cash register.

1 I I I ILL

y = the number of dollar bills in the cash register.

The following two equations are true: 3x = y 4x + y = 70

a. Explain in words the meaning of each equation.

b. Find two pairs of values for x and y that satisfy the first equation.

c. Find two pairs of values for x and y that satisfy second equation.

d. Find pairs of values for x and y that satisfy both equations simultaneously.

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Maryland College and Career Readiness Standards

FI.IF.4 F.BF.1* For a function that models a relationship between two Write a function that describes a relationship between two

quantities, interpret key features of quantities.

the graph and the table in terms of a. Determine an explicit expression, a recursive process, or

the quantities, and sketch the steps for calculation from a context.

graph showing key features given a verbal description of the relationship. F.BF.3

(supporting)(cross-cutting)

(major) Identify the effect on the graph of replacing f(x) by f(x) + k, k

f(x), WO, and f(x + k) for specific values of (both positive and

F.IF.5 negative); find the value of k given the graphs. Experiment with

Relate the domain of a function to its graph and, where applicable, to

cases and illustrate an

explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. (additional) (cross-cutting)

F.IF.7*

Graph functions expressed symbolically and show key features of the graph, by hand in simple cases using technology for more complicated cases.

a. Graph linear functions and show intercepts (supporting)

F.LE.1 Distinguish between situations that can be modeled with

linear functions

a. Prove that linear functions grow by equal

differences over equal intervals

b. Recognize situations in which one quantity

(supporting)

the quantitative relationship it describes. (major)

F.IF.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. (major)

F.LE.2 Construct linear functions, given a graph, a description of

a relationship, or two input-output pairs (include reading

these from a table) (supporting)(cross-cutting)

F.LE.5*

Interpret the parameters in a linear function in terms of a context (supporting)(cross-cutting)

A.RE1.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. (additional) (cross-cutting)

A.REI.11*

Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., technology to graph the functions, make tables of values, or find

successive approximations. Include cases where f(x) and g(x) are linear functions. (major)(cross-cutting)

A.REI.12 Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes. (major)

Scoring Rubric / Success Criteria

Conceptual Understanding

38 Total Points

Part I: Exponential Equations (la, lb, lc, Id, 2, 3, 4a, 4b, 5a, 5b, 5c, 5d)

12

One point for each part of each problem

Part II: Linear Functions (1, 2a, 2b, 2c, 2d, 2e, 3a, 3b, 3c, 3d)

10

One point for each part of each problem

Part Ill: Solving Linear Systems of Equations and Inequalities

(1, 2, 3a, 3b, 3c, 3d, 4, 5, 6, 7a, 7b, 7c, 7d, 8, 9, 10)

16 One point for each part of each problem

Execution of Mathematical Practices

12 Total Points

MPl: Make sense of a problem and persevere in solving them

• Analyze and explain the meaning of the

problem

• Actively engage in problem solving

(Develop, carry out, and refine a plan)

2

one point per bullet

MP2: Reason abstractly and quantitatively

• Represent a problem with symbols

• Explain their thinking

• Examine the reasonableness of their

answers/calculations

3

one point per bullet

MP3: Construct a viable argument and critique the reasoning of others

• Justify solutions and approaches

1

MP4: Model with mathematics 2

• Use representations to solve real life

problems

• Apply formulas and equations where

appropriate

one point per bullet

MP6: Attend to precision

• Calculate accurately and efficiently

• Explain their thinking using mathematics

vocabulary

• Use appropriate symbols and specify units

of measure

3

one point per bullet

MP7: Look for and make use of structure

• Use knowledge of properties to efficiently

solve problems

1

Final Score /50