ASE MA 3 Algebra Functions and ModelingSteve Schmidt, ABSPD, abspd.appstate.edu, schmidtsj@appstate.edu

Workshop Focus: This course will focus on building the algebra knowledge adult students need for adult secondary education, college, and career success.

Workshop Goals:

Understand and use the Standards for Mathematical Practice to develop mathematically proficient students

Understand concepts necessary to successfully complete adult high school courses and pass high school equivalency exams

Identify course content requirements based on adult education standards

Understand what instructors should know and be able to do to deliver effective instruction

Know how to use technology, including the TI 30 XS calculator, to enhance learning

Know how to design and plan instruction based on assessment results

Know where to locate supplemental resources

Workshop Schedule

8:30 - 10:00 Welcome and introductionsAHS/Modeling/GED 2014 Overview Webb’s DOKFunctions Intro

10:00 – 10:15 Break

10:15 – 11:45 Looking at functions 4 ways Functions Matching

Functions Foldables

11:45 – 12:45 Lunch

12:45 – 2:00 Algebra TilesModeling FunctionsWindow Washing Model

2:00 – 2:15 Break

2:15 – 3:40 Modeling Functions Polya Method

3:40 – 4:00 Resource sharingTake AwaysProfessional Development

1

Standards for Mathematical Practice

1. Makes sense of problems and perseveres in solving them

• Understands the meaning of the problem and looks for entry points to solve it

• Analyzes information (givens, constraints, relationships, goals)

• Designs a plan• Monitors and evaluates progress and

changes course as necessary • Checks their answers to problems and asks,

“Does this make sense?”

2. Reason abstractly and quantitatively • Makes sense of quantities and relationships• Represents a problem symbolically • Considers the units involved• Understands and uses properties of

operations

3. Construct viable arguments and critique the reasoning of others

• Uses definitions and previously established results in making arguments

• Makes conjectures and attempts to prove or disprove through examples and counterexamples

• Communicates and defends their mathematical reasoning using objects, drawings, diagrams, and actions

• Listens or reads the arguments of others• Decides if the arguments of others make

sense• Asks useful questions to clarify or improve the

arguments

4. Model with mathematics• Applies reasoning to create a plan or analyzes

a real world problem• Applies formulas/equations

• Makes assumptions and approximations to make a problem simpler

• Checks to see if an answer makes sense and changes a model when necessary

5. Use appropriate tools strategically• Identifies relevant math resources and uses

them to pose or solve problems• Makes sound decisions about the use of

specific tools. Examples may include:• Calculator• Concrete models• Digital Technology• Pencil/paper• Ruler, compass, protractor

• Uses technological tools to explore and deepen understanding of concepts

6. Attend to precision• Communicates precisely using clear

definitions • Provides carefully formulated explanations• States the meaning of symbols, calculates

accurately and efficiently • Labels accurately when measuring and

graphing

7. Look for and make use of structure• Looks for patterns or structure• Recognizes the significance in concepts and

models and can apply strategies for solving related problems

• Looks for the big picture or overview

8. Look for and express regularity in repeated reasoning

• Notices repeated calculations and looks for general methods and shortcuts

• Continually evaluates the reasonableness of their results while attending to details and makes generalizations based on findings

• Solves problems arising in everyday life

Adapted from Common Core Standards for Mathematics: Standards for Mathematical Practice

2

GED 2002 and 2014 Math Test Comparison

GED 2002 GED 2014- There are 50 multiple choice questions

- Each multiple choice question has 5 possible answer choices

- The test is primarily multiple choice with about 46 items

- Each multiple choice question has 4 possible answer choices- There are fill-in-the-blank, drop down, hot spot, and drag and drop items

Test takers are given 90 minutes to complete the test

Test takers are given 90 minutes to complete the test

A formula page is provided in the test booklet

A new formula page has more algebra and surface area but fewer basic formulas

The test is designed to measure high school equivalency

The test measures high school equivalency and college and career readiness

There are 2 sections, one that allows the use of the Casio FX260

There is an approximate 12 minute non- calculator section (first 5 questions) and 78 minute calculator active section using the TI30XS Virtual Calculator

Each of the following content areas are about 25%: Number sense and operations, Measurement and geometry, Data statistics and probability, Algebra functions and patterns

The content is 45% quantitative problem solving (number operations and geometric thinking) and 55% algebraic problem solving that are presented in academic and workforce contexts

410 is the lowest passing score, students need to average 450 to pass the total test battery

Scaled scores will range from 100 to 200, the high school equivalency passing score is 150 per test (600 total score), a higher college and career readiness score will be determined in late 2014

Source: GED Testing Assessment Guide

3

4

5

GED 2014 Technology Skills Keyboarding

Can your students:

☐ Type approximately 25 words per minute, fast enough to complete a 500 word constructed response in 45 minutes?

☐ Use the keys: space bar, return/enter, shift, arrows, delete, backspace, tab? ☐ Use the numbers and punctuation keys?

Using a Mouse

Can your students use a mouse to:

☐ Point to an area or word on screen? ☐ Left click, right click, double click? ☐ Highlight and select text? ☐ Cut, copy, and paste text? ☐ Select an object or window by clicking on it? ☐ Drag an object to a new location? ☐ Open/close a pop-up window? ☐ Select one or more radio and/or check buttons? ☐ Scroll horizontally and vertically?

Navigation

Can your students:

☐ Use the screen navigation commands previous/next? ☐ Navigate multiple windows? ☐ Navigate toolbars and drop-down menus? ☐ Navigate screen tabs? ☐ Use the undo and redo operations? ☐ Mark a section/question as a “Flag for Review” and return to section/question as needed?

Free Technology Resources

GCF Learn FreeThis free site guides students in learning how to use a mouse and computer basics as well as helping

students learn Microsoft Office applications like Word and Excel.

Google: gcf learn free or go to www.gcflearnfree.org

Typing WebThis free site helps students learn keyboarding. Instructors can register students to monitor their

progress. Google: typing web or go to www.typingweb.com

Shifts in the Mathematics Common Core Standards

6

Prior to the common core, state standards were “an inch deep and a mile wide.” Instructors would try to move through a huge amount of material at a quick pace in order to “cover” everything by the end of the year. This led to students with a superficial grasp of a variety of concepts. When they attempted the college placement test and postsecondary math, they performed poorly. A recent study on developmental math in North Carolina showed that only one out of five students completed the developmental math sequence and took college level math classes.

Studies of math instruction in countries that consistently perform well on international math tests showed three major things: focus, coherence, and rigor. Instead of trying to cover everything, these countries focused on a deeper understanding of a smaller set of skills. There is also a coherent progression, a logical pattern where similar topics are explored each year at a greater depth with the ultimate goal of algebra instruction. Developing students’ algebraic thinking skills are a major theme throughout math instruction. Finally, there is increased rigor that encourages students to think more deeply and engage in active problem solving instead of relying on teachers to just give students the answer.

Standards Shift 1: Focus on… Key ideas, understandings, and skills

Deep learning of concepts stressed

Fewer concepts with more depth

Standards Shift 2 - Coherence… Designing learning around coherent progressions level to level

o Coherence for mastery

o All roads lead to algebraic thinking – abstract reasoning

Standards Shift 3 - Rigor… The level of rigor is increasing

Pursuing conceptual understanding, procedural skill and fluency, and application

Other Notes:

Subject

Webb’s Depth of KnowledgeLevel 1 Level 2 Level 3

7

Mat

hem

atic

sRequires students to recall or observe facts, definitions, or terms. Involves simple one-step procedures. Involves computing simple algorithms (e.g. sum, quotient).

Examples:

Recall or recognize a fact, term or property

Represent in words, pictures or symbols in a math object or relationship

Perform routine procedure like measuring

Requires students to make decisions of how to approach a problem. Requires students to compare, classify, organize, estimate, or order data. Typically involves two-step procedures.

Examples:

Specify and explain relationships between facts, terms, properties or operations

Select procedure according to criteria and perform it

Solve routine multiple-step problems

Requires reasoning, planning, or use of evidence to solve problem or algorithm. May involve activity with more than one possible answer. Requires conjecture or restructuring of problems. Involves drawing conclusions from observations, citing evidence, and developing logical arguments for concepts. Uses concepts to solve non-routine problems.

Examples:

Analyze similarities and differences between procedures

Formulate original problem given situation

Formulate mathematical model for complex situation

Source GED 2014 Assessment Guide

Solve each problem and identify the Depth of Knowledge for each question.

Algebra 1 EOC: Webb’s Depth of Knowledge Level _______

1. Katie and Jennifer are playing a game.

Katie and Jennifer each started with 100 points. At the end of each turn, Katie’s points doubled. At the end of each turn, Jennifer’s points increased by 200.

At the start of which turn will Katie first have more points than Jennifer?

GED 2014: Webb’s Depth of Knowledge Level _______

2. Hartley opened a food truck business to sell food on the street. On day 2, the business earned $112. On day 5, the business earned $367. Hartley assumes that the earnings will continue to increase at the same rate. How much will the business earn on day 10?

$GED 2014 Algebra Assessment Targets

8

Common Core State Standards References

Algebraic Problem Solving Assessment Targets Content IndicatorsRange of Depth of

Knowledge (DOK)

9

A.1 Write, evaluate, and compute with expressions and polynomials

7.EE.1 A.1.a Add, subtract, factor, multiply and expand linear expressions with rational coefficients.

1-2

6.EE.2 A.1.b Evaluate linear expressions by substituting integers for unknown quantities. 1-2

6.EE.2; 6.EE.6 A.1.c Write linear expressions as a part of word-to-symbol translations or to represent common settings.

1-2

A-APR.1 A.1.d Add, subtract, multiply polynomials, including multiplying two binomials, or divide factorable polynomials.

1-2

6.EE.2 A.1.e Evaluate polynomial expressions by substitiuting integers for unknown qualities. 1-2

A-SSE.2;A-SSE.3;A-SSE.4

A.1.f Factor polynomial expressions 1-2

6.EE.2; 6.EE.6 A.1.g Write polynomial expressions as part of word-to-symbol translations or to represent common settings.

1-2

6.EE.3 A.1.h Add, subtract, multiply and divide rational expressions. 1-2

6.EE.2 A.1.i Evaluate rational expressions by substituting integers for unknown quantities. 1-2

6.EE.2; 6.EE.6 A.1.j Write rational expressions as part of word-to-symbol translations or to represent common settings.

1-2

A.2 Write, manipulate, and solve linear equations

7.EE.4; 8.EE.7; A-REI.3

A.2.a Solve one-variable linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms or equations with coefficients represented by letters.

1-2

7.EE.4; A-CED.1; A-CED.2

A.2.b Solve real-world problems involving linear equations. 1-2

6.EE.6;A-CED.1;A-CED.2

A.2.c Write one-variable and multi-variable linear equations to represent context. 1-2

8.EE.6;A-REI.6 A.2.d Solve a system of two simultaneous linear equations by graphing, substitution,

or linear combination. Solve real-world problems leading to a system of linear equations.

1-2

A.3 Write, manipulate, solve, and graph linear inequalitiesA-REI.3 A.3.a Solve linear inequalities in one variable with rational number coefficients. 1-2

6.EE.8; 7.EE.4 A.3.b Identify or graph the solution to a one variable linear inequality on a number line. 1-2

7.EE.4;A-CED.1;A-CED.2

A.3.c Solve real-world problems involving inequalities. 1-2

10

6.EE.2;A-CED.1;A-CED.2

A.3.d Write linear inequalities in one variable to represent context. 1-2

A.4 Write, manipulate, and solve quadratic equationsA.REI.4 A.4.a Solve quadratic equations in one variable with rational coefficients and real

solutions, using appropriate methods. (e.g., quadratic formula, completing the square, factoring, inspection)

1-2

A-CED.1 A.4.b Write one-variable quadratic equations to represent context. 1-2

A.5 Connect and interpret graphs and functions6.NS.6 A.5.a Locate points in the coordinate plane. 1

8.F.4 A.5.b Determine the slope of a line from a graph, equation, or table. 1-2

8.EE.5 A.5.c Interpret unit rate as the slope in a proportional relationship. 2

A-CED.2;F-IF.7 A.5.d Graph two-variable linear equations 1-2

8.F.3; 8.F.5;F-IF.5 A.5.e For a function that models a linear or nonlinear relationship between two

quantities, interpret key features of graphs and tables in terms of quantities, and sketch graphs showing key features of graphs and tables in terms of quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior, and periodicity.

1-2

A.6 Connect coordinates, lines, and equationsA-CED.2 A.6.a Write the equation of a line with a given slope through a given point. 1-2

A-CED.2 A.6.b Write the equation of a line passing through two given distinct points. 2

G-GPE.5 A.6.c Use slope to identify parallel and perpendicular lines and to solve geometric problems.

1-2

A.7 Compare, represent, and evaluate functions

8.EE.5 A.7.a Compare two different proportional relationships represented in different ways. Examples include but are not limited to: compare a distance-time graph to a distance-time equation to determine which of the two moving objects has a greater speed.

2

8.F.1; F-IF.1 A.7.b Represent or identify a function in a table or graph as having exactly one output (one element in the range) for each input (each element in the domain)

1-2

F-IF.2 A.7.c Evaluate linear and quadratic functions for values in their domain when 1-2

11

represented using function notation.

8.F.2; F-IF.9 A.7.d Compare properties of two linear or quadratic functions each represented in a different way (algebraically, numerically in tables, graphically or by verbal descriptions). Examples include but are not limited to: given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.

2

Teaching Functions Using a One to One Correspondence

Vacation Winners

Toaster/Function Machine

What is a function?

A relationship where each input (x) has only one matching output (y).

Function Examples in 4 Forms: Table/Equation/Words/Graph

TableThis table shows a function since each input (x) has only one output (y):

X Y0 - 11 22 53 8

12

EquationThis equation shows a function since there can be only one output for every input: y = 3x - 1As we plug in different input values for x, we will always get only one output value for y. Putting in 0 for x will give us - 1 for y, putting in 1 for x will give us 2 for y, and this pattern will continue.

WordsWe can also describe a function in words: y is equal to three times x minus one. A real life example could be: Keisha sells programs at football games. For each program she sells, she earns $3. Her cost to produce each program is $1. Her profit is described by the function, profit (y) is equal to three times the number of programs she sells (x) minus the one dollar production cost.

GraphThe graph for the function y = 3x -1 is shown here:

Function Notation

Name ofthe function The function

f (x) = 3x + 4

Tells what numberto plug in the function

13

f (x) is the same as y: y = 2x + 1 is the same as f(x) = 2(x) + 1. We use f(x) since it makes it easier to see what happens as we put in different inputs for x:

f(x) = 2(x) + 1

f(0) = 2(0) + 1

f(0) = 1

f(1) = 2(1) + 1

f(1) = 3

f(2) = 2(2) + 1

f(2) = 5

f(3) = 2(3) + 1

f(3) = 7

Linear, Quadratic, and Exponential Function Graphs

Linear Quadratic ExponentialDescription: Straight line U Shape (parabola) that

opens up or downGrows/Shrinks fast and levels off on 1 side

14

2

10

3x + 4

Graph looks like:

Equation y = 2x y = 3x2 - 8 y = 2x

How to tell from the equation what the graph looks like:

The x term is to the first power (x)

The x term is to the second power (x2)

The x term is the exponent (2x)

Graph specific information:

In the equation y = mx + b, if m is negative, the line decreases (goes down) as we look from left to right: y = - x + 5

If m is positive, the line increases (goes up) as we look from left to right: y = .5x + 2

If the number with the x2 is negative, the U shape opens down. If the number with the x2 is positive, the U shape opens up:

In the equation y = ax , if a is bigger than 1 the curve will grow fast. If a is between 0 and 1, the curve will shrink fast.

Finding the x and y Intercepts in a Linear FunctionTo find the x intercepts in a linear function like y = 2x + 6, we are going to set y = 0 and solve for x. To find the y intercept, we’ll set x = 0 and solve for y:

15

x InterceptSet y = 0 and solve for x:

y = 2x + 6

0 = 2x + 6

- 6 = 2x

-3 = x

When y = 0, x is -3 so we have the ordered pair: (-3,0)

y InterceptSet x = 0 and solve for y:

y = 2x + 6

y = 2(0) + 6

y = 6

When x = 0, y is 6 so we have the ordered pair: (0,6)

Quadratic Functions - Minimums and MaximumsGraphs of quadratic functions either have minimums and maximums. The maximum is the highest point on the upside down U shape (or parabola) where we have the largest y value. It is also called the vertex. The minimum is the lowest point on the upward facing U shape (or parabola) where we have the smallest y value. It is also called the vertex.

16

Quadratic Functions - SymmetryThe axis of symmetry is the vertical (straight up and down) line through the vertex on the graph of a quadratic function. It cuts the U shape in half.

Quadratic Functions – InterceptsThe x intercept(s) in a quadratic function are the places where the graph crosses the x axis (where y = 0). The y intercept is the place where the graph crosses the y axis (where x = 0).

Increasing/Decreasing/ConstantWe always read graphs like we read a book from left to right. Look to see where the graph is going up, down, or staying the same:

17

Where is this graph increasing, decreasing, and remaining constant?

PeriodicityFunctions that have periodicity repeat their values in regular intervals or periods.

End Behavior

Youtube: How to Describe End Behavior of Functions

How Can We Tell Which of Two Functions Has a Greater Rate of Change?

18

California offers five cents for every aluminum can brought in to recycle. Michigan offers 10 cents for every aluminum can turned in for recycling. If we represent California by the equation y = .05x and Michigan by the following table, which function has the greater rate of change?

Cans (x) Money (y)1 .102 .203 .304 .405 .50

10 1.00

In the table, the rate of change is found by taking the change in y divided by the change in x:

.20 - .10 = .10 = .10 2 – 1 1

In the equation, to find the rate of change, look at the number with the x term (sometimes called the independent variable). In y = .05x , .05 is the rate of change.

So, the function in the table has the greater rate of change (.10 is bigger than .05).

Adapted from Understanding and Comparing Functions Lesson 5 of 6 (Youtube)

Let’s Practice

Jamal can walk a certain number of miles per hour as described in this equation: y = 4x. His friend Marquisha’s speed is shown on the chart below. Who can walk faster?

Time (x)In Hours

Miles (y)

.5 31 6

1.5 92 125 30

Your Turn

Create 2 functions and show them by a graph, in words, in an equation, or in a table on a piece of flip chart paper (or on a whiteboard). Have another group guess which has the bigger rate of change.

Activities with Functions

19

Function Matching

This activity gives students practice in noticing four representations of a function: words, graphs, equations, and tables. You will be given 13 functions and must match the cards representing the function in words, graphs, equations, and tables. Be careful with the graphs as not all the scales are the same! The master for this activity is on the abspd.appstate.edu website.

Adapted from http://mrsreillyblog.wordpress.com

Function Foldables

This activity takes a minimum of prep time and all you need are a few sheets of card stock. Cut the card stock in half and have students fold it in fourths. Give students a function equation and have them show it in words, graphs, and as a table. You could also vary the activity and have students start with either a function in words/graphs/tables and show the other forms.

Adapted from Math Activities for Migrating to the Common Core Classroom (Donna Ware)

Real Life Functions

Another way to develop algebraic thinking in adult learners and answer the, “When are we ever gonna use this?” question is to have students explore situations where linear functions are found. Once students have brainstormed and identified areas where functions are found in real life, they should represent the function in words, as a graph, in an equation, and as a table.

One idea to get you started:

- If gas costs $3.25 a gallon and I buy none, it costs $0. If I buy 1 gallon it costs $3.25, 2 gallons cost $6.50, 3 gallons cost $9.75 etc.

The follow up activity to this is a gallery walk where students look at other students work posted around the classroom.

Adapted from Life Functions by Donna Parrish in The Math Practitioner, Spring 2013.

Algebra Tiles Practice

20

1. Show x – 4 2. Show 5x2- 2x

3. Show 4x2- 2 x + 6 4. Show x2 - 7x + 4

5. Simplify -2x2 + 6x + 3 - 4x + 4 6. Simplify 3x2 - 3x + x2 + 2x - 4

7. Simplify x2 - 2x + 2 + 3x2 – 2x + 4 8. Simplify 2x2 - 2x + 1 + 7x - 4

9. (2x2 – 3x + 2) + (-3x2 – x) 10. (x2 - 3) + (-3x2 – 3x + 1)

11. (-5x2 – x – 3) + (4x – 2) 12. (x2 + 4x + 4) + (-2x2 – 6x + 3)

13. (x2 – 6x – 1) – (-3x2 + 2x + 2) 14. (4x2 + 4x + 2) – (-x2 + 3x – 2)

15. (x2 + 2x + 6) – (-4x2 + 6x – 2) 16. (3x2 – 6x – 1) – (2x2 – x – 4)

17. x – 2 = 5 18. 2x – 1 = 3

19. 3x + 2 = 5 20. 2x + 3 = 9

21. (x – 2) (2x + 1) 22. (3x + 1) (x – 2)

23. (x + 3)(x – 2) 24. (2x - 2) (x + 2)

Polya Problem Solving Method

21

1. Understand the problem

What are you asked to do?Will a picture or diagram help you understand the problem? Can you rewrite the problem in your own words?

2. Create a plan

Use a problem solving strategy:

Guess and check Solve an easier problemMake a list ExperimentDraw a picture or diagram Act it outLook for a pattern Work backwardsMake a table Change your viewpointUse a variable

3. Carry out the plan

Be patient Be persistent Try different strategies

4. Look back and reflect

Does your answer make sense? Are all the questions answered?What other ways are there to solve this problem?What did you learn from solving this problem?

Understand the problem

22

Create a plan

Carry out the plan

Look back and reflect

Problems from NC Algebra 1 End of Course Test

1. What scenario could be modeled by the graph below?

23

A. The number of pounds of apples, y, minus 2 times the number of pounds of oranges, x, is at most 5.B. The number of pounds of apples, y, minus half the number of pounds of oranges, x, is at most 5.C. The number of pounds of apples, y, plus two times the number of pounds of oranges, x, is at most 5. D. The number of pounds of apples, y, plus half the number of pounds of oranges, x, is at most 5.

2. The function of f(t) = -5t2 + 20t + 60 models the approximate height of an object t seconds after it is launched. How many seconds does it take the object to hit the ground?

3. Suppose that the function f(x) = 2x +12 represents the cost to rent x movies a month from an internet movie club. Makayla now has $10. How many more dollars does Makayla need to rent 7 movies next month?

4. The table below shows the cost of a pizza based on the number of toppings.

Number of Toppings (n) Cost (C)1 $122 $13.503 $154 $16.50

Which function represents the cost of a pizza with n toppings?

A. C(n) = 12 +1.5(n - 1)B. C(n) = 1.5n + 12C. C(n) = 12 + nD. C(n) = 12n

Source: ncpublicschools.org

Resources

Professional Development

24

Google: nc basic skills training calendar

The 2013-14 training calendar shows all the professional development opportunities available across the state. Professional development will be held in each region.

Adult High School

Adult High School Algebra Standards

Google: nc dpi algebra 1 unpacked

These are North Carolina’s standards for high school Algebra 1

AHS Algebra 1 Course

http://vlcmoodle.nccommunitycolleges.edu/Search for AHS Algebra 1 under continuing education courses

This is a complete Algebra 1 course that can be used for Adult High School

2003 NC Standard Course of Study and Algebra 1 Crosswalk

Google: nc dpi algebra 1 crosswalk

This site compares the new Common Core Algebra 1 standards with the 2003 NC Standard Course of Study

Algebra Resources

Google: get the math http://www.thirteen.org/get-the-math

Get the Math show how algebra is used in the real world in music, sports, fashion, videogames, restaurants, and special effects.

Manipulatives

Google: national library of virtual manipulativeshttp://nlvm.usu.edu/en/nav/vlibrary.html

This site has algebra manipulatives for instruction across grade levels

Algebra Tiles

Google: algebra 4 all learnport

This site provides an easy to use set of virtual algebra tiles.

25