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FOCUSING ON MATHEMATICAL REASONING: TRANSITIONING TO THE 2014 GED ® TEST

Presenters: Bonnie Goonen

Susan Pittman

Session Objectives

• Identify content of the Mathematics module of the 2014 GED ® test • Explore essential mathematical practices and behaviors • Discuss beginning strategies for the classroom • Identify resources that support the transition to the next generation assessment

Algebraic Sugar Cane

10 factories produce sugar cane. The second produced twice as much as the first. The third and fourth each produced 80 more than the first. The fifth produced twice as much as the second. The sixth produced 40 more than the fifth. The seventh and eighth each produced 40 less than the fifth. The ninth produced 80 more than the second. The tenth produced nothing due to drought in Australia. If the sum of the production equaled 11,700, how much sugar cane did the first factory produce?

Answer

500. First write down what each factory produced in relation to the first factory (whose production is

): 1.

2. 2

+ 80 4.

+ 80 5. 4

6. 4

+ 40 7. 4

- 40 8. 4

- 40 9. 2

+ 80 10. 0

Add them all up and set equal to 11,700 to get the equation: 23

+ 200 = 11,700 Solution is:

= 500

Mathematical Reasoning What’s New?

Mathematical Reasoning Module

Overview Content • 115 minute test • Technology-enhanced items ― Multiple choice ― Drag-drop ― Drop-down ― Fill-in-the-blank ― Hot spot • Calculator (TI 30XS MultiView ™) allowed on all but first five items • • Quantitative problem solving • 25% with rational numbers • 20% with measurement • Algebraic problem solving • 30% with expressions and equations • 25% with graphs and functions Integration of mathematical practices

What’s new in the Mathematical Reasoning domain?

• Identify absolute value of a rational number • Determine when a numerical expression is undefined • Factor polynomial expressions • Solve linear inequalities

What’s new in the Mathematical Reasoning domain?

• Identify or graph the solution to a one variable linear inequality • Solve real-world problems involving inequalities • Write linear inequalities to represent context • Represent or identify a function in a table or graph

What’s

not

directly assessed on the 2014 Mathematical Reasoning Module?

• Select the appropriate operations to solve problems • Relate basic arithmetic operations to one another • Use estimation to solve problems and assess the reasonableness of an answer • Identify and select appropriate units of metric and customary measures • Read and interpret scales, meters, and gauges • Compare and contrast different sets of data on the basis of measures of central tendency • Recognize and use direct and indirect variation

Mathematical Reasoning Item Sampler

MATHEMATICAL TOOLS NEW REALITIES

Mathematical Tools

Mathematics “Symbol Selector” Explanation

It’s Your Turn!

"Calculators can only calculate - they cannot do mathematics." -- John A. Van de Walle http://education.ti.com/en/us/products/calc ulators/scientific-calculators/ti-30xs multiview/classroom-activities/activities exchange

Mathematical Practices:

behaviors that are essential to the mastery of mathematical content

Mathematical Practices

• Practices • Mathematical fluency • Abstracting problems • Building solution pathways and lines of reasoning • Furthering lines of reasoning • Evaluating reasoning and solution pathways • Most practices are not specific to any one particular area of mathematics content

Let’s Get Started!

“Anyone who has never made a mistake has never tried anything new.”

Albert Einstein

Solution Pathways = Problem Solving

Polya’s Four Steps to Problem Solving Understand the problem Look back (reflect) Devise a plan Carry out the plan Polya, George.

How To Solve It,

2nd ed. (1957). Princeton University Press.

Value of Teaching with Problems

• Places students’ attention on mathematical ideas • Develops “mathematical power” • Develops students’ beliefs that they are capable of doing mathematics and that it makes sense • Provides ongoing assessment data that can be used to make instructional decisions • Allows an entry point for a wide range of students

A Few Problem-Solving Strategies

• Look for patterns • Consider all possibilities • Make an organized list • Draw a picture • Guess and check • Write an equation • Construct a table or graph • Act it out • Use objects • Work backward • Solve a simpler (or similar) problem

Explicit instruction matters

Why teach different problem-solving strategies?

• Explicit instruction has shown consistent positive effects on performance with words problems and computation • Students receives extensive practice in use of newly learned strategies and skills

Let’s SOLVE a Math Problem

Even Albert Einstein said:

“Do not worry about your difficulties in Mathematics. I can assure you mine are still greater.”

SOLVE a Problem

tudy the problem (What am I trying to find?)

rganize the facts (What do I know?)

ine up a plan (What steps will I take?)

erify your plan with action (How will I carry out my plan?)

xamine the results (Does my answer make sense? If not, rework.)  Always double check!

S = Study the problem

What is the problem asking me to do?

Find the question.

We are going to practice SOLVE with this one!

A painter rented a wallpaper steamer at 9 a.m. and returned it at 4 p.m. The painter found rental costs as high as $15.00 per hour. He paid a total of $28.84. What was the rental cost per hour?

O = Organize the Facts

What facts are provided in order for you to solve the problem?

• Identify each fact.

• Eliminate unnecessary facts.

• List all necessary facts.

A painter rented a wallpaper steamer at 9 a.m. and returned it at 4 p.m. The painter found rental costs as high as $15.00 per hour. He paid a total of $28.84. What was the rental cost per hour?

L = Line Up a Plan

• Select the operations to use.

• State the plan/strategy that you will use in words.

I will use a multi-step approach. First, I will calculate the elapsed time that I had the steamer in number of hours. Next, I will divide the total rental amount by the number of hours I rented the steamer. This will provide me with the hourly rate.

V = Verify Your Plan

9 – 10 10 – 11 11 – 12 12 – 1 1 – 2 2 – 3 3 – 4 Total = 1 hr.

= 1 hr.

7 hrs.

$28.84 / 7 =

$4.12

E = Examine the Results (Is it reasonable? Does it make sense? Is it accurate?)

$4.12 IS

reasonable because it is substantially less than the total rental amount. Also, the answer is a decimal number because the total rental amount is a decimal number. If one estimates $28.84 is close to 28 and 7 goes into 28 four times. Therefore, the painter paid $4.12 per hour for the machine.

Let’s SOLVE

S O L V E

S How much will the business earn on day 10?

O L

On day 2, the business earned $112.

On day 5, the business earned $367.

Earnings will continue to increase at the same rate 1 .

1 Not all the facts have to be numeric.

There is a relationship between two variables. Number of days (x) is the independent variable, and money earned (y) is the dependent one. Since the rate is constant, there is a linear relationship.

a) Calculate the rate (slope) and the y-intercept.

b) Write the linear equation.

c) Solve for the money earned on day 10.

a) b = 367 – 85(5) = -58 b) y = 85x – 58 c) y = 85(10) – 58 = 850 – 58 = $792

The amount earned after 10 days is greater than the amount earned after 5 days. This is consistent with what happened from day 2 to day 5. Since the slope and the y-intercept are integers, the answer must also be an integer.

Another Way to SOLVE

S – How much will be earned on day 10?

O – day 2 = 112 day 5 = 367 L – 1. find slope – rise run 2. Find linear equation linear equation V – test different ways E – examine by using graph 367 792 1 2 3 4 5 6 7 8 9 10 Days

Problem Solving

• Reflect on the process of problem solving • Emphasis: • mathematical discourse and classroom as a learning community • understanding and engaging with mathematics • extensions of solved problems • Construct problem-solving strategies

Quantitative Reasoning Skills

Some Big Ideas in Quantitative Reasoning (45%) • Number sense concepts (rational numbers, absolute value, multiples, factors, exponents) • Spatial visualization • Dimensions, perimeter, circumference, and area of two dimensional figures • Dimensions, surface area, and volume of three dimensional figures • Interpret and create data displays • Probability, ratio, percents, scale factors • Graphic literacy

Let’s SOLVE!

S O L V E

Focus on Geometry

• • • • • • Students need proficiency in basic measurement and geometric thinking skills: Use concepts Use spatial visualization Select appropriate units of measure Identify and define different types of geometric figures Predict impact of change on perimeter, area and volume of figures Compute surface area and volume of composite 3-D geometric figures, given formulas as needed

The Van Hiele Theory

• Level 1: Visualization • Level 2: Analyze • Level 3: Informal Deduction • Level 4: Formal Deduction • Level 5: Rigor

Visualizing The First Step in Geometric Reasoning

Visualization

• Recognize and name shapes by appearance • Do not recognize properties or if they do, do not use them for sorting or recognition • May not recognize shape in different orientation (e.g., shape at right not recognized as square)

Visualization

Implications for Instruction - Visualization • Provide activities that have students sort shapes, identify and describe shapes (e.g., Venn diagrams) • Have students use manipulatives • Build and draw shapes • Put together and take apart shapes • Make sure students see shapes in different orientations • Make sure students see different sizes of each shape

Analysis The Second Step in Geometric Reasoning

Analysis

• Can identify some properties of shapes • Use appropriate vocabulary • Cannot explain relationship between shape and properties (e.g., why is second shape not a rectangle?)

Analysis

Description 1

• • • • The design looks like a bird with a hexagon body; a square for the head; triangles for the beak and tail; and triangles for the feet.

Analysis

Description 1

Analysis

• • • •

Description 2:

Start with a hexagon. On each of the two topmost sides of the hexagon, attach a triangle.

On the bottom of the hexagon, attach a square. Below the square, attach two more triangles with their vertices touching.

Analysis

Description 2

Implications for Instruction - Analysis

• • • • • • Work with manipulatives Define properties, make measurements, and look for patterns Explore what happens if a measurement or property is changed Discuss what defines a shape Use activities emphasize classes of shapes and their properties Classify shapes based on lists of properties

Right Angle

Informal Deduction to Formal Deduction The Third and Fourth Steps in Geometric Reasoning

Informal Deduction

• Can see relationships of properties within shapes • Recognize interrelationships among shapes or classes of shapes (e.g., where does a rhombus fit among all quadrilaterals?) • Can follow informal proofs (e.g., every square is a rhombus because all sides are congruent)

Deduction

• Usually not reached before high school; maybe not until college • Can construct proofs • Understand the importance of deduction • Understand how postulates, axioms, and definitions are used in proofs A E D B A C F

What do you think?

• Is it possible to draw a quadrilateral that has exactly 2 right angles and no parallel lines? • Try it. While you’re working, ask yourself . . . • What happens if …?

• What did that action tell me?

• What will be the next step?

Geometric Reasoning

• Seeking Relationships • Checking Effects of Transformations • Generalizing Geometric Ideas • Conjecturing about the “always” & “every” • Testing the conjecture • Drawing a conclusion about the conjecture • Making a convincing argument • Balancing Exploration with Deduction • • Exploring structured by one or more explicit limitation/restriction Taking stock of what is being learned through the exploration

Let’s SOLVE!

S O L V E

Algebraic Reasoning Skills

“Algebraic thinking or reasoning involves forming generalizations from experiences with number and computation, formalizing these ideas with the use of a meaningful symbol system, and exploring the concepts of pattern and functions .”

- Van de Walle

Algebraic Reasoning

Think of any number. • Multiply it by 2.

• Add 4.

• Multiply by 3.

• Divide by 6.

• Subtract the number with which you started.

You got 2!

Explain with algebra why this works.

The answer is . . .

Start with the expression that describes the operations to be performed on your chosen number,

: and simplify the expression. You'll end up with 2, regardless of the value of

What we know . . .

People have a “love-hate” relationship with mathematics • Twice as many people hated it as any other school subject • It was also voted the most popular subject Associated Press Poll

Algebra Misconceptions

1) 2) 3) 4) 5) 6) 7) a + a + a + a = 4a 3a x 2b = 5ab c x c = 2c 4d + 2e - d + 3e = 3d + 5e 8f - 3g + 2f - 6g = 10f - 3g 5y - y = 5 Find the perimeter of this shape a 8) 9) 10) b x b x b x b = b 4 3(2k + 3) = 6k + 3 a(a + 3) = a 2 + 3a a 2 b 2 b

Algebraic Thinking in Adult Education

Algebraic Thinking in Adult Education • Create opportunities for algebraic thinking as a part of regular instruction • Integrate elements of algebraic thinking into arithmetic instruction • Acquiring symbolic language • Recognizing patterns and making generalizations • Reorganize formal algebra instruction to emphasize its applications Adapted from National Institute for Literacy,

Algebraic Thinking in Adult Education

, Washington, DC 20006

Examining the Components

Let’s investigate the basics of algebra and see what we can include in our programs.

Some Big Ideas in Algebra

• Variable • Symbolic Notation • Equality • Ratio and Proportion • Pattern Generalization • Equations and Inequalities • Multiple Representations of Functions

Variable

Some students believe that letters represent particular objects or abbreviated words

Symbolic Notation

A Few Examples

Sign

= (equal)

Arithmetic

. . . And the answer is + Addition operation Subtraction operation

Algebra

Equivalence between two quantities Positive number Negative number

Which Is Larger?

2 3 or 3 2 3 4 or 4 3 6 2 or 2 6 8 9 or 9 8

Patterns – Thinking Algebraically

• Finding patterns • Describing patterns • Explaining patterns • Predicting with patterns

Teaching Patterns

Banquet Tables Arrangement 1 Arrangement 2 Arrangement 3 • Arrangement 1 seats four people. How many people can be seated at Arrangement 100?

Teaching Patterns

Build or draw the following sequence of houses made from toothpicks.

• http://www2.edc.org/edc research/curriki/ROLE/lc/sessions/session2/ToothpickHou ses.htm

 1.





















2 









y z



x u

Figure Out the Digits

In this number puzzle, each letter (q - z) represents a different digit from 0-9. Find the correspondence between the letters and the digits. Be prepared to explain where you started, and the order in which you solved the puzzle.

 1.





















2 









y z



x u

q = 9 z = 8

1. 4 x 2 = 8 2. 5 + 0 = 5 3. 2 + 2 + 2 + 2 = 8 4. 3 + 6 = 9 5. 7 x 1 = 7 6. 3 2 = 9 7. 2 + 2 = 4 8. 3 + 4 = 7 9. 6/8 =3/4

r = 2 v = 1 s = 7 w = 0 t = 5 x = 3 u = 4 y = 6

Use Multiple Representations

• Represent problems using symbols, expressions, and equations, tables, and graphs • Model real-world situations • Complete problems different ways (flexibility in problem solving)

Vertical Multiplication of Polynomials

Effective Questions

Ask challenging, well-crafted, open ended questions, such as: • What would happen if . . . ? • What would have to happen for . . . ? • What happens when . . . ? • How could you . . . ? • Can you explain why you decided . . . ?

Teacher Responses

• Phrases to Use • I’m not sure I understand, could you show me an example of ... ? • What do you think the next step should be?

• Where would you use ... ? • Could ____ be an answer?

• How do you know you are correct? • Phrases to Avoid • • • Let me show you how to do this. That’s not correct. I’m not sure you want to do that.

Math journals help students to . . .

• Be aware of what they do and do not know • Make use of prior knowledge • Identify their mathematical questions • Develop their ability to problem solve • Monitor their own progress • Make connections • Communicate more precisely

Mathematics is like a video game; If you just sit and watch, You’re wasting your time.

Algebra Manipulatives (the “C” of CRA) • Students with access to virtual manipulatives achieved higher gains than those students taught without manipulatives.

• Students using hands-on and manipulatives were able to explain the how and why of algebraic problem solving.

Big Ideas Using Algebra Tiles

• Add and subtract integers • Model linear expressions • Solve linear equations • Simplify polynomials • Solve equations for unknown variable • Multiply and divide polynomials • Complete the square • Investigate

Introduction to Algebra Tiles

Positive Tiles

Negative Tiles Remember, they could be called x, y, b, t, etc.

Introduction to Algebra Tiles

x Each tile represents an area.

x Area of large square = x (x) = x 2 x 1 Area of rectangle = 1 (x) = x 1 1 Area of small square = 1 (1) = 1

Note: Tiles are not to scale. 10 little tiles don’t equal 1 big tile!

What’s My Polynomial?

Simon says show me . . .

• 2x 2 • 4x • - x 2 • 2x + 3 • - x 2 + 4 • 2x 2 + 3x + 5 • -2x 2 - 6x - 5

Zero Pairs (Remember Additive Inverses??????) • A combined positive and negative of the same area (number) produces a zero pair. • I have $1.00. I spend $1.00. I have $0 left.) 2 – 2 + 2x – 2x = 0

Use Algebra Tiles to Model Addition of Integers • Addition is “combining.” • Combining involves the forming and removing of zero pairs.

• Remember, an integer is a number with no fractional part.

Use Algebra Tiles to Model Addition of Integers (+3) + (+1) = (-2) + (-1) =

Addition of Integers

(+3) + (-1) = (+4) + (-4) = Don’t forget that a positive and a negative “cancel” each other out!

Use Algebra Tiles to Model Integer Subtraction • Subtraction can be interpreted as “take-away.” • Subtraction can also be thought of as “adding the opposite.”

Use Algebra Tiles to Model Subtraction of Integers (+5) – (+2) = (-4) – (-3) =

Subtracting Integers – It’s Your Turn!

(+3) – (-5) (-4) – (+1) (+3) – (-3)

How About Subtracting Polynomials?

• To use algebra tiles to model subtraction, represent each expression with tiles. Put the second expression under the first.

• (5x + 4) – (2x + 3) 5x + 4 tiles which x x x x x 1 1 1 1 x x 1 1 1 (5x + 4) – (2x + 3)= 3x +1

“Minus times minus results in a plus, The reason for this, we needn't discuss .” —OgdenNash

Use Algebra Tiles to Combine Polynomials • “Simplify” means to combine like terms and complete all operations.

Terms in an expression are

like terms

identical variable parts. if they have You can

combine terms

that are alike.

You

cannot combine

terms that are unalike.

Combining Like Terms

How much do I have here?

I have 5x + 4

Combining Like Terms How much do I have here?

I have x

+ 2x + 6

Notice that we write the largest shape first, then the medium size , then the smallest . We write them in order from largest to smallest.

Let’s Collect Tiles!

The Rules!

• Big squares can’t touch little squares.

• Little squares should all be together.

• Tiles should always be in a rectangular array.

2 x 2 + 7x + 6 Which looks best?

Algebra Tiles – Time to Collect Tiles!

• x 2 + 6x + 8 • x 2 – 4x + 3 • x 2 + 7x + 6 • 2x 2 + 7x + 6

Multiplying Polynomials

It’s just like figuring area!

• Place one term at the top of the grid • Place the second term on the side of the grid • Maintain straight lines when filling in the grid • The inner grid is your answer!

Multiplying Polynomials

(x + 2)(x + 3) =

x 2 + 5x + 6

x + 3

Multiplying Polynomials

(x – 1)(x + 4) =

x 2 + 3x -4

x + 4

Multiplying Polynomials

• (x + 2) (x + 1) = • (x + 5) (x + 3) = • (2x + 2) (2x + 1) =

Dividing Polynomials

You know the inner grid and one of the two sides, so . . .

• Place one term at the top or side of the grid (it doesn’t matter which) • Fill in the inner grid • The missing side is your answer!

Dividing Polynomials

x 2 + 7x + 6 = x + 1

x + 6

Dividing Polynomials

2x 2 + 5x – 3 x + 3 x 2 – x – 2 x – 2

Factoring Polynomials

• Algebra tiles can be used to factor polynomials. Use tiles and the frame to represent the problem.

• Use the tiles to fill in the array so as to form a rectangle inside the frame.

• Be prepared to use zero pairs (when needed) to fill in the array.

• Solve!

Factoring Polynomials

3x + 3

Factoring Polynomials

x 2 + 6x + 8 =

(x +3) (x +2)

Factoring Polynomials

x2 – 5x + 6 =

(x -3) (x -2)

Factoring Polynomials

x 2 – x – 6 =

(x -3) (x +2)

Sometimes the zero property is needed to build a rectangular shape. Remember the balance beam!

Factoring Polynomials

• x 2 – 4 • 2x 2 – 3x – 2 • 2x 2 + 3x – 3 • -2x 2 + x + 6

There’s More!

Investigate

• Use algebra tiles to prove that (x + 1) 2 not equivalent and (x 2 + 1) are

So, why don’t all teachers use manipulatives? • Lack of training or resources • Availability of funds or time • Lessons using manipulatives may be noisier and not as neat • A fear of a breakdown in classroom management • Concerns about connecting concrete representational-abstract • Concerns that students won’t “like” them

It all leads to connecting mathematical concepts with effective mathematical practices/problem solving

Conceptual Teaching

What is conceptual teaching?

• Using schema to organize new knowledge • Developing units around concepts • Providing schema based on students’ prior knowledge • Teaching knowledge/skill/concept in context What it’s not!

• Worksheets • Drill • Memorization of discrete facts

Real-World Algebra

My Ford Bronco was fitted at the factory with 30 inch diameter tires. That means its speedometer is calibrated for 30 inch diameter tires. I "enhanced" the vehicle with All Terrain tires that have a 31 inch diameter. How will this change the speedometer readings? Specifically, assuming the speedometer was accurate in the first place, what should I make the speedometer read as I drive with my 31 inch tires so that the actual speed is 55 mph?

CTL Resources for Algebra. The Department of Mathematics. Education University of Georgia http://jwilson.coe.uga.edu/ctl/ctl/resources/ Algebra/Algebra.html

Real-World Math

The Futures Channel

http://www.thefutureschannel.com/algebra/algebra_real_wo rld_movies.php

Real-World Math

http://www.realworldmath.org/

Get the Math

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

Math in the News

http://www.media4math.com/MathInTheNews.asp

A Few Strategies to Get Started

• Model, explain, and guide • Move towards self-regulation • Provide opportunities for algebraic thinking • Keep it real • Teach often to the whole class, in small groups, and with individuals • Set high expectations

“High achievement always occurs in the framework of high expectation.” Charles F. Kettering (1876-1958)

The IPDAE project is supported with funds provided through the Adult and Family Literacy Act, Division of Career and Adult Education, Florida Department of Education.