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FOCUSING ON MATHEMATICAL REASONING: TRANSITIONING TO THE 2014 GED ® TEST Presenters: Bonnie Goonen Susan Pittman

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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 2

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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?

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Answer 500. First write down what each factory produced in relation to the first factory (whose production is x): 1. x 2. 2x 3. x x x 6. 4x x x x x Add them all up and set equal to 11,700 to get the equation: 23x = 11,700 Solution is: x = 500

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Mathematical Reasoning Whats New?

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Mathematical Reasoning Module 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 Overview Content

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Whats 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

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Whats 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

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Whats 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

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Mathematical Reasoning Item Sampler

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MATHEMATICAL TOOLS NEW REALITIES

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Mathematical Tools

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Mathematics Symbol Selector Explanation

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Its Your Turn! ulators/scientific-calculators/ti-30xs- multiview/classroom-activities/activities- exchange "Calculators can only calculate - they cannot do mathematics." -- John A. Van de Walle

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Mathematical Practices: behaviors that are essential to the mastery of mathematical content

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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 16

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Lets Get Started! Anyone who has never made a mistake has never tried anything new. Albert Einstein

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Solution Pathways = Problem Solving Polyas Four Steps to Problem Solving Polya, George. How To Solve It, 2nd ed. (1957). Princeton University Press.

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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

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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

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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

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Lets SOLVE a Math Problem Even Albert Einstein said: Do not worry about your difficulties in Mathematics. I can assure you mine are still greater.

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SOLVE a Problem S tudy the problem (What am I trying to find?) O rganize the facts (What do I know?) L ine up a plan (What steps will I take?) V erify your plan with action (How will I carry out my plan?) E xamine the results (Does my answer make sense? If not, rework.) Always double check!

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S = Study the problem What is the problem asking me to do? Find the question. 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 $ What was the rental cost per hour? We are going to practice SOLVE with this one! We are going to practice SOLVE with this one!

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O = Organize the Facts Identify each fact. Eliminate unnecessary facts. List all necessary facts. What facts are provided in order for you to solve the problem? 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 $ What was the rental cost per hour?

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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.

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V = Verify Your Plan 9 – 10 = 1 hr. 10 – 11 = 1 hr. 11 – 12= 1 hr. 12 – 1 = 1 hr. 1 – 2 = 1 hr. 2 – 3 = 1 hr. 3 – 4 = 1 hr. Total 7 hrs. Elapsed Time Total # Hours $28.84 / 7 = $4.12 Total Rental Cost Total # Hours Cost per Hour

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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.

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Lets SOLVE SOLVESOLVE

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SHow much will the business earn on day 10? OOn 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. LThere 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. Va) b = 367 – 85(5) = -58 b) y = 85x – 58 c) y = 85(10) – 58 = 850 – 58 = $792 EThe 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.

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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 3. Substitute into linear equation V – test different ways E – examine by using graph Days $ in hundreds

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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

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Quantitative Reasoning Skills

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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

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Lets SOLVE! SOLVESOLVE

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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

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The Van Hiele Theory Level 1: Visualization Level 2: Analyze Level 3: Informal Deduction Level 4: Formal Deduction Level 5: Rigor

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Visualizing The First Step in Geometric Reasoning

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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)

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Visualization

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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

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Analysis The Second Step in Geometric Reasoning

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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?)

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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.

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Analysis Description 1

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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.

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Analysis Description 2

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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

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Triangle Quadrilateral Rotation Reflection Congruent Right Angle Polynomial Parallel

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

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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)

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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

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What do you think? Is it possible to draw a quadrilateral that has exactly 2 right angles and no parallel lines? Try it. While youre working, ask yourself... What happens if…? What did that action tell me? What will be the next step?

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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

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Lets SOLVE! SOLVESOLVE

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Algebraic Reasoning Skills

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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

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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.

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The answer is... Start with the expression that describes the operations to be performed on your chosen number, x: and simplify the expression. You'll end up with 2, regardless of the value of x.

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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

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

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Algebraic Thinking in Adult Education

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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

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Examining the Components Lets investigate the basics of algebra and see what we can include in our programs.

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Some Big Ideas in Algebra Variable Symbolic Notation Equality Ratio and Proportion Pattern Generalization Equations and Inequalities Multiple Representations of Functions

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Some students believe that letters represent particular objects or abbreviated words Variable

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A Few Examples Symbolic Notation SignArithmeticAlgebra = (equal)... And the answer is Equivalence between two quantities +Addition operation Positive number -Subtraction operation Negative number

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Which Is Larger? 2 3 or or or or 9 8

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Patterns – Thinking Algebraically Finding patterns Describing patterns Explaining patterns Predicting with patterns

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

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Teaching Patterns Build or draw the following sequence of houses made from toothpicks. research/curriki/ROLE/lc/sessions/session2/ToothpickHou ses.htm

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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.

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1. 4 x 2 = = = = x 1 = = = = /8 =3/4 q = 9r = 2s = 7t = 5u = 4 z = 8 v = 1w = 0x = 3y = 6

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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)

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Vertical Multiplication of Polynomials

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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... ?

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Teacher Responses Phrases to Use Im 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. Thats not correct. Im not sure you want to do that.

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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

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

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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.

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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

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Introduction to Algebra Tiles Positive Tiles Negative Tiles x2x2 x 1 Remember, they could be called x, y, b, t, etc.

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Introduction to Algebra Tiles Each tile represents an area. x x Area of large square = x (x) = x 2 Area of rectangle = 1 (x) = x Area of small square = 1 (1) = 1 x Note: Tiles are not to scale. 10 little tiles dont equal 1 big tile!

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Whats My Polynomial?

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Simon says show me... 2x 2 4x - x 2 2x x x 2 + 3x x 2 - 6x - 5

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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

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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.

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

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Addition of Integers (+3) + (-1) = (+4) + (-4) = Dont forget that a positive and a negative cancel each other out!

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

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

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Subtracting Integers – Its Your Turn! (+3) – (-5) (-4) – (+1) (+3) – (-3)

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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) xxxxx 1111 xx 111 5x + 4 2x + 3 Now remove the tiles which match in each expression. The answer is the expression that is left. (5x + 4) – (2x + 3)= 3x +1

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

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Use Algebra Tiles to Combine Polynomials Simplify means to combine like terms and complete all operations. Terms in an expression are like terms if they have identical variable parts. You can combine terms that are alike. You cannot combine terms that are unalike.

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How much do I have here? I have 5x + 4 Combining Like Terms

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How much do I have here? I have x 2 + 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. Combining Like Terms

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Lets Collect Tiles! The Rules! Big squares cant 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?

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Algebra Tiles – Time to Collect Tiles! x 2 + 6x + 8 x 2 – 4x + 3 x 2 + 7x + 6 2x 2 + 7x + 6

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Multiplying Polynomials Its 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!

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Multiplying Polynomials (x + 2)(x + 3) = x + 3 x + 2 x 2 + 5x + 6

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Multiplying Polynomials (x – 1)(x + 4) = x + 4 x – 1 x 2 + 3x -4

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Multiplying Polynomials (x + 2) (x + 1) = (x + 5) (x + 3) = (2x + 2) (2x + 1) =

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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 doesnt matter which) Fill in the inner grid The missing side is your answer!

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Dividing Polynomials x 2 + 7x + 6 = x + 1 x + 6

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Dividing Polynomials 2x 2 + 5x – 3 x + 3 x 2 – x – 2 x – 2

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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!

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Factoring Polynomials 3x + 3

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Factoring Polynomials x 2 + 6x + 8 = (x +3) (x +2)

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Factoring Polynomials x2 – 5x + 6 = (x -3) (x -2)

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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!

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Factoring Polynomials x 2 – 4 2x 2 – 3x – 2 2x 2 + 3x – 3 -2x 2 + x + 6

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Theres More! Investigate Use algebra tiles to prove that (x + 1) 2 and (x 2 + 1) are not equivalent

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So, why dont 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 wont like them

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

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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 its not! Worksheets Drill Memorization of discrete facts

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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 Algebra/Algebra.html

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Real-World Math The Futures Channel rld_movies.php Real-World Math Get the Math Math in the News

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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

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High achievement always occurs in the framework of high expectation. Charles F. Kettering ( )

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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.

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