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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 Key Points Review the objectives of today’s session.

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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 What’s New?**

Key Points Mathematical reasoning is the logical thinking skills that students develop while learning mathematics. Through mathematical reasoning, students to learn how to evaluate a problem, develop a plan to solve the problem, execute that plan, and then evaluate the results and make adjustments as needed. These same reasoning skills can then be applied to real-life problems on the job, at home, within the community, or in higher education and training.

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**Mathematical Reasoning Module**

3/31/2017 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 Key Points © Copyright 2013 GED Testing Service LLC. All rights reserved.

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**What’s new in the Mathematical Reasoning domain?**

07/2013 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 Key Points There is a greater emphasis on algebraic reasoning skills. As a result, programs will need to focus on helping students build their skills in algebra. Review the basic assessment targets in algebra from the bulleted list. Discuss with participants any concerns that they have related to the inclusion of more algebraic problem solving questions. © Copyright 2013 GED Testing Service LLC. All rights reserved.

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**What’s new in the Mathematical Reasoning domain?**

07/2013 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 Key Points There is a greater emphasis on algebraic reasoning skills. As a result, programs will need to focus on helping students build their skills in algebra. Review the basic assessment targets in algebra from the bulleted list. Discuss with participants any concerns that they have related to the inclusion of more algebraic problem solving questions. © Copyright 2013 GED Testing Service LLC. All rights reserved.

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**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 Key Points There are some skills that were assessed on the 2002 test that are not assessment targets on the 2014 version. Review the skills that will not be included on the test in 2014. Explain that these are foundational skills that students need in order to be effective problem solvers. However, these skills will not be directly assessed as part of the test. Foundational skills will be used in the process of problem-solving that is being assessed at a higher level. Emphasize the importance of teachers providing instruction in higher-order mathematics skills.

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**Mathematical Reasoning Item Sampler**

Key Points Review the functionality of the 2014 GED® test by accessing the Item Sampler.

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**Mathematical Tools New Realities**

07/2013 Mathematical Tools New Realities Key Points Mathematical reasoning is the logical thinking skills that students develop while learning mathematics. Developing mathematical reasoning skills enables students to: solve the problems they encounter on the test and carry those skills over to problem solving in all areas of their lives. Through mathematical reasoning, students learn how to evaluate a problem, design a plan to solve the problem, execute that plan, and then evaluate the results and make adjustments as needed. These same reasoning skills can then be applied to real-life problems on the job, at home, within the community, or in higher education and training. © Copyright 2013 GED Testing Service LLC. All rights reserved.

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**Mathematical Tools Key Points**

Review the different tools that students will have access to on the 2014 GED® test.

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**Mathematics “Symbol Selector” Explanation**

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**Let’s Have Some Fun! It’s Your Turn!**

"Calculators can only calculate - they cannot do mathematics." -- John A. Van de Walle It’s Your Turn! Let’s Have Some Fun! Key Points Have participants use the calculator to complete an activity, such as those available from the Texas Instrument site (free).

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

Key Points Mathematical Practices are behaviors that are essential to the mastery of mathematical content. These practices are skills that have been identified from both the Common Core State Standards for Mathematical Practices, and the Principles and Standards for School Mathematics developed by the National Council of Teachers of Mathematics.

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**Mathematical 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 Key Points Mathematical Practices focus on the reasoning skills that test-takers need in order to be effective problem solvers. Mathematical Practices are not specific to one content area and should be used irrespective of whether teachers are helping students build foundational skills or algebraic problem solving skills. Mathematical practices go beyond memorizing multiplication tables or knowing how to divide fractions. Students must build the skills and habits of thinking logically and mathematically so that they can use those skills in real-life problem solving situations.

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

Let’s Get Started! “Anyone who has never made a mistake has never tried anything new.” Albert Einstein

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**Solution Pathways = Problem Solving**

Polya’s Four Steps to Problem Solving Understand the problem Devise a plan Carry out the plan Look back (reflect) Key Points George Polya’s Four Steps to Problem Solving has stood the test of time and today is still one of the most widely used process for teaching students the process for solving problems. These steps provide test-takers a framework for building solution pathways when working with mathematical content. Students in adult education classrooms often want to just “do” the problem. This is analogous to the approach students often taken to writing – they just “write.” They do not want to read, plan, organize, edit and revise – they just want to write and move to the next task. Students need to understand that solving problems requires planning, drafting (trying the plan), checking results, and making “edits and revisions” if the answer is not correct or reasonable. Polya outlines four basic steps. Understand the Problem: Determine what is being asked, what information is known, what information is unnecessary to addressing the problem, and what information is missing or not known. Devise a Plan: Find a strategy to help in solving the problem. Sample strategies include: Look for a Pattern; Make a Table; Examine a Simpler Case Identify a Sub-goal; Examine a Related Problem; Work Backwards; Write an Equation; Draw a diagram; Guess and Check; Use Indirect Reasoning. Carry Out the Plan: Attempt to solve the problem with the chosen strategy. If the strategy does not work, try another strategy. Look Back: Check to see if the answer “makes sense.” A good way to look back is to use another strategy to see if the same answer results. 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 Key Points Review the research-based positives of using actual mathematical problems when teaching concepts, rather than just computational skills.

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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 Key Points There are many different strategies that can be used to solve mathematical problems. See if participants can think of other strategies that they or their students use.

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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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**Let’s SOLVE a Math Problem**

Sure-Fire Steps to Becoming a Math Genius! Even Albert Einstein said: “Do not worry about your difficulties in Mathematics. I can assure you mine are still greater.” Key Points The next slides take you through a problem solving process where students can “SOLVE” a math problem. Using the script on each of the slides, explain the step-by-step process of mathematical problem solving.

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

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! Key Points Review the following steps for solving mathematical problems: “S” stands for State The Problem. When you state or study the problem you need to: Highlight or underline the question. Answer the question, “What is the problem asking me to find?” “O” stands for Organize The Facts. When you organized the facts you need to: Identify each fact Eliminate unnecessary facts (By putting a line through it) List all necessary facts “L” stands for Line Up a Plan. When you line up a plan you need to: Choose the operations to be used (Add, Subtract, Multiply, Divide) Tell in words how you are going to solve the problem “V” stands for Verify Your Plan With Action. When you verify your plan with action, you need to: Estimate your answer Carry out your plan “E” stands for Examine The Results. When you examine the results you need to ask yourself: Does your answer make sense? (Check what the problem was asking you to find) Is your answer reasonable? (Check you estimate) Is your answer accurate? (Check your work and rework if needed)

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**What is the problem asking me to do?**

07/2013 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! © Copyright 2013 GED Testing Service LLC. All rights reserved.

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

07/2013 O = Organize the 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? Identify each fact. Eliminate unnecessary facts. List all necessary facts. © Copyright 2013 GED Testing Service LLC. All rights reserved.

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07/2013 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. © Copyright 2013 GED Testing Service LLC. All rights reserved.

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**V = Verify Your Plan $28.84 / 7 = $4.12 9 – 10 = 1 hr. 10 – 11 = 1 hr.**

Total hrs. Elapsed Time Total # Hours $28.84 / 7 = $4.12 Cost per Hour Total Rental Cost Total # Hours

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07/2013 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. © Copyright 2013 GED Testing Service LLC. All rights reserved.

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**Let’s SOLVE S O L V E 07/2013 Key Points**

Have the participants work in small groups and use the SOLVE strategy in order to solve the problem. Provide groups with chart paper, so they can post their results. Have groups share the strategy they used to solve the problem. Did they construct a chart, table, or graph, set up a formula, etc.? © Copyright 2013 GED Testing Service LLC. All rights reserved.

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S How much will the business earn on day 10? O On day 2, the business earned $112. On day 5, the business earned $367. Earnings will continue to increase at the same rate1. 1Not all the facts have to be numeric. L 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. V a) b = 367 – 85(5) = -58 b) y = 85x – 58 c) y = 85(10) – 58 = 850 – 58 = $792 E 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.

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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 112 367 792

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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**

Key Points Mathematical Practices are behaviors that are essential to the mastery of mathematical content. These practices are skills that have been identified from both the Common Core State Standards for Mathematical Practices, and the Principles and Standards for School Mathematics developed by the National Council of Teachers of Mathematics.

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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 Key Points Forty-five percent of the items on the 2014 Mathematics module will focus on quantitative problem solving, including: Number operations/number sense Geometric thinking Statistics and data representations (also included in Social Studies and Science)

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**Let’s SOLVE! S O L V E Key Points**

Have the participants work in small groups and use the SOLVE strategy in order to solve the problem. Provide groups with chart paper, so they can post their results. Have groups share the strategy they used to solve the problem. Did they construct a chart, table, or graph, set up a formula?

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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 Key Points Geometric thinking and problem solving will be included in Quantitative Problem Solving. Students will need proficiency in basic measurement and geometric thinking. Students will need to know the basic formulas for calculating the area of a square or perimeter of a circle. Students will also need to know how to apply higher level formulas, such as those associated with surface area and volume. Discuss with participants the importance of students having geometric reasoning skills, not only for the test, but also in real-life.

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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 Key Points The work of two Dutch educators, Pierre van Hiele and Dina van Hiele-Geldof, has given us a vision around which to design geometry curriculum. Through their research they have identified five levels of understanding spatial concepts through which students move sequentially on their way to geometric thinking. Students pass through all prior levels to arrive at any specific level. • The levels are not age-dependent in the way Piaget described development. • Geometric experiences have the greatest influence on advancement through the levels. • Instruction and language at a level higher than the level of the student may inhibit learning. Discuss with participants the importance of understanding the Van Hiele Theory and implementing instruction that respects where students are within the five levels. For more information on the Van Hiele Theory visit one of the following sites:

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**in Geometric Reasoning**

Visualizing The First Step in Geometric Reasoning Key Points Visualization is the first step in geometric reasoning. If students are unable to visualize or recognize shapes, the students will not be able to progress through to the next level. This is the foundational level at which all students need to begin in order to develop geometric reasoning skills.

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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) Key Points In visualization, students can name and recognize shapes by their appearance, but cannot specifically identify properties of shapes. Although they may be able to recognize characteristics, they do not use them for recognition and sorting. At this level, students manipulate physical models, e.g., lay one shape on top of another. Picturing, both on paper and in your mind, is an important part of geometric reasoning. You can learn the mathematics of making accurate drawings; drawings from which you can reason. You can also learn to pay more attention to the geometry you see and to visualize with your mind. Activity In this activity, you will display three different objects to participants. Each object should be displayed for no more than 3 seconds. Make sure that participants are all looking at the screen when the object is displayed. Click on the next slide. Count to. Click again. The figure will disappear. Have participants draw the figure. After participants draw the figure, have them describe what they drew. Click one more time and the figure will reappear.

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**Visualization Key Points**

This activity works best when done in groups. The object will be shown for three seconds. Ask the following questions For each image, what did you notice the first time you saw the shape? What features were in your first pictures? What did you miss when you first saw each shape? How did you revise your pictures? Answers will vary. For example, In the first image, did you notice where the bisectors met? Did you notice that each of the sides is the same length? Did you see this shape as a hexagon with two bisectors or as two triangles and two diamonds? Did you see this as a flat or a three-dimensional shape (a cube with a top and side missing)?

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**Visualization Key Points**

Repeat the process. Ask participants to look at screen. Display image for 3 seconds. Have participants draw the image. Discuss afterwards and then display the image again.

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**Visualization Key Points**

Repeat the process. Ask participants to look at screen. Display image for 3 seconds. Have participants draw the image. Discuss afterwards and then display the image again. 42

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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 Key Points After activity is complete, discuss the following: How can you help students better understand geometric concepts? How can you help students become better problem solvers? Share your ideas with your group. What are the geometric concepts that you feel are necessary in order to provide a full range of math instruction in the GED classroom? How will you incorporate the areas of geometry identified by GEDTS as most problematic into the math curriculum? If your students have little background knowledge in geometry, how could you help them develop and use such skills in your classroom? Discuss each of the bulleted items.

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**in Geometric Reasoning**

Analysis The Second Step in Geometric Reasoning Key Points At the analysis level, students begin to identify attributes of shapes and learn to use appropriate vocabulary related to attributes, but do not make connections between different shapes and their properties. Using an example of a area of a rectangle, students would be able to count the component square units.

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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?) Key Points Review each of the bulleted items. Discuss the importance of students being able to master this step in order to build their geometric problem solving skills.

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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. Key Points At the analysis level, students can identify some properties of objects and generally are able to use geometric vocabulary. Activity Have participants draw the figure based on the description provided. Do not respond to specific questions that they participants may have, but rather let them use their own knowledge and skills to construct the figure.

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**Analysis Description 1 Key Points**

Have participants share their figures with their peers. Discuss similarities and differences. Identify any areas where the participants may have added items that were not included in the description, i.e., lines for legs, circle for eye, etc.

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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. Key Points At the analysis level, students can identify some properties of objects and generally are able to use geometric vocabulary. Activity Have participants draw the figure based on the description provided. Do not respond to specific questions that participants may have, but rather have them use their own knowledge and skills to construct the figure.

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**Analysis Description 2 Key Points**

Have participants share their figures with their peers. Discuss similarities and differences. Discuss difficulties participants had in drawing this figure. Discuss why this figure was more difficult to construct that the last figure. 49

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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 Key Points Students need to understand not only “how” to do something but also “why” it works the way that it does. When possible use manipulatives in the classroom to provide students with hands-on learning activities. Manipulatives can be purchased through educational vendors or can be made of card-stock paper using templates that are available on the web. Have students create figures, make measurements, and look for patterns. The key to helping students build their analysis skills is to have ample time for exploration and discovery.

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**Use the Language of Geometry! It’s Geometry Match! Congruent**

Triangle Quadrilateral Use the Language of Geometry! Polynomial Parallel Key Points Emphasize to instructors the importance of using the language of mathematics during instruction. Students need to be able to recognize terms, what they mean, and in the case of geometry be able to recognize a diagram of each, i.e., students should be able to recognize parallel lines and understand that parallel lines do not intersect. A geometry matching game has been provided as a resource for trainers. Emphasize to instructors that they should slowly introduce each of the terms included in the geometry matching game. The matching game can be used as a fun, evaluative tool. However, it is recommended that students be asked to match no more than terms at a time (less if they are at an adult basic education level). It’s Geometry Match! Reflection Right Angle Rotation

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**Informal Deduction to Formal Deduction The Third and Fourth Steps **

in Geometric Reasoning Key Points To be successful on the 2014 GED® Mathematics test, students need to be competent in informal deduction. Students who intend to pursue postsecondary education and training should have some basic knowledge and understanding of formal deduction.

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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) Key Points Level 3 is Informal Deduction in which students are able to recognize relationships between and among properties of shapes or classes of shapes and are able to follow logical arguments using such properties. Students at this level can also see the relationship between length, width, and area for all rectangles.

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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 Key Points Level 4 is Formal Deduction in which students go beyond just identifying characteristics of shapes and are able to construct proofs using postulates or axioms and definitions. A typical high school geometry course should be taught at this level.

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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 you’re 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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**Let’s SOLVE! S O L V E Key Points**

Have the participants work in small groups and use the SOLVE strategy in order to solve the problem. Provide groups with chart paper, so they can post their results. Have groups share the strategy they used to solve the problem. Did they construct a chart, table, or graph, set up a formula, etc.

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**Algebraic Reasoning Skills**

6/2013 Algebraic Reasoning Skills © Copyright 2013 GED Testing Service LLC. All rights reserved.

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07/2013 Key Points © Copyright 2013 GED Testing Service LLC. All rights reserved.

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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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6/2013 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 Key Points According to a study from the Associated Press Poll, people have a “love-hate” relationship with mathematics. Twice as many people hated it as any other school subject. The rest of the group surveyed voted that math was their favorite subject. It’s all in one’s perception. © Copyright 2013 GED Testing Service LLC. All rights reserved.

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**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 a2b2 8) b x b x b x b = b4 9) 3(2k + 3) = 6k + 3 10) a(a + 3) = a2 + 3a a b

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**Algebraic Thinking in Adult Education**

Key Points This excellent research article is provided as a supplemental resource for you. As you move forward to help instructors address the need for algebra instruction, you will want to the article produced by the National Institute for Literacy in September 2010. Algebraic Thinking in Adult Education was written by Myrna Manly, Numeracy Consultant, and Lynda Ginsburg, Senior Research Associate for Mathematics Education at Rutgers University.

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**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 Key Points Algebraic Thinking in Adult Education outlines the steps that adult education programs can take in order to develop algebraic thinking as part of the regular instructional program. If you are able to provide participants with the article prior to the workshop, assign it as pre-work so that you can discuss various elements within the article and address any issues or concerns that instructors may have. Adapted from National Institute for Literacy, Algebraic Thinking in Adult Education, Washington, DC 20006

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**Examining the Components**

Let’s investigate the basics of algebra and see what we can include in our programs. Let’s take the model apart and dissect each component.

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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 Key Points Ask participants what they are currently teaching from this list. Chances are very good that they will indicate ratio and proportion and possibly variables or symbolic notation. However, rarely will participants indicate that they are currently teaching all of these “big” ideas. Explain that over the next few slides, they will have an opportunity to look at some of these “big” ideas and identify ideas that students need to learn if they are going to build algebraic reasoning skills.

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Variable Some students believe that letters represent particular objects or abbreviated words Students have difficulty discriminating among the different ways letters may be used. First, in arithmetic letters are first encountered in formulas (for example,), which are generally provided as procedural guides. The student is asked to substitute the appropriate quantities to determine the perimeter, area, or volume. Letters in the formulas are actually variables, but they are rarely discussed as such. Second, a letter can represent a specific number that is currently unknown but should be determined or “found” (e.g., ). Third, a letter can represent a general number (such as), which is not one particular value. Finally, letters can represent variables (such as in), each of which represents a range of unspecified values along with a systematic relationship among them (Kieran 1992, p. 396). Teachers may assume that students can easily navigate among these different uses and meaning of letters, but interviews with adult students indicate this is not necessarily so (Jackson and Ginsburg 2008). Understanding the use of variables is crucial for students’ success in algebra. Some students have been found to believe that letters represent particular objects or abbreviated words because of their alphabetic connection (e.g., that represents “three dogs”) or that a letter is a general referent (John’s height is 10 inches more than Steve’s height: ) (MacGregor and Stacey 1997). Again, teachers need to pay close attention to such errors.

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**Symbolic Notation Sign Arithmetic Algebra = (equal)**

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

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**Which Is Larger? 23 or 32 34 or 43 62 or 26 89 or 98 Key Points**

Many students still use an exponent as a multiplier, i.e., 32=3 x 2 = 6 rather than 32=3 x 3 = 9. Students who continue to use an exponent as a multiplier rather than as how many times a number is multiplied times itself, they will continue to miss questions and struggle with basic scientific notation. This activity is a fun way to have students predict which is larger and then do the calculations either by hand or using a calculator. Answers 32 34 26 98

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**Patterns – Thinking Algebraically**

3/31/2017 Patterns – Thinking Algebraically Finding patterns Describing patterns Explaining patterns Predicting with patterns Key Points As students learn algebra, they need to develop different procedures to use. Being able to recognize a pattern is an important critical thinking skill in solving certain algebraic problems. Finding patterns involves looking for regular features of a situation that repeats. Describing patterns involves communicating the regularity in words or in a mathematically concise way that other people can understand. Explaining patterns involves thinking about why the pattern continues forever, even if one has not exhaustively looked at each one. Predicting with patterns involves using your description to predict pieces of the situation that are not given.

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**Teaching Patterns Banquet Tables**

Arrangement Arrangement Arrangement 3 Arrangement 1 seats four people. How many people can be seated at Arrangement 100? Key Points Have participants solve the pattern problem. Discuss how they determined the answer.

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Teaching Patterns Build or draw the following sequence of houses made from toothpicks. Key Points Have participants solve the pattern problem. Discuss how they determined the answer.

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Figure Out the Digits In this number puzzle, each letter (q - z) represents a different digit from 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. Key Points Math benders can be helpful when teaching students how to reason when solving a math problem. This is one example of a math bender (brain teaser) that can be used in the classroom.

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1. 4 x 2 = 8 = 5 = 8 = 9 5. 7 x 1 = 7 = 9 = 4 = 7 9. 6/8 =3/4 Key Points Hide this slide. Answers q = 9 r = 2 s = 7 t = 5 u = 4 v = 1 w = 0 x = 3 y = 6 z = 8 q = 9 r = 2 s = 7 t = 5 u = 4 z = 8 v = 1 w = 0 x = 3 y = 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) Key Points In middle school, students learn that there are multiple ways to represent problems. However, rarely in adult education mathematics classes do we use the same process and represent problems in multiple ways. Problems can be represented using symbols, expressions, equations, tables, and graphs. Review the chart on the slide. Remind instructors that presenting information through real-world situations is one of the best ways to allow students to make the connection and thus retain the information.

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**Vertical Multiplication of Polynomials**

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**Use Effective Questioning Techniques**

Key Points Effective questioning techniques are an important part of teaching mathematical reasoning skills.

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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 ? Key Points Review open-ended questions that require students to reason through the process, rather than just finding the answer.

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**Teacher Responses Phrases to Use Phrases to Avoid**

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. Key Points Instructional feedback or responses are also an important part of teaching students how to reason through a process, rather than providing them with the answer.

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Use Journaling

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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 Be more aware of what they do and do not know Make use of prior knowledge when solving new problems Identify their mathematical questions Develop their ability to think through a problem, organize their thoughts, and identify possible methods for solving a problem Monitor their own progress Make connections between mathematical ideas as they write about various problem-solving strategies Communicate more precisely about how they think

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**Application of Basic Math Principles to Calculation**

Make It Real! Mathematics is like a video game; If you just sit and watch, You’re wasting your time. Key Points Active learning is essential if students are to gain the mathematical reasoning skills they need to be successful on the 2014 GED® test. Students should have the opportunity to work in teams or small groups and in large groups whenever possible. Students need to do more than merely review a process or algorithm and then complete sample problems. They must have the opportunity to practice new skills and learn how to apply those skills to new situations – especially real-life situations, they may encounter at home, at work, or in postsecondary education and training.

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**Algebra Manipulatives (the “C” of CRA)**

6/2013 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. Key Points One strategy in teaching students why things work is the use of manipulatives in the mathematical classroom. Review the research on the use of manipulatives and their positive effect on problem solving and higher-order thinking skills. © Copyright 2013 GED Testing Service LLC. All rights reserved.

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**Big Ideas Using Algebra Tiles**

6/2013 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 Key Points Review how algebra tiles can be used to represent the various “big ideas” of algebraic thinking skills, including quadratic equations and polynomials. © Copyright 2013 GED Testing Service LLC. All rights reserved.

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**Introduction to Algebra Tiles**

Application of Basic Math Principles to Calculation Introduction to Algebra Tiles Positive Tiles Negative Tiles x2 x 1 Algebra tiles can be used to model algebraic expressions and operations with algebraic expressions. There are three types of tiles: 1. Large square with x as its length and width. 2. Rectangle with x and 1 as its length and its width 3. Small square with 1 as its length and width. Remember, they could be called x, y, b, t, etc.

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**Introduction to Algebra Tiles**

Each tile represents an area. x Area of large square = x (x) = x2 x 1 Area of rectangle = 1 (x) = x 1 Area of small square = 1 (1) = 1 Note: Tiles are not to scale. 10 little tiles don’t equal 1 big tile!

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What’s My Polynomial?

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**Simon says show me . . . 2x2 4x - x2 2x + 3 - x2 + 4 2x2 + 3x + 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**

Application of Basic Math Principles to Calculation Use Algebra Tiles to Model Addition of Integers (+3) + (+1) = (-2) + (-1) = Addition can be viewed as “combining”. Combining involves the forming and removing of all zero pairs. For each of the given examples, use algebra tiles to model the addition.

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**Application of Basic Math Principles to Calculation**

Addition of Integers (+3) + (-1) = (+4) + (-4) = Don’t 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**

Application of Basic Math Principles to Calculation Use Algebra Tiles to Model Subtraction of Integers (+5) – (+2) = (-4) – (-3) =

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**Subtracting Integers – It’s Your Turn!**

Application of Basic Math Principles to Calculation Subtracting Integers – It’s 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) Now remove the tiles which match in each expression. 5x + 4 x x x x x The answer is the expression that is left. 1 1 1 1 x x 2x + 3 1 1 1 (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. Key Points Review basic information on simplifying polynomials.

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

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**I have x2 + 2x + 6 Combining Like Terms How much do I have here?**

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

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**Application of Basic Math Principles to Calculation**

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 x2 + 7x + 6 Which looks best?

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**Algebra Tiles – Time to Collect Tiles!**

Application of Basic Math Principles to Calculation Algebra Tiles – Time to Collect Tiles! x2 + 6x + 8 x2 – 4x + 3 x2 + 7x + 6 2x2 + 7x + 6

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**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!

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**Multiplying Polynomials**

Application of Basic Math Principles to Calculation Multiplying Polynomials (x + 2)(x + 3) = x2 + 5x + 6 x x

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**Multiplying Polynomials**

Application of Basic Math Principles to Calculation Multiplying Polynomials (x – 1)(x + 4) = x2 + 3x -4 x x –

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**Multiplying Polynomials**

Application of Basic Math Principles to Calculation Multiplying Polynomials (x + 2) (x + 1) = (x + 5) (x + 3) = (2x + 2) (2x + 1) = Your Turn!

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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 doesn’t matter which) Fill in the inner grid The missing side is your answer!

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Dividing Polynomials x2 + 7x + 6 = x + 1 x + 6

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Dividing Polynomials 2x2 + 5x – 3 x + 3 x2 – 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**

x2 + 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**

x2 – 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**

x2 – 4 2x2 – 3x – 2 2x2 + 3x – 3 -2x2 + x + 6

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**There’s More! Investigate**

Use algebra tiles to prove that (x + 1) 2 and (x2 + 1) are not equivalent Key Points If time permits, show different concepts that can be taught using algebra tiles.

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**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

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It all leads to connecting mathematical concepts with effective mathematical practices/problem solving SOLVE UNRAVEL S T A R

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**Conceptual Teaching What is conceptual teaching? What it’s not!**

07/2013 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 Key Points Discuss what conceptual teaching is and what it is not. © Copyright 2013 GED Testing Service LLC. All rights reserved.

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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? Script: [Insert different voice to read the problem on the screen, then begin the actual script.] Even with the best teaching strategies available, students may still ask “Why do I need to know this?” A strategy to use in the classroom is to connect algebra to real-world situations. Show students how they use algebra both in their daily lives, as well as in the workplace. There are many different resources to get you started. The Department of Mathematics Education at the University of Georgia provides numerous faculty and student created problems, like the one on the screen, with sample student answers that provide different ways to solve the real-world problem. Note: The following is the link for CTL (Career-Technical Learning) Resources for Algebra. CTL Resources for Algebra. The Department of Mathematics. Education University of Georgia

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Real-World Math The Futures Channel Real-World Math Get the Math Math in the News Script: [Insert different voice to read the problem on the screen, then begin the actual script.] Even with the best teaching strategies available, students may still ask “Why do I need to know this?” A strategy to use in the classroom is to connect algebra to real-world situations. Show students how they use algebra both in their daily lives, as well as in the workplace. There are many different resources to get you started. The Department of Mathematics Education at the University of Georgia provides numerous faculty and student created problems, like the one on the screen, with sample student answers that provide different ways to solve the real-world problem. Note: The following is the link for CTL (Career-Technical Learning) Resources for Algebra.

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**A Few Strategies to Get Started**

6/2013 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 Key Points As you get started, you will want to start with a few basic strategies. [Review each of the strategies on the slide.] © Copyright 2013 GED Testing Service LLC. All rights reserved.

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Q & A

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**“High achievement always occurs in the framework of high expectation.”**

6/2013 “High achievement always occurs in the framework of high expectation.” Charles F. Kettering ( ) © Copyright 2013 GED Testing Service LLC. All rights reserved.

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**Thank you for being with us today!**

3/31/2017 Thank you for being with us today! Bonnie Goonen Susan Pittman 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. Goonen & Pittman-Shetler

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