Presentation on theme: "Focusing on Mathematical Reasoning: Transitioning to the 2014 GED® Test Presenters: Bonnie Goonen firstname.lastname@example.org."— Presentation transcript:
1Focusing on Mathematical Reasoning: Transitioning to the 2014 GED® Test Presenters: Bonnie GoonenSusan Pittman
2Session ObjectivesIdentify content of the Mathematics module of the 2014 GED® testExplore essential mathematical practices and behaviorsDiscuss beginning strategies for the classroomIdentify resources that support the transition to the next generation assessmentKey PointsReview the objectives of today’s session.
3Algebraic Sugar Cane10 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?
4Answer500. 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
5Mathematical Reasoning What’s New? Key PointsMathematical 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.
9What’s not directly assessed on the 2014 Mathematical Reasoning Module? Select the appropriate operations to solve problemsRelate basic arithmetic operations to one anotherUse estimation to solve problems and assess the reasonableness of an answerIdentify and select appropriate units of metric and customary measuresRead and interpret scales, meters, and gaugesCompare and contrast different sets of data on the basis of measures of central tendencyRecognize and use direct and indirect variationKey PointsThere 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.
10Mathematical Reasoning Item Sampler Key PointsReview the functionality of the 2014 GED® test by accessing the Item Sampler.
14Let’s Have Some Fun! It’s Your Turn! "Calculators can only calculate - they cannot do mathematics." -- John A. Van de WalleIt’s Your Turn!Let’s HaveSome Fun!Key PointsHave participants use the calculator to complete an activity, such as those available from the Texas Instrument site (free).
15Mathematical Practices: behaviors that are essential to the mastery of mathematical content Key PointsMathematical Practices are behaviors that are essential to the mastery of mathematical content.These practices are skills that have been identified from both theCommon Core State Standards for Mathematical Practices, and thePrinciples and Standards for School Mathematics developed by the National Council of Teachers of Mathematics.
16Mathematical Practices Mathematical fluencyAbstracting problemsBuilding solution pathways and lines of reasoningFurthering lines of reasoningEvaluating reasoning and solution pathwaysMost practices are not specific to any one particular area of mathematics contentKey PointsMathematical 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.
17“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
18Solution Pathways = Problem Solving Polya’s Four Steps to Problem SolvingUnderstand the problemDevise a planCarry out the planLook back (reflect)Key PointsGeorge 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 CaseIdentify 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.
19Value of Teaching with Problems Places students’ attention on mathematical ideasDevelops “mathematical power”Develops students’ beliefs that they are capable of doing mathematics and that it makes senseProvides ongoing assessment data that can be used to make instructional decisionsAllows an entry point for a wide range of studentsKey PointsReview the research-based positives of using actual mathematical problems when teaching concepts, rather than just computational skills.
20A Few Problem-Solving Strategies Look for patternsConsider all possibilitiesMake an organized listDraw a pictureGuess and checkWrite an equationConstruct a table or graphAct it outUse objectsWork backwardSolve a simpler (or similar) problemKey PointsThere 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.
21Explicit instruction matters Why teach different problem-solving strategies?Explicit instruction has shown consistent positive effects on performance with words problems and computationStudents receives extensive practice in use of newly learned strategies and skills
22Let’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 PointsThe 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.
23S tudy the problem (What am I trying to find?) SOLVE a ProblemS 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 PointsReview 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 factEliminate 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 answerCarry 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)
30SHow 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 rate1.1Not 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) = -58b) y = 85x – 58c) y = 85(10) – 58 = 850 – 58 = $792EThe amount earned after 10 days is greater than the amount earned after 5 days. This is consistent with what happened from day 2 to day 5. Since the slope and the y-intercept are integers, the answer must also be an integer.
31Another Way to SOLVES – How much will be earned on day 10? O – day 2 = 112 day 5 = 367 L – 1. find slope – rise run 2. Find linear equation 3. Substitute into linear equation V – test different ways E – examine by using graphDays$ in hundreds112367792
32Problem Solving Reflect on the process of problem solving Emphasis: mathematical discourse and classroom as a learning communityunderstanding and engaging with mathematicsextensions of solved problemsConstruct problem-solving strategies
33Quantitative Reasoning Skills Key PointsMathematical Practices are behaviors that are essential to the mastery of mathematical content.These practices are skills that have been identified from both theCommon Core State Standards for Mathematical Practices, and thePrinciples and Standards for School Mathematics developed by the National Council of Teachers of Mathematics.
34Some Big Ideas in Quantitative Reasoning (45%) Number sense concepts (rational numbers, absolute value, multiples, factors, exponents)Spatial visualizationDimensions, perimeter, circumference, and area of two-dimensional figuresDimensions, surface area, and volume of three-dimensional figuresInterpret and create data displaysProbability, ratio, percents, scale factorsGraphic literacyKey PointsForty-five percent of the items on the 2014 Mathematics module will focus on quantitative problem solving, including:Number operations/number senseGeometric thinkingStatistics and data representations (also included in Social Studies and Science)
35Let’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?
36Focus on GeometryStudents need proficiency in basic measurement and geometric thinking skills:Use conceptsUse spatial visualizationSelect appropriate units of measureIdentify and define different types of geometric figuresPredict impact of change on perimeter, area and volume of figuresCompute surface area and volume of composite 3-D geometric figures, given formulas as neededKey PointsGeometric 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.
37The Van Hiele Theory Level 1: Visualization Level 2: Analyze Level 3: Informal DeductionLevel 4: Formal DeductionLevel 5: RigorKey PointsThe 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:
38in Geometric Reasoning VisualizingThe First Stepin Geometric ReasoningKey PointsVisualization 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.
39Visualization Recognize and name shapes by appearance Do not recognize properties or if they do, do not use them for sorting or recognitionMay not recognize shape in different orientation (e.g., shape at right not recognized as square)Key PointsIn 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.ActivityIn 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.
40Visualization Key Points This activity works best when done in groups. The object will be shown for three seconds.Ask the following questionsFor 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)?
41Visualization 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.
42Visualization 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
43Implications for Instruction - Visualization Provide activities that have students sort shapes, identify and describe shapes (e.g., Venn diagrams)Have students use manipulativesBuild and draw shapesPut together and take apart shapesMake sure students see shapes in different orientationsMake sure students see different sizes of each shapeKey PointsAfter 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.
44in Geometric Reasoning AnalysisThe Second Stepin Geometric ReasoningKey PointsAt 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.
45Analysis Can identify some properties of shapes Use appropriate vocabularyCannot explain relationship between shape and properties (e.g., why is second shape not a rectangle?)Key PointsReview 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.
46Analysis Description 1 The design looks like a bird with a hexagon body;a square for the head;triangles for the beak and tail; andtriangles for the feet.Key PointsAt the analysis level, students can identify some properties of objects and generally are able to use geometric vocabulary.ActivityHave 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.
47Analysis 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.
48Analysis 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 PointsAt the analysis level, students can identify some properties of objects and generally are able to use geometric vocabulary.ActivityHave 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.
49Analysis 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
50Implications for Instruction - Analysis Work with manipulativesDefine properties, make measurements, and look for patternsExplore what happens if a measurement or property is changedDiscuss what defines a shapeUse activities emphasize classes of shapes and their propertiesClassify shapes based on lists of propertiesKey PointsStudents 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.
51Use the Language of Geometry! It’s Geometry Match! Congruent TriangleQuadrilateralUse the Languageof Geometry!PolynomialParallelKey PointsEmphasize 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!ReflectionRight AngleRotation
52Informal Deduction to Formal Deduction The Third and Fourth Steps in Geometric ReasoningKey PointsTo 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.
53Informal 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 PointsLevel 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.
54DeductionUsually not reached before high school; maybe not until collegeCan construct proofsUnderstand the importance of deductionUnderstand how postulates, axioms, and definitions are used in proofsKey PointsLevel 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.
55What 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?
56Geometric Reasoning Seeking Relationships Checking Effects of TransformationsGeneralizing Geometric IdeasConjecturing about the “always” & “every”Testing the conjectureDrawing a conclusion about the conjectureMaking a convincing argumentBalancing Exploration with DeductionExploring structured by one or more explicit limitation/restrictionTaking stock of what is being learned through the exploration
57Let’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.
60“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
61Algebraic 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.
62The 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.
64Algebra Misconceptions 1) a + a + a + a = 4a2) 3a x 2b = 5ab3) c x c = 2c4) 4d + 2e - d + 3e = 3d + 5e5) 8f - 3g + 2f - 6g = 10f - 3g6) 5y - y = 57) Find the perimeter of this shapea2b28) b x b x b x b = b49) 3(2k + 3) = 6k + 310) a(a + 3) = a2 + 3aab
65Algebraic Thinking in Adult Education Key PointsThis 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.
66Algebraic Thinking in Adult Education Create opportunities for algebraic thinking as a part of regular instructionIntegrate elements of algebraic thinking into arithmetic instructionAcquiring symbolic languageRecognizing patterns and making generalizationsReorganize formal algebra instruction to emphasize its applicationsKey PointsAlgebraic 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
67Examining 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.
68Some Big Ideas in Algebra VariableSymbolic NotationEqualityRatio and ProportionPattern GeneralizationEquations and InequalitiesMultiple Representations of FunctionsKey PointsAsk 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.
69VariableSome students believe that letters represent particular objects or abbreviated wordsStudents 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.
70Symbolic Notation Sign Arithmetic Algebra = (equal) A Few ExamplesSignArithmeticAlgebra= (equal). . . And the answer isEquivalence between two quantities+Addition operationPositive number-Subtraction operationNegative number
71Which 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.Answers32342698
72Patterns – Thinking Algebraically 3/31/2017Patterns – Thinking AlgebraicallyFinding patternsDescribing patternsExplaining patternsPredicting with patternsKey PointsAs 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.
73Teaching Patterns Banquet Tables Arrangement Arrangement Arrangement 3Arrangement 1 seats four people. How many people can be seated at Arrangement 100?Key PointsHave participants solve the pattern problem. Discuss how they determined the answer.
74Teaching PatternsBuild or draw the following sequence of houses made from toothpicks.Key PointsHave participants solve the pattern problem. Discuss how they determined the answer.
75Figure Out the DigitsIn 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 PointsMath 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.
761. 4 x 2 = 8= 5= 8= 95. 7 x 1 = 7= 9= 4= 79. 6/8 =3/4Key PointsHide this slide.Answersq = 9 r = 2 s = 7 t = 5 u = 4v = 1 w = 0 x = 3 y = 6 z = 8q = 9 r = 2 s = 7 t = 5 u = 4z = 8 v = 1 w = 0 x = 3 y = 6
77Use Multiple Representations Represent problems using symbols, expressions, and equations, tables, and graphsModel real-world situationsComplete problems different ways (flexibility in problem solving)Key PointsIn 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.
79Use Effective Questioning Techniques Key PointsEffective questioning techniques are an important part of teaching mathematical reasoning skills.
80Effective QuestionsAsk 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 PointsReview open-ended questions that require students to reason through the process, rather than just finding the answer.
81Teacher 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 AvoidLet me show you how to do this.That’s not correct.I’m not sure you want to do that.Key PointsInstructional feedback or responses are also an important part of teaching students how to reason through a process, rather than providing them with the answer.
83Math journals help students to . . . Be aware of what they do and do not knowMake use of prior knowledgeIdentify their mathematical questionsDevelop their ability to problem solveMonitor their own progressMake connectionsCommunicate more preciselyBe more aware of what they do and do not knowMake use of prior knowledge when solving new problemsIdentify their mathematical questionsDevelop their ability to think through a problem, organize their thoughts, and identify possible methods for solving a problemMonitor their own progressMake connections between mathematical ideas as they write about various problem-solving strategiesCommunicate more precisely about how they think
84Application 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 PointsActive 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.
87Introduction to Algebra Tiles Application of Basic Math Principles to CalculationIntroduction to Algebra TilesPositive TilesNegative Tilesx2x1Algebra 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 width3. Small square with 1 as its length and width.Remember, they could be called x, y, b, t, etc.
88Introduction to Algebra Tiles Each tile represents an area.xArea of large square = x (x) = x2x1Area of rectangle = 1 (x) = x1Area of small square = 1 (1) = 1Note: Tiles are not to scale. 10 little tiles don’t equal 1 big tile!
91Zero 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
92Use 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.
93Use Algebra Tiles to Model Addition of Integers Application of Basic Math Principles to CalculationUse 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.
94Application 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!
95Use Algebra Tiles to Model Integer Subtraction Subtraction can be interpreted as “take-away.”Subtraction can also be thought of as “adding the opposite.”
96Use Algebra Tiles to Model Subtraction of Integers Application of Basic Math Principles to CalculationUse Algebra Tiles to Model Subtraction of Integers(+5) – (+2) =(-4) – (-3) =
97Subtracting Integers – It’s Your Turn! Application of Basic Math Principles to CalculationSubtracting Integers – It’s Your Turn!(+3) – (-5)(-4) – (+1)(+3) – (-3)
98How 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 + 4xxxxxThe answer is the expression that is left.1111xx2x + 3111(5x + 4) – (2x + 3)= 3x +1
99“Minus times minus results in a plus, The reason for this, we needn't discuss.”—OgdenNash
100Use 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 PointsReview basic information on simplifying polynomials.
101Combining Like TermsHow much do I have here?I have 5x + 4
102I 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.
103Application 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 + 6Which looks best?
104Algebra Tiles – Time to Collect Tiles! Application of Basic Math Principles to CalculationAlgebra Tiles – Time to Collect Tiles!x2 + 6x + 8x2 – 4x + 3x2 + 7x + 62x2 + 7x + 6
105Multiplying Polynomials It’s just like figuring area!Place one term at the top of the gridPlace the second term on the side of the gridMaintain straight lines when filling in the gridThe inner grid is your answer!
106Multiplying Polynomials Application of Basic Math Principles to CalculationMultiplying Polynomials(x + 2)(x + 3) =x2 + 5x + 6xx
107Multiplying Polynomials Application of Basic Math Principles to CalculationMultiplying Polynomials(x – 1)(x + 4) =x2 + 3x -4xx –
108Multiplying Polynomials Application of Basic Math Principles to CalculationMultiplying Polynomials(x + 2) (x + 1) =(x + 5) (x + 3) =(2x + 2) (2x + 1) =Your Turn!
109Dividing PolynomialsYou 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 gridThe missing side is your answer!
112Factoring 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!
118There’s More! Investigate Use algebra tiles to prove that (x + 1) 2 and (x2 + 1) are not equivalentKey PointsIf time permits, show different concepts that can be taught using algebra tiles.
119So, why don’t all teachers use manipulatives? Lack of training or resourcesAvailability of funds or timeLessons using manipulatives may be noisier and not as neatA fear of a breakdown in classroom managementConcerns about connecting concrete-representational-abstractConcerns that students won’t “like” them
120It all leads to connecting mathematical concepts with effective mathematical practices/problem solvingSOLVEUNRAVELS T A R
122Real-World AlgebraMy 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
123Real-World MathThe Futures Channel Real-World Math Get the Math Math in the NewsScript: [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.
127Thank you for being with us today! 3/31/2017Thank you for being with us today!Bonnie GoonenSusan PittmanThe IPDAE project is supported with funds provided through theAdult and Family Literacy Act, Division of Career and Adult Education,Florida Department of Education.Goonen & Pittman-Shetler