1 Mathematical Reasoning07/2016 Mathematical Reasoning © Copyright GED Testing Service LLC. All rights reserved
2 Mathematical Reasoning07/2016 © Copyright GED Testing Service LLC. All rights reserved
3 Mathematical Reasoning07/2016 Session Objectives Review expectations of the Mathematical Reasoning test Explore and use reading strategies to improve problem solving Explore thinking routines to teach the steps in mathematical reasoning Share resources and ideas Key Points Review the objectives of the session. Emphasize the importance of exploring strategies and using those strategies during activities included in the session. Explain that during the session, participants will have an opportunity to review a number of resources that they can use in the classroom. At this point, you may want to briefly walk participants through the contents of their materials. Explain that they should feel free to write on the activities in the book, since they have clean copies in the resource section of the workbook that they can copy and use in the classroom. © Copyright GED Testing Service LLC. All rights reserved
4 Mathematical Reasoning07/2016 An Overview Stuff to Teach Key Points This section will provide a very brief overview of the Mathematical Reasoning test and PLDs and HIIs. If participants are new to the field, additional time may be needed in this area. © Copyright GED Testing Service LLC. All rights reserved
5 Mathematical Reasoning07/2016 Mathematical Reasoning Overview Item Types One test with calculator allowed on most items Content 45% - Quantitative Problem Solving 55% - Algebraic Problem Solving Texas Instruments - TI 30XS Multiview™ Integration of mathematical practices Technology-enhanced items Multiple choice Drag-drop Drop-down Fill-in-the-blank Key Points Review the information on the slide. For veteran instructors, this will be a quick review. New instructors may need explanation of some of the items. © Copyright GED Testing Service LLC. All rights reserved
6 Mathematical Reasoning07/2016 Math Practices Overarching Habits of Mind 1. Make sense of problems and persevere in solving them. 6. Attend to precision. Reasoning and Explaining 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. Modeling and Using Tools 4. Model with mathematics. 5. Use tools strategically. Seeing Structure and Generalizing 7. Look and make use of structure. 8. Look for and express regularity in repeated reasoning. Key Points Review the information on the slide. Discuss how these practices are integrated into different question types. Note: The following is background information on math practices. Standards for Mathematics Mathematical practices are not standards that are taught in isolation. Instead, these standards connect with content standards at each level. The standards of mathematical practice are the “how” of math – how students become participatory practitioners of math. Overarching Habits of Mind The eight practices fall into four basic categories. First, you have the “habits of mind” that students need as effective mathematical thinkers: Make sense of problems and persevere in solving them. In this practice, students know how to explain a problem and solve it. Attend to precision. One of the most important elements within this practice is the attention to precision. In this practice, students understand math terms, recognize symbols, and apply concepts to new situations. Reasoning and Explaining 2. Reason abstractly and quantitatively. In other words, mathematically proficient students know how to reason in mathematics, not just compute. Many students can compute or do basic math calculations. However, they don’t know how to use those skills to determine how to solve problems. 3. Construct viable arguments and critique the reasoning of others. Mathematically proficient students analyze situations and determine what they need to do in order to solve a problem. Modeling and Using Tools 4. Model with mathematics. This practice is the student’s ability to analyze the relationships among numbers and other information and solve a word problem. They know how to transfer knowledge from one problem to another. They know how to solve a problem and determine whether or not the answer is reasonable. 5. Use tools strategically. It is essential that students know how to use the tools of mathematics. Students need to be able to use technology whether that is a calculator or a spreadsheet to solve problems. Seeing Structure and Generalizing 7. Look and make use of structure. An essential practice is the ability to see patterns. Students need to see these patterns from basic number properties, such as the distributive or commutative property, to more complex patterns found in algebraic expressions. 8. Look for and express regularity in repeated reasoning. Mathematically proficient students see not only the process but also the shortcuts, and they understand the concept that is behind the shortcut. It is not enough for students to learn a shortcut. They have to understand the underlying concept and how it can be used in different types of problems. Last, but certainly not least, they can evaluate the problem and determine if their solution is reasonable. Workbook p. 2 © Copyright GED Testing Service LLC. All rights reserved
7 Mathematical Reasoning07/2016 Different Versions Official Version uses the official language of the GED® test assessment targets and indicators. Student-friendly Version translates the official assessment target language into simpler language that is less technical and more understandable by students. Key Points For both the GED® test and the GED Ready® practice test PLDs, there are two versions that you can use: The official version that includes the official language of the GED® test assessment targets and indicators The student-friendly version that translates the official assessment target language into simpler language that is less technical and more understandable by students. The student-friendly PLDs are the basis for the study recommendations received after taking the GED® test or GED® Ready practice test. Note: You may wish to show participants the two different versions and where to locate them on the GEDTS website. Providing participants with the opportunity to see the similarities and differences will assist them in determining which format is most useful for them to use. © Copyright GED Testing Service LLC. All rights reserved
8 Mathematical Reasoning07/2016 For example . . . Official Version Use the Pythagorean theorem to determine unknown side lengths in a right triangle at a satisfactory level. Student-Friendly Version Use the Pythagorean theorem (𝑎2+𝑏2=𝑐2) to determine unknown side lengths in a right triangle at a satisfactory level. Key Points This is one example of how the same indicator is written in the official version versus the student-friendly version. Notice how the student-friendly version provides the equation for the Pythagorean theorem, in case the student doesn’t recognize the name. © Copyright GED Testing Service LLC. All rights reserved
9 Too Much to Teach – Too Little Time?Mathematical Reasoning 07/2016 Too Much to Teach – Too Little Time? Key Points We always seem to have too much content to teach and too little time. Plus, we are often faced with multi-level and multi-area classrooms. So, how do we use the PLDS? © Copyright GED Testing Service LLC. All rights reserved
10 How to Use PLDs in the ClassroomMathematical Reasoning 07/2016 How to Use PLDs in the Classroom Use PLDs to: Tip 1: Assess student’s current skill level Tip 2: Determine when students are ready to test Tip 3: Shape learning activities Tip 4: Add perspective to lesson plans Key Points There are many different ways that we can use PLDs to assist us in developing an effective curriculum. Here are four tips to get started. Tip 1: Use PLDs to assess a student’s current skill level Identify where to focus in order to develop the skills students need to move to the next performance level Tip 2: Use PLDs to determine when students are ready to test Determine if a student should take GED Ready® or the GED® test Use with the Enhanced Score Report's personalized study plan to create a plan for developing student skills Tip 3: Use PLDs to shape learning activities Set learning objectives in your classroom based on the PLDs Determine if you need to adjust how you're approaching the content Work one-on-one with students to help develop needed skills Tip 4: Use PLDs to add perspective to lesson plans Determine how prepared your students are and create lesson plans accordingly Identify the gaps in your students’ skills and develop focused lesson plans to address those gaps © Copyright GED Testing Service LLC. All rights reserved
11 High Impact IndicatorsMathematical Reasoning 07/2016 High Impact Indicators Important skills that are widely applicable May currently receive light coverage during GED® test preparation Lend themselves to straightforward instruction Key Points Have you heard about High Impact Indicators? While all of the indicators listed in the Performance Level Descriptors describe skills that are essential for test-taker success, GEDTS has provided us with specific PLDs that are termed High Impact Indicators (HII). What are High Impact Indicators and how were they determined? The GEDTS® extensively analyzed test-taker performance on the operational GED® test. They examined differences in performance between those who scored between and those who scored in each content area. Those results were then examined based on the following factors: Were the skills widely applicable? How much coverage do the skills typically receive during GED® test preparation? Could these skills be taught in a straightforward manner in instructional programs? These are definitely important skills to that should be taught in our classroom. © Copyright GED Testing Service LLC. All rights reserved
12 High Impact Indicator for MathMathematical Reasoning 07/2016 High Impact Indicator for Math Indicator What to look for in student work. Students’ work shows they have… Q.4 Calculate dimensions, perimeter, circumference, and area of two-dimensional figures Q.5 Calculate dimensions, surface area, and volume of three-dimensional figures identified the dimensions of a geometric figure from a diagram, then substituted the values for those dimensions into the appropriate formula for geometric measurement, then calculated the resulting numerical expression. calculated the perimeter of polygons. identified the shapes that comprise a composite figure. Key Points Before we look at strategies, let’s first look at one example of a High Impact Indicator. We know that in Mathematical Reasoning students need to have a basic understanding of two- and three-dimensional figures and the ability to calculate such things as perimeter, circumference, surface area, and volume. To assist us in assessing this indicator, students should be able to: Identify the dimensions of a geometric figure from a diagram, then substitute the values for those dimensions into the appropriate formula for geometric measurement, then calculate the resulting numerical expression. Calculate the perimeter of polygons. Identify the shapes that comprise a composite figure.. © Copyright GED Testing Service LLC. All rights reserved
13 Sequencing – It’s Essential in Math!Mathematical Reasoning 07/2016 Sequencing – It’s Essential in Math! R.3.1: Order sequences of events in texts. Primarily measured with literary texts. SSP.3.a: Identify the chronological structure of a historical narrative and sequence steps in a process. SP.3.b Reason from data or evidence to a conclusion. SP.3.c Make a prediction based upon data or evidence MP.1 a. Search for and recognize entry points for solving a problem. MP.1 b. Plan a solution pathway or outline a line of reasoning MP.3.a Build steps of a line of reasoning or solution pathway, based on previous steps or givens. Key Points Another High Impact Indicator is sequencing. Yes, the chart shows that this is a High Impact Indicator for RLA. But, where else do we see sequencing on the GED® test? Is there a relationship between this indicator and indicators in other content areas? We know that students need be able to determine the sequence of events in a literary work. Being able to determine sequence also allows students to follow a historical narrative or see how a series of events led to a certain outcome. Think about science. The scientific method is a sequence of steps taken to prove or disprove a hypothesis. Most importantly having the ability to sequence properly allows students to identify the point at which they can begin solving a problem in mathematics. Students need to be able to set up a plan or a sequence of steps in order to solve a problem. Think about the challenges many students have in identifying the correct number of steps required to complete a multi-step problem. How many students don’t take the correct steps to the logical and correct conclusion of a problem? It is important to remember that the GED® test is integrated across the content areas. There is a relationship that occurs between many of the different indicators from one area to another. That is why as we look at different strategies, we should always assess them for usability across different content areas. Workbook p. 3 © Copyright GED Testing Service LLC. All rights reserved
14 Reading and Reasoning in MathematicsMathematical Reasoning 07/2016 Reading and Reasoning in Mathematics Steps to Success Key Points One area that crosses all content areas is the skill of close reading. While we don’t often think about teaching reading in a mathematics class, being an effective reader is an essential skill in becoming an effective problem solver. © Copyright GED Testing Service LLC. All rights reserved
15 Mathematical Reasoning07/2016 "Mathematics is no more computation than literature is typing." Key Points Review the quote from Dr. Paulos and emphasize the importance of looking beyond basic computation, a level at which many students are comfortable and fairly competent, to conceptual understanding which can be used to solve problems academically and in real-life. Note: The following information is provided as background information regarding Dr. Paulos. John Allen Paulos is a well-known author, popular public speaker, and former monthly columnist for ABCNews.com, the Scientific American, and the Guardian. Professor of Math at Temple University in Philadelphia, Dr. Paulos earned his Ph.D. from the University of Wisconsin. His recent book (November, 2015) is A Numerate Life - A Mathematician Explores the Vagaries of Life, His Own and Probably Yours. Other writings of his include Innumeracy (NY Times bestseller for 18 weeks) and A Mathematician Reads the Newspaper (on the Random House Modern Library's compilation of the 100 best nonfiction books of the century). John Allan Paulos, Ph.D. Temple University © Copyright GED Testing Service LLC. All rights reserved
16 We need to help students . . .Mathematical Reasoning 07/2016 We need to help students . . . Build their reading skills Build math vocabulary Depend less on rote memory Key Points So what can we do to help our students become more effective math problem solvers? Think for a moment about the skills that effective readers have. Now, think how these skills would impact your students’ ability to read, analyze, and solve math problems. Each of the items listed is essential in students being able to understand what the math problem is telling them, as well as what they need to do to solve that problem. Note: Animations are included with this slide. Review the animations prior to conducting the training, so that you are aware of each and can comment it again. Build their reading skills. Students need practice in closely reading math problems. They need to be able to slow down and look at the details of a problem. The details of this problem focus on the numbers and days as well as the question that is being asked. Build math vocabulary. If students are going to be able to read and understand problems, they need to know the vocabulary of math – not just words, but symbols as well. In this case, students need to know what the phrase “to increase at the same rate” means. Depend less on rote memory and increase conceptual knowledge. There are many rules in math. However, memorizing is not the best approach. In this case, the equation that would be need to be known is complex. The student may or may not be able to recall the equation. However, if the student has conceptual understanding of a linear relationship, the student can solve the problem whether through use of a formula or through another heuristic. Students need conceptual understanding in mathematics. Increase conceptual knowledge © Copyright GED Testing Service LLC. All rights reserved
17 Two Essential StrategiesMathematical Reasoning 07/2016 Two Essential Strategies Helping students learn how to learn is critical to aiding the development of higher-order thinking skills Modeling is one way to teach students how to learn Scaffolding allows students to practice with diminishing support—to build confidence and competence Key Points Discuss how two strategies, modeling and scaffolding, can support visual thinking in the classroom and help student learn how to learn through development of higher-order thinking skills. Emphasize how both are dependent on the teacher using a more direct approach to instruction. Modeling requires the teacher to clearly demonstrate how he/she thinks through a problem and works to a solution. This requires that teachers work slowly and thoughtfully through a problem so students can see how the “thinking process” is used to arrive at a reasonable solution. Scaffolding requires that teachers understand where students are in the learning process in order to help build students confidence and competence. In scaffolding instruction, the concept or skill to be taught is broken in smaller parts with each part building on the previous part. © Copyright GED Testing Service LLC. All rights reserved
18 The Payoff for StudentsMathematical Reasoning 07/2016 The Payoff for Students When higher-order thinking skills are used, students carry the knowledge longer. Knowledge gained from higher-order thinking processes is more easily transferrable…and that knowledge becomes accessible for solving new problems. Key Points What is the payoff for modeling and scaffolding? Review bulleted items. © Copyright GED Testing Service LLC. All rights reserved
19 Reading and Reasoning ProcessMathematical Reasoning 07/2016 Reading and Reasoning Process First Read: Read for Understanding Second Read: Identify a Problem-Solving Process Third Read: Solve the Problem and Check for Reasonableness Key Points Miler and Koesling in Mathematics Teaching for Understanding: Reasoning, Reading, and Formative Assessment outline a three-step process for helping students use their reading and reasoning skills in order to solve problems. The process involves multiple readings of the problem. Each step in the process is essential if students are going to build the reading and reasoning skills necessary to be effective problem solver. Miller, P. and Koesling, D. “Mathematics Teaching for Understanding: Reasoning, Reading, and Formative Assessment. Danvers, MA Workbook p. 4 © Copyright GED Testing Service LLC. All rights reserved
20 First Read: Read for UnderstandingMathematical Reasoning 07/2016 First Read: Read for Understanding Read through the problem aloud – noting your reactions to what you’re reading. What vocabulary do you not know? What’s the real-world context of the problem? Is there a picture that can help you visualize the problem? What questions are being asked? Key Points Briefly review the questions that students should be asking themselves as they do a first read of the problem. Miller, P. and Koesling, D. “Mathematics Teaching for Understanding: Reasoning, Reading, and Formative Assessment.” Danvers, MA © Copyright GED Testing Service LLC. All rights reserved
21 Mathematical Reasoning07/2016 First Read: Read for Understanding Key Points First Read: Read for Understanding The first read sets the stage for solving the problem. In a first read, students make sure that they understand the vocabulary that is used. If there are visual representations, such as a graph, they should also look at it to see if there is information that might be useful as they read for understanding. Note: Take on the role of the classroom teacher and walk the participants through the word problem using the following text as a reference. Read the math problem. Are there any words with which students may not be familiar? What about the word “respectively”? Would students know that respectively provides them important information about the costs of a one-bedroom versus a two-bedroom apartment? Maybe they don’t know the word, but they can use their own background knowledge to make the assumption that a one-bedroom apartment would cost less than a two-bedroom apartment. Now that you’ve read the problem, what is the real-world context of the problem? Could your students explain what this problem is about in their own words? In this case, the problem is about apartments and how much they cost per month. What does the problem ask you to do? It asks you to determine how many of the 12 units are two-bedroom apartments. As an effective problem solver, you would ask yourself these questions subconsciously. You set the stage for problem solving by gathering information and getting a “feel” for the problem. Unfortunately, most students head straight for the numbers in a problem and try to attempt a solution from there without understanding the context or in some cases what is being asked. Answer - 7 Workbook p. 5 © Copyright GED Testing Service LLC. All rights reserved
22 Mathematical Reasoning07/2016 Tiered Vocabulary Workbook p. 6 Tier 3 Domain-specific academic vocabulary Tier 2 High-Utility academic vocabulary found in many content texts, cross-curricular terms Tier 1 Everyday words, familiar to most students primarily learned through conversation Absolute value Additive inverse Algorithm Attribute Constant Distance formula Exponent Function Dependent variable Independent variable Linear Numerical expression Profit Property Proportional gain Rate of change Strategy Value Analyze Compare Contrast Demonstrate Describe Argument Conclusions Evidence Determine Develop Evaluate Explain Identify Infer Draw Distinguish Suggest Interpret Organize Illustrations Predict Key Points Note: This slide has two animations. Click to reveal the Tier 2 and Tier 3 vocabulary words. In the problem that you just read, most of the vocabulary was very clear. There were no high-level math vocabulary words. However, the word “respectively” may have posed a problem for students. That word is considered a high-utility word – or a word that can be used in many different contexts. When looking at vocabulary, there are basically three tiers of words that students need as they grow as learners. Tier 1 – Everyday words, familiar to most students primarily learned through conversation. These are the words that students learned early in their homes and in school. You may be familiar with words from the Fry or Dolch list. Tier 2 – High-utility academic vocabulary found in many content texts, cross-curricular terms. If you are interested in identifying even more of these words, you may want to download the Academic Word List handout included in the resources for this course. The Academic Word List was developed at the University of New Zealand and represents 500 words that are most commonly found in postsecondary text. Tier 3 – Domain-specific academic vocabulary. Now, look at this list of words – notice how they are domain-specific, i.e., tied directly to mathematics. Would your students recognize and understand these words? Most adult learners need significant work in building their mathematical vocabulary. © Copyright GED Testing Service LLC. All rights reserved
23 Mathematical Reasoning07/2016 Building Vocabulary Definition (In Own Words) Facts/Characteristics Characteristics or Facts Word or Symbol Examples Non-examples Non-Examples Key Points Sometimes, students need a graphic organizer that they can use in order to learn a math term and to build conceptual understanding related to that term. This is the Frayer Model. Students add information to the organizer as they work with the term or symbol. This graphic organizer can be used for terms that students find most challenging. Students begin by writing the term or symbol in the oval. Next, students should include a definition that is in the student’s own words – not a textbook definition. Next, students include facts or characteristics of the term or symbol. While some graphics organizers, ask students to provide examples, most do not include non-examples. The Frayer Model, with its inclusion of both, helps students have a better understanding of the term and the concepts behind it. Frayer Model – (Barton and Heidema, 2002) Workbook p. 7 © Copyright GED Testing Service LLC. All rights reserved
24 Tools for Building VocabularyMathematical Reasoning 07/2016 Tools for Building Vocabulary An equation is a mathematical statement that shows that two expressions are equal. Always has one equal sign The left side is equivalent to the right side Some equations have 0, 1, 2, or more solutions Some are algebraic Facts/Characteristics Equation ab = ba (an identify) F = 1.8C +32 (a formula) 5 = 6 = 11 (a number statement) x = 3 (statement of value) Non-examples 2x + 3y (expression) 3 (number) Perimeter (word) x
25 Mathematical Reasoning07/2016 Second Read: Identify a Problem-Solving Process What is the pertinent information in this problem? What problem-solving strategies could I use? Which of those problem-solving strategies is best suited for this problem? How will I represent the problem in the symbolic language of mathematics? What mathematical details will I select as I reason and solve this problem? Key Points Briefly review the questions that students should be asking themselves as they do a second read of the problem. Miller, P. and Koesling, D. “Mathematics Teaching for Understanding: Reasoning, Reading, and Formative Assessment.” Danvers, MA © Copyright GED Testing Service LLC. All rights reserved
26 Second Read: Identify a Problem-Solving ProcessMathematical Reasoning 07/2016 Second Read: Identify a Problem-Solving Process Key Points Note: There are three animations on this slide. Click to view each graphic. The second read of a word problem requires that students identify the FACTS of the problem. Without the facts relevant to solving the problem, the student will arrive at an incorrect answer. Likewise, if irrelevant facts are used the student will also arrive at an incorrect answer. Also important in the second read is the student’s determination of an appropriate thinking strategy or heuristic that he/she can use to approach solving the problem. Does the student need to draw a table or picture? Would it be helpful to look for a pattern or make a list? There are a number of different approaches that can be taken. Sadly, many students turn to the infamous “guess and check” strategy too often. They need to broaden the number of strategies at their disposal. Last, but not least in this read, the student needs to determine the math operations that will be needed. Without this second read, students will not have everything they need in order to solve a problem – and solve it correctly. Miller, P. and Koesling, D. “Mathematics Teaching for Understanding: Reasoning, Reading, and Formative Assessment.” Danvers, MA © Copyright GED Testing Service LLC. All rights reserved
27 Mathematical Reasoning07/2016 Begin with the Basics Noticing Wondering Allows all students to participate Work independently or in groups What is stated in the problem What are the “givens” of the problem Is the planning part Talk about strategies to use Restate the problem Pose questions about what they noticed Allows students to slow down and think Brainstorm, list, and discuss ideas Key Points For strong problem solvers, the basics of “noticing” and “wondering” are second nature. In fact, it is often done subconsciously. For students who are not strong problem solvers, these two activities need to be taught. They are part of the questioning that happens during the first and second read of a problem. Noticing generally occurs during that first read. What students notice in a problem leads them to asking questions or “wondering”. These two activities assist the student in identifying the strategies or heuristics to be used and the computations that should be made to solve the problem. Teachers need to practice these skills with students so the skills become second nature. © Copyright GED Testing Service LLC. All rights reserved
28 Mathematical Reasoning07/2016 Try It! What do you notice? What do you wonder? Key Points As a group, have participants use this problem and determine what they notice and what they wonder. Start with what they notice about the problem. Then proceed to what they wonder based on what was noticed. There is no need to solve the problem. Simply have participants determine a process for doing so. Discuss with teachers the importance of integrating these two activities into the problem-solving process – more practice leads to greater success. Workbook p. 8 © Copyright GED Testing Service LLC. All rights reserved
29 Mathematical Reasoning07/2016 It’s better to solve one problem five different ways than to solve five different problems. Understand the problem Devise a plan Carry out the plan Look back (reflect) Key Points George Polya can rightly be called the father of problem solving in mathematics education. His four step process serves as the basis for the mathematical problem solving. Understand the problem Devise a plan Carry out the plan Look back These steps provide test-takers with 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.” 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. The primary issue is to ensure that students have other approaches at their disposal. As Dr. Polya says, “It’s better to solve one problem five different ways than to solve five different problems.” Note: As you move through the next series of slides, the participants will have a chance to learn about a number of heuristics and see how they can incorporate them into the problem solving process. George Polya, Mathematician Stanford University © Copyright GED Testing Service LLC. All rights reserved
30 Mathematical Reasoning07/2016 Must-Have Heuristics Workbook p. 9 Key Points Explain that as you move through the next series of slides, the participants will have a chance to learn about a number of heuristics and show they can incorporate them into the problem-solving process. Explain that the following problems are not examples of GED® test items. Rather, they are examples that enable students to clearly see how a thinking strategy or heuristic can be used. After students learn the basic strategies, teachers may want to identify specific items from the GED® Mathematical Reasoning Item Sampler to help students identify different heuristics that could be used to solve those problems. This would make a great small or large group activity providing students with a chance to talk through problems and discover that many problems can be solved in multiple ways. © Copyright GED Testing Service LLC. All rights reserved
31 Strategy 1 - Guess and CheckMathematical Reasoning 07/2016 Strategy 1 - Guess and Check Copy the figure below and place the digits 1, 2, 3, 4, and 5 in the circles so that sums across (horizontally) and down (vertically) are the same. Key Points Have participants use the strategy to solve the problem. Ask if there is more than one solution. Discuss the results and the thinking process used to derive the results. Workbook p © Copyright GED Testing Service LLC. All rights reserved
32 Mathematical Reasoning07/2016 Guess and Check Copy the figure below and place the digits 1, 2, 3, 4, and 5 in the circles so that sums across (horizontally) and down (vertically) are the same. Key Points Display the results and the fact that 2 and 4 cannot be placed in the middle circle. Have participants discuss why. © Copyright GED Testing Service LLC. All rights reserved
33 Mathematical Reasoning07/2016 Strategy 2 - Make a List Three darts hit this dart board and each scores a 1, 5, or 10. The total score is the sum of the scores for the three darts. There could be three 1’s, two 1’s and one 5, one 5 and two 10’s, and so on. How many different possible total scores could a person get with three darts? Key Points Have participants use the strategy to solve the problem. Ask if there is more than one solution. Discuss the results and the thinking process used to derive the results. © Copyright GED Testing Service LLC. All rights reserved
34 Mathematical Reasoning07/2016 Make a List # of 1’s # of 5’s # of 10’s Score 3 2 1 7 12 11 16 21 15 20 25 30 Key Points There are 10 possible scores. Ask what would happen if they just started calling out different scores without being organized. The answer – they would probably repeat scores. © Copyright GED Testing Service LLC. All rights reserved
35 Strategy 3 - Draw a DiagramMathematical Reasoning 07/2016 Strategy 3 - Draw a Diagram In a stock car race, the first five finishers in some order were a Ford, a Pontiac, a Chevrolet, a Buick, and a Dodge. The Ford finished seven seconds before the Chevrolet. The Pontiac finished six seconds after the Buick. The Dodge finished eight seconds after the Buick. The Chevrolet finished two seconds before the Pontiac. In what order did the cars finish the race? What strategy did you use? Key Points Have participants use the strategy to solve the problem. Ask if there is more than one solution. Discuss the results and the thinking process used to derive the results. © Copyright GED Testing Service LLC. All rights reserved
36 Mathematical Reasoning07/2016 Draw a Diagram In a stock car race, the first five finishers in some order were a Ford, a Pontiac, a Chevrolet, a Buick, and a Dodge. The Ford finished seven seconds before the Chevrolet. The Pontiac finished six seconds after the Buick. The Dodge finished eight seconds after the Buick. The Chevrolet finished two seconds before the Pontiac. In what order did the cars finish the race? What strategy did you use? Key Points Discuss the use of number lines as a way for students to determine the results of the race. Ask if there are any other strategies that could be used. Discuss those strategies. © Copyright GED Testing Service LLC. All rights reserved
37 Strategy 4 - Make a Table or ChartMathematical Reasoning 07/2016 Strategy 4 - Make a Table or Chart South Point Amusement Park has a special package for large groups: a flat fee of $20 and $6 per person. If a club has $100 to spend on admission, what is the largest number of people who can attend? Key Points Have participants use the strategy to solve the problem. Ask if there is more than one solution. Discuss the results and the thinking process used to derive the results. © Copyright GED Testing Service LLC. All rights reserved
38 Mathematical Reasoning07/2016 Make a Table or Chart South Point Amusement Park has a special package for large groups: a flat fee of $20 and $6 per person. If a club has $100 to spend on admission, what is the largest number of people who can attend? Key Points Discuss the advantages and disadvantages of using a table. Ask if there are any other strategies that could be used. Discuss those strategies. © Copyright GED Testing Service LLC. All rights reserved
39 Strategy 5 - Find a PatternMathematical Reasoning 07/2016 Strategy 5 - Find a Pattern Continue these numerical sequences by finding the next three numbers for each group. 1, 4, 7, 10, 13, ___, ___, ___ 19, 20, 22, 25, 29, ___, ___, ___ 2, 6, 18, 54, ___, ___, ___ Key Points Have participants use the strategy to solve the problem. Ask if there is more than one solution. Discuss the results and the thinking process used to derive the results. © Copyright GED Testing Service LLC. All rights reserved
40 Mathematical Reasoning07/2016 Find a Pattern Continue these numerical sequences by finding the next three numbers for each group. 1, 4, 7, 10, 13, ___, ___, ___ (add 3 to the previous term) 1, 4, 7, 10, 13, 16, 19, 22 19, 20, 22, 25, 29, ___, ___, ___ ( add 1 to the previous term, then add 2 to that term, then add three to that term) 19, 20, 22, 25, 29, 34, 40, 47 2, 6, 18, 54, ___, ___, ___ (multiply the previous term by 3 to generate the next term) 2, 6, 18, 54, 162, 486, 1458 Key Points Discuss how students could best approach determining the sequence of numbers and the use of patterns to do so. © Copyright GED Testing Service LLC. All rights reserved
41 Strategy 6 - Make it SimplerMathematical Reasoning 07/2016 Strategy 6 - Make it Simpler The houses on Main Street are numbered consecutively from 1 to 150. How many house numbers contain at least one digit 7? Key Points Have participants use the strategy to solve the problem. Ask if there is more than one solution. Discuss the results and the thinking process used to derive the results. © Copyright GED Testing Service LLC. All rights reserved
42 Mathematical Reasoning07/2016 Make it Simpler The houses on Main Street are numbered consecutively from 1 to 150. How many house numbers contain at least one digit 7? Break the problem down. First determine how many houses have a seven in the units place. (15) 7, 17, 27, 37, 47, 57, 67, 77, 87, 97, 107, 117, 127, 137, 147 Next, determine how many houses have a seven in tens place. (10) 70, 71, 72, 73, 74, 75, 76, 77, 78, 79 Take out any duplicates (1) Answer – 24 houses contain at least one digit 7. Key Points Discuss how students might get on the wrong track in identifying the number of houses containing at least one digit 7. © Copyright GED Testing Service LLC. All rights reserved
43 Strategy 7 - Act It Out or Use ObjectsMathematical Reasoning 07/2016 Strategy 7 - Act It Out or Use Objects The figure shows twelve toothpicks arranged to form three squares. How can you form five squares by moving only three toothpicks? Key Points Have participants use the strategy to solve the problem. Ask if there is more than one solution. Discuss the results and the thinking process used to derive the results. © Copyright GED Testing Service LLC. All rights reserved
44 Act It Out or Use ObjectsMathematical Reasoning 07/2016 Act It Out or Use Objects The figure shows twelve toothpicks arranged to form three squares. How can you form five squares by moving only three toothpicks? Answer: One of the squares is formed by the outer boundary of the arrangement. There was no requirement that each of the five squares must be congruent to each of the others. Key Points Working with manipulatives can be a very effective way for students to solve problems. Discuss how teachers could use more manipulatives to help students improve their thinking strategies when solving problems. Remind teachers that while students can’t take manipulatives into the GED® Mathematical Reasoning test, they can certainly draw an object on the laminated cards provided for their use. © Copyright GED Testing Service LLC. All rights reserved
45 Strategy 8 - Work BackwardsMathematical Reasoning 07/2016 Strategy 8 - Work Backwards Brady was trying to decide when to get up in the morning. He needs 45 minutes to get ready for the workshop he plans to attend. It takes him 25 minutes to drive to the adult center where the workshop will be held. He wanted to get to the center 20 minutes early to stop by his classroom and pick up some materials. If the session starts at 7:30 a.m., what time should he get up, if he wants to give himself 10 extra minutes in case the traffic is bad? Key Points Have participants use the strategy to solve the problem. Ask if there is more than one solution. Discuss the results and the thinking process used to derive the results. © Copyright GED Testing Service LLC. All rights reserved
46 Mathematical Reasoning07/2016 Work Backwards Brady was trying to decide when to get up in the morning. He needs 45 minutes to get ready for the workshop he plans to attend. It takes him 25 minutes to drive to the adult center where the workshop will be held. He wanted to get to center 20 minutes early to stop by his classroom and pick up some materials. If the session starts at 7:30 a.m., what time should he get up, if he wants to give himself 10 extra minutes in case the traffic is bad? Start with the time he has to be at the workshop – 7:30 He needed 20 minutes to go by the classroom – 7:10 He drove for 25 minutes – 6:45 He needed 45 minutes to get ready – 6:00 To allow for 10 minutes in case the traffic was bad – 5:50 Key Points Working with time can be challenging for students, especially elapsed time. This is an excellent example of a problem where students can work backwards in order to derive a correct response. © Copyright GED Testing Service LLC. All rights reserved
47 Strategy 9 - Brainstorm and Write an EquationMathematical Reasoning 07/2016 Strategy 9 - Brainstorm and Write an Equation Two apples weigh the same as a banana and a cherry. A banana weighs the same as nine cherries. How many cherries weigh the same as one apple? Key Points Have participants use the strategy to solve the problem. Ask if there is more than one solution. Discuss the results and the thinking process used to derive the results. © Copyright GED Testing Service LLC. All rights reserved
48 Brainstorm and Write an EquationMathematical Reasoning 07/2016 Brainstorm and Write an Equation Two apples weigh the same as a banana and a cherry. A banana weighs the same as nine cherries. How many cherries weigh the same as one apple? You will need to use three variables. A = the weight of an apple B = the weight of a banana C = the weight of a cherry 2A = B + C B = 9C Substituting: 2A = 9C + C 2A = 10C A = 5C Answer: 5 cherries weigh the same as 1 apple Key Points Students need to be able to show the relationship between each item within the problem. Using variables and the substitution allows them to do so. © Copyright GED Testing Service LLC. All rights reserved
49 Strategy 10 – Use Logical ReasoningMathematical Reasoning 07/2016 Strategy 10 – Use Logical Reasoning Three apples and two pears cost 78 cents. However, two apples and three pears cost 82 cents. What is the total cost of one apple and one pear? Key Points Have participants use the strategy to solve the problem. Ask if there is more than one solution. Discuss the results and the thinking process used to derive the results. © Copyright GED Testing Service LLC. All rights reserved
50 Strategy 10 – Use Logical ReasoningMathematical Reasoning 07/2016 Strategy 10 – Use Logical Reasoning Three apples and two pears cost 78 cents. However, two apples and three pears cost 82 cents. What is the total cost of one apple and one pear? By combining the two clues given, one can conclude that five apples and five pears cost 78 plus 82 cents, or 160 cents. Divide that by five and you can conclude that one apple and one pear costs 32 cents. Remember – you are not looking for the cost of an apple or a pear, but the combined cost of both. Key Points While this is not an example of a GED® Mathematical Reasoning test question, it is an excellent example of students having to apply logical reasoning to determine a solution. This type of problem requires that students not jump to conclusions about looking for the cost of a single item, but rather at the combined costs. In a problem like this, students have to focus on the details of the problem. © Copyright GED Testing Service LLC. All rights reserved
51 Mathematical Reasoning07/2016 Third Read: Solve the Problem and Check for Reasonableness Now that I understand the problem’s content, how can I best use my math skills to solve the problem? Am I answering the right question? How should the answer to the question be expressed? Key Points It’s time for a third read, where you will solve the problem and then go back to see if it is reasonable and most importantly does it answer the right question. Sadly, some students get a “right” answer, but it isn’t for the question that is asked. Remember, multiple choice items contain distractors – answers that could be easily derived, but are based on faulty reasoning, inclusion of irrelevant information, or use of the wrong output or units. Students need to see their teachers model the three-read process. They need to practice using the process both as their teacher guides them and then independently. Effective problem solvers do not become so overnight. It requires time and patience. However, the payoff is well worth the personal investment as students transfer their problem solving from the classroom to higher education and ultimately to the workplace. Miller, P. and Koesling, D. “Mathematics Teaching for Understanding: Reasoning, Reading, and Formative Assessment.” Danvers, MA © Copyright GED Testing Service LLC. All rights reserved
52 Routines for Problem SolvingMathematical Reasoning 07/2016 Routines for Problem Solving Applying to Mathematical Problem Solving in the Classroom Key Points So far, we have looked at using a three-read process and multiple heuristics to help students become better problem solvers. Now, it is time to put these pieces together in a routine that students can use now and into the future. © Copyright GED Testing Service LLC. All rights reserved
53 Mathematical Reasoning07/2016 Goals and Givens Active Reading Strategy for Problem Solving Read problem closely Identify the goal – the task(s) to be completed Paraphrase what author wants to be done Write in own words Identify the givens – information relevant to solving the task Look for key terms Key Points Goals and Givens is an active reading strategy for problem solving. It incorporates all of the different steps that have been discussed. It pulls those steps into a clearly defined plan that requires students to: Read the problem closely so they can: Identify the goal or task(s) to be completed Identify the givens of the problem – the relevant information needed to meet the goal or the task Workbook p © Copyright GED Testing Service LLC. All rights reserved
54 Goals and Givens TemplateMathematical Reasoning 07/2016 Goals and Givens Template Key Points Have participants look at each element of the Goals and Givens graphic organizer provided by Phoenix Union High School Literacy Resource Guide - Read Like a Mathematician. Briefly discuss each of the elements. Emphasize the link between the three-read process and the heuristics included in the “Plan” portion of the graphic organizer. © Copyright GED Testing Service LLC. All rights reserved
55 Mathematical Reasoning07/2016 How Does It Work? Let’s Start Easy A bag of M&Ms has 96 pieces in three colors, red, blue, and yellow. The bag has twice as many red M&Ms as blue and five times as many blue as yellow. How many M&Ms of each color are in the bag? Key Points Have participants closely read the problem. Use the following slides to take them through the key components of the Goals and Givens graphic organizer. © Copyright GED Testing Service LLC. All rights reserved 55
56 Mathematical Reasoning07/2016 Goals and Givens Goals Givens Find out how many M&Ms of each color are in the bag. Total of 96 pieces 3 colors – red, blue, yellow 2x red = blue 5x blue = yellow What strategies will you use? May have multiple checked. __Draw/label Diagram __Guess and Check __Make it Simpler __Look for patterns _X_Make a table __Act out or use objects _X_Write an equation __Word backwards __Other_______ Key Points Discuss each element and how the information is used to first address the Goal and then the Givens of the problem. Discuss the heuristics identified. Ask if there are other heuristics or strategies that could be used to solve the problem. © Copyright GED Testing Service LLC. All rights reserved
57 Mathematical Reasoning07/2016 Solve It! Make a Table Write an Equation Write an equation. Use substitution. r + b + y = 96 r + b + y = 96 (r= 2b and b = 5y) 2b + b + y = 96 (substitute r = 2b) 2(5y) + 5y + y = 96 (substitute b = 5y) 10y + 5y + y = 16y = 96 so y = 6 y = 6, b = 30, r = 60 Red Blue Yellow Total 20 10 2 32 30 15 3 48 40 4 64 50 25 5 80 60 6 96 Key Points Show the participants how the problem can be solved by using the two selected heuristics – make a table and write an equation. Discuss the advantages and disadvantages of each. © Copyright GED Testing Service LLC. All rights reserved
58 Mathematical Reasoning07/2016 Use Goals and Givens! Workbook p. 16 Key Points Have participants use the Goals and Givens graphic organizer and solve the problem. Discuss the heuristics identified to solve the problem and why a specific heuristic was selected. © Copyright GED Testing Service LLC. All rights reserved
59 Mathematical Reasoning07/2016 Content A Few Tips and Strategies for the Classroom Key Points Teachers always want to know where to focus their attention when it comes to specific content within mathematics. This section includes some quick tips and strategies that teachers can use in various areas of mathematics to solve some of the “little” problems that can turn into bigger issues. © Copyright GED Testing Service LLC. All rights reserved
60 Mathematical Reasoning07/2016 Common Concerns with the TI-30XS Multiview Calculator Key Points Share with participants that students need to be proficient using mathematical tools. The calculator is one such tool. The next slides will provide some basics that all students should know. © Copyright GED Testing Service LLC. All rights reserved
61 Mathematical Reasoning07/2016 How do I reset the TI-30XS MultiView™ Calculator? To reset the TI-30XS MultiView™ Calculator, Press [ON] and [CLEAR] simultaneously. The calculator will reset immediately. MEMORY CLEARED should appear on the screen. Clear Key Points Explain that students often have problems with some of the most basic elements of working with the TI-30XS MultiView™. In most cases, this comes back to students not having adequate time to work with the calculator. In addition, students often fail to take advantage of the calculator tutorial, which is easily accessible to them. Briefly review the points on the slide. Model for the participants how to use each of the keys. Use of an emulator or a document camera (such as an Elmo) are great ways to model the different functions or operations of the calculator. If these are not available, use of a calculator chart can also be used. You may wish to have participants bring their own calculators to the training. Note: The calculator tutorial from GEDTS is located at: On © Copyright GED Testing Service LLC. All rights reserved
62 Mathematical Reasoning07/2016 Home Screen & Important Keys Overview Solar Panel Home Screen Displays a Max of 4 lines 16 Characters per line Scrolling Press or to place the cursor horizontally over the expression. Press or to scroll through previous entries. Key Points Briefly review the points on the slide. Show participants how each of the keys work, specifically the scrolling function. Number Keys © Copyright GED Testing Service LLC. All rights reserved
63 Mathematical Reasoning07/2016 More Important Keys Key Points Briefly review the points on the slide. Show participants how to change the Mode, as well as how the toggle key works. The standard setup Mode of DEG, NORM, FLOAT and MATHPRINT are sufficient for our students in their preparation for the test. The toggle key changes the form of the answer. For example, from fraction to decimal. The enter key displays the answer. © Copyright GED Testing Service LLC. All rights reserved
64 Mathematical Reasoning07/2016 Classic vs. MathPrint Mode Key Points Briefly review the points on the slide and model the two different print modes. Classic and MathPrint mode displays the output differently. MathPrint mode allows our students to check their work with more accuracy. © Copyright GED Testing Service LLC. All rights reserved
65 Mathematical Reasoning07/2016 Calculator Resources Key Points Don’t forget the resources available on the GED Testing Service® website, including the calculator tutorial, the YouTube video on the calculator, and the TI-30XS Calculator Reference Sheet. Note: Show participants where each of the resources is located on the GED Testing Service® website and the GED Testing Service® YouTube Channel. © Copyright GED Testing Service LLC. All rights reserved
66 Mathematical Reasoning07/2016 Getting a Grip on surface Area and Volume Key Points One of the High Impact Indicators focuses on issues related to geometric reasoning, specifically surface area. The following slides provide a very effective strategy for working with surface area. © Copyright GED Testing Service LLC. All rights reserved
67 Mathematical Reasoning07/2016 Formulas Figure SA Formula V Formula Rectangular prism SA = ph + 2B V = Bh Right prism SA = ph +2B Cylinder SA = 2rh + 2r2 V = r2h Pyramid SA = ½ps + B V = 1/3Bh Cone SA = rs + r2 V = 1/3r2h Sphere SA = 4r2 V = 4/3r3 p = perimeter of base with area B; = 3.14 Key Points Students often see the formulas for surface area particularly challenging. They get lost in the symbols used. Think about the fact that students are used to seeing area designated with an “A”. However in working with surface area, the formula includes the use of area “B”. As a result, students are just not sure what to do. © Copyright GED Testing Service LLC. All rights reserved
68 Mathematical Reasoning07/2016 Quick Draw Key Points This activity requires that you show the drawing for 3-5 seconds, hide the drawing, and ask participants to replicate it in their workbook. To hide the drawing, just click the mouse. To reveal the drawing again, click a second time. After participants complete the quick draw., ask them to identify the figure. Discuss how they came to that conclusion. The figure is a cube. Reference: Grayson Wheatley, formerly of Florida State University and Purdue University, feels strongly about mental imagery in math. He believes that “All meaningful mathematics learning is imaged-based.” His book Quick Draw provides teachers with short activities to engage students in mental imagery, thus helping to improve their spatial sense. This activity has been drawn from his book. Workbook p. 17 © Copyright GED Testing Service LLC. All rights reserved
69 Mathematical Reasoning07/2016 Quick Draw Key Points This activity requires that you show the drawing for 3-5 seconds, hide the drawing, and ask participants to replicate it in their workbook. To hide the drawing, just click the mouse. To reveal the drawing again, click a second time. After participants complete the quick draw., ask them to identify the figure. Discuss how they came to that conclusion. The figure is a rectangular prism. © Copyright GED Testing Service LLC. All rights reserved
70 Mathematical Reasoning07/2016 Quick Draw Key Points This activity requires that you show the drawing for 3-5 seconds, hide the drawing, and ask participants to replicate it in their workbook. To hide the drawing, just click the mouse. To reveal the drawing again, click a second time. After participants complete the quick draw., ask them to identify the figure. Discuss how they came to that conclusion. The figure is a square pyramid. © Copyright GED Testing Service LLC. All rights reserved
71 Mathematical Reasoning07/2016 Two Ways to Do It Surface area can be computed by using the formula or by finding the area of each surface and adding them up! Key Points Explain that surface area can be computed two ways. Using a formula Finding the area of each surface or face of the figure and adding them up © Copyright GED Testing Service LLC. All rights reserved
72 Mathematical Reasoning07/2016 Example: 7 cm 4 cm 8 cm Surface Area Top/bottom 2(8)(4) = 64 Left/right 2(4)(7) = 56 Front/back 2(8)(7) = 112 Add them up! SA = 232 cm² V = lwh V = 8(4)(7) V = 224 cm³ Key Points Discuss the computations made to find the area of each face of this figure and the final calculation of the surface area. Explain that volume does not create the same type of problem for students in that they are used to each of the symbols included in the formula l=length w = width h = height © Copyright GED Testing Service LLC. All rights reserved
73 Use Nets to “Catch” Some SkillsMathematical Reasoning 07/2016 Use Nets to “Catch” Some Skills A net is the shape that is formed by unfolding a three-dimensional figure. In other words, a net is composed of all of the faces of the figure. Key Points Review the definition of a net. © Copyright GED Testing Service LLC. All rights reserved
74 Mathematical Reasoning07/2016 Using Nets to Find Surface Areas prism Math Interactives Key Points If time permits and access is available, visit the Math Interactives site so participants can see how the process of using a net works to determine surface area. © Copyright GED Testing Service LLC. All rights reserved
75 Mathematical Reasoning07/2016 Using Nets to Find Surface Areas Find the surface area of the rectangular prism by using a net. prism Key Points Have participants use the graph paper provided in the workbook and draw a net that represents the figure shown. Have them then calculate the surface area for the figure. Workbook p © Copyright GED Testing Service LLC. All rights reserved
76 Mathematical Reasoning07/2016 Using Nets to Find Surface Areas prism Key Points Discuss how this activity can be used with adult basic education students, as well as those in a GED® preparation class. Discuss how students could literally count the square units on the graph paper to determine the area for each rectangle and then add then up. The surface area is 160 cm2 © Copyright GED Testing Service LLC. All rights reserved
77 Mathematical Reasoning07/2016 Surface Area of a Cylinder prism Imagine that you can open up a cylinder like so: You can see that the surface is made up of two circles and a rectangle. Key Points Discuss how a net can be used to find the surface area of a cylinder. The length of the rectangle is the same as the circumference of the circle! © Copyright GED Testing Service LLC. All rights reserved
78 Mathematical Reasoning07/2016 EXAMPLE: Round to the nearest TENTH. Top or bottom circle A = πr² A = π(3.1)² A = π(9.61) A = Rectangle C = length C = π d C = π(6.2) C = Now the area A = lw A = 19.5(12) A = 234 Now add: = SA = in² Key Points Review the steps taken to determine the surface area of a cylinder by using nets. © Copyright GED Testing Service LLC. All rights reserved
79 Mathematical Reasoning07/2016 Using a Formula prism SA = 2πrh + 2πr² SA = 2π(3.1)(12) + 2π(3.1)² SA = 2π (37.2) + 2π(9.61) SA = π(74.4) + π(19.2) SA = SA = in² Key Points Compare the differences in the results, as well as the complexity of working with the formula. The answers are REALLY close, but not exactly the same. That’s because we rounded in the problem. © Copyright GED Testing Service LLC. All rights reserved
80 Mathematical Reasoning07/2016 Let’s Try It! I want to paint the outside of a decorative pillar that has a height of 48 inches and a diameter of 16 inches. One small canister of paint will cover about 200 square inches. How many small canisters of paint will I need to paint the cylinder? What do I need to know? What type of net can I draw? What formula can I use? Key Points If time permits, have participants actually draw a net and solve the problem. If time is limited, discuss the problem and how it could be solved using nets. Note: The following is provided as reference material for the trainer. Closed cylinder Surface Area = 2*Base area + Rectangle area 2*Area of base (circle) = 2*r2 Area of rectangle = Circle circumference * height Surface Area of Closed Cylinder = (2r2 + 2rh) sq units Find the area of the tube by multiplying the circumference of the circle base by the height of the cylinder. Find the area of one of the circle bases. Double the area of the circle for the top and bottom of the cylinder. Add the area of the two circles and the area of the tube surface. Answer: 14 small canisters Workbook p © Copyright GED Testing Service LLC. All rights reserved
81 Mathematical Reasoning07/2016 Getting Down to Basics With Algebraic Reasoning Key Points Sometimes it really does come down to some of the most basic knowledge that creates problems for students. The following slides provide information on some “quick fixes” related to algebraic reasoning. © Copyright GED Testing Service LLC. All rights reserved
82 Mathematical Reasoning07/2016 Remember . . . Arithmetic is doing something to numbers to get an answer. Algebra is exploring the relationships between numbers. Key Points There is a difference between arithmetic and algebra. Review the bulleted items. © Copyright GED Testing Service LLC. All rights reserved
83 Mathematical Reasoning07/2016 Variable Some students believe that letters represent particular objects or abbreviated words Key Points Students have difficulty discriminating among the different ways letters may be used. In arithmetic, letters are first encountered in formulas, 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. A letter can also represent a specific number that is currently unknown but should be determined or “found” (e.g., ). A letter can also represent a general number, which is not one particular value. Finally, letters can represent variables, each of which represents a range of unspecified values along with a systematic relationship among them (Kieran 1992, p. 396). Teachers often 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 student success in algebra. Some students believe that letters represent particular objects or abbreviated words because of their alphabetic connection. For example: students may see 3d as representing three dogs if the problem referenced dogs. Review the two interpretations provided. Discuss the differences between them and why students need to learn how to correctly interpret an expression or equation. © Copyright GED Testing Service LLC. All rights reserved
84 Confusion About the Equal SignMathematical Reasoning 07/2016 Confusion About the Equal Sign The equal sign stands for balance or equality. The concept of balance can be used to reinforce the idea of equality – both sides of the number sentence need to be the same, the equation needs to balance. Key Points Review the information on the slide. Explain that students need to understand that an equal sign does not announce an answer, but rather shows a relationship. Students need to fully understand equivalency. If they learn this at an early age or when working with very basic mathematical computation skills, the concept will flow easily to algebra. © Copyright GED Testing Service LLC. All rights reserved
85 Mathematical Reasoning07/2016 Symbolic Notation A Few Examples Sign Arithmetic Algebra = (equal) . . . And the answer is Equivalence between two quantities + Addition operation Positive number - Subtraction operation Negative number Key Points Review the table of symbols and how each appear to students, whether working in arithmetic and algebra. © Copyright GED Testing Service LLC. All rights reserved
86 Mathematical Reasoning07/2016 Which Is Larger? 23 or 32 34 or 43 62 or 26 89 or 98 Key Points Students still struggle with exponents. This is problematic in so many ways. It impacts the answer they get when working with formulas, such as the Pythagorean Theorem. It also impacts them when working with scientific notation. The key is to work with students to understand that 23 or 32 do not represent the same value. The issue appears to occur when students encounter 22 as the first exponent with which they deal. If possible, a different base, such as 102 or 52, to start the process. Answers for each example. 32 34 26 98 © Copyright GED Testing Service LLC. All rights reserved
87 Use Multiple RepresentationsMathematical Reasoning 07/2016 Use Multiple Representations Start with the concrete 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, graphs, etc. 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. © Copyright GED Testing Service LLC. All rights reserved
88 Would you teach multiplication . . .Mathematical Reasoning 07/2016 Would you teach multiplication . . . This Way? 3,452 x 267 This Way? 3,452 x 267 = Key Points Note: This slide includes animation. Ask the teachers if they would teach basic multiplication in a vertical or horizontal manner. Discuss why they would use the vertical method. © Copyright GED Testing Service LLC. All rights reserved
89 Use Vertical Multiplication of PolynomialsMathematical Reasoning 07/2016 Use Vertical Multiplication of Polynomials Key Points Ask teachers how they most often teach the multiplication of polynomials. Would they use a vertical or horizontal manner? Discuss why the vertical method would be helpful for students. Note: You may wish to model the process of vertical multiplication of polynomials on chart paper so that participants can see the process in action. © Copyright GED Testing Service LLC. All rights reserved
90 Mathematical Reasoning07/2016 From Words to Symbols Translating Word Problems Key Points Students may recognize symbols when they see them. However, they often have difficulty when faced with a word problem in translating words to symbols. © Copyright GED Testing Service LLC. All rights reserved
91 What would your students do?Mathematical Reasoning 07/2016 What would your students do? Guess Select “C” because they haven’t used it in a while Skip it Sign up for a retest Key Points Have teachers view the problem and then determine what they think students would most likely do. Discuss how some students would just get confused by the wording in the problem and guess or even skip it. Explain that students don’t have to skip these types of problems; they just need to have a strategy for reading the problem and translating it. Workbook p. 22 © Copyright GED Testing Service LLC. All rights reserved
92 What students need to do!Mathematical Reasoning 07/2016 What students need to do! Read the problem carefully and determine what you are trying to find Assign a variable to the quantity that must be found Write down what the variable represents Write an equation for the quantities given in the problem Solve the equation Answer the question Check the solution for reasonableness Key Points Briefly review each of the steps that students need to take when translating a problem from words to symbols. © Copyright GED Testing Service LLC. All rights reserved
93 Mathematical Reasoning07/2016 Practice Translating Jennifer has 10 fewer DVDs than Brad. j – 10 = b (common answer, but incorrect) Insert the words and see the difference in the equation. j (has) = b (fewer) – 10 so j = b – 10 Key Points Use this slide to demonstrate how students often work from left to right in translating a problem, thus ending up with the wrong interpretation (and answer). © Copyright GED Testing Service LLC. All rights reserved
94 Use a Math Translation GuideMathematical Reasoning 07/2016 Use a Math Translation Guide English Math Example Translation What, a number x, n, etc. Three more than a number is 8. n+ 3 = 8 Equals, is, was, has, costs = Danny is 16 years old. A CD costs 15 dollars. d = 16 c = 15 Is greater than Is less than At least, minimum At most, maximum > < Jenny has more money than Ben. Ashley’s age is less than Nick’s. There are at least 30 questions on the test. Sam can invite a maximum of 15 people to his party. j > b a < n t 30 s 15 More, more than, greater, than, added to, total, sum, increased by, together + Kecia has 2 more video games than John. Kecia and John have a total of 11 video games. k = j + 2 k + j = 11 Less than, smaller than, decreased by, difference, fewer - Jason has 3 fewer CDs than Carson. The difference between Jenny’s and Ben’s savings is $75. j = c – 3 j – b = 75 Of, times, product of, twice, double, triple, half of, quarter of x Emma has twice as many books as Justin. Justin has half as many books as Emma. e = 2 x j or e = 2j j = c x ½ j = e/2 Divided by, per, for, out of, ratio of __ to __ Sophia has $1 for every $2 Daniel has. The ratio of Daniel’s savings to Sophia’s savings is 2 to 1. s = d 2 s = d/2 d/s = 2/1 Key Points Explain that one way to help students move past problems with translation is to use a Math Translation Guide. This guide can help students learn to associate specific words with math symbols and enable them to avoid some of the mistakes that they commonly make when working with word problems. Workbook p. 24 © Copyright GED Testing Service LLC. All rights reserved
95 Posing Purposeful QuestionsMathematical Reasoning 07/2016 Posing Purposeful Questions Final Thoughts for the Classroom Key Points In this presentation, you have learned about the importance of incorporating close reading into mathematics, how to help students expand their thinking strategies or heuristics to give students more options for problem solving, and a routine that incorporates a graphic organizer. Each of these areas requires that students gain more effective questioning skills. In this last section, it is time to think about the questions that teachers pose to their students. © Copyright GED Testing Service LLC. All rights reserved
96 Posing Purposeful QuestionsMathematical Reasoning 07/2016 Posing Purposeful Questions Effective teaching of mathematics uses purposeful questions to assess and advance students’ reasoning and sense making about important mathematical ideas and relationships. Four Types of Questions Gathering Information Probing Thinking Making the mathematics visible Encouraging reflection and justification Key Points Discuss the different types of purposeful questions. The next slides identify some samples. Share with participants that they will want to include these types of questions into their math classroom to assist students in building their mathematical reasoning skills. Effective Teaching and Learning. (2014). In Principles to Actions : Ensuring mathematical success for all (p. 36). Reston, VA: NCTM. © Copyright GED Testing Service LLC. All rights reserved
97 Mathematical Reasoning07/2016 Purposeful Questions Question type Description Examples Gathering information Students recall facts, definitions, or procedures. When you write an equation, what does the equal sign tell you? What is the formula for finding the area of a rectangle? Probing thinking Students explain, elaborate, or clarify their thinking, including articulating the steps in solution methods or the completion of a task. As you drew that number line, what decisions did you make so that you could represent 7 fourths on it? Can you show and explain more about how you used a table to find the answer to the Smartphone Plans task? Key Points Review the types of questions for each of the areas. Effective Teaching and Learning. (2014). In Principles to Actions : Ensuring mathematical success for all (p. 36). Reston, VA: NCTM. Workbook p. 23 © Copyright GED Testing Service LLC. All rights reserved
98 Mathematical Reasoning07/2016 Purposeful Questions Question type Description Examples Making the mathematics visible Students discuss mathematical structures and make connections among mathematical ideas and relationships. What does your equation have to do with the band concert situation? How does that array relate to multiplication and division? Encouraging reflection and justification Students reveal deeper understanding of their reasoning and actions, including making an argument for the validity of their work. How might you prove that 51 is the solution? How do you know that the sum of two odd numbers will always be even? Key Points Review the types of questions for each of the areas. Effective Teaching and Learning. (2014). In Principles to Actions : Ensuring mathematical success for all (p. 36). Reston, VA: NCTM. Workbook p. 23 © Copyright GED Testing Service LLC. All rights reserved
99 Mathematical Reasoning07/2016 The Challenge Increase instruction on problem-solving strategies Increase emphasis on geometric and algebraic thinking Provide instruction in higher-order mathematics Shift focus from “rules or processes” of mathematics to deeper understanding of “why” Incorporate close-reading strategies into the math classroom Have high expectations of all students Key Points Review each of the bulleted items. © Copyright GED Testing Service LLC. All rights reserved
100 Mathematical Reasoning07/2016 Resources Workbook p Key Points Take time to briefly review the online resources provided in the workbook. Note: You may wish to access some of the websites and show participants the resources. An actual demo of web resources provides a great incentive for participants to “check them out” after the training. © Copyright GED Testing Service LLC. All rights reserved
101 Mathematical Reasoning07/2016 © Copyright GED Testing Service LLC. All rights reserved
102 Mathematical Reasoning07/2016 Thank you! © Copyright GED Testing Service LLC. All rights reserved