Mathematical Reasoning: Transitioning from ABE to GED Skills October 207 Debi K. Faucette, Senior Director
Session Objectives Discuss Performance Level Descriptors (PLDs) at Levels and 2 Identify selected skill sets students need to demonstrate on calculator prohibited items Identify selected skill sets students need to successfully transition from ABE to GED preparation Explore online resources to aid students in developing mathematical reasoning skills
Where are the problems? Students at the Adult Basic Education level Have limited but developing proficiency Lack consistency in applying skills Need to strengthen foundational skills Need to develop additional skills 3
Understanding Skills Students Have Low Intermediate Basic Education (4-5.9 GLE) Students can perform with high accuracy all four basic math operations using whole numbers up to three digits and can identify and use all basic mathematical symbols. High Intermediate Basic Education (6-8.9 GLE) Students can perform all four basic math operations with whole numbers and fractions; can determine correct math operations for solving narrative math problems and can convert fractions to decimals and decimals to fractions; and can perform basic operations on fractions. Low Adult Secondary Education (9-0.9 GLE) Students can perform all basic math functions with whole numbers, decimals, and fractions; can interpret and solve simple algebraic equations, tables, and graphs and can develop own tables and graphs; and can use math in business transactions. 4
C-R-A Essential for Understanding 5
Performance Level Descriptors Focusing Instruction Level to Level 2
Targets Indicators Application Assessment Targets describe the general concepts that are assessed on the GED test Indicators are fine-grained descriptions of individual skills contained within an assessment target Application describes what to look for in student work 7
Performance Level Descriptors (PLDs) Helpful tool for the classroom Explain in detail the skills students need to demonstrate to pass the test Two formats Official Version Test-taker Version 8
Different Versions 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 (a 2 + b 2 = c 2 ) to determine unknown side lengths in a right triangle at a satisfactory level. 9
Where to find the PLDs 0
Where to find the PLDs
PLD for Mathematical Reasoning Indicator MP. d. Recognize and identify missing information that is required to solve a problem. MP.5 c. Identify the information required to evaluate a line of reasoning. What to look for in student work. The student can: Deconstruct word problems Identify missing information Determine information needed to solve a problem Problem solve through a step-by-step process 2
How to Use PLDs in the Classroom Use PLDs to: Tip : 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 3
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Calculator- Prohibited Items 5
GED Calculator-Prohibited Indicators https://www.gedtestingservice.com/uploads/files/09738c2fe4 e4accd9a6bab7cb99a3c.pdf 6
Sample Items Ordering Fractions and Decimals Place the following numbers in order from greatest to least: 0.2, -/2, 0.6, /3,, 0, /6 Factors and Multiples Find the LCM that is necessary to perform the indicated operation. 7/6 /4 = Rules of Exponents Simplify the following: (x 3 ) 5 Distance on a Number Line Find the distance between two points -9 and -3 on a number line. 7
Sample Items Workbook page 3 Operations on Rational Numbers Solve: 3 (½) 3 ½ = Squares and Square Roots of Positive Rational Numbers Find 9 Find 24 Cubes and Cube Roots of Rational Numbers Find (-4) 3 Undefined Value Over the Set of Real Numbers Solve (2x 3) (x + 2) = 0 8
Quick Tip 9
Numerators and Denominators On My! A fraction is a way of representing division of a 'whole' into 'parts'. It has the form where the Numerator is the number of parts chosen and the Denominator is the total number of parts 20
Number Operations and Number Sense Teaching Fractions Let students use physical materials to create fractional amounts (draw, fold, cut, shade) to explore and develop concepts Use fraction words: two-thirds of a candy bar, a third + a third Relate unknown fractions to well known fractions, such as /2 or /4: It s more than a fourth, but less than a half. It s smaller than a quarter Use language that emphasizes relationship of fractional quantity to unit instead of number of pieces How many of this piece would fit into the whole candy bar? instead of How many pieces is the candy bar cut into? Relate fractions to real-life entities, such as money 2
2 2 3 3 3 5 5 5 5 5 6 6 6 6 6 6 7 7 7 7 7 7 7 9 0 0 0 0 0 0 0 0 0 0 8 8 8 8 8 8 8 8 4 4 4 4 9 9 9 9 9 9 9 9 What is more, /4 or /3? What is more, /9 or /0? Fraction Tiles 22
Simplify Fractions 2 3 4 5 6 7 8 9 0 2 4 6 8 0 2 4 6 8 20 3 6 9 2 5 8 2 24 27 30 4 8 2 6 20 24 28 32 36 40 5 0 5 20 25 30 35 40 45 50 6 2 8 24 30 36 42 48 54 60 7 4 2 28 35 42 49 56 63 70 8 6 24 32 40 48 56 64 72 80 9 8 27 36 45 54 63 72 8 90 0 20 30 40 50 60 70 80 90 00 2 28 45 72 The fraction 4/8 can be reduced on the multiplication table as /2. 23
Make Sure Students Can Use a Number Line 24
Check Students Understanding of Absolute Value Absolute Value indicates how far a number is from 0. Remove any negative sign and think of all numbers as positive Recognize symbol used to represent absolute value 25
Operations on Rational Numbers Recommendations for Test-Takers Be able to: Multiply and divide with decimals Compute With fractions, mixed numbers, and negative numbers Using order of operations 26
Get Rid of Misconceptions about Order of Operations Misconception - All multiplication should happen before division. Misconception 2 All addition comes before subtraction. Remember: M/D have the same precedence. Evaluate as they appear from left to right. Same with A/S. 27
Squares and Square Roots of Positive Rational Numbers Recommendations for Test-Takers Memorize the first 2 perfect squares (, 4, 9,..., 44) Understand inverse relationship between pairs of squares and square roots (2 = 44 and 44 = 2) Understand difference in squaring a negative number and the negative of a square number, i.e. (-3) 2 = 9 -(-3) 2 = -9 Practice computing with squares and square roots that include fractions and decimals 28
Rules of Exponents workbook page 5 29
Rules of Exponents Made Easier The Math Dude Law of Exponents - https://www.youtube.com/watch?v=g4bkgsc2ioy 30
Cubes and Cube Roots of Rational Numbers Recommendations for Test-Takers Memorize the first 6 perfect cubes (, 8, 27,..., 26) Perform and understand recommendations for squares and square roots, but with cubes rather than squares. 3
Undefined Value Over the Set of Real Numbers Recommendations for Test-Takers Reinforce skills on questions that involve Zero in the denominator Fractions with expressions equivalent to zero in the denominator Square roots of negative numbers Expressions that when simplified result in square roots of negative numbers Substitution with linear expressions 32
Quantitative Problem Solving Skills A Few Tips and Strategies for the Classroom 33
Geometric Reasoning Seeking relationships Checking effects of transformations Generalizing geometric ideas Conjecturing about the always & every Testing the conjecture Drawing a conclusion about the conjecture Making a convincing argument Balancing exploration with deduction Exploring structured by one or more explicit limitation/restriction Taking stock of what is being learned through the exploration 34
Focus on Geometric Reasoning Van Hiele Theory Level : Visualization Level 2: Analyze Level 3: Informal Deduction Level 4: Formal Deduction Level 5: Rigor 35
Visualization Recognize and name shapes by appearance Do not recognize properties or if they do, do not use them for sorting or recognition May not recognize shape in different orientation (e.g., shape at right not recognized as square) 36
Visualization 37
Visualization 38
Visualization 39
Implications for Instruction - Visualization Provide activities that have students sort shapes, identify and describe shapes (e.g., Venn diagrams) Have students use manipulatives Build and draw shapes Put together and take apart shapes Make sure students see shapes in different orientations Make sure students see different sizes of each shape 40
Analysis Can identify some properties of shapes Use appropriate vocabulary Cannot explain relationship between shape and properties (e.g., why is second shape not a rectangle?) 4
Analysis Description The design looks like a bird with a hexagon body; a square for the head; triangles for the beak and tail; and triangles for the feet. 42
Analysis Description 43
Analysis Description 2 Start with a hexagon. On each of the two topmost sides of the hexagon, attach a triangle. On the bottom of the hexagon, attach a square. Below the square, attach two more triangles with their vertices touching. 44
Analysis Description 2 45
Implications for Instruction - Analysis Work with manipulatives Define properties, make measurements, and look for patterns Explore what happens if a measurement or property is changed Discuss what defines a shape Use activities that emphasize classes of shapes and their properties Classify shapes based on lists of properties 46
Mathematical Reasoning The Challenges of Math 47
Analysis of Math Challenges In Mathematical Reasoning, items require: Application and development of quantitative and algebraic reasoning skills Grounded in real-world examples Beyond rote application of formulas and/or procedural steps The why and how of math Strong critical reading and thinking skills What is the question asking? What heuristics can I use? Is the answer reasonable? 48
Notice, Name, and Highlight Thinking What kind of thinking do we want our students to do? Make connections Reason with evidence Observe closely and describe Consider different viewpoints Capture the heart and form conclusions Build explanations and interpretations Solve problems in different ways??? 49
Routines for Problem Solving Applying to Mathematical Problem Solving in the Classroom 50
Understand the problem It s better to solve one problem five different ways than to solve five different problems. Look back (reflect) Devise a plan George Polya, Mathematician Stanford University Carry out the plan 5
Must-Have Strategies for Problem Solving 52
How Do We Teach Thinking Skills? Research and Support 53
Understanding the Basics Matters! Students can move beyond area to surface area 54
Formulas Figure SA Formula V Formula Rectangular prism SA = ph + 2B V = Bh Right prism SA = ph +2B V = Bh Cylinder SA = 2 rh + 2 r 2 V = r 2 h Pyramid SA = ½ps + B V = /3Bh Cone SA = rs + r 2 V = /3 r 2 h Sphere SA = 4 r 2 V = 4/3 r 3 p = perimeter of base with area B; = 3.4 55
What is this? 56
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. All students need to do is add up the value of the area of each face. 57
Using Nets to Find Surface Areas Math Interactives http://www.learnalberta.ca/content/mejhm/index.html?l=0&id=ab.math.jr.shap&id2=ab.math.jr.shap.surf&less on=html/object_interactives/surfacearea/use_it.html 58
Using Nets to Find Surface Areas Find the surface area of the rectangular prism by using a net. 59
Using Nets to Find Surface Areas The surface area is 60 cm 2 60
From Words to Symbols Translating Word Problems 6
What students need to do!. Read the problem carefully and determine what you are trying to find 2. Assign a variable to the quantity that must be found 3. Write down what the variable represents 4. Write an equation for the quantities given in the problem 5. Solve the equation 6. Answer the question 7. Check the solution for reasonableness 62
Practice Translating Jennifer has 0 fewer DVDs than Brad. j 0 = b (common answer, but incorrect) Insert the words and see the difference in the equation. j (has) = b (fewer) 0 so j = b 0 63
Use a Math Translation Guide wkbk pg 29 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 6 years old. A CD costs 5 dollars. Is greater than Is less than At least, minimum At most, maximum More, more than, greater, than, added to, total, sum, increased by, together Less than, smaller than, decreased by, difference, fewer Of, times, product of, twice, double, triple, half of, quarter of Divided by, per, for, out of, ratio of to > < 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 5 people to his party. + Kecia has 2 more video games than John. Kecia and John have a total of video games. - Jason has 3 fewer CDs than Carson. The difference between Jenny s and Ben s savings is $75. x Emma has twice as many books as Justin. Justin has half as many books as Emma. Sophia has $ for every $2 Daniel has. The ratio of Daniel s savings to Sophia s savings is 2 to. d = 6 c = 5 j > b a < n t 30 s 5 k = j + 2 k + j = j = c 3 j b = 75 e = 2 x j or e = 2j j = c x ½ or j = e/2 s = d 2 or s = d/2 d/s = 2/ 64
65 The Challenge Provide ample practice in the basics to ensure consistency Increase emphasis on geometric reasoning Shift focus from rules or processes of mathematics to deeper understanding of why Help students learn how to translate from words to symbols Have high expectations of all students 65
Resources 66
67 Resources 67
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Thank you! Debi Faucette Debi.Faucette@GEDTestingService.com 202-302-6658 Communicate with GEDTS 7 communications@gedtestingservice.com