mathematical reasoning - let's goaalrc.org/adminteachers/conferences/math reasoning.abe...
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Mathematical Reasoning:Transitioning from ABE to GED® Skills
October 2017
Debi K. Faucette, Senior Director
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Session Objectives
• Discuss Performance Level
Descriptors (PLDs) at Levels 1
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
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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
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Understanding Skills Students Have
Low Intermediate Basic
Education
(4-5.9 GLE)
High Intermediate
Basic Education
(6-8.9 GLE)
Low Adult Secondary
Education
(9-10.9 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.
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.
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.
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C-R-A – Essential for Understanding
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Performance Level
Descriptors
Focusing Instruction – Level 1 to Level 2
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Targets Indicators Application
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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
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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
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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
(𝑎2 + 𝑏2 = 𝑐2) to determine unknown side
lengths in a right triangle at a satisfactory level.
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Where to find the PLDs
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Where to find the PLDs
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PLD for Mathematical Reasoning
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Indicator What to look for in student work.
The student can:
MP.1 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.
Deconstruct word problems
Identify missing information
Determine information needed to solve a problem
Problem solve through a step-by-step process
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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
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Calculator-
Prohibited Items
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GED Calculator-Prohibited Indicatorshttps://www.gedtestingservice.com/uploads/files/09738c12fe4
e4accd9a16bab7cb99a3c.pdf
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Sample Items
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• Ordering Fractions
and Decimals
• Factors and
Multiples
• Rules of
Exponents
• Distance on a
Number Line
Place the following numbers in order from
greatest to least: 0.2, -1/2, 0.6, 1/3, 1, 0, 1/6
Find the LCM that is necessary to perform the
indicated operation. 7/6 – 1/4 =
Simplify the following: (x3)5
Find the distance between two points -9 and -3
on a number line.
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Sample Items
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• Operations on Rational
Numbers
• Squares and Square Roots of
Positive Rational Numbers
• Cubes and Cube Roots of
Rational Numbers
• Undefined Value Over the Set
of Real Numbers
Solve: 3 (½) ÷ 3 ½ =
Find √ 9 Find √ 24
Find (-4)3
Solve (2x – 3) (x + 2) = 0
Workbook page 3
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Quick Tip
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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
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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 1/2 or 1/4:
– 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
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1 1 2
1 2
1 3
1 3
1 3
1 5
1 5
1 5
1 5
1 5
1 6
1 6
1 6
1 6
1 6
1 6
1 7
1 7
1 7
1 7
1 7
1 7
1 7
1 9
1 10
1 10
1 10
1 10
1 10
1 10
1 10
1 10
1 10
1 10
1 8
1 8
1 8
1 8
1 8
1 8
1 8
1 8
1 4
1 4
1 4
1 4
1 9
1 9
1 9
1 9
1 9
1 9
1 9
1 9
What is more, 1/4 or 1/3? What is more, 1/9 or 1/10?
Fraction Tiles
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1 2 3 4 5 6 7 8 9
2 4 6 8 10 12 14 16 18
3 6 9 12 15 18 21 24 27
4 8 12 16 20 24 28 32 36
10
20
30
40
6 12 18 24 30 36 42 48 54 60
7 14 21 28 35 42 49 56 63 70
8 16 24 32 40 48 56 64 72 80
9 18 27 36 45 54 63 72 81 90
10 20 30 40 50 60 70 80 90 1
5 10 15 20 25 30 35 40 45 50
00
The fraction 4/8 can be reduced on the multiplication table as 1/2.
2128
45
72
Simplify Fractions
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Make Sure Students Can Use a
Number Line
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Check Students Understanding of
Absolute Value
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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
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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
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Get Rid of Misconceptions about Order of
Operations
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Misconception 1 - 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.
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Squares and Square Roots of Positive
Rational Numbers
Recommendations for Test-Takers
• Memorize the first 12 perfect squares (1, 4, 9, . . ., 144)
• Understand inverse relationship between pairs of squares and
square roots (12 =√144 and √144 = 12)
• 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
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Rules of Exponents workbook page 5
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Rules of Exponents Made Easier
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The Math Dude – Law of Exponents -
https://www.youtube.com/watch?v=g4bKGsC2IoY
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Cubes and Cube Roots of Rational
NumbersRecommendations for Test-Takers
• Memorize the first 6 perfect cubes (1, 8,
27, . . ., 216)
• Perform and understand recommendations
for squares and square roots, but with
cubes rather than squares.
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Undefined Value Over the Set of Real
NumbersRecommendations 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 expressions32
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Quantitative
Problem Solving SkillsA Few Tips and Strategies for the Classroom
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Geometric Reasoning
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• Seeking relationships
• Checking effects of transformations
• Generalizing geometric ideas
– Conjecturing about the “always” & “every”
– Testing the conjecture
– Drawing a conclusion about the conjecture
– Making a convincing argument
• Balancing exploration with deduction
– Exploring structured by one or more explicit
limitation/restriction
– Taking stock of what is being learned through the exploration
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Focus on Geometric ReasoningVan Hiele Theory
• Level 1: Visualization
• Level 2: Analyze
• Level 3: Informal Deduction
• Level 4: Formal Deduction
• Level 5: Rigor
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Visualization• Recognize and name
shapes by appearance
• Do not recognize properties
or if they do, do not use
them for sorting or
recognition
• May not recognize shape in
different orientation (e.g.,
shape at right not
recognized as square)
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Visualization
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Visualization
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Visualization
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Implications for Instruction -
Visualization
• Provide activities that have students
sort shapes, identify and describe
shapes (e.g., Venn diagrams)
• Have students use manipulatives
• Build and draw shapes
• Put together and take apart shapes
• Make sure students see shapes in
different orientations
• Make sure students see different
sizes of each shape
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Analysis• Can identify some
properties of shapes
• Use appropriate
vocabulary
• Cannot explain
relationship between
shape and properties
(e.g., why is second shape
not a rectangle?)
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Analysis
Description 1
The design looks like a bird with
• a hexagon body;
• a square for the head;
• triangles for the beak and tail; and
• triangles for the feet.
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Analysis
Description 1
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Analysis
Description 2
• Start with a hexagon.
• On each of the two topmost sides of the
hexagon, attach a triangle.
• On the bottom of the hexagon, attach
a square.
• Below the square, attach two more triangles
with their vertices touching.
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Analysis
Description 2
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Implications for Instruction - Analysis
• Work with manipulatives
• Define properties, make
measurements, and look for patterns
• Explore what happens if a
measurement or property is changed
• Discuss what defines a shape
• Use activities that emphasize classes
of shapes and their properties
• Classify shapes based on lists of properties
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Mathematical Reasoning
The Challenges of Math
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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?
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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
– ? ? ?
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Routines for Problem Solving
Applying to Mathematical Problem Solving in the Classroom
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It’s better to solve one
problem five different
ways than to solve five
different problems.
— George Polya, Mathematician
Stanford University
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Understand the problem
Devise a plan
Carry out the plan
Look back (reflect)
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Must-Have Strategies for Problem Solving
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How Do We Teach
Thinking Skills? Research and Support
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Understanding the Basics
Matters!Students can move beyond area to surface area
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Formulas
Figure SA Formula V Formula
Rectangular prism SA = ph + 2B V = Bh
Right prism SA = ph +2B V = Bh
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
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p = perimeter of base with area B; = 3.14
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What is this?
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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.
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Math Interactives
http://www.learnalberta.ca/content/mejhm/index.html?l=0&ID1=AB.MATH.JR.SHAP&ID2=AB.MATH.JR.SHAP.SURF&less
on=html/object_interactives/surfaceArea/use_it.html
Using Nets to Find Surface Areas
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Using Nets to Find Surface Areas
Find the surface area of the rectangular prism by
using a net.
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The surface area is 160 cm2
Using Nets to Find Surface Areas
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From Words to SymbolsTranslating Word Problems
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1. 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
What students need to do!
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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
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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 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
or
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
or
s = d/2
d/s = 2/1
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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
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Resources
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Resources
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Stay Current - Sign up for
InSession, be the “first” to know
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2017 Webinars & Teleconferences
Month Tuesdays for Teachers3:30 p.m.-5:00 p.m. EST
Test Talk12:00 p.m.-1:00 p.m. EST
Train the Trainer1:30 p.m.-3:00 p.m. EST
January 24
February 28
March 28 28
April 25
May 23
June 27 27
July 18*
August 22
September 26 26
October 24
November 28
December 12 12
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Thank you!Debi Faucette
202-302-6658
Communicate with GEDTS