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Mathematical Reasoning:Transitioning from ABE to GED® Skills

October 2017

Debi K. Faucette, Senior Director


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


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)

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.

4


C-R-A – Essential for Understanding

5


Performance Level

Descriptors

Focusing Instruction – Level 1 to Level 2


Targets Indicators Application

7

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


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

(𝑎2 + 𝑏2 = 𝑐2) to determine unknown side

lengths in a right triangle at a satisfactory level.

9


Where to find the PLDs

10


Where to find the PLDs

11


PLD for Mathematical Reasoning

12

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


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

13

GEDtestingservice.com • GED.com14


Calculator-

Prohibited Items

15


GED Calculator-Prohibited Indicatorshttps://www.gedtestingservice.com/uploads/files/09738c12fe4

e4accd9a16bab7cb99a3c.pdf

16


Sample Items

17

• 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.


Sample Items

18

• 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


Quick Tip

19


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 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

21


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

22


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

23


Make Sure Students Can Use a

Number Line

24


Check Students Understanding of

Absolute Value

25

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


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

27

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.


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

28


Rules of Exponents workbook page 5

29


Rules of Exponents Made Easier

30

The Math Dude – Law of Exponents -

https://www.youtube.com/watch?v=g4bKGsC2IoY

Science/MATH DUDE Unit6-1 Laws Of Exponents.avi

Science/MATH DUDE Unit6-1 Laws Of Exponents.avi

https://www.youtube.com/watch?v=g4bKGsC2IoY


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.

31


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


Quantitative

Problem Solving SkillsA Few Tips and Strategies for the Classroom

33


Geometric Reasoning

34

• 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


Focus on Geometric ReasoningVan Hiele Theory

• Level 1: 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?)

41


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.

42


Analysis

Description 1

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


It’s better to solve one

problem five different

ways than to solve five

different problems.

— George Polya, Mathematician

Stanford University

51

Understand the problem

Devise a plan

Carry out the plan

Look back (reflect)

51


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 = 2rh + 2r2 V = r2h

Pyramid SA = ½ps + B V = 1/3Bh

Cone SA = rs + r2 V = 1/3r2h

Sphere SA = 4r2 V = 4/3r3

55

p = perimeter of base with area B; = 3.14


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


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



Find the surface area of the rectangular prism by

using a net.

59


The surface area is 160 cm2



From Words to SymbolsTranslating Word Problems

61


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!

6262


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

6363


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

6464


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

65


Resources

66


Resources

67

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Stay Current - Sign up for

InSession, be the “first” to know

68


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

69


Thank you!Debi Faucette

[email protected]

202-302-6658

Communicate with GEDTS

[email protected]

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