©2014 IPDAE. All rights reserved. All content in this presentation is the proprietary property of The Institute for the Professional Development of Adult

©2014 IPDAE. All rights reserved. All content in this presentation is the proprietary property of The Institute for the Professional Development of Adult Educators

A Look at the Most-Missed Items on the GED® Mathematical Reasoning Test

Bonnie Goonen – [email protected]


Algebraic Sugar Cane

10 factories produce sugar cane. The second produced twice as much as the first. The third and fourth each produced 80 more than the first. The fifth produced twice as much as the second. The sixth produced 40 more than the fifth. The seventh and eighth each produced 40 less than the fifth. The ninth produced 80 more than the second. The tenth produced nothing due to drought in Australia. If the sum of the production equaled 11,700, how much sugar cane did the first factory produce?


The Answer

500. First write down what each factory produced in relation to

the first factory (whose production is x):

1. x

2. 2x

3. x + 80

4. x + 80

5. 4x

6. 4x + 40

7. 4x - 40

8. 4x - 40

9. 2x + 80

10. 0x

Add them all up and set equal to 11,700 to get the equation: 23x + 200 = 11,700

Solution is: x = 500

“The essence of mathematics is not to make simple things complicated, but to make complicated things simple.”



2014 GED® Test

GED ® Mathematical Reasoning Test• One test with calculator allowed (except for the first

five items)• 115 minutes in length• Content

• 45% - Quantitative Problem Solving• Number operations• Geometric thinking

• 55% - Algebraic Problem Solving• Texas Instruments - TI 30XS Multiview™• Integration of mathematical practices


2014 GED® Test

Status Math RLA Science Social Studies

Passed 56% 77% 73% 68%

National Passing 56% 77% 73% 69%

Passed with

Honors3% 6% 3% 6%

GEDTS Data Retrieved 11/2014


2014 GED® Test

GEDTS Data Retrieved 11/2014


2014 GED® Test

Area of Concern

• Math continues to be most difficult

• Different test takers have gaps

• Test takers are not accessing

support available

(materials/ tutorials/practice tests)

MOST MISSED ITEMS ANALYSIS

Quantitative Reasoning


Most-Missed Items

Quantitative Reasoning• Compute area/circumference of circles

o Find radius or diameter when given area or circumference


Most-Missed Items

Quantitative Reasoning• Compute

perimeter/area of polygonso Find side length

when given perimeter or area


Most-Missed Items

Quantitative Reasoning• Compute perimeter/area of 2-dimensional

composite shapes


Most-Missed Items

Quantitative Reasoning• Use scale factors to determine magnitude of

a size change


Most-Missed Items

Quantitative Reasoning• Solve two-step real world problems involving

percent, such as simple interest, tax, markups/markdowns, gratuities, commissions, percent increase/decrease

QUANTITATIVE PROBLEM SOLVING SKILLS


Geometric Reasoning

• Seeking relationships • Checking effects of transformations• Generalizing geometric ideas

o Conjecturing about the “always” & “every”

o Testing the conjecture

o Drawing a conclusion about the conjecture

o Making a convincing argument

• Balancing exploration with deductiono Exploring structured by one or more explicit

limitation/restriction o Taking stock of what is being learned through

the exploration

Triangle

Quadrilateral

RotationReflectio

n

Congruent

Right Angle

PolynomialParallel


Tangrams

• Originated in China• Its invention is unrecorded in history• The word “tangram” may have come from

the Tanka people who were traders who played this puzzle and past it on via sailors.

• The word might come from an obscure English word “trangram” which means puzzle or trinket.

• Became popular during the 19th century in Europe and America.


Tangrams

Tangram puzzles are made of seven shapes:• 5 triangles• 1 square• 1 parallelogram


What can I do with Tangrams?

Teach math in an interesting manner• Recreate given shapes• Create new shapes• Integrate mathematical skills

using manipulatives (e.g., What fraction of the original shape are each of the 7 pieces?)

http://www.google.com/url?sa=i&rct=j&q=&esrc=s&source=images&cd=&cad=rja&docid=rWUUdTFwGOCRgM&tbnid=ADuw3Z_gdSFbJM:&ved=0CAUQjRw&url=http://www.thepcmanwebsite.com/media/tangram_puzzles/tangram_puzzles_1.shtml&ei=DF0VU43OHoiIyAH5t4CIDQ&psig=AFQjCNGfn4PxHAvjHgaYyTfjEXfCjBV_GQ&ust=1393995363985626



Reinforce geometric principles and ratios• Given a specified side length, find

the area of each of the smaller shapes making up the tangram.

• Determine different ways all 7 shapes be used to make a square.

• Given a specified side length, determine individual side lengths (and/or perimeters) of each of the 7 pieces.

• Find the smallest perimeter using all 7 shapes. The maximum perimeter.



Can you make a square using all of your tangram pieces?



Can you make a square using all of your tangram pieces?


Create the Angel fish


Create the Angel fish


Create the sitting cat


Create the sitting cat


It’s All About Geometric Reasoning

Sample problemIf the entire tangram is worth $120, how much is each piece worth?

What skills will a student use with this problem?



What fractional part of the original shape are each of the 7 pieces? Can you determine the decimal equivalency?



Activity 1How many different lengths are involved?Number the lengths in order from smallest to largest. Start by labeling the smallest side 1, the next largest side 2, etc.

http://www.google.com/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&docid=osupEBKZ7Dz0fM&tbnid=rqazk82HlJZFJM:&ved=0CAUQjRw&url=http://www.challenge.gov.sg/2010/05/the-worlds-first-creativity-test/&ei=7i1LUpT8E6nb4AO_0IDACw&bvm=bv.53371865,d.dmg&psig=AFQjCNHqa_oDYdDXfHx_G2yhPd-EuyIpNg&ust=1380745048002494



Answer to Activity 1

How many different

lengths are involved?

There are four different

measurements.

Number the lengths in

order from smallest to

largest. Start by labeling

the smallest side 1, the

next largest side 2, etc.

1

1

1

11 1

1

1

1

12

3

2

2

2

2

3

3

3

4

4

























Activity 2If the area of the parallelogram is equal to 4 square units find the area of each piece.

4



Answer to Activity 2If the area of the parallelogram is equal to 4 square units find the area of each piece.

4

4

4

2

2

8

8



Activity 3If the area of the small square is equal to 4, find the length of each side of each piece.

4





Answer to Activity 3If the area of the small square is equal to 4, find the length of each side of each piece.

4 2 5.7

2 2 2.82

2 2

2

2

22

2

2

2 2

2 2

2 2

2 2

4

44

4

4 2

4 2

4
























Activity 4Place the three isosceles right triangles on top of each other.

• Compare the corresponding angles.• What is the relationship between the

three triangles?• Suppose the length of the leg on the

smallest triangle is equal to 2 units. How long is each of the nine sides of the three triangles?

MOST MISSED ITEMS ANALYSIS

Algebraic Reasoning


Most-Missed Items

Algebraic Reasoning• Locate points in

coordinate plane

• Determine slope of a line from graph, equation, or table


Most-Missed Items

Algebraic Reasoning• Graph two-variable linear equations


Most-Missed Items

Algebraic ReasoningSolve• One-variable linear

equations, and formulas with multiple variables

• Solve linear inequalities with one variable

• Solve one-variable quadratic equations with real solutions

•


Most-Missed Items

Algebraic Reasoning• Create linear expressions as part of word-to-

symbol translations

On an algebra test, the highest grade was 42 points higher than the lowest grade. The sum of the two grades was 138. Find the lowest grade.


Most-Missed Items

Algebraic Reasoning• Create one- or two-variable linear equations

to represent situations given

The cost of admission to a popular music concert was $162 for 12 children and 3 adults. The admission was $122 for 8 children and 3 adults in another music concert. How much was the admission for each child and adult?


Most-Missed Items

Algebraic Reasoning• Create one-variable linear inequalities to

represent situations you have been given

The perimeter of a square must be less than 160 feet. What is the maximum length of a side in feet?

x = length of a side4x < 160

x < 40Each side must be less than 40 feet.


Algebra

• Students need to be exposed to the underlying structure of algebra while working with arithmetic.

• Students need to recognize that the total number of items in two sets can be written as 6 + 9, rather than only 15.

Algebra should begin at the ABE level


Where do we begin?

• Create opportunities for algebraic thinking as a part of regular instruction

• Integrate elements of algebraic thinking into arithmetic instructiono Acquiring symbolic language o Recognizing patterns and making

generalizations

• Reorganize formal algebra instruction to emphasize its applications

Adapted from National Institute for Literacy, Algebraic Thinking in Adult Education, Washington, DC 20006


Some Big Ideas of Algebra

• Variable• Symbolic Notation• Equality• Ratio and Proportion• Pattern Generalization• Equations and Inequalities• Multiple Representations of Functions

EQUIVALENCE AND VARIABLES

Two Fundamental Problems


The Problem with Equivalence

Students believe In realityThe = sign is the announcement of the result or answer of an arithmetic operation.

3 + 4 = (ta da) 7

The = sign is a symbol of equivalence (a symbol that denotes a relationship between two quantities).



• Teach students to think of a balance scale when they are performing basic arithmetic operations

• Avoid the words makes, equals, and totals

• Use the phrase “is equivalent to” or “is the same as”



• Flip number sentences so the numbers and operations are on the right rather than the left side

• Insert a letter instead of a blank or box

•


The Problem with Variables

Students have difficulty understanding variables.

Variables • play a central role in

understanding algebra• are hard to define • vary dependent on the

context in which they are used



• Misconception 1: Variables can be combined



• Misconception 2: Variables have Defined Values



Misconception 3: Variables can be any number•



Misconception 4: Variables = Objects


How can the problem be solved?

• Make sure students understand the difference between abbreviations and variables.

“6 meters” instead of “6m”

• Have students write out the literal translation of the equation/expression.

6P = SSix times the number of professors equals

the number of students.



Use a Math Translation Guide



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



Use Multiple Representations• Represent problems using

symbols, expressions, and equations, tables, and graphs

• Model real-world situations• Complete problems different

ways (flexibility in problem solving)



Which would you rather do?

5861 x 1246 5861

x 1246



Use Vertical Multiplication of Polynomials

ALGEBRA MATCH GAME

http://www.google.com/url?sa=i&rct=j&q=&esrc=s&source=images&cd=&cad=rja&docid=Akv6ukskfWrPvM&tbnid=ACfLw85zisaOUM:&ved=0CAUQjRw&url=http://www.branfordseven.com/blogs/laura_maniglia_-_on_education/article_c31ff31a-1961-11e3-9cde-001a4bcf6878.html?mode=image&photo=0&ei=4EkVU7aEKorIyAGHsoHoDw&bvm=bv.62286460,d.aWc&psig=AFQjCNG2SwK71mAaeNRJ7TT8ALo07IbFIQ&ust=1393990484720718

PROBLEM SOLVINGDeveloping Quantitative and Algebraic Reasoning Skills

“Mathematical problem solving skills are

critical to successfully function in today’s

technologically advanced society. Solving

word problems requires understanding the

relationships and outcomes of problems. You

must make connections between the different

meanings, interpretations, and relationships

to mathematical operations.”Van de Walle, 2004



It’s All About Problem Solving

Example of “Student Problem Solving” without guidance



Problem Solving Strategies• Look for patterns• Consider all possibilities• Make an organized list• Draw a picture• Guess and check• Write an equation• Construct a table or graph• Act it out• Use objects• Work backward• Solve a simpler (or similar) problem



Polya’s Four Steps to Problem Solving

Understand the problem

Devise a plan

Carry out the plan

Look back (reflect)

Polya, George. How To Solve It, 2nd ed. (1957). Princeton University Press.



Explicit Instruction Matters

• Shows consistent positive effects

on performance

• Places students’ attention on

mathematical ideas

• Develops “mathematical power”

• Develops students’ beliefs that they

are capable of doing mathematics

• Provides ongoing assessment data



“Anyone who has never made a mistake has never tried anything new.”

Albert Einstein

Remember, once is not enough when teaching problem solving.

Meaning Making

The single greatest

tool/instructional

method we can use

in our classrooms


Resources


Thank You

By Educators For Educators

www.floridaipdae.org

Thank You!Bonnie Goonen

[email protected]

Documents

©2014 IPDAE. All rights reserved. All content in this presentation is the proprietary property of The Institute for the Professional Development of Adult