Fraction Contraption
Math Intervention Model
Measurement, Fractions, Probability, Decimals, & Percent
Workshop
August 10, 2011
Contents
WORKSHOP OVERVIEW ..................................................................................................................................... 3
FRAMEWORK ......................................................................................................................................................... 4
Seven Essentials of Math................................................................................................................................ 4
State Standards .................................................................................................................................................. 5
Seven Essentials of Math Related to the Common Core State Standard Domains .................. 5
Student Learning Objectives ........................................................................................................................ 6
Supplies/Equipment ....................................................................................................................................... 6
FRACTION CONTRAPTION LESSONS ............................................................................................................ 7
Measurement...................................................................................................................................................... 7
Playing Fraction Contraption with Integers ................................................................................... 12
Workshop Discussion Notes: Measurement & Integer Game ................................................. 13
Probability ........................................................................................................................................................ 14
Workshop Discussion Notes: Probability ....................................................................................... 18
Fractions ........................................................................................................................................................... 19
Workshop Discussion Notes: Fractions ........................................................................................... 23
Decimals ............................................................................................................................................................ 24
Workshop Discussion Notes: Decimals ........................................................................................... 27
Percent ............................................................................................................................................................... 28
Workshop Discussion Notes: Percent .............................................................................................. 29
APPENDIX ............................................................................................................................................................. 30
Vocabulary........................................................................................................................................................ 31
How to Play the Fraction Contraption Game ...................................................................................... 33
Using the Ruler to Aid Calculating Fractions ...................................................................................... 34
Using the Ruler to Aid Calculating Decimals ....................................................................................... 35
Fraction Contraption Variations .............................................................................................................. 44
Making the Fraction Contraption ............................................................................................................ 45
Fraction Contraption Evaluation Plan ................................................................................................... 48
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Fraction Contraption Workshop – August 10, 2011
WORKSHOP OVERVIEW
This workshop will prepare participants to teach measurement, probability, fractions, decimals, and
percent in their classrooms using the Fraction Contraption game. The theme running through the
workshop is that fractions are fun, useful, and have to be practiced.
How to teach each skill using the Fraction Contraption game will be presented, including teacher
procedures, student activities, assessments, and a discussion regarding how to fit the game into the
classroom flow. The game can be used in the K-12 setting, including any Career Technical Education
class that uses math. The primary ingredient of Fraction Contraption’s success is that is provides time to
play/practice in a non-threatening environment.
Sierra College, through its Center for Applied Competitive Technologies (CACT) and National Science
Foundation grant, is partnering with Jonathan Schwartz, to present this workshop. Over the past year,
Jonathan, who teaches Mathematics and Project Lead the Way (PLTW) at Colfax High School, has been
researching essential math skills that students must have to pass the California High School Exit Exam
(CAHSEE) and college-level Mathematics assessment tests. He has been developing lesson plans to
engage students through hands-on, applied learning.
Workshop Objectives

Provide professional development for teachers to use the Fraction Contraption game and associated
measurement, probability, fraction, percent, and decimal lesson plans as a mathematics prevention
tool (grades 3-6) or an intervention tool (grades 7-12) and

Prepare teachers to pilot the Fraction Contraption across multiple schools in the K-12 arena.
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Fraction Contraption Workshop – August 10, 2011
FRAMEWORK
Basic number sense is fundamental to every year of math education and success in school and life. To
solve problems arising in everyday life such as buying groceries; keeping track of expenses; constructing
something; figuring sports scores; or following a recipe requires applying math. Playing the Fraction
Contraption game every day coupled with student/teacher dialog about the game helps math make sense
to students. Early indicators demonstrate playing the Fraction Contraption game engages students while
they reinforce math skills that are the foundation for success in advanced math coursework.
The Fraction Contraption game is designed to address seven of the most essential math skills. The Seven
Essentials of Math were developed by a team of educators, business leaders, and college representatives
in the Placer/Nevada region of California
Seven Essentials of Math
1.
2.
3.
4.
Measurement
Fractions
Ratios/Proportions
Probability
5. Decimals
6. Percent
7. Geometric Reasoning
These essential skills are assessed on the California High School Exit Exam. Sample questions include:
Fractions/Proportions
1. John uses 2/3 of a cup of oats per serving to make oatmeal. How many cups of oats does he need
to make 6 serving?
A. 2 2/3
B. 4
C. 5 1/3
D. 9
2. If Freya makes 4 of her 5 free throws in a basketball game, what is her free throw shooting
percentage?
A. 20%
B. 40%
C. 80%
D. 90%
Ratio/Percent
3. Some students attend school 180 of the 365 days in a year. About what part of the year do they
attend school?
A. 18%
B. 50%
C. 75%
D. 180%
C. 2.67
D. 3.7
Fractions/Decimals
4. What number equals
A. 0.267
?
B. 0.375
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Fraction Contraption Workshop – August 10, 2011
The California State University System’s Entry Level Mathematics examination (ELM) devotes 35
percent of the exam to numbers and data. This section, in part, tests students’ abilities to carry out basic
arithmetic calculations; understand and use percent in context; compare and order rational numbers
expressed as fractions and/or decimals; solve problems involving fractions and/or decimals in context;
and interpret and use ratio and proportion in context.
State Standards
In addition to the Seven Essentials, the Fraction Contraption game supports teaching the Common Core
State Standards for Mathematics. Strands of mathematical proficiency in the Common Core include:
―…adaptive reasoning, strategic competence, conceptual understanding (comprehension of mathematical
concepts, operations and relations), procedural fluency (skill in carrying out procedures flexibly,
accurately, efficiently and appropriately), and productive disposition (habitual inclination to see
mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own
efficacy).‖ (Common Core State Standards for Mathematics, 10/18/10 Common Core State Standards
Initiative, page 6.) The Common Core State Standards for Mathematics domains covered by the aspects
of the Seven Essentials of Math are noted below.
Seven Essentials of Math Related to the Common Core State Standard Domains
Grade 1 Overview
Operations and Algebraic Thinking
Number and Operations in Base Ten
Measurement and Data
Geometry
Grade 2 Overview
Operations and Algebraic Thinking
Number and Operations in Base Ten
Measurement and Data
Geometry
Grade 3, 4 and 5 Overview
Operations and Algebraic Thinking
Number and Operations in Base Ten
Number and Operations—Fractions
Measurement and Data
Geometry
Grade 6 and 7 Overview
Ratios and Proportional Relationship
The Number System
Statistics and Probability
Grade 8 Overview
The Number System
Probability
Geometry
Algebra Overview
Creating Equations
Reasoning with Equations
Geometry Overview
Congruence
Similarity, Right Triangles, and Trigonometry
Circles
Geometric Measurement and Dimension
Modeling with Geometry
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Fraction Contraption Workshop – August 10, 2011
Student Learning Objectives
There are numerous points where the Fraction Contraption lessons align with the Common Core State
Standards for Math. After practicing math using the Fraction Contraption lessons on measurement,
fractions, decimals, percent, and probability students will be able to perform to the following
standards. Notation after each objective denotes the grade level (2), domain (MD), and standard number
(4).
1. Measure to determine how much longer one object is than another, expressing the length difference in
terms of a standard length unit. (2.MD, 4)
2. Generate measurement data by measuring lengths using rulers marked with halves and fourths of an
inch. Show the data by making a line plot, where the horizontal scale is marked off in appropriate
units – whole numbers, halves, or quarters( 3.MD,4)
3. Understand that the probability of a chance event is a number between 0 and 1 that expresses the
likelihood of the event occurring.(7.SP, 5)
4. Demonstrate a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal
parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.(3.NF, 1)
5. Represent fractions on a number line diagram.(3.NF, 2)
6. Compare two fractions with different numerators and different denominators, e.g., by creating
common denominators or numerators, or by comparing to a benchmark fraction such as 1/2.
Recognize that comparisons are valid only when the two fractions refer to the same whole.(4.NF,2)
7. Add and subtract fractions by joining and separating parts referring to the same whole.(4.NF, 3a)
8. Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the
quantity); solve problems involving finding the whole, given a part and the percent.(6.RP, 3b)
9. Demonstrate a fraction with denominator 10 as an equivalent fraction with denominator 100, and use
this technique to add two fractions with respective denominators 10 and 100.(4.NF, 5)
10. Express that in a multi-digit number, a digit in one place represents 10 times as much as it represents
in the place to its right and 1/10th of what it represents in the place to its left.(5.NBT, 1)
11. Use decimal notation for fractions with denominators 10 or 100. For example, rewrite 0.62 as 62/100;
describe a length as 0.62 meters; locate 0.62 on a number line diagram.(4.NF,6)
Supplies/Equipment
Fraction Contraption game and regular, fraction, and percent dice
Ruler with at least 8ths
Pencil and Paper
Optional – student ―chalk board‖ and markers
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Fraction Contraption Workshop – August 10, 2011
FRACTION CONTRAPTION LESSONS
Measurement
Fractions
Decimals
& Play with
Integers
& Play with
Fractions
& Play with
Decimal Dice
Probability
Play with
Percent
& Play with
Integers
Fraction Dice &
Decimal Tiles
& Play with
Percent Dice
Measurement
Teacher Procedure – Anticipatory Set
Ask students what one of their parents or a relative does in their job; relate the job to using math.
Students could text using http://www.polleverywhere.com/ what their parent/relative does for a living.
Teacher Procedure - Lecture
Measurement is the ability to find the magnitude of something. It can be weight, time, length; anything
that has value has a measurement attached to it. The United States uses the English measurement system
(feet, miles, gallons, etc.) The rest of the world uses the metric system (meters, kilometers, liters).
Fraction Contraption concentrates on the U.S. system of linear measurement using a ruler.
The rule is a basic measuring tool. Carpenters, machinists, architects, tailors, interior designers, and other
workers in related fields are required to know how to read one. Also, knowing how to read a rule is
important for everyone who might need to measure when ordering materials or making home repairs.
A rule is divided into equal parts such as inches and feet. A foot is divided into twelve (12) inches and
each inch is divided into equal fractional parts. The inch on a typical rule is divided into sixteen (16) parts
expressed as sixteenths (16ths). The length of rules may vary. They may be 6, 12, or 36 inches long.
Measuring tapes may be 6 to 25 feet in length and longer. When reading a rule, fractional measurements
are always reduced to the lowest terms. A measurement of would be read as ;
as ; and as .
The Ruler on the Fraction Contraption is split into 1/8 inch increments. The fractional parts on the
Fraction Contraption are in halves ( ), quarters ( ), and eighths ( ). The lower numbers (called the
denominator) of a fraction indicate the total number of like spaces of that size that are found in an inch.
For example, there are eight same-sized spaces on the Fraction Contraption ruler.
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Fraction Contraption Workshop – August 10, 2011
Student Activity
Students label the halves on the halves diagram; quarters on the quarter diagram, and eighths on the
eighths diagram.
Components of a Rule
Inches
1
2
3
Halves
1
Eighths
1
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Fraction Contraption Workshop – August 10, 2011
4
Quarters
1
Student Activity
Provide a ruler that is divided into eighths. Ask the students to find the length of the line using a ruler to
the nearest eighth of an inch and check their work.
1. _________________________________________________
2. _____________________________________
3. _________________________________
Student Activity
Ask the students to sketch the Fraction Contraption below. Measure the dimensions of the game and the
pieces and put them on your sketch:
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Fraction Contraption Workshop – August 10, 2011
Student Assessment – Measurement Quiz
1. What is the basic measuring tool?
A. Triangle
B. Machinist Rule
C. Rule
D. Try square
2. What workers are required to read a rule?
A. Machinist
B. Carpenter
C. Tailor
D. All of the above
3. Why should homeowners learn to read a rule?
A. They may need to make home repairs
B. They may need to measure ingredients
C. They may need to measure the weights of objects
D. all of the above
4. In this class, who is responsible for learning to read a rule?
A. Most students
B. Every student
C. Beginning students
D. Advanced students
5. How many inches is a foot divided into?
A. Six
B. Twelve
C. Twenty four
D. 6, 18, or 36 inches
6. How many parts is the inch divided into on the Fraction Contraption?
A. Fractions of an inch
B. Eight
C. Twenty four
D. Sixteen
7. What part of a fraction indicates the number of same-sized spaces in an inch?
A. Lower
B. Upper
C. Both lower and upper
D. None of the above
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Fraction Contraption Workshop – August 10, 2011
Student Assessment –Fraction Contraption Ruler Quiz
Fill in the measurements on the ruler below.
1
2
3
4
1
2
3
4
Extension: Have students fill in the ruler in 16ths in the Appendix
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Fraction Contraption Workshop – August 10, 2011
Playing Fraction Contraption with Integers
Teacher Procedure - Lecture
The best way to introduce the game is by playing with integers first. The goal is to move all tiles down or
have the lowest score. The game is in 8ths just as the ruler has 8 inches.
The Game with Integers

The game can be played with one or two players

Move all tiles (numbered 1 through 8) up

Roll the 2 dice and add the two numbers

Whatever you roll, move that same value on the tiles down (for example, when you roll a 6 and a 4
(sum of 10) you can move down the 6 and 4 or the 7 and 3 and so on)

When all tiles are down, you win the game

If there are remaining tiles, your score is the sum of the remaining tiles (for example, if the only tile
remaining is a six and you roll a five then play is over and your score is a six)

In 2 person play, low score wins
Student Activity – Play with Integers
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Fraction Contraption Workshop – August 10, 2011
Workshop Discussion Notes: Measurement & Integer Game
Teaching Measuring Tips/Lesson Extensions
Extended Student Activity: use a tape measure to estimate the size of object like a table, book, pencil,
short pencil, white board, door, or other common objects.
Other
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Fraction Contraption Workshop – August 10, 2011
Probability
Measurement
Fractions
Decimals
& Play with
Integers
& Play with
Fractions
& Play with
Decimal Dice
Probability
Play with
Percent
& Play with
Integers
Fraction Dice &
Decimal Tiles
& Play with
Percent Dice
Introduce probability when students have a solid understanding of measurement.
Teacher Procedure - Lecture
The probability of an event in an experiment is the proportion (or frequency) of that event when the same
exact experiment is repeated many times. When you have a six-sided die, you have six potential
outcomes – one through six. With six possible outcomes you have one out of six (1/6) chances of rolling
a particular number; this works on average over time and is dependent on the experiment being random.
Student Activity - One Die Roll
Have students roll one die 10 times and record the outcome.
Try 1
Try 2
Try 3
Try 4
Try 5
Try 6
Try 7
Try 8
Try 9
Try 10
Teacher Procedure – Group Discussion
Have a discussion around the results and what students think about what happened. Have students call
out their results to see results from the whole class. Plot the class results.
Plot Chart
1
2
3
4
5
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Fraction Contraption Workshop – August 10, 2011
6
Teacher Procedure - Lecture
Demonstrate the sample space of a two dice toss by showing all possible outcomes.
Sample Space: All Possible Outcomes of a Two Dice Toss
1
1
1,1
2
1,2
3
1,3
4
1,4
5
1,5
6
1,6
2
2,1
2,2
2,3
2,4
2,5
2,6
3
3,1
3,2
3,3
3,4
3,5
3,6
4
4,1
4,2
4,3
4,4
4,5
4,6
5
5,1
5,2
5,3
5,4
5,5
5,6
6
6,1
6,2
6,3
6,4
6,5
6,6
Review the sums of a two dice toss as illustrated below. The sum of 7 has the highest frequency out of 36
possible outcomes.
Possible Sums of a Two Dice Toss
1
2
3
4
5
6
1
2
3
4
5
6
7
2
3
4
5
6
7
8
3
4
5
6
7
8
9
4
5
6
7
8
9
10
5
6
7
8
9
10
11
6
7
8
9
10
11
12
Student Activity
Have students roll two dice and find their sum. Repeat 10 times and record the outcomes.
Try 1
Try 2
Try 3
Try 4
Try 5
Try 6
Try 7
Try 8
Try 9
Try 10
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Fraction Contraption Workshop – August 10, 2011
Have students call out their results and add the whole classroom’s sums together from 2-12. Plot the class
results.
Plot Chart
1
2
3
4
5
6
1
2
3
4
5
6
Student Activity – Play with integers and introduce competition.
Have students play the game alone with integers for one or two rounds and then have them play with a
partner in a competition.
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Fraction Contraption Workshop – August 10, 2011
Student Assessment – Probability Quiz
1. If there are 30 students in a class and 13 of them are girls, what it the probability that a student picked
randomly will be a boy?
2. If two dice are rolled, what is the probability the sum will be 12?
3. If two dice are rolled, what is the probability the sum will be 7?
4. What sum of two dice is most likely to come up when rolling 6-sided dice?
5. What is the probability of rolling any one number on a die?
6. What is the probability of rolling a 5 or a 2 on a die?
7. What is the probability of rolling a 3 and then a 6?
8. What is the probability of rolling a total of 7 with two dice?
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Fraction Contraption Workshop – August 10, 2011
Workshop Discussion Notes: Probability
Questions:
How do you see tying probability and playing the game with integers into your classroom?
How should probability instruction be differentiated by grade?
Grade School
Middle School
High School
How can the probability lessons and play with Fraction Contraption fit in the classroom flow?
Amount of Time to Practice/When
Frequency
Relating probability to the other Fraction Contraption Games
Competition type
Brackets
List every student in a ladder where they can challenge up to five spaces above.
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Fraction Contraption Workshop – August 10, 2011
Fractions
Measurement
Fractions
Decimals
& Play with
Integers
& Play with
Fractions
& Play with
Decimal Dice
Probability
Play with
Percent
& Play with
Integers
Fraction Dice &
Decimal Tiles
& Play with
Percent Dice
Move on to fractions when students have a solid understanding of measurement and playing the game
with integers.
Teacher Procedure – Lecture and Demonstration Use white or chalk board to demonstrate
Fractions are a way to measure parts of the whole. If a pizza is cut into 8 slices, the whole pizza has 8
parts (slices). The 8 slices are the bottom number called the denominator. If 3 slices are eaten, they are
3 parts of the whole. The 3 parts are the top number called the numerator.
3
8
Numerator
Denominator
How much of the pizza was eaten? A total of
Whole= 8/8
of the pizza was eaten.
Three Slices= 3/8
Adding and Subtracting Fractions with Like (Common) Denominators & Finding the Lowest
Denominator
A common denominator means that the denominators in two (or more) fractions are the same as in the
pizza example above. To add or subtract fractions with the same denominators, add or subtract across the
top and keep the bottom number the same.
For example:
+ =
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Fraction Contraption Workshop – August 10, 2011
Usually you reduce a fraction such as
to its lowest terms by the highest number that can divide the
numerator and the denominator exactly; in this fraction you can divide 6 by 2 and 8 by 2:
Add/ Subtract Fractions with Dissimilar Denominators & Find the Least Common Denominator
Adding and subtracting fractions with denominators that are not the same requires finding a least common
denominator. For example to add
; requires a common denominator; to do this find the least
common multiple of 8 and 4 by listing multiples of each number. For example: for 8, multiples are 16,
32, and for 4, multiples are 8, 16, and so on. The least multiple the two have in common is 8. Therefore,
the common denominator will be 8.
The first fraction is ready. The second fraction’s denominator needs to become an 8. However,
multiplying by 2 will change the value of the fraction. To change the denominator into an 8, multiply by
(this is the equivalent of 1 and therefore does not change the value of the fraction). To multiply
fractions, multiply across the numerators (top number) and multiply across the denominators (bottom
number). For example
=
When you have a common denominator add the numerators (top numbers) and keep the denominators
(bottom number) the same because they are parts of the same whole.
Example:
+
Convert to
, multiply
x
=
results in:
Student Activity – Fractions Exercise
Have students practice fractions before they play the game.
1.
2.
+ =
+
=
3.
-
=
4.
-
=
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Fraction Contraption Workshop – August 10, 2011
+
=
Student Activity- Play Fraction Contraption
Play the Fraction Contraption Game with fractions. The game is in 8ths. The tiles are yellow and white;
yellow provides a visual signal that a fractional number must be converted to 8ths to be added; this
conversion makes the fractions alike and parts of the same whole.

The game can be one or two players.

Move all tiles (numbered 1 through 8) down

Roll the eight-sided fraction dice

Whatever you roll, move the sum on the tiles up (e.g. roll
; convert to
you can move up
the 7/8).

When all tiles are up, you win the game,

If there are remaining tiles, your score is the sum of the remaining tiles (e.g., if the only tile remaining
is
and you roll
and
then play is over and your score is the sum of the whole number tiles or
).

In 2 person play, low score wins.
If a student has trouble with adding fractions they can use the built in ruler to add the fractions –
see the Appendix, page 34 for instructions.
Student Assessment: Fractions Quiz
Add, subtract, multiply, divide the following fractions:
1.
+ =
2.
+
3.
-
=
=
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Fraction Contraption Workshop – August 10, 2011
4. What is the measurement on the indicated ruler?
1
2
5. A recipe calls for 1 cup flour,
3
cup of sugar, and
4
cup brown sugar. Lauren would like to triple
the recipe. How much of each ingredient will she need?
Student Assessment: Fractions Quiz
Add, subtract, multiply, divide the following fractions:
1.
+ =
2.
+
3.
-
=
=
4. A recipe calls for 4 cups of flour, 1 and cups of sugar, and
cup brown sugar. Lauren would like to
cut the recipe in half. How much of each ingredient will she need?
5. Find the length of the line below to the nearest
inch:
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Fraction Contraption Workshop – August 10, 2011
Workshop Discussion Notes: Fractions
Teaching Fractions Tips
How do you see using the number line to play the game with lower grade students as a ―starter‖ or older
students that are having trouble with fractions?
How do you see helping students transfer what they have learned playing with 8ths to other fractional
numbers?
How you see using the Fraction Contraption and fractions in the classroom/fitting into the weekly flow?
Amount of Time to Practice/When
Frequency
Relating to the other Fraction Contraption Games
Competition
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Fraction Contraption Workshop – August 10, 2011
Decimals
Measurement
Fractions
Decimals
& Play with
Integers
& Play with
Fractions
& Play with
Decimal Dice
Probability
Play with
Percent
& Play with
Integers
Fraction Dice &
Decimal Tiles
& Play with
Percent Dice
Move on to decimals when students have solid understanding of measurement and fractions.
Student Activity – Play Fraction the Contraption Game
Play the game using the decimal tiles with the fractional dice before lecturing on decimals. This
experience provides decimal recognition. (See instructions using the number line in the Appendix at page
35.)
Teacher Procedure – Lecture/Demonstration
Decimals are the same as fractions except the denominator is always a factor of ten.
For example
.1 is
.25 is
5.365 is 5
The first part of understanding decimals is to comprehend place value.
Place Value
23.237
Tens
Ones
Decimal Point
Tenths
Hundredths
Thousandths
2
3
.
2
3
7
Student Activity: Match the names from the chart below to the number 18.459
Tens
Ones
Decimal Point
Tenths
Hundredths
.
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Fraction Contraption Workshop – August 10, 2011
Thousandths
Teacher Procedure - Lecture
Converting Decimals to Fractions:
To convert a decimal to a fraction look at the place value. For example for .50, the 0 is in the hundredths
place so it becomes
Tens
Ones
Decimal Point
Tenths
Hundredths
.
5
0
Thousandths
After you convert the decimal to a fraction, reduce it if you can. If both numbers are even, divide them
in half;
becomes
. If both are still even divide them in half again. If you can’t divide in half any
more – or both numbers were not even - check to see if there is a number that can divide the numerator
and the denominator exactly; if so, continue to reduce. For example ( ) can be reduced to
and then to
its lowest term; it can’t be reduced further.
Converting Fractions to Decimals:
Like fractions, decimals are not whole numbers. Converting fractions to decimals involves division.
Every fraction represents a numerator divided by its denominator.
For example:
is the same as 3 divided 8 or 8 √
which becomes .375
Adding and Subtracting Decimals:
The key to adding or subtracting decimals is that the place value order must line up vertically. Zeros put
in as place holders can help. For example:
30.5
+0.3
30.8
3.46
+5.22
8.68
9.345
+17.200
26.545
2.03
-1.24
0.79
Adding and Subtracting Money:
The United States money system is based on 100 cents to the dollar. Adding and subtracting money is
using decimals. For example: $20 + $42.25 is the same as:
$20.00
+$42.25
$62.25
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Fraction Contraption Workshop – August 10, 2011
Measurement
Fractions
Decimals
& Play with
Integers
& Play with
Fractions
& Play with
Decimal Dice
Probability
Play with
Percent
& Play with
Integers
Fraction Dice &
Decimal Tiles
& Play with
Percent Dice
Student Activity - Play the Fraction Contraption Game
Use the decimal tiles and decimal dice.
Student Assessment - Decimals
1. Write the following decimals as a fraction:
a) .25
b) .375
c) .65
d) .625 e) .22
2. Write the following fractions as a decimal:
a)
b)
c)
d)
3. Add or subtract the following decimals:
a) 1.25+.45
b) .34+1.25
c) .452-.234
d) .625-1.25
4. Multiply the following decimals:
a) .25 x .45
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Fraction Contraption Workshop – August 10, 2011
Workshop Discussion Notes: Decimals
Teaching Decimals Tips
How do you see working with students as they use the fractional dice to play the game with decimals
using the number line before you teach decimals? How will you reinforce sight recognition of decimals?
Using the Fraction Contraption in the Classroom/Fitting into the Weekly Flow
Amount of Time to Practice
Frequency
When
Competition
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Fraction Contraption Workshop – August 10, 2011
Percent
Measurement
Fractions
Decimals
& Play with
Integers
& Play with
Fractions
& Play with
Decimal Dice
Probability
Play with
Percent
& Play with
Integers
Fraction Dice &
Decimal Tiles
& Play with
Percent Dice
Move on to percent when your students have a solid understanding of measurement and fractions.
Teacher Procedure - Lecture
The ―cent‖ in the word percent means hundred. A percent is a ratio. A ratio compares two different
numbers and is sometimes expressed as a fraction. An example is if you have 10 students and 1 pizza, the
ratio of pizza to students is 1 to 10 or
or 1:10.
A percent expressed as a fraction always has a denominator of 100.
For example: 35% is
Percents are similar to decimals, except the decimal point is moved over two places to the right and a %
symbol is put on the end.
For example: .35 is 35%
.534 is 53.4%
To convert percents to decimals, reverse the process; insert a decimal point in two places to the left.
For example: 25% is .25, it is 25 parts of a hundred, 37.5% is .375 parts of a hundred, and125% is 1.25
parts of a hundred.
To convert percents to fractions, first convert the percent to a decimal and then convert to a fraction as
explained above.
For example: 25% is
is when reduced to its lowest terms.
Student Activity – Play the Fraction Contraption Game
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Fraction Contraption Workshop – August 10, 2011
Workshop Discussion Notes: Percent
Teaching Percent Tips
Using the Fraction Contraption in the Classroom/Fitting into the Weekly Flow
Amount of Time to Practice
Frequency
When
Competition
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Fraction Contraption Workshop – August 10, 2011
APPENDIX
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Fraction Contraption Workshop – August 10, 2011
Vocabulary
Decimal is based on the number 10 in which smaller units are related to the principal units as powers of
ten (tenths, hundredths, thousandths, etc.)
Fractions
Fractions have two numbers:
3
8
Numerator
The number above the line showing how many of the equally sized parts indicated
by the denominator are taken
Denominator The number below the line showing the number of equally sized parts that make a
whole
The numerator and denominator of a fraction can be multiplied by the same number without
changing its number value
and
have the same number value
Improper Fraction has a numerator greater than (or equal to) the denominator
Mixed Fraction is a whole number and a proper fraction
Proper Fraction has a top number less than its bottom number
Lowest or Least Common Denominator: least common multiple of two or more denominators
become
Simplify Fractions: divide the top and bottom by the highest number that
can divide both numbers exactly
Multiply fractions
1. Multiply the numerators
2. Multiply the denominators
3. Simplify the fraction if needed.
Divide Fractions:
Turn the second fraction (the one you want to divide by) upside-down (now a reciprocal)
Multiply the first fraction by that reciprocal
Simplify the fraction (if needed)
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Fraction Contraption Workshop – August 10, 2011
Frequency is the number of times an outcome occurs
Numbers
Integers are like whole numbers, but they also include negative numbers
Counting Numbers are whole numbers without the zero
Rational Numbers are written as a ratio (fractions, decimals)
Whole Numbers are the numbers 0, 1, 2, 3, 4, 5, … (and so on)
.
6
4
7
Tenths
Hundredths
Thousandths
Tens
5
Decimal
2
Ones
3
Hundreds
Percent means per one hundred; a ratio in which the denominator is always100
Place Value is the value a digit represents depending on its place in the number
Probability is the extent to which an event is likely to occur measured by the ratio of favorable cases to
the whole number of cases possible. Example, a six-sided die presents six potential outcomes – one
through six or 1 in 6 stated 1:6
Ratio is a comparison of any two quantities; may be expressed as a:b or a to b or ab
Sample Space is the set of all possible outcomes of an experiment.
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Fraction Contraption Workshop – August 10, 2011
How to Play the Fraction Contraption Game
Anticipatory set
First Learn to play the game with Integers and introduce probability.
The Game with Integers

The game can be one or two players.

Move all tiles (numbered 1 through 8) up

Roll 2 dice

Whatever you roll, move that same value on the tiles down (e.g. roll a 9 move down the 8 and 1 or the
7 and 2 and so on).

When all tiles are down, you win the game,

If there are remaining tiles, your score is the sum of the remaining tiles (e.g., if the only tile remaining
is a six and you roll a five then play is over and your score is a six).

In 2 person play, low score wins.
The Game with Fractions

The game can be one or two players.

Move all tiles (numbered 1 through 8) down

Roll the eight-sided fraction dice

Whatever you roll, move that same value on the tiles up (e.g. roll 3/8 and ½ move up the 7/8; or roll
5/8 and 5/8 move up the 1 and ¼, and so on).

When all tiles are up, you win the game,

If there are remaining tiles, your score is the sum of the remaining tiles (e.g., if the only tile remaining
is a 5/8 and your roll a ¼ and an ½ then play is over and your score is a 5/8).

In 2 person play, low score wins.
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Fraction Contraption Workshop – August 10, 2011
Using the Ruler to Aid Calculating Fractions
If the student has trouble adding fractions they can use the built-in ruler to add the fractions.
Roll 1:
and
Number line
on ruler
1
2
3
4
5
6
7
8
1
Fraction
=
Convert to eighths to have a common denominator:
Add
+
Notice on the number line that there is a 6 and a 7; these match the fractions’
numerators and equal 13, the sum of the two fractions’ numerators. Or look for any numbers on the
number line that add up to 13, e.g. 8 and 5 - move up the 1 and the .
1
2
3
4
5
6
7
8
1
Roll 2:
Notice on the number line that the 7 is taken. Look for other numbers that add
up to 12, such as 3, 4, and 5. Move those tiles up
1
2
3
4
5
6
7
8
1
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Fraction Contraption Workshop – August 10, 2011
Roll 3:
Notice on the number line that 8 is available and the fraction, reduced, equals 1.
Move the tile up.
1
2
3
4
5
6
7
8
1
Roll 4:
Find the common denominator
Add the fractions
There are no numbers left that add up to 5 on the number line. The game is over - add up the number line
remainders. The score is 3 or 3/8
1
2
3
4
5
6
7
8
1
Using the Ruler to Aid Calculating Decimals
If the student has trouble adding fractions and converting to decimals they can use the built-in ruler.
Roll 1:
1
Number line
on ruler
and
2
3
4
5
6
7
8
1
.125
.25
.375
.5
.625
.75
.875
1
Decimal
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Fraction Contraption Workshop – August 10, 2011
=
Convert to eighths to have a common denominator:
Add
Notice on the number line that there is a 6 and a 7 that the fractions’ numerators add up
+
to – these numbers match the fractions
and
. The number line provides sight recognition between fractions on the
dice and decimals.
1
2
3
4
5
6
7
8
1
.125
.25
.375
.5
.625
.75
Refer to the fraction example using the number line for further help.
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Fraction Contraption Workshop – August 10, 2011
.875
1
Components of a Rule
Inches
1
2
3
Halves
1
Eighths
1
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Fraction Contraption Workshop – August 10, 2011
4
Quarters
1
Probability – Dice Toss
Try 1
Try 2
Try 3
Try 4
Try 5
Try 6
Try 7
Try 8
Try 9
Try 10
1
2
3
4
1
2
3
4
5
6
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Fraction Contraption Workshop – August 10, 2011
5
6
Fraction Contraption Ruler Quiz in Eights
Fill in the measurements on the ruler below. Reduce fractions
1
2
3
4
1
2
3
4
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Fraction Contraption Workshop – August 10, 2011
Ruler Quiz in Sixteenths
Fill in the measurements on the ruler below. Reduce fractions.
1
2
3
4
1
2
3
4
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Fraction Contraption Workshop – August 10, 2011
Tens
Ones
Decimal Point
Tenths
Hundredths
.
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Fraction Contraption Workshop – August 10, 2011
Thousandths
Student Assessment: Fractions Quiz
Add, subtract, multiply, divide the following fractions:
1.
+ =
2.
+
3.
-
=
=
4. What is the measurement on the indicated ruler?
1
2
5. A recipe calls for 1 cup flour,
cup of sugar, and
3
4
cup brown sugar. Lauren would like to triple the
recipe. How much of each ingredient will she need?
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Fraction Contraption Workshop – August 10, 2011
Student Assessment: Fractions Quiz
Add, subtract, multiply, divide the following fractions:
1.
+ =
2.
+
3.
-
=
=
4. A recipe calls for 4 cups of flour, 1 and cups of sugar, and
cup brown sugar. Lauren would like to
cut the recipe in half. How much of each ingredient will she need?
5. Find the length of the line below to the nearest
inch:
_______________________________________
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Fraction Contraption Workshop – August 10, 2011
Fraction Contraption Variations
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Fraction Contraption Workshop – August 10, 2011
Making the Fraction Contraption
Note: Instructional videos for many of these operations are available at www.fractioncontraption.com.
Safety First: Only use the tools on which you have been trained; never build this without consent of your
instructor.
1. Make the base of 3/4‖ hardwood or MDF. Cut wood to 6‖ x 10‖.
2. Cut the main pocket in the base 4‖ by 8‖ at a depth of ½‖ using a CNC router (technical drawing
and visit http://www.fractioncontraption.com/how_its_made for .dxf format)
Wood base before routing
Wood base after routing
3. Round edges of wood base with router and sand to finish.
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Fraction Contraption Workshop – August 10, 2011
4. Making the melamine tiles
a) The ruler is made of ¼‖ melamine cut to size 1‖ x 10‖. Laser engrave the ruler, starting the first
line 1‖ from the left of the piece of material; this means there will be 1 inch of blank material on
either side of the first and last lines (See picture of game).
b)
The tiles are made from ¼‖ melamine. Start with a piece cut 2 ½‖ x 9‖. Laser engrave, CNC, or
otherwise mark both integer numbers and fractional numbers. The tiles are 1‖ wide when
finished, but add an additional 1/8‖ to each tile before cutting to account for blade width when
tiles are cut apart.
c)
Add groove to tiles to make it easier to slide tiles up and down by making a cut ¼‖ in from each
side of the uncut tile piece. Using a 1/8‖ blade, make a straight cut 1/16‖ deep across the uncut
tiles (a table saw is best, see ―Step 3 - Laser Engrave Tiles, Assembly video on
www.fractioncontraption.com/how_its_made. Making the groove cut starts 44 seconds into the
video.)
d) Cut tiles apart using a table saw or chop saw, which will remove the 1/8 offsets. As your fingers
will come very close to the chop saw blade, finish cutting the last two tiles on a band saw.
e)
Sand edges of tiles so they don’t fit together too tightly (a table belt sander is easiest). Round the
corners of only the left side of the 1 tile and the right side of the 8 tile (these corners meet the
corners of the pocket). Make sure the tiles fit well.
5. Lay tiles and ruler in place to make sure everything fits (ruler and tiles shown are injection
molded plastic not melamine). If the ruler slot is too narrow, use a chisel to widen slightly. Drill
pilot holes for screws using a 1/8‖ bit. Go through both the ruler and wood to make sure the holes
will align later.
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Fraction Contraption Workshop – August 10, 2011
6.
Turn the board over. Use a 1/8‖ counter sink bit to drill into the holes from the other side.
7. Apply wood glue to the underside of two 6/32 x 5/8‖ length flat head machine screws. This is to
prevent the bolts from spinning when the ruler is affixed with a nut. Try not to get glue on the
threads. Use drill to sink screws into wood.
8. Put tiles back in place, and guide the holes in the ruler over bolts. If the fit is too tight, drill a
larger hole (an 11/64ths bit works well). Finger tighten with thumb nuts or another 6/32 nut; do
not use a nut that sits too high, or the game sets won’t be able to stack for storage.
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Fraction Contraption Workshop – August 10, 2011
Fraction Contraption Evaluation Plan
The Fraction Contraption evaluation plan will measure three aspects: student engagement, math
confidence, and technical competence.
This plan will be implemented at multiple sites in multiple settings. Two High Schools (Colfax and
Rocklin) will use this methodology to assess their introductory freshman career and technical education
(CTE) classes. Fraction Contraption will also be implemented in the K-8 setting. Having multiple
settings will allow researchers to identify efficacy differences based on grade/age, i.e. is the Fraction
Contraption better for intervention or primary instruction? General demographic and school wide
performance data will be collected and used to compare results between sites as well as teacher
perceptions.
STUDENT ENGAGEMENT
The student engagement evaluation will concentrate on the cognitive/intellectual/academic engagement as
defined by the High School Survey of Student Engagement (HSSSE) developed by the Center for
Evaluation and Education Policy (CEEP) at Indiana University in Bloomington. The evaluation will focus
on instructional time and instruction-related activities to capture the work students do and the ways they
go about their work. Student engagement will be measured in three ways.
1) Classroom observations
The observations will be conducted using the methodology and materials as prescribed by Dr.
Richard Jones, in his writings: Student Engagement – Teacher's Handbook (a companion to Student
Engagement – Creating a Culture of Academic Achievement). Please see the Student Engagement
Classroom Observation Checklist.
2)
Student surveys to assess student's perception of their engagement
Please see Student Engagement & Math Perception Pre & Post Survey.
3)
Conduct a reflection/debrief to determine teachers’ perceptions of student engagement
The first debrief will be midway through the project, and the second will be at the end of the project.
Teachers will also be asked to keep a log and post comments about their experience and any questions
at www.sierraschoolworks.com See Teacher Perceptions of the Fraction Contraption Project.
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Fraction Contraption Workshop – August 10, 2011
MATH CONFIDENCE
Students’ math confidence will be assessed by using a survey. This survey is incorporated into the Student
Engagement & Math Perception Pre & Post Survey.
TECHNICAL COMPETENCE
Technical competence will be measured by administering pre- and post-tests. Please see Pre/Post
Diagnostic Tests.
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Fraction Contraption Workshop – August 10, 2011
Student Engagement Classroom Observation Checklist
Very High
High
Medium
Low
Very Low
Positive body
language
Students’ body postures indicate they are paying attention to the teacher and/or other students.
Consistent
Focus
All students are focused on the learning activity with minimum disruptions.
Verbal
Participation
Students express thoughtful ideas, reflective answers, and questions relevant or appropriate to learning.
Student
Confidence
Students are confident, can initiate and complete a task with limited coaching and can work in a group.
Fun and
Excitement
Students exhibit interest and enthusiasm and use positive humor.
General Notes for Observer
All walkthroughs should have a common set of criteria. The purpose of this activity is not to evaluate the
teacher, but to make classroom observations to obtain specific information about the level of engagement.
During a walkthrough, the observer should avoid disturbing the classroom lesson. Once walkthroughs are
common practice, teachers and students will accept them as routine.
Before the walkthrough, observers should introduce themselves briefly to the teacher and obtain any
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Fraction Contraption Workshop – August 10, 2011
background information that will help them understand in what the students are engaged at that particular
time. As the observation is not an evaluation, observers make no judgments nor give any feedback to
students or the teacher. The focus should be on students interacting with their peers or how engaged
students are in the work they are doing. A good rule of thumb is to conduct walkthroughs in classrooms
once a month.
Optional Student Feedback Survey
To get feedback on students’ engagement for a specific class period, use the following five-point scale.
1. Low level of engagement: class was boring, time moved slowly.
2. Low to moderate level of engagement: class was OK.
3. Moderate level of engagement overall or high level for a short time: class was good.
4. High level of engagement for a major portion of the class period: class was very good.
5. High level of engagement for the entire class period: wish we had more time.
Before students give feedback, explain that. a class is highly engaging if:

The work is interesting and challenging

Students are inspired to do high-quality work

Students understand why and what you are learning

Time seems to pass quickly
Have all students give their rating simultaneously and anonymously. Have students write a rating number
on a card or individual whiteboard and then collect them.
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Fraction Contraption Workshop – August 10, 2011
Student Engagement & Math Perception Pre & Post Survey
SOME
YES
1. I feel comfortable asking my teacher for help on fractions
2. I know why math is important
3. I like learning fractions
4. I can add and subtract fractions without help from my teacher
5. I always try to work the problem and get the right answer
6. Fractions are difficult for me
7. Math makes sense to me
8. I understand percent
9. I can add and subtract decimals without much trouble
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Fraction Contraption Workshop – August 10, 2011
TIMES
NO
Teacher Perceptions of the Fraction Contraption Project
1. What were the best and worst parts of this project?
2. How did your students perform?
3. What were common errors that students encountered during this project?
4. Is there something that you tried and would like to share about this project?
5. Do you feel your students benefited from this project? If so, how... If not, why?
6. Would you do this project again if given the option?
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Fraction Contraption Workshop – August 10, 2011
Fraction Contraption Log
Week
Activity
Student engagement & self-image perception pre-survey administered
Student pre-diagnostic test administered
Student engagement & self-image perception post-survey administered
Student post-diagnostic test administered
Pre/Post Diagnostic Tests
Fraction Test Generator: http://www.homeschoolmath.net/worksheets/fraction.php
Fraction tests can be generated from this site by grade level.
Ruler Test Generator: http://themathworksheetsite.com/read_tape.html
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Fraction Contraption Workshop – August 10, 2011