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Fraction Contraption

Math Intervention Model

Measurement, Fractions, Probability, Decimals, & Percent

Workshop

August 10, 2011

2 Fraction Contraption 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


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.


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 Math1. Measurement

2. Fractions

3. Ratios/Proportions

4. 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%

Fractions/Decimals

4. What number equals

?

A. 0.267 B. 0.375 C. 2.67 D. 3.7


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




Geometry

Grade 3, 4 and 5 Overview



Number and Operations—Fractions


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


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


FRACTION CONTRAPTION LESSONS

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.

Measurement

& Play with Integers

Probability


Fractions

& Play with Fractions

Play with

Fraction Dice & Decimal Tiles

Decimals

& Play with Decimal Dice

Percent

& Play with Percent Dice

http://www.polleverywhere.com/


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

1 2 3 4

Inches

1 Halves 1 Quarters

1 Eighths


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:


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


Student Assessment –Fraction Contraption Ruler Quiz

Fill in the measurements on the ruler below.

Extension: Have students fill in the ruler in 16ths in the Appendix

1 2 3 4

1 2 3 4


Playing Fraction Contraption with Integers


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


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


Probability

Introduce probability when students have a solid understanding of measurement.


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


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 6

Measurement


Probability


Fractions


Play with


Decimals


Percent




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 2 3 4 5 6

1 1,1 1,2 1,3 1,4 1,5 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 6



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.


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?


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.


Fractions

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

of the pizza was eaten.

Whole= 8/8 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:

+

=

Measurement


Probability


Fractions


Play with


Decimals


Percent



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.

-

=


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.

-

=


4. What is the measurement on the indicated ruler?

5. A recipe calls for 1

cup flour,

cup of sugar, and

cup brown sugar. Lauren would like to triple

the recipe. How much of each ingredient will she need?



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:

1 2 3 4


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


Decimals

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


.

Measurement


Probability


Fractions


Play with


Decimals


Percent




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


. 5 0

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


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

Measurement


Probability


Fractions


Play with


Decimals


Percent



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


Percent

Move on to percent when your students have a solid understanding of measurement and fractions.


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

Measurement


Probability


Fractions


Play with


Decimals


Percent



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


APPENDIX


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)

http://www.mathsisfun.com/reciprocal-fraction.html

http://www.mathsisfun.com/fractions_multiplication.html

http://www.mathsisfun.com/simplifying-fractions.html


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)

Percent means per one hundred; a ratio in which the denominator is always100

3 2 5 . 6 4 7

Hundre

ds

Ten

s

Ones

Dec

imal

Ten

ths

Hundre

dth

s

Thousa

ndth

s

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.


How to Play the Fraction Contraption Game

Anticipatory set

First Learn to play the game with Integers and introduce probability.

The Game with Integers


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,


is a six and you roll a five then play is over and your score is a six).


The Game with Fractions


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,


is a 5/8 and your roll a ¼ and an ½ then play is over and your score is a 5/8).



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

1 2 3 4 5 6 7 8

1

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

Number line

on ruler

Fraction


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:

and

1 2 3 4 5 6 7 8

1

.125

.25

.375

.5

.625

.75

.875

1

Number line

on ruler

Decimal


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

.875

1

Refer to the fraction example using the number line for further help.


Components of a Rule

1 2 3 4

Inches

1 Halves 1 Quarters

1 Eighths


Probability – Dice Toss

Try 1


Try 6


1 2 3 4 5 6

1

2

3

4

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


Ruler Quiz in Sixteenths

Fill in the measurements on the ruler below. Reduce fractions.

1 2 3 4

1 2 3 4



.




1.

+

=

2.

+

=

3.

-

=

4. What is the measurement on the indicated ruler?

5. A recipe calls for 1

cup flour,

cup of sugar, and

cup brown sugar. Lauren would like to triple the

recipe. How much of each ingredient will she need?

1 2 3 4




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:

_______________________________________


Fraction Contraption Variations


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.

http://www.fractioncontraption.com/

http://www.fractioncontraption.com/how_its_made


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.

http://www.fractioncontraption.com/how_its_made


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.


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.


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.


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


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.


Student Engagement & Math Perception Pre & Post Survey

YES

SOME

TIMES

NO

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


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?


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

http://www.homeschoolmath.net/worksheets/fraction.php

http://themathworksheetsite.com/read_tape.html

fraction contraption - sierra school...

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