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Fantastic Fractions and Delightful Decimals 1

Fantastic Fractions and Delightful Decimals

Chelsea Webb

TESL 220

Cross-cultural Learning and Communication: Understanding the Needs of English Language

Learners in Elementary Schools

Spring 2013


Table of Contents

Introduction……………………………………………………………………………..pages 3-4

ELL Learning Outcomes……………………………………………………………….pages 5-6

Instructional Strategies…………………………………………………………..……pages 7-11

Work Samples………………………………………………………………….….....pages 12-18

Annotated Bibliography……………………………………………………………..pages 19-20

Summary Reflection……………………………………………………………………...page 21

References…………………………………………………………………………………page 22


Introduction

This Instructional Resource Guide will cover the topic of addition and subtraction of

fractions and decimals. The objectives for this guide are found in the Virginia Standards of

Learning (SOLs). The SOL focused on is Grade Four Math 4.5 which says “the student will…b)

add and subtract fractions having like and unlike denominators that are limited to 2, 3, 4, 5, 6, 8,

10, 12,…, using common multiples…; c) add and subtract with decimals…” (Mathematics). The

goal for this guide is to help with teaching fractions, a math topic that many students struggle

with, not just English Language Learners (ELLs). Hopefully, the guide will help to serve a

purpose for all students.

I know that in my classroom I may have students who know little to no English or are in

a developmental stage of learning English. Being in TESL 220, I have a better understanding of

how ELLs develop over time. Upon arrival to the United States, ELLs may be overwhelmed and

going to a new school where they will have to interpret everything that is said can be stressful.

As a teacher, I am aware of how ELLs will develop in the classroom over time. In the

Preproduction Stage or Silent Period (0-6 months), the student will have small amounts of

comprehension and most likely will not verbalize (Hill & Flynn). The student will however nod

“yes” or “no” and use drawings to communicate (Hill & Flynn). In the Early Production Stage (6

months-1 year), the student will have some comprehension but this is limited and instead of only

nodding he or she may give one or two word responses (Hill & Flynn). In the Speech Emergence

Stage (1-3 years), the student will have better comprehension and will use simple sentences with

grammar and/or pronunciation errors (Hill & Flynn). In the Intermediate Fluency Stage (3-5

years), the student will have outstanding comprehension and will make small amounts of

grammatical errors. In the Advanced Fluency Stage (5-7 years), the student will be speaking at


an almost native level (Hill & Flynn). Understanding these stages of development will help me to

create differentiated instruction for my students, so they can all learn the material in a way that

works for them.

I chose to focus on fractions and decimals because it is a topic that many students

struggle with in school today. Many students do not enjoy math classes at all mainly because

they had one bad math teacher, and then they just give up. My goal as a teacher is to be in a

kindergarten classroom, but I would not mind teaching math to higher level elementary grade

levels. I am a math major, so I enjoy doing math. I want students to enjoy math too and not dread

to go to the class every day.


ELL Learning Outcomes

Learning Outcomes How to Achieve These Outcomes

Add fractions with like denominators-Review addition of whole numbers-Review what fractions are-Teach to add the numerators and leave denominators alone

Subtract fractions with like denominators

-Review subtraction of whole numbers-Review how to add fractions-Teach to subtract the numerators and leave denominators alone (similar to addition of fractions)

Add fractions with unlike denominators(2, 3, 4, 5, 6, 8, 10, 12)

-Review addition of fractions-Review multiplication facts-Teach how to find a common denominator by listing multiples of each denominator-Show that once there is a common denominator the problems are just like the ones we did before

Subtract fractions with unlike denominators(2, 3, 4, 5, 6, 8, 10, 12)

-Review subtraction of fractions-Review how to find a common denominator by listing multiples of each denominator-Show that once there is a common denominator the problems are just like the ones we did before

Add decimals-Review what a decimal is-Teach to line up the decimal point and then add as normal

Subtract decimals-Review how to add decimals-Teach to line up the decimal point and then subtract as normal

The chart on above illustrates what I would like for students to know in my fourth grade

classroom after we complete a unit on fractions and decimals. The chart is a rough sketch on how

I plan to approach each outcome. I go into more detailed instructional strategies and activities on


pages seven through eleven. The learning outcomes presented are mathematical concepts that all

students need to know so they can succeed in high level math courses.

Mathematics is hard subject to teach to ELLs because math itself is a language

(Moschkovich). As an ELL, I can imagine it being difficult to learn the English language in the

classroom to communicate, read, and write and then also have to learn the language of

mathematics. To get ELLs to engage more in math, I can “focus on students’ mathematical

reasoning, not accuracy in using language” (Moschkovich). Rather than focusing on the fact that

the student did not describe the problem using correct terminology, focus on whether or not they

executed the problem correctly. Also, do not have a list of math-related vocabulary words and

definitions (Moschkovich). Students will learn the terminology as they get more into the content.

I will go into more differentiated instruction for ELLs in the next section.


Instructional Strategies

Add fractions with like denominators:

To begin teaching addition of fractions, I will first go over addition of whole numbers to

make sure that all of the students are comfortable with addition. I will pass out small dry erase

boards and dry erase markers to each student. I will then write a problem on the board and say it

aloud, and they will be responsible for writing down the answer. Then, I will ask them to hold up

their dry erase board, and I will quickly scan the room to make sure everyone has written the

correct answer. For example, I would write “5+3=?” on the board, the students would write

down their answer, I would ask for them to hold them up, and I would check for correct answers.

If a student gets one wrong, I will have another student explain to him or her how to get the

correct answer.

After reviewing addition of whole numbers, we will review fractions and what they are. I

will ask students what they do remember about fractions before I show them the “Fraction Wall”.

The “Fraction Wall” is a diagram of how fractions are divided and put together to make a whole.

I will post this on the board for our fraction unit. The “Fraction Wall” will look similar to this:

From http://pinterest.com/pin/179018153909202165/


I will then demonstrate on the board that if the bottom numbers of the fractions,

denominators, are the same number, then the top numbers, numerators, can simply be added like

whole numbers and the denominator stays the same. I will do a few more examples, and then I

will pass out a practice sheet for the students to do and hand in after they have completed it.

Subtract fractions with like denominators:

To teach subtraction of fractions, I will go through a similar process to addition of

fractions. I will begin by reviewing subtraction of whole numbers. I will have the class count off

into groups. Each group member will be asked to take out a sheet of paper and a pencil. I will

write on the board a problem, and they are to work the problem on their own paper. Once

everyone in the group has finished, each student will share his or her answer. If someone gets a

problem wrong, the group will need to help explain to that student what the correct answer is and

why. I will be available for questions at any time.

After reviewing subtraction of whole numbers, I will ask in anyone remembers how to

add fractions. I will write an addition of fractions problem on the board and ask a student to

come up to the board and write the correct answer. If no one has any questions about adding

fractions, then we will continue on to learn about subtracting fractions.

I will write a problem on the board and show the students that subtracting fractions with

like denominators is the similar to adding fractions with like denominators. The numerators are

subtracted and the denominators remain the same. To illustrate this concept, I will give each

group construction paper and one subtraction problem. Their goal is to cut the paper into the

appropriate number of pieces, and then they will use the pieces to show the subtraction problem.

I will observe each group. For example, group one will have the problem “45−2

5=?”. They will


cut their paper into five equal sections, lay out 4 pieces and then take away 2 pieces to get an

answer of 25 .

Add fractions with unlike denominators:

Teaching addition of fractions with unlike denominators is a bit tougher than teaching

addition of fractions with like denominators. I will begin by reviewing how to add fractions with

like denominators. I will write a few problems on the board and ask students to raise their hand if

they know the answer. I will also review fraction multiplication and multiples of numbers.

After reviewing, I will begin to teach how to add fractions with unlike denominators. We

need to find a common denominator between the two fractions. For example, if the problem said

37+ 2

4=?, then we would need to list the multiples of 4 and 7 not including 1.

4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40

7: 7, 14, 21, 28, 35, 42, 49, 56, 63, 70

After we list the multiples of 4 and 7 going up to 10, we see that both of them have the number

28 in common. For the number 4, 28 is the 7th term so we need to multiply our fraction 24 by 7.

For the number 7, 28 is the 4th term so we need to multiply our fraction 37 by 4. We end up with

1428

+1228

=2628 . Once the denominators are the same, we can add the numerators just like we did

before and leave the denominators alone. We will go over simplification at a later date.

Subtract fractions with unlike denominators:

Subtracting fractions with unlike denominators is the same concept as adding fractions

with unlike denominators. We will review how to subtract fractions with like denominators first.


To review students will be asked to write down a few problems and draw a visual depiction of

the problem. I will also review with them how to add fractions with unlike denominators.

After we review, I will teach them a different way to find a common denominator.

Another way to find like denominators is simply by multiplying one fraction by the opposite

denominator and vice versa. For example, 58−1

2=? So, we would take

58∗2=10

16 and 12∗8= 8

16 .

Since we have found our common we can subtract the numerators just like we did before. So,

1016

− 816

= 216 . This method of finding a common denominator is faster than writing out all of the

multiples, however, this is not the least common denominator between 8 and 2. We will talk

more about this when we go over simplification of fractions.

Add decimals:

To teach addition of decimals we will review what a decimal is first. I will have the

students tell me how to go from decimal to a fraction. They will also have a “greater than/less

than” worksheet to complete as review. The worksheet will ask the students to determine

whether or not the fraction in the problem is greater than or less than the decimal. For example,

“Is 13 >, <, = .5?” The student would need to circle the correct answer, in this case “<”.

Now that we have reviewed decimals, we can talk about how to add them together.

Decimals are much easier to think about than fractions are when it comes to addition. All we

have to do is line up the decimal points, add some zeros where we have blank spaces, and add

like normal. For example, 04 . 23

+ 24 . 80 29 . 03

Adding decimals is as simple as that. The students will have some take-home problems to turn in

the next day.


Subtract decimals:

Subtracting decimals is the same as adding decimals except we must remember all of our

rules of subtraction. We will review addition of decimals and the rule about lining up our

decimals before we are able to add or subtract. I will also tell the students to reference the

subtraction poem I will have posted in the classroom. The subtraction poem will look something

like this:

From http://theinspiredapple.blogspot.com/search?q=subtraction+poem

I will have the students say the poem with me aloud a couple times so we remember how to

subtract correctly.

We can use the subtraction poem when subtracting decimals as well. We just have to

make sure to line up our decimals and we can subtract just like we do with whole numbers. I will


have students work in groups on a set of problems, and then have them share their group answers

with the class to make sure no one had any trouble.


Work Samples:

Student Work Sample #1:


The student did will with the subtraction of the numbers but he or she is not clearly

visualizing the equal parts that the whole rectangle or circle needs to be divided into. To make a

clearer visual for the student, we can make a “Fraction Flip Book”. On each page of the flip

book, the student will have the name of the fraction (i.e. “thirds” or “fifths”), how many equal

parts it takes to make a whole, a circle diagram, and a rectangular diagram. The flip book will

help to visualize the equal parts we are looking for. The “Fraction Flip Book” will look like this:

From http://buzzingwithmsb.blogspot.com/search?q=fractions



This student did not understand how to find multiples of each denominator. Instead, he or

she cross-multiplied and could not find any common number between the two denominators. So,


the student added straight across the top and bottom of the fraction. A more visual way of

teaching how to add and/or subtract fractions with unlike denominators does exist. They are

called “Butterfly Fractions”.

From http://www.moveitmaththesource.com/realfractions/butterflyfractio.html

“Butterfly Fractions” are a great way to teach addition/subtraction of fractions with unlike

denominators. The diagram above shows every step that needs to be taken. First, draw the

“wings” crossing over in a diagonal. Then, draw some antenna on the top of the butterfly. On the

inside of the antenna, we will write the answer to the multiplication of the diagonals. Then, we

draw a small tail on the butterfly and multiply the denominators. Then, we add or subtract the

two antenna numbers and leave the denominator the same. Simplify when necessary (we will get

to simplification later on).



This student lined the digits of each decimal up instead of lining up the decimal point. He

or she also did not add zeros into the blank spaces which could have helped to visualize the

problem more. I have come up with an acronym to help students remember the steps to take


when adding decimals. The acronym is “Llamas Are not Angry”. “Llamas” stands for “line up

the decimal points,” “Are not” stands for “add zeros in blank spaces,” and “Angry” stands for

“add like normal.” I will have a poster in the classroom with the saying on it, and the poster will

have pictures to illustrate what the saying means.

Image from http://iwastesomuchtime.com/on/?i=33774

Another way I can help this student and other students who may be struggling is to do a

visual place holder activity. Each student will have a small bag of Skittles and a place holder

table. Students will be asked the same problems as above and will use the Skittles in the place

holder table to represent each problem. For example,

Problem #1

Hundreds(100’s)

Tens(10’s)

Ones(1’s)

Decimal(.)

Tenths(10th’s)

Hundredths

(100th’s)

Thousandths(1000th’s)

27.1 .

.05 .

Llamas line up the decimal points

Are not add zeros in blank spaces

Angry add like normal

HAPPY!

4 . 25+ 74 . 069

04 . 250+ 74 . 069

04 . 250+ 74 . 069 78 . 319

+

2 7 . 1 5


Hopefully, by visualizing this, the concept of adding decimals will be better understood by all

student not just ELLs.


Annotated Bibliography:

1. Butterfly fractions (n.d.). Retrieved from http://www.moveitmaththesource.com/

realfractions/butterflyfractio.html

2. Hill, J., & Flynn, K. (n.d.). Retrieved from http://www.ascd.org/publications/books

/106009/chapters/The-Stages-of-Second-Language-Acquisition.aspx

This website provides a very nice simplified chart of the developmental stages of English

Language Learners. The chart helps to simplify everything we have learned in class into

one nice document to glance at and refresh my memory.

3. Mathematics standards of learning for Virginia public schools - grade 4. (2009,

February). Retrieved from http://www.doe.virginia.gov/testing/sol/standards

_docs/mathematics/2009/stds_math4.pdf

This article is the fourth grade math SOLs. I am focusing the most on fractions and

decimals for my project. Understanding the SOLs will help me to find some cute, fun

ways to teach fractions and decimals. Also, reading through the descriptions helps me to

understand what a fourth grader should understand at this level of math.

4. Moschkovich, J. (n.d.). Mathematics, the common core, and language: Recommendations


for mathematics instructions for ELs aligned with the common core.

Understanding Language: Language, Literacy, and Learning in the Content

Areas, Retrieved from http://www.sccoe.com/depts/ell/accountability/13thannual/

11_KenjiUL Stanford Final 5-9-12 w cover.pdf

This article shows that mathematics is another language all in itself. The article tries to

make teachers aware that math can be difficult for English Language Learners because

they are already learning the English language and will have to learn the language of

math as well. The article gives great recommendations on how to connect math content to

language.


Summary Reflection:

This assignment was a lot of fun to do. I enjoyed coming up with fun creative ways to

teach ELLs material that they may not fully understand in a regular classroom setting. The

assignment was somewhat challenging because I did have to make sure that I had enough visuals

in the teaching of the material so that ELLs could follow along even if they did not know what I

was saying. Being able to come up with an Instructional Resource Guide will be very helpful in

the future when I am in the classroom. If I have students that do not understand the topics I am

covering, I can make a guide to help myself re-evaluate my current lesson plans.

As a teacher, being culturally and linguistically diverse is very important. I need to know

my students and how they learn in the classroom. I want to get to know my students on a

personal level as well because they can teach me so much about different cultures and customs

that I can use later on in my life or in the classroom. I want to make my classroom a community,

and I hope that my students will want to engage with others about their lives so we can all learn

from one another. Being in TESL 220 has opened my eyes to many situations that may occur in

the classroom. I would not have expected to encounter some students like the examples that Dr.

Stallions has given us in class. I do not know how I would have responded in some of the

situations that she described, but hearing her stories has helped me get a grasp on how to handle

situations with not just ELLs but any student that is struggling.

Overall, the assignment bettered my understanding of ELLs and how they develop

throughout their time in the classroom. I also got to be creative in planning lessons and making

them easy to understand for all students. I have also learned that being culturally and

linguistically diverse is important in the classroom as well in everyday life.


References:

Butterfly fractions (n.d.). Retrieved from http://www.moveitmaththesource.com/realfractions

/butterflyfractio.html

Hill, J., & Flynn, K. (n.d.). Retrieved from http://www.ascd.org/publications/books /106009/

chapters/The-Stages-of-Second-Language-Acquisition.aspx

Mathematics standards of learning for Virginia public schools - grade 4. (2009, February).

Retrieved from http://www.doe.virginia.gov/testing/sol/standards _docs

/mathematics/2009/stds_math4.pdf

Moschkovich, J. (n.d.). Mathematics, the common core, and language: Recommendations

for mathematics instructions for ELs aligned with the common core. Understanding

Language: Language, Literacy, and Learning in the Content Areas. Retrieved from

http://www.sccoe.com/depts/ell/accountability/13thannual/11_KenjiUL Stanford Final 5-

9-12 w cover.pdf

Documents

webapps.roanoke.eduwebapps.roanoke.edu/courses/educ/cfwebb/IRG Final.d… · Web viewHopefully, the guide will help ... limited and instead of only nodding he or she may give one