Assessment and Teaching Project Kara and Jeanne (2) grammar and spelling checked

Assessment and Teaching Project

MAFS.3.NF.1.2

Jeanne N Asberry and Kara D Wakley

University of South Florida Sarasota-Manatee

Dr. Patricia Hunsader MAE 6117

April 2, 2016

2

Fraction Interview Protocol

Hello, my name is ________ and what is your name, age, and grade? <<Student

response>> Today’s activity will be placing fractions on a number line. This is a quick check

and I will not teach anything today. The purpose of this quick check is to see can you correctly

place fractions on a number line if I give you the numbers zero, one, and two. To help you with

placing the fractions on the number line, there are pie graphs on top of the written fractions.

However, you can relax, be comfortable, and have fun, as this activity is not graded. Thank you

for your help today.

Before we begin the activity, I want to find out how much you know about fractions. My

first question is what is a fraction? <<Student response>> what is the bottom number below the

line of a fraction called? <<Student response>> what is the top number above the line of the

fraction called? <<Student response>> Do you know what the top number means? The bottom

number means. <<Student response>>How can you tell if one fraction is larger than the other?

<<Student response>> Are you familiar with pie graphs? Can you point them out to me?

<<Student response and/or gesture>> During the activity, I would like to hear your strategy.

What thinking are you using to place the fractions on the number line? <<Student Response>>

3

Jeanne’s Interview Report

Below is an example of the number line, fractions, and pie graphs Niccolo was given:

Before conducting the assessment, the following assumptions were made:

a. The student has a basic understanding of a fraction as a number (MAFS.3.NF.1).

b. The student can recognize a fraction as a/b.

c. The student has a basic understanding of 1/b where 1= 1 part of a whole and b= the

whole that is partitioned into equal parts. (MAFS.3.NF.1.1)

d. The student has basic understanding that the top number is part of a whole called the

numerator (MAFS.3.NF.1.AP.1a). The bottom number of a fraction is the whole (total

number of parts) called the denominator (MAFS.3.NF.1.AP.1b).

e. The student is familiar with a number line.

f. The student is familiar with a pie graph and can use the pie graph to place the fraction

correctly on the number line.

After conducting the assessment, the following expectations were:

a. The student would correctly place at least 3/6 or 50% of the fractions on the number line

correctly given that fraction knowledge is beginning to develop in the third grade

(MAFS.3.NF.1).

b. Identify equivalent fractions on a number line (MAFS.3.NF.1AP.3a)

0 1 2

1/11 1/4 2/6 4/12 5/8 6/11

3

4

c. The student can utilize and notice patterns from the pie graphs. The more pieces of the

pies that are filled or the closer the numerator is to the denominator, the fraction will be

closer to one on the number line.

d. The students would notice that if the numerator were greater than the denominator, than

the fraction will be greater than one on the number line.

Niccolo is in the third grade and is eight years old at Booker Elementary School. After the

introductions and a discussion of today’s activity, Niccolo was asked to identify what the top

number and bottom number of a fraction is called. He was able to correctly identify and use the

correct terminology-numerator and denominator. However, when asked about what the

numerator and denominator means, Niccolo just replied that it was the top and bottom number of

a fraction. He could not conclude that the numerator was a part of a whole and the denominator

was the whole. When Niccolo was asked about his familiarity with pie graphs, he hesitated a

little before shaking his head “yes.” However, given the six visuals, Niccolo could identify the

pie graphs without any prompting from the interviewer. As Niccolo begins the assessment, he is

verbalizing his strategy. Niccolo begins by placing 5/8 as the third number after two on the

number line and he places 1/1 a little after 1 and 5/8 somehow became 5/3. The question for

Niccolo at this time was how did 5/8 become 5/3? Niccolo replied, “He was subtracting the

numerator from the denominator.” The interviewer intervened and asked the question, would

you want to start with eight slices of pizza or three slices as your whole? Niccolo replied, “Eight

slices.” Niccolo then knew his denominator had to remain the same. Besides 5/8 becoming 5/3,

Niccolo’s reasoning for placing 5/8 as the third number after two, was that he used the

numerators to order the fractions on the number line. Two-sixths was placed after the two on the

5

number line. When Niccolo was asked why he placed 2/6 beside the two, he stated that there is a

two in the numerator.

Niccolo’s number line looked like the following:

Niccolo was then asked what reasoning he used to place ¼ next to 1/1 on the number line. He

stated that the four in the denominator of ¼ was more than the one in 1/1. Again, the interviewer

used some verbal prompting and had the student review the pie graph of 1/1. Niccolo then

noticed that 1/1 was a whole pie, so using that he placed 1/1 in the same position as one on the

number line. The interviewer then asked Niccolo, what about the other fractions. If we look at

their denominators and compare the numerators, would they be more or less than one? Niccolo

could not answer this question. However, he did review the fractions and 2/6 was moved next to

¼. When asked by the interviewer, what made him move 2/6 next to ¼? Niccolo stated that he

skip counted in his head. When skip counting by twos, six comes after four. Niccolo could not

successfully place the other five fractions on the number line. In fact, he identified ¼ and 2/6 as

halves. The other fractions 4/12, 5/8, and 6/11 became fourths. The misunderstanding for

Niccolo was that he was still treating fractions as whole numbers, ignoring the denominator.

However, he could identify the denominator and numerator; he did not know what these terms

meant. During the assessment, he even subtracted the numerator and denominator or skip

counted by twos. Furthermore, Niccolo was not able to use the pie graphs to aid him in correct

placement of the fractions. It is clear that Niccolo would need a review of the basics for

fractions. First, while using some basic fractions such as thirds, halves, and fourths; the teacher

will review the terminology in detail. I will use rainbow circles so that Niccolo can manipulate

0

1

2 311

14

26

412

611

58

6

parts and their whole. The essential questions will be how many parts does it take to make one

whole and even greater? Where do these fractions belong on the number line?

Kara’s Interview Report

The student, Kealynn, is nine and in third grade. She has already been immersed into

fractions and should have the understanding that fractions are not whole numbers. I expect her

to understand what a number line is, where the benchmark fractions belong, and how these all

relate to each other. After speaking with Kealynn, I have learned that she works with fractions at

home with her father and has spent the better part of the past year in class working with them as

well. My expectations were high. I assumed that Kealynn would be able to place the fractions

on the number line with no issue. When asked whether she knew what the number on the top of

a fraction was called the student explained that she was able to remember it was the numerator

because north/down. She explained that north means up and starts with N and then explained

how down is obviously down and starts with D, she then explained numerator is on top and

denominator on the bottom.

The assessment was that the student was given was going to be placing fractions on a

number line. My partner and I agreed that on the first phase of the assessment we would show

students the number line that showed 0-1/2-1, this way students were not confused. My student

was given ½, 3/5, 2/3, 4/9, 5/7, and 8/10 pie graphs on clothespins for the first assessment. My

partner and I agreed to use random fractions because this was just a formal assessment to test the

student’s knowledge. The student was verbally given the directions for the assessment and on

the number line; I had the directions written so she could read them herself. At first, the student

was putting the numbers into numerical orders. This led me to believe she may not understand

what a fraction meant in terms of it when compared to a whole number. When I explained the

7

directions to her one more time she went right to work. But the student was stuck on 4/9. I let

the student try to figure out where 4/9 would go for a minute or two before I intervened. I

prompted the student to explain what she thought 4/9 was. She took a second to think about it

and told me that she thought it was close to ½ but was not exactly one half. I asked the student

to explain why she thought this. The student explained that she thought about what half of nine

was and knew it was 4.5 so she knew that 4/9 was close to 1/2.

The image below is a picture of the first line graph used and the circle graphs the students would have seen.

½ 4/9 2/3 3/5 5/7 8/10

For the next section of the assessment, my partner and I agreed to use a larger number

line and to give the students mixed fractions to see their understanding of how to place these. I

wanted to make sure my student would be able to work through this portion so I asked her to

explain what 8/8 and 1/1 were. She explained correctly that they were equal to one whole. I

gave my student three blank clothespins and explained to her that these will represent fractions

5/4, 6/4 and 3/2. The student was able to put 5/4 and 6/4 on the number line easy but when I

asked her to explain what 3/2 meant she was stumped. I prompted the student to think about

what 2/2 was. The student easily explained one whole. I then questioned the student on what the

left over numbers meant. She explained the remaining portion of the fraction of ½ so she

inferred correctly that the fraction was equal to 1 ½.

I concluded that the student understands the meaning of what fractions are but still needs

work on understanding a number line correctly. To do this, in the lessons she needs to be taught

8

about where benchmark fractions belong on a number line and their relationship to each other.

This student, in particular, needs to be taught with direct instruction. Direct instruction along

with gradual release is the best part in teaching math. The student will need to be kept engaged

in the learning process as well.

Kara and Jeanne’s Recommendation of Lessons For Students

Jeanne and I realized that both of our students needed work on the number line

extensively. We realized that our students did not understand the relationship between fractions

and the number line particularly fractions between zero and one. In discussing our interviews,

we realized that students did not understand that fractions were numbers on a number line; they

were not whole numbers and could not be categorized that way. With the first student, Nicco,

we both concluded that he needed extra work with fractions and what their parts represented. He

could name the parts of a fraction such as the denominator and numerator, but still did not grasp

what the terms meant. With Kealynn, we both came to the same conclusion that she needed a

little bit more work with the number line, but she understood fractions really well. Jeanne

decided to address the benchmark fractions, which were fractions equaling one-half, fractions

that are less than one away from being whole, and all the fractions with one in the numerator.

The premise was that if students could locate these basic fractions on a number line, they could

locate other fractions.

With Jeanne’s approval, Kara took the approach of following up Jeanne’s lesson. Jeanne

did a great job of laying the groundwork for Kara’s lesson. Kara realized that the students still

needed a lesson on splitting number lines. The splitting number lines exercise aided both

students in conceptualizing equal parts of a number line. If they could split the number line into

equal parts according to the denominator, then the students would gain further understanding of

9

locating fractions on a number line. With Jeanne’s lesson, the students had experience with the

number line, but not with splitting it. Both Kara and Jeanne realized that without knowledge on

how to place fractions on the number line, the students did not have a full understanding of what

fractions were.

10

Lesson Plan

USFSM FORMAL LESSON PLAN FORMAT

Jeanne N Asberry Date: 4/5/16Lesson Title Benchmark Fractions Grade 3

Florida Standards

MAFS.3.NF.1.1Understand 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.

MAFS.3.NF.1.2Understand a fraction as a number on the number line; represent fractions on a number line diagram.

a. Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line.

b. Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line.

MAFS.3.G.1.2Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole. For example, partition a shape into 4 parts with equal area, and describe the area of each part as 1/4 of the area of the shape.

Arts Standards n/a

Essential Question

How many parts does it take to make one whole and even greater and where do these parts belong on a number line? The importance of mastering this skill is so the students can extend understanding of fraction equivalence and ordering, use equivalent fractions to add and subtract fractions. The student will later build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers, apply and extend understandings of multiplication and division to multiply and divide fractions, and understand decimal notation for

11

fractions, and compare decimal fractions and percent.

Type of lesson New knowledge and review

AssessmentSummative (include a tool or data you’ll gather)

Student will complete the fraction number line to 1 exercise

Formative (include specific ways to measure and document progress)

The teacher will have the student demonstrate his understanding of 1/2, 1/3, and ¼ at the beginning of the exercise with rainbow circles. The student will also demonstrate knowledge by playing a fraction game on the Illumination website.

Learning Objectives

Objective(s)

After a miniliesson on the fraction halves, thirds, fourths, and benchmark fractions, the student will be able to locate and place at least 4/6 of fractions correctly on a number line.

Prior Knowledge

The student is already aware of halves, thirds, and fourths and can demonstrate this knowledge without much guidance. The student can identify the denominator and numerator and is already familiar with pie graphs.

Complexity

Student may not understand that fractions are numbers and that the closer the numerator is to the denominator, the closer the fraction is to one on the number line.

Level 2 on Webb’s Depth of Knowledge Basic Application of Skills and Concepts

Vocabulary

Vocabulary such as pie graphs, numerator and denominator. By defining numerator (part) and denominator (whole) for the student, this will help them to locate and place the numbers between the intervals of 0 to 1 on the number line.

Instructional Design Framework

 Direct Instruction, Predict-Explain-Observe-Explain, Guided Inquiry (Level 3), Open Inquiry

Differentiation How will you handle progressing, proficient, and exemplary knowledge and skill levels? How will you implement appropriate and allowable instructional accommodations:

Varying Exceptionalities-pie graphs and rainbow circles will be utilized throughout the minilesson and exercise. The lesson will begin in direct instruction with guided practice from the teacher.

12

Diverse Learners- Kinesthetic-can form a human number line and fractions. Visual-use of the computer and the Illuminations website can provide other visuals. The teacher will also provide a paper copy of the number line and rainbow circles. Musical-create a song about fractions on a number line or watch a YouTube video.

English Language Learners-label parts of the fraction with numerator and denominator. These learners will also receive modeling and guided practice from the teacher.

Accelerated Learners- is given more complex fractions for intervals between one and two. Teacher will still assist if needed.

Technology Integration

LCD ProjectorComputer Game- http://illuminations.nctm.org/Activity.aspx?id=4148

Materials and Resources

Rainbow circlesPencilCrayon or colored pencilsFraction number lines to 1 exercise (math-salamanders.com)Blank number line sheet (Math-Aids.Com)Drawing paper

Approximate Time Frame

30-45 min

Instructional Delivery and Facilitation Gradual Release ProcessOpening

(2-3 minutes)Let us pretend we are running a pizza restaurant. I have some rainbow circles that will be my pizzas. I need your help cutting these three orders: 1) a ½ pepperoni pizza, 2) 1/3 anchovies pizza, and 3) ¼ spinach pizza

The teacher will also hand out the completed copy of Fraction Number Lines to 1 so that the student can begin to see where fractions are located on the number line. The student can use this sheet as a reference during the exercise.

The student should be demonstrating with the rainbow circles what the fractions ½, 1/3, and ¼ look like.

The students may not know that given 1/2, there are two parts to a whole (denominator) and only one of those parts is needed (numerator) for the first pizza. Subsequently, the student may not be aware that 1/3 and ¼ are three and four whole with only one part needed.

13

Development Guided: If the student cannot demonstrate the basic fractions ½, 1/3, and ¼. Then the teacher will provide a review.

Discuss vocabulary: numerator and denominator

Question 1: “How many parts make up the whole?

Question 2: How many parts are needed?

If student has demonstrated all three fractions, have student look at the Fraction Number Line to 1 sheet to see where the basic fractions belong on the number line. Let the student explore the sheet to see what patterns he/she can see on the fraction number lines. The student may also use the rainbow circles at any time during the lesson as a visual for the fractions.

If students cannot make connections right away, the following questions should help.

Questions 1: How many parts make up the whole (denominator) on each number line?

Transition: Now let us look again at our reference sheets. Pay close

For review if needed: The student should be able to demonstrate 2, 3 and 4 whole parts(denominator) and show that only one part is needed(numerator). The student can use drawing paper to trace the rainbow circles and then color the one part that is needed.

Student-generated responses:“2”, “3”, or “4” for ½, 1/3, and 1/4

Student-generated response:“1”

If review was not needed, the student should be looking at the reference sheet and noticing the various fractions.

The student may respond with the following:

“2, 3, 4, 5, 6, 7, 8, 10, 12” are all good student-generated responses.

“When the top number (numerator) is one away from the bottom number

14

attention to fractions such as ¾, 4/5, 5/6, 7/8, 9/10, and 11/12.

Question 2: What do you notice about those fractions and their numerators and denominators? Where are they located on the number line?

Transition: We see the fractions that are close to one and their pattern, but what about ½, 2/4, 3/6, 4/8, 5/10, and 6/12.

Question 3: Where are these fractions located on the number line? What pattern do you notice with these fractions?

Transition: Now we see that ½, 2/4, 3/6, 4/8, 5/10, and 6/12 are all halves and in the center of 0 and 1. Can we turn our attention to 1/3, ¼, 1/5, 1/6, 1/8, 1/10, and 1/12?

Question 4: Where are these fractions located on the number line?

Probing Question: Why are these fractions near zero?

Transition: Using your reference sheet, I will give you the following fractions-1/1, ¼, 2/6, 4/12, 5/8, and 6/11. You will also be given a blank number line chart. Please place the fractions correctly on the number line. Remember you can use your reference sheet as a guide.

(denominator), it is closer to one.”

“1/2 is the same as 2/4, 3/6, 4/8, 5/10, and 6/12”

“½ is in the middle of 0 and 1”

“1/3, ¼, 1/5, 1/6, 1/8, 1/10, and 1/12 are closer to 0 because the top number (numerator) is 2, 3, 4, 5, 7, 9, and 11 parts away from being one whole.

Student should be able to locate and place 4/6 of the fractions on the number line.

15

Closing the Lesson

(3-5 minutes)

Show student computer game on the Illuminations website. This is a reinforcement of the fraction on a number line concept

Let student explore the game as a review of what was just taught.

Questions From Blooms Taxonomy of the Cognitive DomainorWebb’s Depth of Knowledge

Webb’s Depth of Knowledge: Level 2 Basic Application of Skills and ConceptsLevel 1: Knowledge of Blooms Taxonomy

Home Extension Illuminations website

16

17

18

Candidate Name: Kara D Wakley Date 4/8/2016Lesson Title Teaching Students to Split the Number Line Grade 3

Florida Standards

MAFS.3.NF.1.2Understand a fraction as a number on the number line; represent fractions on a number line diagram.Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line.Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line.MAFS.3.NF.1.AP.2aLocate given common unit fractions (i.e., 1/2, 1/4) on a number line or ruler.

Arts Standards Include if appropriate to this lessonEssential Question

How do you place fractions onto the number line?

Type of lesson Where does this lesson fit into the content (new knowledge, guided practice, skill/drill for fluency and mastery, conceptual understanding, review, etc.)?

AssessmentSummative (include a tool or data you’ll gather)

The students will be able to say whether the fractions placement on the number line are “true” or “false”

Formative (include specific ways to measure and document progress)

Students will practice splitting their number lines into the given numbers

Learning Objectives

Objective(s)After learning about splitting number lines, students should be able to differentiate between true or false number lines that have fractions listed.

Prior Knowledge

MAFS.2.MD.2.6Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2... and represent whole-number sums and differences within 100 on a number line diagram.MAFS.3.NF.1.1Understand 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.

Complexity The skill fits into level 1 of DOK, recall. Students have been immersed in fractions and what fractions represent in the past couple months.

Vocabulary Numerator Denominator Number line

19

Fraction True False

The knowledge of vocabulary will help students to understand what the whole, part means, and what that means in relation to the number line. This also bring into account that students will need to understand number line and true/false for the assessments.

Instructional Design Framework

Direct instruction is used with gradual release.

Differentiation

ELL: Provision of visuals, big tasks broken down to small tasks, small discussions, and verbal directions.Varying Exceptionalities: Provision of visuals, connect content to their lives, given students an overview of what is expected

Technology Integration

N/A

Materials and Resources

Graphic Organizers with number lines (self-created)Writing Utensils

Approximate Time Frame

30-45 minutes with assessments included

20

21

22

Jeanne’s Lesson Reflection

When I first began my lesson, I did a pretest. Niccolo and Kealynn used the rainbow

circles to demonstrate knowledge of basic fractions such as ½, 1/3, and ¼. This area went well

as they both knew that out of two, three, or four parts; only 1 part is needed (numerator). During

the lesson, the students were able to make great connections from the Fractions Number Line to

One reference sheet. Some of the initial observations were: Kealynn noticed a kind of triangular

shape with the placement of the fractions on each corresponding number line. She also noticed

that for all the fractions in the center of the number line such as ½, 2/4, 3/6, 4/8, and so forth the

top number is half of the bottom and is located in the middle of the number line. Niccolo kept

quiet at first, but eventually made the observation that all the fractions in the center of zero and

one equal ½. In discussing numbers such as ¾, 5/6, and 7/8, I asked the students to compare the

numerators and denominators. Niccolo noticed the numerator was “one away from” the

denominator. Kealynn was having difficulty with fractions that were one away from the being

whole. She was still treating fractions like whole numbers. Kealynn kept counting the

denominators, but leaving the numerator alone. I spent a little more time to with Kealynn, as I

wanted her to make the leap from the whole number concept to thinking about the numerator and

denominator. As Kealynn was just beginning to grasp this concept, Niccolo made another

discovery. I had him write the discovery down on a piece of paper, as I did not want Niccolo to

lose the thought. After finishing with Kealynn, I looked at what Niccolo wrote. In his own

words, he wrote the following: “I noticed that on the other side it’s taking away and going close

to the 0 side!” He was referring to the 1/3, ¼, 1/5, 1/6, 1/7, 1/8, 1/10, and 1/12. He noticed that

they all just had one part out of their wholes.

23

During the formative assessment, both students could use the Fractions Number Line to

One reference sheet given to locate fractions. The summative assessment was not conducted

according to plan. I wanted the students to be able to place 1/1, 1/4, 2/6, 5/8, and 4/12 on a blank

number line using the benchmark fractions discussed during their observations. Instead, Niccolo

and Kealynn looked at the reference sheet and placed 1/4, 2/6, 5/8, and 4/12 on the blank number

line according to their general location on the reference sheet. The blank number line form that I

used was confusing. Next time, I will use a form with just one number line instead of five or six

on the page and have them turn the reference sheet over. I would also reiterate that we are using

benchmark fractions like 1/3, ¼, 1/5, 1/6, fractions that equal ½, and fractions that are one away

from being whole to locate where other fractions belong on the number line. This concept does

take time to develop as students in 3rd grade are just beginning to recognize fractions as numbers.

I finished the lesson with a review that went well. Niccolo and Kealynn were able to tell me in

their own words that 1) Fractions are numbers 2) ½ is in the middle of the number line 3)If a

number is one away from whole it is closer to one. I did not cover the Illuminations website and

I did not have an art component. Even though my assessment did not go well, I was excited that

Niccolo and Kealynn made connections.

24

Observation of Mrs. Wakley

Before the lesson-

The question was asked how you place fractions on a number line. Do you place them wherever

you want? Niccolo and Kealynn responded “no.” My thoughts were that Mrs. Wakley was

trying to add a sense of humor and make the lesson fun. Mrs. Wakley then asked Niccolo and

Kealynn do you know what a fraction is. Their hands went up and then Kealynn and Niccolo

both stated that a fraction is part of a shape and how many parts it is. My only comment here is

that Mrs. Wakley could have asked both students to give her examples of fractions. Mrs.

Wakley next question addressed the parts of a fraction- do you know about the numbers in the

fractions? Both students were able to respond with numerator for the top number and

denominator for the bottom number. Kealynn responded north and down for numerator and

denominator as her connection to the terminology. The thinking here is Mrs. Wakley should

have phrased the question differently. Instead of “do you know,” a better beginning would be

what is the top number and bottom number of a fraction called?

During Lesson-

Niccolo and Kealynn are given copies of a number line exercise and asked where ½ would be

placed? Niccolo responded with down the middle. He then used his marker to hop from zero to

½ and from ½ to one. Kealynn noticed the spaces on the number line having two parts. Mrs.

Wakley asked both students what would 1/3 look like? Kealynn drew a pie graph with three

parts and then shaded one part. She then stated that the number line for 1/3 is split into three

parts. Niccolo also agreed that the number line must be split into three parts for 1/3 and the

fraction would be close to zero.

25

Given ¾ as a fraction to locate, Kealynn split the line into four pieces and stated that ¾ is almost

a whole and 4/4=one whole. Niccolo was able to respond that ¾ meant that the number line had

four equal parts. Mrs. Wakley asked the students did they know any of these concepts before.

They replied “no” and that their knowledge was derived from the benchmark fractions lesson.

Mrs. Wakley displayed great teacher affect and she looked energized that they were making their

connections.

Kealynn who made the connection that 4/4=one whole asked about 5/4 on the number line, but

Mrs. Wakley explained to Kealynn that this would require a larger number line greater than one.

She then looked over at Kealynn and asked her “correct?” My thinking was maybe given a little

more think time, Kealynn would have made the connection that 5/4 is greater than one and

another number line is needed. Mrs. Wakley then gave Niccolo and Kealynn 4/12 and both

students were able to split the number line into 12 parts to locate 4/12. For the problem 4/10,

Niccolo and Kealynn began by marking the halfway point on the number line. Then they made

their tick marks and located 4/10.

Assessment-True/False

Niccolo and Kealynn were given problems like the one above. They were to identify if the

location of the fraction was true or false. Niccolo used a green marker and miscounted the tick

marks. He quickly was told to use the red marker to make new tick marks. This caused some

confusion; however, Niccolo was able to identify the location as true. Kealynn immediately

noticed that 3/6 equals ½ and she circled true on her paper. Mrs. Wakley went over strategies

used. Kealynn stated she only wanted to look at the top number and realized she could not. My

0 136

26

thoughts were that the students were engaged and Mrs. Wakley used their prior knowledge and

went over strategies to have a successful lesson. In fact, Niccolo was able to self-correct

whereby during the initial assessment, he could not.

27

Jeanne’s Assignment Reflection

I enjoyed this assignment as it allowed me to assess the student’s needs, plan a lesson, and then

teach that lesson. As a substitute teacher, I walk into the classroom and all the lessons are ready.

During this project, it felt different. I felt connected and the lesson was more meaningful. As I

was teaching the lesson, I became filled with this incredible energy. This energy came from

Niccolo and Kealynn making connections or having their “aha” moments. In fact, Kealynn

stated she is moving, and will take her Fractions on a number line to one reference sheet with

her. Niccolo, who could not identify at least two fractions, was now able to locate three

fractions. I even watched as Kealynn and Niccolo pointed toward me remembering the

benchmark fraction material. It was amazing that one hour can have that much impact. There

was some miscommunication between Kara and me in the beginning, which caused the

assignment to become stagnant. The collaboration needed was absent. However, in the end, the

collaboration was strong and I felt exhilarated with Kara’s lesson. My lesson ended with equal

parts of a number line from zero to one and Kara was able to hone in on this idea and create a

great lesson.

28

Kara’s Lesson Reflection

To preface my lesson reflection, I will say that I think it went very well. I followed my

lesson plan thoroughly. I began with humor and it loosened the children up and let them feel

more comfortable in the lesson. I began by making sure that both students knew what a fraction

was, what a numerator was, what a denominator was, and a number line. I made sure that I set

the expectation that the students should raise their hand and not popcorn out the answer and this

seemed to work well and continue through the lesson. The students knew all the fraction

terminology. This allowed me to continue the lesson without review on the verbiage.

I handed my students their graphic organizer that I made that had blank number lines

with zero on one end and one on the other. I also used one to model what I wanted the students

to learn to do. I asked both Niccolo and Kealynn what ½ looked like. They both drew circle

graphs split into 1/2s. I asked them to show me on the number line how this would look and they

both did a great job and split the number line in half. Then I asked the students to show me 1/3.

Both students went right to work with drawing a pie graph that was split into thirds. Niccolo did

have trouble with this but he was able to wrap his mind about splitting the circle into thirds.

Then I asked the students to show me this on a number line. Both students were able to do this

perfect and I showed them that I was also modeling this. The students split their number line into

thirds and then counted the spaces in between to double check. The next fraction I wanted to test

was ¾. I tested ¾, because this is a fraction that students often encounter. I asked the students to

first split the number line first and then show me where ¾ went. Kealynn was excited because

she knew exactly where it went without splitting her number line. I asked her to explain her

thinking and she told me that ¾ is a part away from a whole, which is 4/4. I was excited that

29

Kealynn remembered this concept. Niccolo was able to split his number line into fourths as well

and then “hopped” to the mark that is ¾. Niccolo was very engaged and showed enthusiasm;

this made me happy.

I then asked my students to pick a fraction that we can use to split the number line and

find its location. Kealynn picked 5/4. I explained to her that maybe that was not such a great

idea. I asked her why it would not be but she could not think of it right away. I explained to her

that maybe we would need a bigger number line and I asked her what 5/4 represented. She told

me that 4/4 was one whole so this was 4/4 plus a part, so 5/4. Niccolo then picked 4/12. I

laughed and told them that this will be a fun fraction to split on the number line. The students

were able to this correctly and fast! I was very excited because the higher a number, it becomes

more difficult to place. The students enjoyed this so much that they asked if we could turn the

paper over and do it one more time. I told them sure but we had to pick a fraction. Niccolo

picked 4/10 and both students were again able to do this correctly.

I told the students that we would do a quick assessment and Kealynn was immediately

discouraged. I told her that this was a fun and non-graded assessment and not to worry. I

handed the students their assessments and they immediately went to work. Both students first

circled their answers and then both students went back and corrected what they thought was

wrong. Niccolo had not split his number line at first but was then upset because he had made

incorrect marks in splitting his number line but I gave him a new marker and he was able to use

the new color to make his marks. After he split his number line, he got all of his answers correct.

Kealynn went over her answers as well. Both students immediately did not mark their number

lines and I noticed that when they did not mark them, they got their answers incorrect but when

they marked their number lines, both students got their answers correct. I went over the answers

30

with the students and asked each student how they figured out what their answers. I enjoyed this

lesson and would use it again. I would make sure to explain to students that even if there are

already ticks for the ½ or the fraction that they should make their own splitting of the number

line to ensure that they are not confused. I also thought it was great that when I asked my

students questions on their concepts they were able to point at Ms. Asberry and say that they

remember certain things about the number line from her lesson. The students have definitely

shown growth and improvement.

31

Kara’s Observation

Observation of Ms. Asberry-

Opening-

Ms. Asberry began her lesson by telling the students she was the owner of a pizza parlor!

The students were immediately intrigued. She wanted to make sure her students got the concept

of the fractions ½, 1/3, and ¼. I noticed that because she was allowing the students to popcorn

their answers, Kealynn, the more knowledgeable on fractions student, was answering first and

then the student who needed more work based on the interview, Niccolo, would just go along

with what Kealynn said. It was hard to tell if he actually knew that a ½ was one part of a whole

of the fractions. While Niccolo did struggle at one point, Ms. Asberry was able to go back and

work through the pie graph showing thirds with him. She used proper terminology,

denominator, and numerator to ensure that students were not saying “top number” or “bottom

number.” To make this portion a little more interactive, the students could have had their own

set of fraction circles but I think this portion went well. The students (at least Kealynn) were

able to answer the different questions about the fractions in the form of pizza and toppings. This

would be a fun activity to use a lesson for beginner fraction learners so I think it was a great way

to review.

During lesson-

The lesson began by naturally moving on. Ms. Asberry was sure to use proper

vocabulary and this allowed Kealynn to explain how she remembers denominator and numerator.

Ms. Asberry praised her for her thoughts using great teacher affect and this led to her asking the

students what the words “mean.” Ms. Asberry asked the students explicitly what the numerator

means and again, Kealynn answer. Ms. Asberry asked Niccolo “if he agreed with it or not.” She

32

then let Niccolo know that he should be giving his thoughts aloud like Kealynn. I found that,

again, Niccolo was following Kealynn’s lead. At this point, Ms. Asberry also realized this and

let Niccolo know that she wants to hear from him as well. At this point, Ms. Asberry allowed the

students to discuss what they both thought about the numerator and denominator. She made sure

that her students understood part and whole and went over the concept a few more times. Ms.

Asberry then handed the students a “Fractions reference sheet.” The sheet contained number

lines with different fractions in their correct order.

Ms. Asberry asked the students to look over the sheet and let her know they noticed

anything make some discoveries. I think this was a great idea because that way students did not

feel the need to look at intensely later. The students began to popcorn out numerous different

patterns that did have to do with fractions; Kealynn also used the term skip counting which I

found to be a great use of vocabulary on her part. Ms. Asberry focused on trying to get students

to understand that 1/2, 3/6, 4/8, 5/10, and 6/12 are all the same fraction essentially equaling ½

and go straight down the center. The students were encouraged to “go a little deeper” in their

thinking when they were not making the connection. I understand that Ms. Asberry wanted her

students to figure it out herself, but this may have been the wrong way to go about it. When she

does tell her students what the connection is, she states “all the denominators are equal, it is all

equal to one half.” I feel like this was misleading because Ms. Asberry meant that the fractions

are all equal, not the denominator. The students had a great “ah-hah” moment when they

realized that Ms. Asberry was correct. It was great to see that. When Ms. Asberry tried to

explain to students 2/3, ¾, 4/5, 5/6, 7/8, 9/10, and 11/12 and how it is one numerator away from

one. I feel like this was too broad to teach to the students. The students would grasp at any

pattern they could think of to try to give an answer. It was at this point that Ms. Asberry noticed

33

that Kealynn was acting as if the fractions were whole numbers, Ms. Asberry took a small

amount of time out to explain to Kealynn, and then Niccolo made a discovery. He wrote his

answer down to remember it, that on the one side of the number line the fractions are closer to

zero and on the other side the fractions are now closer to 1. Kealynn also realized this with the

help of Ms. Asberry. Overall, during this portion of the lesson I did noticed Kealynn was bored

and would often start trying to draw in her binder. A lot of the time, I felt like she did not have

enough interaction. This portion was great but would be better taught with more time.

Assessment-

This portion of the lesson was a summative assessment that was completing a graphic

organizer with numerous number lines that were 0-1 on them. Ms. Asberry handed the students

the assessment and then handed them some of the pie graphs we had used for the interview

assessment. This portion did not go as planned. The students were allowed to use their

reference sheet and it ended up being that the students were just looking at their reference sheet

and placing the fractions in the correct place. They were not actually using their knowledge

learned. When Ms. Asberry did ask them to put their reference sheet away, the students rather

bemoaned it.

Overall-

I think Ms. Asberry did a great job teaching her lesson. It is hard to try to get both

students to be engaged in the learning process when they are both different kinds of learners.

This is something we will continue to work on as students though. I did notice we ran out of

time and were not able to use the technology portion of the lesson. I also feel that if given a do-

over Ms. Asberry should and would use a different method of assessment. Ms. Asberry did a

34

great job of using teacher affect when appropriate and she tried to keep her students engaged in

the learning process.

Kara’s Assignment Reflection

I was very overwhelmed with the idea of coming up with a lesson to teach to students

based on an interview. After learning more about lesson planning and understanding more about

what I was going to teach, the number line and fractions, I felt more in tune with the assignment.

It was interesting to me that I was able to come up with a lesson plan based on meeting and

speaking with the students. I was able to conclude what each student needed to learn and the

best way to instruct him or her. Without knowing the difference between direct instruction and

indirect instruction, I would have been lost. Education on my part has really helped me with this

lesson.

Working with Kaelynn and Niccolo was enjoyable and a different experience than I had

ever had. I experienced such joy and excitement when the students would finally get an “ah-

hah” moment. The energy from the students when they finally got a concept was contagious. I

was excited for them and by them. I could not have this type of feeling for anything else. Math

may be a “hard” subject but I think that it depends on how it is taught. If students are properly

engaged and are taught math in an enjoyable way it will affect how the students learn. Math is

not so scary when the lessons are not scary.

35

References

Fractions Number Lines to 1. (n.d). Retrieved from www.math-salamanders.com

Number Lines. (n.d). Retrieved from www.math-aids.com