Introduction

Comparing Fractions

The activity aims at assisting students to develop flexibility with a number of

strategies. It sets expectations for various procedures of ordering fractions. Although

determining a common denominator of comparing fractions is a good strategy, it is inefficient

for numerous pairs of fractions (Ebby, Sirinides, Supovitz, & Oettinger, 2013). For instance,

in relation to the assignment provided, descriptions below show a way of comparing fractions

by an analysis of their size. The learner may share extra strategies. The first one involves

comparing to one-half. One-half is less than five-eighths, and one-half is greater than two-

fifths; thus, one-half is smaller. The second one revolves around common numerators; fifths

are smaller than fourths, so three-fifths is less than three-fourths. The third one entails

comparing to one. Both the provided fractions are one fraction less than one. One-sixth is

greater than one-eighth, so seven eighths lack a lesser part and is closer to 1. The fourth

activity is comparing same numerators. Two-sixths is equal to one-third (Ebby et al., 2013).

Sixths are lesser than fifths, and, as such, two-sixths are less than two-fifths.

Although it is usually overused, determining common denominators is a helpful

approach for some pairs of fractions. Learners, who make an analysis of the problem, give

and use a suitable and well-organized strategy for contrasting portions are likely to have a

better knowledge of fractions. Learners, who determine common denominators when other

effective strategies are sensible, may not understand fractions (Ebby et al., 2013).

Determining a common denominator is vital during additions and subtraction of a number of

pairs of fractions.

Problems on Sameness and Ordering of Numbers

LESSON/UNIT PLAN AND RATIONALE 3

The activity of solving provided problems is aimed at deepening the learners’

knowledge of fraction equivalency. It also acts as a demonstration to participants on the

diversity of problems that they might encounter while gaining knowledge of fraction

equivalency and order (Iowa Core Mathematics, n. d). The activity helps the teacher

completely appreciate the depth of knowledge required to solve problems and prepare

questions to assist learners if they experience difficulty.

Assessment

Criteria

A B C D

Analysis of

Questions

asked

(Iowa Core

Mathematics

, n. d)

Crucial

application of a

wide range of

relevant

strategies in

answering the

questions

provided. The

learner

demonstrates

great

understanding

and application

of concepts

learned in the

classroom.

Great level of

understanding

strategies taught

in the classroom

and good outlook

of the work.

Sound

understanding of

strategies taught

in the classroom.

Some proof of

understanding

and application

of strategies

learnt in class.

LESSON/UNIT PLAN AND RATIONALE 4

Question 4

The table below gives sample teacher responses and the scoring rationale from the

students’ grades.

Score Teacher Response Explanation

4

(Ebby et al., 2013).

Learners require an

understanding of the meaning

of fractions. As earlier

taught, a fraction is a portion

of a whole number. On the

contrary, a whole number can

be a collection of things or a

single thing. According to

question number 4, 30

mangoes is a whole number.

Similarly, 12/12 represents a

whole number. Learners also

need to recall that when

adding fractions, one does

not add the denominator.

Students need to either find

1/3 and 2/4 of 24 or

determine a common

denominator.

This learning trajectory

response references

numerous fundamental

concepts.

3 (Ebby et al., 2013). The learners need to

understand that 2/4 is the

This conceptual response

centres on fundamental

LESSON/UNIT PLAN AND RATIONALE 5

same as 1/2. They also

require knowing that thirds

are bigger than fourths.

concepts and understanding

2C

(Ebby et al., 2013).

Learners understand the

meaning of fractions, as

portions of a whole, and they

know how to add.

This reply mentions the

part/whole idea but only in a

broad way

2P

(Ebby et al., 2013).

The students have to reduce

fractions and have the ability

to determine common

denominators and add

fractions.

This answer is procedural

since it centres only on a

precise procedure.

1

(Ebby et al., 2013).

The learner possesses the

capability to count up to 20.

The student can add.

This broad response applies

general terms and only

indicates sub-skills.

Examination of a Learner’s thinking: Sample responses, scores, and rationale are

assuming John, a student, answered that 2/4 is equal to ½. Students also argue that since 2/3

and ½ do not make a whole number, Mary and Paul do not fill the box. The student also drew

a circle to show that 2/3 and ½ do not make a whole number.

Considering John's solution process in relation to what the work proposes about his

knowledge of numbers and operations, a response, scores, and rationale, approach can be

represented as shown below:

Score Teacher’s Response Explanation

4

(Ebby et al., 2013).

John has an

understanding that the

This learning method

answer references

LESSON/UNIT PLAN AND RATIONALE 6

denominator

determines the size of

fractions. He also

knows that portions

involve breaking the

total into equal parts.

He also has knowledge

of equivalent fractions.

As a result, he can

contrast the two

fractions, and,

eventually, compare his

solutions to one whole.

Highly developed

knowledge trajectory

orientation.

numerous fundamental

concepts in the

learner’s work.

3

(Ebby et al., 2013).

John shows that he has

understanding of the

ideas of the fractional

part of a whole

number.

A proof of learning

trajectory course.

This conceptual answer

centres on fundamental

concepts and

understanding.

2C

(Ebby et al., 2013).

He has the

understanding on how

Although unarticulated,

this answer references

LESSON/UNIT PLAN AND RATIONALE 7

fractions compose a

whole part.

Ability to recognize

accurate and inaccurate

reasoning.

theoretical

understanding, but it is

not.

2P

(Ebby et al., 2013).

He drew two pies and

figured out that the two

diverse fractions did

not make a whole

together.

Lack of prominence on

learner’s analysis or

prioritizing.

The answer is

procedural since it

shows what the learner

did. This general

response does not

provide any specific

evidence.

1

(Ebby et al., 2013).

John possesses a basic

knowledge of fractions.

Use of explicit method

instead of

understanding of

concepts or procedures.

This general answer

fails to provide any

precise evidence.

Additionally, the conceptual category can be divided into broad and articulated

answers. If the reference to understanding concepts remains at the general level, the answer is

given the full conceptual code CP. If the answer articulates the theoretical understanding of

the student, it is assigned the articulated conceptual code CP1 (Ebby et al., 2013). For one to

LESSON/UNIT PLAN AND RATIONALE 8

obtain a rubric score of 3, the answer should have a minimum of one articulated conceptual

code in order.

Kindergarten: Unit: Lesson/Unit Plan and Rationale

Lesson Plan

Teacher

Student

Level of Class

Date

Unit/Subject:

Instructional Plan

Title

Teacher X

Grades 6-8

May 11, 2015

Maths

Fractions

1. Planning

Lesson

Summary and

Focus:

This session involves introducing learners to the application of visual and

practical practice in working out problems related to fractions. Through the use

of the provided coloured nuts, every student will work both as an individual, as

well as a group. The learners will apply varied techniques while working with

segments, and exercise their awareness, in addition to, subtraction,

multiplication, and division (Spear, n. d). They require a previous information

of the elementary increase charts. Common multiples are vital in this

activity. They will also create models to symbolize a fraction. By the time the

lesson ends, the learner should be capable of carrying out calculations that

entail fractions (Spear, n. d). The use of coloured nuts will occur throughout the

learning process. The learner will examine numbers between 1 and 100.

Classroom There exist three key factors in the learning environment, which create learner

LESSON/UNIT PLAN AND RATIONALE 9

and student

factors:

engagement: the content, the teacher, and the student. The student will

comprehend and learn (Student Engagement Trust 2015). The teacher will

instruct students. The content embodies the knowledge learners and the teacher

will learn.

National /

State

Learning

Standards:

Grades 6-8 j

Precise learning goal(s) /

purposes:

Learners will gain the ability to

carry out arithmetic calculations

on fractions.

Teaching notes:

A fraction is a number of the form x/y where x and y are

whole numbers, and y cannot be zero.

Agenda:

The teacher will direct learners in

reviewing whole numbers (Davis,

n. d). The teacher will also

request a volunteer to give

examples of whole numbers. The

use of coloured nuts will occur

right away.

Formative evaluation:

The teacher will jog the students’ memory on multiples of

numbers and the use of basic multiplication tables.

Academic

Language:

Key vocabulary:

Fractions

Function:

Students will demonstrate their

skills in using the provided

resources to perform calculations

Form:

The teacher will

provide questions,

which learners will

LESSON/UNIT PLAN AND RATIONALE 10

on fractions. answer.

Instructional

Material and

Technology:

Coloured nuts (quantity given according to the sizes of group)

Baggies (sufficient for every student)

Calculators

Notepad Paper

Pencils

Grouping: Grades 6

II. TEACHING INSTRUCTIONS

A. The Beginning

Prior

knowledge

connection:

The lesson is a continuation of students’ knowledge of basic multiplication

tables. Thus, learners will require having previous understanding of the

multiplication tables. They also require skills on multiples of numbers (Davis, n.

d). However, an introduction of the different concept related to the various ways

of working with fractions, especially entailing addition, subtraction,

multiplication, and division will occur.

Anticipatory

set:

The session is of great use to learners since it is the foundation of calculations of

mathematics in the learning environment as well as the outside environment

(Davis, n. d). Furthermore, there are a different field of applications of the skills

LESSON/UNIT PLAN AND RATIONALE 11

gained in diverse life aspects.

B. Teaching and Learning Activities:

The Teacher

Does

The Students Do Method of Differentiation

1. The teacher

will introduce

the lesson

through

asking the

learners the

number of

fractions they

can create

from their

bags by

assembling

nuts in

relation to

their colours

(Davis, n. d).

The teacher

ought to

explain that

the number of

coloured nuts

1. A small number of students

should assist in distributing

learning materials to other

students.

2. The students will organize a

group of nuts preferably of

different colours.

3. Furthermore, students together

with the teacher will form

different fractions from the nuts

provided.

4. Students will also carry out

calculations with fractions as

the teacher guides them.

5. The different groups will

compare their results.

6. Students determine the greatest

fractions and verify it using

1. 1. The teacher will differentiate the activities

of learners that have learning disabilities and

those who are disabled emotionally, and give

them a smaller number of coloured nuts to

count. Secondly, the teacher will read the

texts that pose a challenge to the students

aloud and give additional exercises to these

students (Davis, n. d). They will also guide

students individually and in small groups.

2. 2. The teacher will:

3. Be very keen on the behaviourally challenged

learners’ activities and make corrections

where needed.

Give students opportunities to ask and

answer questions.

Encourage group work and give more work.

Provide with more cubes and apples

4. For those who finish early, the teacher will:

Give them more assignment to carry out and

more challenging questions like a greater

number of coloured nuts.

LESSON/UNIT PLAN AND RATIONALE 12

in each bag

forms the

denominator.

In addition,

the learner

needs to

understand

that the

diverse

individual

colours will

form the

numerator.

2. 2. The teacher

requests

learners to

note down

their diverse

fractions on

their sheet of

paper

provided

(Davis, n. d).

3. 3. The teacher

will instruct

their calculators.

7. Students outline the importance

of skills learned.

Request the students to volunteer in

enlightening the other students on the

concepts they have learned.

Provide them with relevant tasks and

encourage them to do part of it while in

school.

Instruct them to guide the other members of

the class to do the assignments given.

LESSON/UNIT PLAN AND RATIONALE 13

each learner to

carry out the

division,

multiplication,

and addition

problems with

fractions

learners have

recorded on

the sheet of

papers.

4. 4. The teacher

guides

learners in

contrasting

their answers

from problems

solved.

5. 5. The teacher

guides each

group in

assembling a

list of all the

diverse

fractions

LESSON/UNIT PLAN AND RATIONALE 14

obtained from

every

individual and

collective.

The teacher

will divide the

group,

allowing some

learners to

solve fractions

with

multiplication,

others with

division, and

yet others

with addition

and

subtraction

(Davis, n. d).

6. 6. The teacher

will allow

learners to

contrast

fractions and

order all

LESSON/UNIT PLAN AND RATIONALE 15

responses to

group

problems in

from smallest

to largest.

7. 7. Contrast the

diverse

answers with

every group.

The teacher

will ask the

learners to

determine the

most popular

colour from

their replies.

They will also

evaluate the

problem that

gave the

biggest

fraction for an

answer. The

teacher will

enquire how

LESSON/UNIT PLAN AND RATIONALE 16

one can

determine the

largest

fraction.

8. 8. The teacher

instructs

students to use

calculators to

verify their

answers

(Davis, n. d).

9. The teacher

and students

discuss the

importance of

learning

fractions and

their

applications in

life.

III. ASSESSMENT

Cumulative

Assessment:

The students will do computations

involving fractions again to assess their

level of understanding of the concepts

taught (Davis, n. d). The teacher will also

The teacher will differentiate the

students in relation to their

capabilities and understanding of

mathematics. He/she will also lend

LESSON/UNIT PLAN AND RATIONALE 17

give learners relevant tasks related to the

lesson taught; particularly one that

requires them to perform individually.

a hand to the motivationally

challenged students through the

provision of more attention. The

teacher will also give learners who

are motivationally weak more

opportunities to enquire on the

concepts they did not understand

and answer their questions.

Conclusion: The teacher advices the learners on the

ways to tackle questions on lessons

taught, and the concepts that some

students might have failed to understand.

The teacher will also request students to

outline the way they will apply the

knowledge they acquired in class.

Assignments: The teacher will give the students

relevant tasks to perform during and

before the next mathematics lesson. The

teacher will also use the drill and practice

assignments, and learners will provide

answers to the question in their books

(Davis, n. d). The teacher will also

provide assistance to students, who failed

to understand.

LESSON/UNIT PLAN AND RATIONALE 18

How the Lesson Reflects each of the 12 Process

1. Observation. Learners will identify colours of nuts provided.

2. Inferring. The teacher will introduce the lesson by asking students the number of

fractions they can create from their bags by assembling nuts in relation to their colours.

3. Measuring. Students will determine the number of the varied colours they have.

4. Communicating. Students will provide answers to questions contained in the sheet of

papers given.

5. Classifying. Students will order coloured nuts according to their colours.

6. Predicting. Students will envisage the problem that will give the biggest fraction before

carrying out actual calculations.

7. Controlling variables. Learners will only deal with a maximum of a hundred nuts.

8. Defining operationally. Students will determine the greatest number of colours selected

through counting.

9. Formulating hypothesis. Students will evaluate the problem that gave the biggest fraction

for an answer. They will then use a calculator for verification.

10. Interpreting data. Students will use the experimental data obtained from visual aids to

determine other strategies applied in solving problems provided.

11. Carrying out an experiment. Learners will use coloured nuts to carry out calculations

related to fractions. Observations made will be used to highlight other strategies used in

calculations involving fractions.

12. Formulating models. Learners will create a pie chart and related shapes to determine

relationships (equivalence) of fractions.

LESSON/UNIT PLAN AND RATIONALE 19

References

Deane, D. (n. d). K-12 Teaching and Learning from the UNC School of Education: Fruit

Loops with Fractions. Retrieved from

< http://www.learnnc.org/lp/pages/3612?ref=search >

LESSON/UNIT PLAN AND RATIONALE 20

Ebby C. B., Sirinides P., Supovitz, J., Oettinger A. (2013). Consortium for Policy Research

in Education: Teacher Analysis of Student Knowledge (TASK): A Measure of

Learning Trajectory-Oriented Formative Assessment.

Retrieved from

< http://www.cpre.org/sites/default/files/workingpapers/1534_tasktechreport.pdf >

Iowa Core Mathematics (n. d). Professional Development Module: Teaching Fractions in

Grades 3-6. Iowa Department of Education.

Retrieved from

< https://www.educateiowa.gov/sites/files/ed/documents/8011z%20Session%206_Inst

ructor%20Notes.pdf >

Spear, C. (n. d) Beacon Lesson Plan Library:A Model Project. Polk County Schools.

Retrieved From:

< http://www.beaconlearningcenter.com/Lessons/4608.htm >

Student Engagement Trust. (2015). The Classroom Interaction Model of Student

Engagement.

Retrieved from

< http://www.studentengagementtrust.org/engagementModel/