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/