Fraction Aliases Caper (Fraction Equivalence)
Grade level: Fourth grade general education (classroom of twenty-three students)
LESSON RATIONALE
New York State Learning Standards and Key Ideas
Mathematics, Science, and Technology Learning Standard 1 (Analysis, Inquiry, and Design): Students will use mathematical analysis, scientific inquiry, and engineering design, as appropriate, to pose questions, seek answers, and develop solutions.
Key idea 1: Abstraction and symbolic representation are used to communicate
mathematically.
Key idea 2: Deductive and inductive reasoning are used to reach mathematical conclusions.
Mathematics, Science, and Technology Learning Standard 3 (Mathematics): Students will understand mathematics and become mathematically confident by communicating and reasoning mathematically, by applying mathematics in real-world settings, and by solving problems through the integrated study of number systems, geometry, algebra, data analysis, probability, and trigonometry.
Key idea 1 (Mathematical Reasoning): Students use mathematical reasoning to
analyze mathematical situations, make conjectures, gather evidence, and construct an
argument.
Key idea 2 (Number and Numeration): Students use number sense and numeration to
develop an understanding of the multiple uses of numbers in the real world, the use of
numbers to communicate mathematically, and the use of numbers in the development of
mathematical ideas.
Key idea 4 (Modeling/Multiple Representation): Students use mathematical
modeling/multiple representation to provide a means of presenting, interpreting,
communicating, and connecting mathematical information and relationships.
Key idea 7 (Patterns/Functions): Students use patterns and functions to develop
mathematical power, appreciate the true beauty of mathematics, and construct
generalizations that describe patterns simply and efficiently.
English Language Arts Learning Standard 1 (Language for Information and Understanding): Students will read, write, listen, and speak for information and understanding.
Key idea 1 (Listening and Reading): Listening and reading to acquire information
and understanding involves collecting data, facts, and ideas; discovering relationships,
concepts, and generalizations; and using knowledge from oral, written, and electronic
sources.
Key idea 2 (Speaking and Writing): Speaking and writing to acquire and transmit
information requires asking probing and clarifying questions, interpreting information in
one’s own words, applying information from one content to another, and presenting the
information and interpretation clearly, concisely, and comprehensibly.
Instructional Objectives
Students will be able to use fraction bars to compare different fractions in order to see
whether the fractions are equivalent or not. (analysis)
Students will be able to use fraction bars to visualize addition of fractions by lying one
fraction bar on top of another and locate a third fraction bar that is equal to these
first two. (application)
Adaptations
Each activity will be no more than fifteen minutes in length, which will ideally be short
enough to hold the students’ attention.
Group work will allow for peer collaboration/tutoring. This allows the teacher to act as a facilitator, promoting this student-centered learning, and assisting all students when necessary.
Case Specific Disabilities
Attention Deficit-Hyperactive Disorder: Students who have a significant inability
to attend, excessive motor activity, and/or impulsivity. Students identified as ADHD will
have their own set of tactile, hands-on materials to work with. Thus they should remain
focused while other members of a group are involved in the experimentation. Students with
ADHD are still expected to participate in group activities. Small tasks/exercises may also
be implemented. Students will be allowed to move about the classroom during the
lesson if they deem necessary as long as they are still engaged in the lesson and are not
disturbing other students. These students will also be considered for the position
of teacher’s assistant, along with students exhibiting behavioral disorders.
Visual Impairments: Condition in which an individual has an inability or limited ability to receive information visually, so much so that it interferes with learning. Students with visual impairments will be placed close to the front of the classroom/board. Written forms of notes will also be available to students to copy following the lesson if they are needed.
Behavior/Emotional Disorders: Condition in which an individual displays serious and persistent age-inappropriate behaviors that result in social conflict, personal unhappiness, and school failure. Any student identified with a behavior disorder will be considered for the daily position as the teacher's helper. These students will be placed in groupings that do not excite them. The student will be verbally praised for his/her attentiveness and appropriate behavior that includes, but is not limited to, staying on task. A main goal is to keep the students involved in the class activities and make sure they feel needed/wanted in the classroom.
Materials
Students will need to use their fraction bars that were constructed at the beginning of
the unit.
The provided worksheet and pencil will also be needed to record one’s findings.
Students will need their math journals to write in.
LESSON OPENING
Anticipatory Set
"Today we are going to get involved in a rather mysterious investigation. Right in
front of you, some fractions are disguising themselves as other fractions! Although the
numbers in these fractions look different, they are really the same. You are going to work
in groups with each of you using your own set of fraction bars. Together you are going to
find the different-looking fractions that are, in all actuality, the same. I challenge you
to find the pattern, so that their aliases (define if necessary) do not fool you
anymore."
LESSON BODY
Assemble heterogeneous groupings based on the class list, but keep aware of students who need adaptations, most notably behavioral issues. (Do not place two behavioral children within the same group.) Remind students of appropriate group work behaviors, which includes allowing everyone’s opinion to be recognized. Once groups are settled in what they consider a "comfortable learning environment" (desk cluster, floor, window area, etc.), have one student pass out the worksheet.
Go through one example with the class to clarify any questions that the students may have. Specific example: I have one orange bar that represents the fraction 4/12. I will show the students how a bar of a different color, yellow, is shaded as 1/3, and yet is equal to the first fraction. (modeling) Inquire: "Why are these the same?" Review the directions on the provided worksheet. (guided practice) Then proceed to start them on their own investigations, reminding them to record their findings on their paper, for they will be writing about their investigation in their journals following the lesson.
Allow students to constructively play with the fraction bars. As the teacher moves from group to group, he should check for understanding to see if the students are matching up the appropriately shaded areas that correspond to one another. For example, if a child has written that 2/3 is equivalent to 8/12, he understands the concept. Inquire "How did you come to this conclusion?" "Why are the two equal to one another?" However, if a child has written that 1/3 equals 1/4 based merely on the fact that one area is shaded on each bar, the student has yet to quite grasp the concept. The teacher can figure out how the child arrived at this answer by asking the student his/her thought process. Include questions such as "Why do you think these two are equal?" or "Can you tell me what does not make these two fraction bars (say 1/4 and 4/12, for example) equal?" (guided practice) Encourage struggling students to keep trying, and asking their peers for assistance.
Stop the class momentarily. Inquire via "What have you discovered thus far about fraction equivalence?" and "Is there any pattern that you have noticed?" to see what the whole group thinks. Record these on the blackboard or on a chart. Discuss these results as a large group. Then draw a square and a rectangle on the blackboard. Shade 1/3 of the rectangle and 1/4 of the square. Ask students to recopy the diagrams in their journal, and respond to "Do these two pictures have shaded parts that are equal to one another? Why or why not?" Then ask them to "fix" one of the two drawings so that they are both equal. They must record what fractional part is now shaded in both drawings. Allow approximately 5 minutes for students to write their response. (check for understanding)
"Now I have a greater challenge for you. I would now like you to take two bars, each of a different color. I would like you to place one of the shaded areas on one of the bars up to the shaded area on the other bar, like this." Demonstrate activity with following example: yellow 1/3, combined with a red 3/6, which equals a red 5/6. (modeling) "I would now like you to find one bar that matches these two put together. You might not be able to find one bar to fit all combined fractions, or you may be able to find many, but let's try. Maybe you can think about what the bar would look like. The findings from this activity will be listed on the back of your recording sheet under ‘Combinations.’ Let's see how many combinations you kids can come up with!"
Walk around the room checking the students' progress. You might notice that some are having trouble. If so, guide them with some more group-individualized examples. Start with a simple problem, such as comparing the orange 6/12 to a blue 2/4 as an equal answer (to 3/12 + 1/4 for example). (guided practice) If they seem to be on the correct track, allow them to continue as they are. Check to see if they are finding multiple answers to what one bar equals the other two combined. If not, push them in this direction. For any accelerated group that appears to have mastered this, ask them what they think this relates to, and have them record it in their journal.
If they have not done so, also have the students record their thoughts on whether all combinations are possible with the given materials (i.e. Is there one bar which will represent all combinations of two bars? The answer is no, since there is no one bar in this set that represents the combination of 6/12 and 2/3.). Then have them write a three to five sentence response to "What have you learned today?" and "What do you think about using fraction bars to help you understand and manipulate fractions?" in their journals. (check for understanding)
Have the students interact with the teacher in a discussion about what is meant by equivalent fractions. Inquire "Do you believe there is an infinite amount of equivalent fractions?" Talk with students about the combinations of fractions, as they just finished with the physical fraction bars. (Although they may not have realized it, they have just had their first experience with fraction addition, however informal is may have been.)
CLOSURE
"Do you remember how some words, such as new and knew, sound alike, but are spelled differently? Today we discovered that just because fractions might be written differently, they too can mean the same thing. That is what is meant by fraction equivalence. What is another way to call/label fractions that look the same? (equal fractions) All of you did a great job finding the fractions that were trying to fool you! Be sure to visit the math center for some extra work on fraction equivalence. There are some problems over there that will challenge you even more! You may not realize it, but you didn't only learn about fraction equivalence today, for you even started to work on adding fractions together in the last activity. If you did not quite grasp it all today, don't worry, for this was just the beginning. We will be continuing with this idea and the fraction bars tomorrow. Now I would like you to place your math journals and the worksheets in the upper right-hand corner of your desk so they can be collected." (Have a student helper collect the materials.)
LESSON FOLLOW-UP
Independent Practice
Continuing with the "real-world" aspect of fractions covered on the initial day
of the unit, my independent learning activity will incorporate real-world materials into
the learning. My plan is to take pictures, preferably from teacher manuals/idea books (or
handmade reproductions), and color particular sections of the object. For instance, there
would be a flower with ten petals, five of which are one color. The students would have to
pair this up with another object that is divided differently but is still fractionally
equivalent to the first. In this example, there might be a square divided into two halves,
one of which is colored. Hence the flower and the square both have the same fractional
part, 1/2. This will also show that geometric shapes are not the only things that can be
divided into portions.
Not only pictures will be available for investigation. Physical objects, such as dice and cards, might also be used. This might stimulate thought since the students have to figure out what "might be the case" given the objects. This allows for no one real right answer. For example, what is the chance one might get an even number on the die? Out of the selected cards, how many are black? Red?
Additionally, students will also be given "higher order thinking" to stimulate their previous knowledge and apply fractions to things which are not always thought of as fractional, even though we have covered it in the opening lesson. This can be illustrated using money, for example, a dime is equal to 1/10, a quarter, 1/4, and so forth.
All of the student observations/solutions will be written on a separate answer sheet. They must also write one sentence, which is listed in the directions, which explains why they believe these two objects match. Since there will be many possibilities, there is no specific right or wrong way to pair up the objects correctly by equivalence; only wrong answers will be marked incorrect unless there is some degree of vagueness.
This activity will also allow the teacher to check how the students have applied the concept covered in the class to other objects.
Students will be able to check their work with the teacher, with other students, and a key with sample ideas.
Evaluation
What did you want the children to learn?
My goal was to have the children realize that some fractions, regardless of their looks,
are actually the same (i.e. 1/2 is equal to 4/8, 3/6, etc.). The last activity in the
lesson uses these "aliases" to begin building the idea of fraction addition.
This may not be mastered however, for it is the focus of an upcoming lesson. The children
may also note that there are limitations to the usefulness of some manipulatives,
dependant upon their availability (i.e. they do not have large enough bars which allow for
all combinations to be matched up in their set). This, however, may not be expressed as
clearly as this; I was looking for a general idea that touches upon any component of this
idea.
How will you know that they learned it?
For one, I will be able to determine whether the students grasp the idea of fractional
equivalence by observing them in their groups, manipulating the fraction bars. Observing
any group work/peer "tutoring" will allow me to key in on who seems to have
ingrained the concepts at hand. I will also be able to determine their depth of
understanding by reading the journal entries that address the day’s procedures,
activities, and their thought process/realizations. The records from their investigation
using the fraction bars will also allow me to analyze how well they understood the
concept.
LESSON RESOURCES
References for student use:
Instructor and any other teachers/paraprofessionals/assistants.
Other students.
Fraction bars.
Mathematics center and corresponding diagrams/illustrations/etc.
Textbook.
References for teacher use:
Burns, M. (1998). About teaching
mathematics: A K-8 resource. Sausalito, CA: Math Solutions.
Cooper, J. D. (2000). Literacy:
Helping children construct meaning (4th ed.). Boston, MA: Houghton Mifflin.
Eggen, P., & Kauchak, D.
(1999). Educational psychology: Windows on classrooms (4th ed.). Upper
Saddle River,
NJ: Merrill/Prentice Hall.
Friend, M., & Bursuck, W.
D. (1999). Including students with special needs (3rd ed.). Boston, MA:
Allyn & Bacon.
Mercer, C. D., & Mercer,
A. R. (1998). Teaching students with learning problems (5th ed.). Upper
Saddle River,
NJ: Merrill/Prentice Hall.
Vacca, R. T., & Vacca, J.
A. L. (1999). Content area reading: Literacy and learning across the curriculum (6th
ed.). New York: Longman.