Summary
Students will learn about fractions.
Materials
- Region, set, and line
model manipulatives
- Jelly beans (12 beans
per pair; use no more
than 3 or 4 colors of
beans per cup)
- Small paper cup for
each pair
- Activity Record Sheet
- Colored pencils
- Journals
- Art paper
- Jelly beans
- Cup
Background for Teachers
Fractional numbers can be demonstrated with three types of models:
the area model, length model, and set model. Creating physical models
generates a concrete representation, and in turn, establishes a long-lasting
nonlinguistic image of this knowledge in the student's mind. In this
activity, students use a variety of different methods to record their
fractional set. Students should learn to equate the numerator of a fraction
with the "count"--number of items in the whole set--and the
denominator with the "size"--as in how big is this portion. In other
words, a fraction is the count/size. So a fraction such as four-fifths means
we have four of something and each one of the pieces or sections is sized
at one-fifth of the unit whole. Throughout this activity, student pairs record
and summarize their information on a structured activity sheet.
This graphic organizer allows students to manipulate new ideas and see
how these ideas are related to concepts they already know. The brain has
a natural capacity to organize, and the graphic organizer allows us to
teach to that ability.
Intended Learning Outcomes
2. Become mathematical problem solvers.
3. Reason mathematically.
4. Communicate mathematically.
Instructional Procedures
Invitation to Learn
Provide a variety of materials for groups to sort into three groups--
region, set, and line models (e.g., ruler, length of yarn, piece of material,
sheet of paper, strip of paper, geoboard, base ten rods, golf tees, erasers,
2-colored chips, centimeter cubes, etc.).
Instructional Procedures
- Demonstrate how each
pair will draw the specified number of
beans from the cup and record the selections on the Activity
Record Sheet by coloring in their beans. Then color the graph to
show the number and color of each bean in the group.
- Distribute the cups,
Activity Record Sheets, and colored pencils
to each group.
- Advise the students that in the last round, they will be
using all
12 jelly beans (so don't eat any).
- The students can either return the beans
to the cup after each
round, or simply draw the necessary number of new beans to
complete the next round.
Example: When moving on to round four, the students could
draw just one bean and add it to the beans used in round three
to equal 5 total beans, or they could return all the beans to the
cup and draw out 5 new beans to record.
- After the beans have been colored
and the circle graph has been
constructed, the students should summarize the data by writing
the fractional portion each color represented of the entire group of
beans (the unit whole).
Example: If they drew 2 white and 1 yellow in round two, the
group would write 2/3 to represent the white beans and 1/3 to
represent the yellow bean, with 3 being the unit whole.
- In round six, the
students need to group their beans by color and
then color them on the strip. The strip will work as a template to
construct the circle graph. After they have completed the
coloring, they can cut it out and curl one end around the
circumference of the circle. Use this as a guide to mark the
sectors on the circle graph.
- Students should compare and generalize the
data in their math
journals.
- Discuss why 1 red bean in round one equals 1/2, but after
drawing 1 red bean in round three, its value drops down to
equal only 1/4.
- What was the relationship between the colors most often
drawn to the color mix in their cup?
Part 2
- Partner A puts a selected number of beans
in the cup, without
letting partner B know how many beans are in the cup.
- Partner A then draws a few of the beans out and gives them to
partner B along with a clue and a challenge. Partner A gives the
clue of telling partner B what fraction of the beans are now out on
the desk.
- Partner B is then challenged to calculate how many total beans
are in the cup.
- Partner B must prove their answer is correct by illustrating
the
whole set, showing the fractional part of the set which was drawn
out of the cup.
- Partners change roles and continue.
Extensions
- Small groups draw quick stick
figure drawings of their whole
group, calling attention to specific attributes that some members
share. Attributes should be observable, or verifiable by
conversation, such as wearing watches or owning a dog.
- As a group, write
four observational sentences about the group,
using fractions to express the findings.
Example: 3/4 of the students in
our group are wearing shoes with
shoelaces.
- Discuss how the complementary, or shadow, fraction (1/4 of the
students are not wearing shoes with shoelaces) relates to the unit
whole.
- Create equations by listing a portion of the students with an
observable trait that is equal to 1/2 of the students with that trait.
How many total students share that attribute?
Example: These three students have braces, that is 1/2 of our
group that has braces. How many total students have braces? Continue by changing
the fractional part (1/4 of our group,
1/6 of our group, etc.).
Family Connections
- Students can discuss inherited
traits with family members and
then recreate a group picture using exaggerated attributes as
outlined in the extension. Be sensitive to students who may not
be living with birth families and instead suggest that they find
other common attributes (3/4 of our family members made their bed this morning).
- Using
items found in the home, draw fractional parts of sets used
to prepare the family meal and write a sentence describing the
fractional part and a question/answer about the unit whole.
Example: A student would draw three forks and a sentence such
as "This is 1/3 of the forks used to set the table for dinner.
How many forks did we use all together?" (9)
Assessment Plan
- Provide students with a collection of related items
that includes at
least one variable (color, length, etc.). Items could include cm
cubes and base ten rods, tangram pieces, several different crayons
in two or three colors, etc. Have students sort the items, write
statements about the similarities and differences using fractions in
their statements, and represent the group with fractions and a set
illustration.
Sample answer: There are 5 cm cubes and 3 base ten rods, so 3/8
of the set are base ten rods and 5/8 of the set are cm cubes
- Show students
an area model that is divided into three sections.
Two of the parts are colored, one is not. Have students write an
explanation on how to find the fraction that describes the shaded
part of the area model. Explain that you don't want to know the
fraction, but how they would decide what the fraction is.
Sample answer: First you count how many total parts there are.
That is the denominator. Then count how many parts are
different (shaded). That is the numerator. Write the fraction
by putting the numerator on top and the denominator on the
bottom.
- Using the school population, have the students write three to five
questions that could be used to highlight certain portions of the
population. They could survey and find which students fit their
highlighted category, and represent that information in fractional
form and illustrated form.
Sample question: How many fifth graders are left handed?
Bibliography
Research Basis
Barton, M., Heidema, C., (2002) Teaching
Reading in Mathematics. Aurora, CO. McREL
This supplement to Teaching Reading in the Content Areas explains
the terminology of "reading mathematics" and the skills needed to
comprehend the words, symbols and text structures associated with
mathematics. The manual also presents suggestions and strategies to help
students become more proficient in mathematics literacy.
Marzano, R., Pickering, D., Pollock, J., (2001) Classroom
Instruction that Works, Alexandria,
VA. ASCD.
This K-12 guide provides extensive research evidence, statistical data,
and case studies that support nine critical teaching strategies, one of
which is nonlinguistic representations. The chapter on nonlinguistic
representations stresses the use of a variety of activities and elaboration
on acquired knowledge.