Students will work in small groups to analyze, record, and demonstrate the concepts of fractions.
Main Curriculum Tie:
Mathematics Grade 4
Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers. 3.
Understand a fraction a/b with a > 1 as a sum of fractions 1/b
For each group:
For each student:
- 3” x 5” index card
- Colored pencils
- Colored pencils/markers
- Construction paper
pattern block pieces or
pattern block stickers
- Large chart paper/student journals
For each pair:
- Tub of several pattern
- 5” x 8” index card
- National Library of Virtual Manipulatives
Many parts of this lesson could also be adapted for use with technology by assessing online pattern block math manipulatives. On the home page, access the pattern blocks by typing in “pattern blocks” in the search window and then clicking on any of the options listed.
Background For Teachers:
One of the foundational steps in working with fractions is to
understand the concept of the unit whole. If the unit whole is divided
into four equal sized pieces, then each piece is 1/4 of the unit whole. A
common misunderstanding that can occur is that the fractional piece
always remains the same, when in fact, there is a direct relationship and
if the unit whole changes, the fractional piece changes as well. In order
for students to gain a working knowledge of this abstract concept, they
must first begin with concrete lessons, and bridge to the abstract with
pictorial representations. This is especially true for English Language
Learners. For some students, it is helpful to think of fractions as equal
portions, or fair share/fair trade. After students gain a mastery of the concept
of the unit whole and its component parts, they will then be
ready to tackle addition and subtraction operations with fractions.
Components of the unit whole objective include an understanding of
equivalency as well as an understanding of key vocabulary terms,
including mixed number, improper fraction, and proper fraction. Just like
a foreign language, for students to master the language of mathematics,
they must be given numerous opportunities to practice using it in context.
Students need to develop an appreciation of the need for precise
definitions and for the communicative power of conventional
mathematical terms by first communicating in their own words.
Allowing students to grapple with their ideas and develop their own
informal means of expressing the information can be an effective way to
foster engagement and ownership. In this light, all students are MLL
(math language learners) and should be combined together in pairs or
small groups to analyze, record, and demonstrate the concepts of
fractions. As they work together, they will be forced to use the
Intended Learning Outcomes:
1. Demonstrate a positive learning attitude toward mathematics.
4. Communicate mathematically.
6. Represent mathematical situations.
Invitation to Learn
Provide each small group a copy of the yin
yang symbol. This example has 1/2 of the circle
in black and 1/2 of the circle in white. Ask each
student to take a 3” x 5” card and have them
select a colored pencil. They are to color exactly
1/2 of their card, but the challenge is to do it in
an interesting and creative way. At the
conclusion of the activity, display all the cards
and draw attention to the many different ways the objective was
- Have each pair of students
select one larger pattern block piece
(yellow or red) and several smaller pieces that are all the same
color and that will cover the larger pattern block piece.
- Continue selecting
and covering pieces until a number of
relationships between pieces is discovered.
- Using the markers and large
chart paper, have each pair record
these relationships and write the fraction that corresponds with
it. It is suggested that you demonstrate with the first
relationship using the overhead pattern blocks and provide a
model for the students to follow.
Example: How many green equilateral triangles are in one
The students can then go on and discover how many triangles
are in the red trapezoid, how many blue parallelograms are in the
yellow hexagon, etc. You may even have a student discover that they can put
two green equilateral triangles in a blue. As new
relationships are found, stress the equivalency by noting and writing comments
such as “3 blue parallelograms are in 1 yellow
hexagon, so 3 parallelograms equal 1 yellow hexagon. 3/3 =1.”
- Have students
express the mathematical relationships in their
journals using colored pencils.
- Assign the yellow hexagon the value of
1. This makes it the “unit
whole.” Express the value of the other pieces in fractional form,
in relation to the yellow hexagon (unit whole).
Example: If the yellow hexagon is the unit whole, then 1 red
trapezoid equals 1/2.”
Go over the pieces several times until the students are very
familiar with the value of each piece when yellow is the unit
whole. Make sure you include the phrase, “if 1 yellow hexagon
equals the unit whole, then…” After the students have mastered
the relationships, pick up one purple right triangle and ask them
the value. They will probably say 1/12, but let them know that
they must add the relationship to the unit whole. In other words,
the correct answer is “if the unit whole equals 1 yellow hexagon,
then the purple right triangle equals 1/12.” Otherwise, they
should have responded, “What is the unit whole?”
- Once the students are
comfortable with pieces and their
relationship to the unit whole being the yellow hexagon, select a
new piece as the unit whole and express the revised values of the
Example: If the red trapezoid equals the unit whole, what
value of the green equilateral triangle? (1/3)
If the red trapezoid is the unit whole, what is the value of the yellow
- Again using the yellow hexagon as the unit whole, select several
pieces of the same color and determine the fractional value.
Example: 5 red trapezoids = 5/2. But we can combine them
together to equal 2 1/2. This shows that 5/2 = 2 1/2.
Use explicit instruction to introduce the math vocabulary terms
Fractions (5/2 = 2 1/2)
- Improper Fraction (5/2)
- Mixed number (2 1/2)
Have students work with their tables to find other
between pattern block pieces that are greater than 1. Show both
the improper and the mixed number. Once they have found their
own relationships, have them find pieces that match your
specifics, such as “How can you demonstrate 2 and 1/3 using just
one kind of fraction piece?” Or, “What piece would you need to
use to equal 2 1/2 and uses 10 of the same color?”
- Distribute a 5” x 8”
index card to each pair. Have them create a
design using the construction paper pattern blocks or pattern
block stickers that equals a specific total, with the yellow hexagon
representing the unit whole.
Example: “If this yellow hexagon equals the unit whole, I would
like you to select pieces that would total 3 1/2 and lay them
on your paper. Write the value of each piece in relation to the
unit whole, then write the corresponding mathematical
equation, which should equal 3 1/2. Demonstrate that if you
took 1 yellow hexagon, it would equal 1. If you took 3 red
trapezoids, that would equal 1/2 + 1/2 + 1/2 or 1 1/2, and if
you had 6 green equilateral triangles, it would equal
1/6 + 1/6 + 1/6 + 1/6 + 1/6 + 1/6 = 6/6 or 1. So our equation
would be 1 + 1/2 + 1/2 + 1/2 + 1/6 + 1/6 + 1/6 + 1/6 + 1/6 +
1/6 = 3 1/2.”
- To add a challenge for accelerated
learners, you might require that
they use an odd number of pieces, or at least three different
colors, or at least seven pieces. Have them write a math sentence
on their card that proves their total.
- For students who struggle with this
activity, you could adapt it by
having them select all their pieces of the same color. For example
the red trapezoid. They would then find out that the relationship
to the hexagon equals 1/2. They could place three yellow
hexagons on their desk, cover them with six red trapezoid pieces, and then
add one more. They could then count the seven
trapezoid pieces and express them as the improper fraction
7/2 = 3 1/2.
- Have early finishers use the pattern block pieces to solve
algebraic equations such as, “If 1 yellow hexagon + 1 blue
parallelogram equals the unit whole, then what is the value of a
green equilateral triangle?” Or, “If 1 yellow hexagon + 1 green
equilateral triangle equal 1, then what is the value of the red
For an even more challenging activity: “If 1 green equilateral
triangle and 1 red trapezoid = 2/3, then what equals 1?” Or, “If 1 yellow
hexagon – 1 blue parallelogram equals 1 1/3, then what
For each student:
- Construction paper
pattern block pieces or
pattern block stickers
- Black construction
paper 3 1/2” x 5”
- Glue sticks
- Distribute a 3” x 5” piece of
black construction paper to each
- Have four students work together on one card to create a design
using the pattern block pieces.
(For better coverage, use combinations of red, brown, purple, and
yellow...or blue, green, purple, and yellow.)
- Once the initial pattern
piece has been designed, replicate it on
each of the other three black papers and then join them together to
form a large design, similar to a quilt block.
- Calculate the total of the
large quilt block if the yellow hexagon
equals the unit whole.
Have a variety of different types of sheet music available. Look
specifically for different time signatures.
- Duplicate a piece of sheet music
for each student. With a quarter
note representing the unit whole, what is the value of each
- Compare your findings with a variety of different selections.
What generalizations can be observed?
- Discover a variety of different ways
that whole, half, quarter and
eighth notes can be rearranged to equal the four beats in a
Find items at home that are
Example: 2 juice glasses equals 1 water glass; 2 sessions of piano
practicing equals 1 T.V. news program.
- With the students using the pattern blocks and teacher
overhead pattern blocks, ask the students to respond by drawing
the answer to questions similar to the following:
- If 1 red trapezoid equals
1 unit, then the yellow hexagon
- If the red trapezoid equals 1 unit, then
the blue parallelogram
- If the green triangle equals 1/2 of the unit
whole, then draw
the unit whole ________________
- If the green triangle is 1/2 of the unit
whole, then draw 2 1/2
- If the blue parallelogram equals 2, then
draw 1 unit
- If the yellow hexagon equals 3 units,
then the blue
parallelogram equals _________________
- If the red trapezoid equals 1/2,
then draw 1 unit
- Students could also be assessed
by responding in writing to:
- Explain how you know that 1 unit whole
- Write an equation showing a fraction that is equivalent to
- Draw three pictures showing an improper fraction.
- Explain why pattern
block pieces have different values in
- Create a design and ask the students to find the
value of the total
if the unit whole equals 1 red trapezoid. Or find the value if the
unit whole equals 1 green triangle.
National Council of Teachers of Mathematics. (2000) Principles and standards
for school mathematics. Reston, VA.
A comprehensive volume that establishes the guiding standards and principles
that should be included in all mathematics programs. It describes particular
features of a high quality mathematics curriculum, as well as sets forth a
forward thinking vision of what instruction could become.
Texas Education Agency. (2000) The Texas Successful Schools Study: Quality
Education for Limited English Proficient Students. ERIC Document Reproduction
Service No. ED479 179
A study by the Texas Education Agency examined the variables contributing
to the academic success of economically disadvantaged and language minority
students. Data was collected from seven highachieving elementary schools with
high poverty rates and high percentages of Limited English Proficient (LEP)
Created Date :
Jan 27 2006 08:52 AM