IDEATE: Unit Design
General approach
Our general teaching approach is to serve as coaches and “question posers” for
students to allow them to construct and extract understanding from the situations they are
presented with. We would like to let kids experience working through fraction problems
themselves in order for the learning to be meaningful to the students. Our “general
routine” is to begin each lesson with a situated problem or an ambiguous question
(occasionally involving manipulatives) and have the kids engage with and interact with
these problems and pull out meaning from these lessons. In other cases, we may begin
with a less “problem-oriented” activity but start with warm-ups that may help to activate
students’ prior knowledge or help to hook students and engage them in the content. In
each case, our approach is to provide the students with situations that yield opportunities
to see the use or need for particular techniques and methods, and then to provide practice
using those techniques. Following each "discovery," we generally include a group
discussion - or some direct instruction, if necessary - in order to formalize and
operationalize the learning that the students uncovered during inquiry activities.
As some of our learning goals centers around overall dispositions toward fractions
and mathematics, we aim to maintain an open-ended, non-judgmental learning
environment to foster a healthy perception of math education. We encourage diverse
opinions and multiple approaches to solving problems, and have embedded in our
activities questions that allow the student to consider various strategies or representations
for solving the same problem. At the same time, we want the students to not only hold the
belief that fraction comparison (and math, more broadly speaking) is tractable and
flexible, but that it is also interesting and relevant to the students’ lives. Current students
(as evidenced through our novice interviews) believe that fractions are boring and
irrelevant to their lives, so our teaching approach tries to combat that mindset by framing
the insightful and deep questions we are asking in terms that are exciting and fun for the
students.
Lastly, it should be noted that although some of our target understandings are
specified in terms of fraction comparison problems, we have included general fraction
problems (e.g. what is a fraction? how can we represent fractions? where do we see
fractions?) in the curriculum for two main reasons. The first reason is that we feel it is
important for students to work with fractions and feel comfortable with the concept of
fractions before engaging in specific types of fraction problems (i.e. fraction
comparisons). Secondly, we feel that revisiting general fraction
understanding periodically allows us to continually build on students' prior knowledge
and experiences. As fraction comparison can’t be done without understanding the
fundamental concepts underlying fraction representations, we have included in our
sequence various ways of priming students with what they already know about fractions
so that they can more readily transfer what they’ve previously learned to the problems
they are currently trying to solve.
Alignment of goals
As we developed our instructional sequence, we ensured that our activities
aligned with the goals and desired understandings for our unit. One way we did that was
by creating a column in our activity spreadsheet in order to check which activities map
onto which understandings. Many of our activities address understandings 4 and 5 - that
fractions are parts of wholes and can be represented in different ways. The first is
addressed early in the unit and then implicit in many future activities. Thus, if students
struggle with that understanding, activities throughout the unit will give them a chance to
reflect on how fractions are parts of wholes. Understanding that fractions can be
represented in a variety of ways is a topic that is revisited throughout our unit. Students
continually have opportunities to represent fractions in different ways, especially as
they consider various ways of comparing fractions (understanding 3). We include 4
particular strategies in our unit - with at least 2 lessons on each, plus additional review
and reflection via
journals/assessments - thus giving students sufficient opportunity to consider our third
desired understanding. Understandings 1 and 3 (that people can interact with and
understand fractions in their own ways and that fractions are relevant to life) are
addressed in many of our activities, as students repeatedly have an opportunity to work
with fractions in the context of familiar objects and to choose their own
representations/problem-solving methods for fraction comparison problems.
Our two main performance tasks are included in our schedule of instructional
activities. Mid-way through the unit, students are asked to complete our first performance
task - designing an end-of-the-year party. This is an appropriate time for this task since
the first half of our unit is particularly focused on understandings 1 and 3, as is this
performance task. The second task, designing a handbook for incoming fifth grade
students, is included as an end-of-unit assessment, after students have had the opportunity
to learn about, use, and discuss a variety of fraction comparison activities. Throughout
the unit, periodic formative assessments are used to
determine students' current levels of understanding and dispositions toward
fractions/math.
Justification using learning principles
Throughout our instructional unit, we bring in many examples of fractions from
students’ everyday lives and have the students participate in problem-based fraction
comparison exercises as a way to both a) activate and utilize students’ prior knowledge
and experience with the representations they are manipulating, and b) provide adequate
practice with the tasks that will actually be relevant to the students’ experiences outside
of the classroom. Furthermore, the examples and activities the students will be engaging
with are designed so that they appeal to the students’ interests and encourage them to
actually connect with the material and make meaning
out of the strategies they are using. We also tried to frequently include activities that are
sufficiently open in order to account for individual variability in learning styles and
preferences, as well as prior experiences. We found this to be important since several of
our enduring understandings focus on the diversity of representation and meaning
regarding fraction comparison. We capitalize on difference in student interest and
learning styles not only by presenting a variety of methods, representations, and activity
types to work with, but also by creating assignments that ask students to consider their
own interest in fractions and reflect on their awareness of what works for them and what
doesn’t. (HLW Principles 1 and 3)
Additionally, the activities and problems we provide to students are framed by
provocative and stimulating questions that force students to rethink their assumptions
about the material (and math, more broadly) and to reflect on what they have learned and
what they will need to learn to solve problems efficiently. Our approach encourages selfdirected learning by providing students with opportunities to evaluate their
strengths/weaknesses (e.g. activity
21), as well as monitor their own performance and organize their knowledge (e.g. fraction
journal). Many of the activities center around problem setups that present students with
new/varied ways of thinking about fractions and fraction comparison, providing a bridge
between reflecting on misconceptions and conceptualizing/applying new understandings.
Although the main topic of our unit is fraction comparison, we are equally concerned that
students have a solid understanding of fractions themselves and feel comfortable "in the
presence" of fractions. Thus, we frequently provide students with activities to help
solidify their understanding of fractions and continue to question prior assumptions. We
believe that providing students with more practice, as well as tailored feedback, will help
break down students' misconceptions and strengthen their understanding of new concepts.
(HLW Principles 2, 4, 5, and 7)
Unit Calendar
1 Warm-up: ½
2 Introduce
3 Warm-up:
4 Warm-up:
5 Warm-up:
6 Parent guest
7 ½ day (2)!!
vs.
5,000/10,000
(4); Meaning of
the denominator
+ cookie cutting
activity (5)
fraction journal
(1); Fraction
representations
(7); Fractions in
house hw
Play Doh
activity (6);
Familiar
fractions
number line
activity (9)
Backpack
Weight (8);
Human number
line activity
and discussion
(10)
Food fractions
(11); Fractions
 decimals
money activity
(12)
(3); Fractions
 decimals
sports activity
(13); Assign
mid-unit
assessment
8 Warm-up
9 ¼ day (2)!
10 Parent
11 Warm-up
12 Warm-up:
13 Parent
reflection:
Lemon activity
(14); Reference
point
comparison (16)
Warm-up:
discussion of ¼;
A Day in the
Life worksheet
– to be
continued for
hw (15);
guest (3);
introduction
and discussion
of LCD method
(19)
reflection: jelly
bean activity
(18); Continue
discussion/
uses of LCD
method (19);
Mid-unit
assessment due
fraction
reflection (21);
Begin letter to
niece/nephew
– finish for hw
(17)
guest (3); Fruit
rollup LCD
activity (20);
Assign end-ofunit
assessment
Warm-up:
discussion of
1/2; Review of
first two
methods;
Journal writing
time
14 You’re the
fraction
activity (22);
discussion of
fraction
comparison
methods;
journal writing
time
15 Efficiency
16 Warm-up:
17 Students’
18
19 Fraction
Timing Activity
(23); Segue into
whole class
discussion of
difference
methods
food choosing
activity (24);
Using Multiple
Methods
activity (25);
Vote on
teachernominated
parties (from
mid-unit
assessment)
choice fraction
day! Wrap-up
discussion of
lessons learned,
journal
findings, etc.
Presentation of
final projects
party!! (using
chosen
students’ plans
from mid-unit
assessment)
*See chart below for corresponding numbers and descriptions of activities.
Activities
WHERETO
HLW
1. Fraction journal - Have kids keep a fraction journal where they record different
methods they have learned, new representations they can use to represent fractions,
or things that are surprising/difficult/exciting to them, etc. The journal will contain a
set of these prompts on the first page, but the students are encouraged to simply
write their thoughts about the lesson content throughout the unit (at least once a
week)
R, E2
Monitoring progress,
identifying
strengths/weaknesses;
revealing knowledge
organization
2. Fraction days - Have class days themed around certain fractions (e.g. ¼ day, ½ day).
The students will bring in/wear representations of that fraction, or bring in objects
from their home where the day's fraction is prevalent. This will occur on 2 or 3 days
throughout the unit.
H, T
Enhance motivation
3. "Career Panel" - Adults will be invited to the classroom to talk about how they use
fractions in their lives and careers. The students will also be able to ask the guests
about how they use and apply fractions, and math more generally, in their jobs and
daily business. Guests will be invited periodically throughout the unit.
4. "1/2 vs. 5,000/10,000" - Warm-up question: The teacher will write ½ and
5,000/10,000 on the board and ask the students which is bigger. It is anticipated that
there will be either some debate, or that the students will assume 5,000/10,000 is
larger. Students are encouraged to explain their reasoning.
4a. After discussion, the teacher will explain that they will learn why this question
may seem confusing, and that they will learn how to answer problems like this during
the unit. The teacher will emphasize that the class will revisit this question at the end
of the unit.
5. Denomi-what? Ask students to volunteer anything they know about denominators.
Make a list of students' ideas on the board. Then put up two fractions: 1/8 and 1/4.
Ask what the difference between them is and how you could draw those fractions.
Then transition into discussion of numerators and how the fractions will change if you
change the numerator. What stays the same and what changes in your picture? Will
eighths always be smaller than fourths, regardless of the numerator? Show a couple
more examples to demonstrate this point.
5a. To further illustrate this point, give each group of kids two soft cookies (or
another similar object) and have them cut each cookie into a different number of
pieces (they can choose their denominators, essentially). Then have them make a
chart comparing different fractions with those two denominators using their
representations. (e.g. What is bigger, 2/3 or 1/4? 1/3 or 1/4? 3/3 or 3/4?)
5b. Transition into whole group discussion of cookies with larger pieces versus
smaller pieces. Is one always bigger than the other? How can we tell which is larger?
What role do the numerator and denominator play in our decision?
6. Play-Doh Warm-up - Give pairs of students two pieces of Play Doh - one whole
block and one block half its size. If the large block represents 1 whole, then what
portion of it is the second block? Once students decide 1/2, tell them to pick their
favorite shape or object and to make both pieces of Play-Doh into that object. Now,
what portion of the larger object is the smaller one? Still 1/2 or has it changed?
E
Establish value
(motivation)
H
Gauge and address
insufficient/inaccurate
prior knowledge
W
Promote productive
climate and encourage
questioning/exploration
W, E
Gauge prior knowledge;
address misconceptions;
reinforce component
skills
R
Address misconceptions;
reinforce component
skills; build fluency
E, R
Reflecting on one's
approach; evaluating
understandings
H, R
Gauge and address
insufficient prior
knowledge
7. Representation Comparison - The teacher will show the students one fraction
represented two different ways and ask the students to discuss with their tablemates
whether they are the same or different. Students will also be asked to share whether
or not they have a preference for one fraction representation versus the other.
7a. The teacher will then discuss with the class why and how fractions can be
represented in multiple ways. The teacher will also emphasize how the students will
be able to reason about what the different representations are useful for.
7b. The students will then write a similar question and present it to fourth
graders. This is designed so that the students can learn whether other students have
misconceptions about this
7c. As a homework assignment, the students will bring in to class the next day a
list of different things around their homes that can be represented using fractions, and
the students will be asked to justify which representation they have learned thus far
fits the objects they have listed.
8. Backpack Weight - Warm-up activity: Each table group will select one student's
backpack to use as the object of discussion. Each group will have a chance to weigh
the backpack and its constituent parts. The students will then be asked to determine
what fraction of the whole backpack certain components represent and record this
information in a table and determine which components represent the largest and
smallest components. The students will be able to decide what parts they compare,
but will be provided examples such as books, folders, lunch, etc.
R, T
Assessing prior
knowledge; Planning an
appropriate approach;
Monitoring one's own
learning; allowing for
diverse learning
preferences
W
Reinforce component
skills; Reflecting on one's
own approach
E
Correct inaccurate
knowledge; building
fluency
H, T
Establishing value;
facilitating transfer
H
Build fluency and
facilitate integration
9. Familiar fractions - The students will be asked to name common fractions, or things
that they think might be fractions; the students will be asked to explain where they
have heard those fractions used before. These fractions will be written horizontally on
the board in the order of mention. The students will then be asked to come up to the
board and place the listed fractions on a number line.
E, T
10. Human Number Line - Allow students to become a part of a number line in order
to better understand how number lines work, how to plot fractions on number lines,
etc. Begin this activity by writing a variety of fractions on index cards and having each
student choose one. Then go into the hallway and use masking tape to draw a long,
straight line. Label the left end 0 and the right end 1. Then ask students how we might
go about placing ourselves in the right location on the number line. If students don't
suggest the need to mark several important places (i.e. 1/2, 1/4, 3/4) or make tick
marks, ask questions to scaffold them and help them realize the importance of dividing
the line. (e.g. How do you know that you're in the right place? How will this student
know where to go? How will she know what fractions are already in place? Feel free to
use the tape to make markings on the floor.)
E
11. Equal Pieces? A Food Warm-up Activity - Give each group of students a different
food (e.g. a piece of popcorn, a fruit rollup, et). Be sure that the types of food vary in
their ability to be split evenly into 4 pieces. Ask the groups to break the food into
fourths using that piece of food. (Discuss the language of "fourths" as opposed to
"one-fourth" if necessary.) After students are finished, pose the following questions:
Were you able to break your food into fourths? Was this easy/hard? Does the size of
the pieces matter? Why?
E, E2
Activate accurate prior
knowledge
Expose and reinforce
component skills
Address misconceptions;
Encourage reflection;
Promoting productive
climate
12. Money Math - Show students a quarter and ask them to write on an index card
how they would represent the amount of money a quarter is worth. Have them hold
up the index cards for you to look (quick diagnostic assessment of students'
decimal/money knowledge). Then ask them if they notice anything interesting about
the name of that coin - and then how we would write that name as a fraction. Once
.25 and 1/4 are written on the board, pose the question: Are these numbers
equivalent? Why or why not? Ask students to suggest ways we could figure out
whether those numbers are the same. If it doesn't come up organically, introduce the
idea that 1/4 is the same as saying 1 divided by 4 (the fraction bar is the same as a
division sign). Allow students to ask questions about this and discuss the relationship.
Then demonstrate how to find decimal equivalents of fractions using long division. Let
students practice that method with their groups and then discuss how it might be
useful in the future.
13. Free-throw Fractions (& more) - Introduce a commonly heard sports statistic shooting 5 for 9 from the free-throw line. Allow a basketball fan in your class explain to
the other students what this means and where these numbers come from. Then ask
whether we could represent that statistic using a fraction. Give students a couple
minutes to discuss this in groups and then report back to the class. Write suggestions
on the board. Then introduce another sports example - baseball batting average. Ask
students what a batting average usually looks like and write an example on the board
(e.g. .250). Then tell the class that people sometimes talk about a player's batting
average as "1 in 4," particularly when talking about a single game. Are these
representations equivalent? Does 1/4 mean the same thing as .25? Again, how can we
tell?
E
H, E, R
13a. To close out the activity, ask students to draw on an index card the different
ways they can represent 1 in 4. Then have them discuss with partners which
representation they prefer (or which makes the most sense to them) and why.
E2, T
14. Lemon Reflection - Do as a group activity in front of the whole class. Cut a lemon
in half two different ways (lengthwise and crosswise). What fraction does each piece
represent? So are the two halves equal, even though they are different shapes? Why
or why not? Help students understand that the same fractions can be formed by
cutting objects in different ways, as long as all of the pieces in a single example are the
same size.
R
15. A Day in the Life worksheet - This worksheet is included to provide students with
another opportunity to relate fractions to their own lives and to think about fractions
in multiple ways. Students can start working on this in class and continue it for
homework. Sample questions are as follows:
T
-In which 1/2 of the year is your birthday? In which 1/4? In which third? How did
you figure it out?
E
-Now round your age to the nearest whole number… What fraction of the way to
50 years old are you? How about to 20 years old?
-What fraction of your parents' ages are you? How about your grandparents?
-Challenge: Look at a calendar. Is each month actually exactly 1/12 of year? Why
or why not?
Establish value (connect
material to student
interest)
Promoting student
development and
productive climate
(resisting single right
answer)
Establish value (connect
material to student
interest)
-What fraction of the day do you spend….. (in school, playing sports, watching
TV, etc.) ?
-How old are you? (give answer as a mixed number - e.g. 10 3/5) How did you
figure it out?
Facilitate transfer
(conditions of
applicability)
E, E2
E
E
H, R
16. Reference Points - Write ½ in middle of board. Have kids take turns coming up to
the board and writing fractions that they know are either greater than (to the right of)
or less than (to the left of) 1/2. Ask students to tell the class how they knew where to
put each fraction. After 5-10 fractions are on the board, ask students to discuss in
groups how they could tell whether a fraction is greater than or less than 1/2. Provide
them with 3 fractions that are not on the board, and ask them to decide where they
belong. After groups share their findings/ideas with the class, talk to the class about
which strategy they think makes the most sense and is the most efficient. Have them
record the strategy in their fraction journals.
E, R, T
Student development
and productive climate
W, R
Address value and
expectancies; encourage
reflection
18. Jelly Bean Count Comparison - The students are presented with a large bag of Jelly
Belly jellybeans and are asked to estimate which flavor represents the largest fraction
of all of the jellybeans. The teacher then poses the question of whether or not
students need fractions to figure that out. Once (and if) the students realize that
fractions are not essential to this problem, the students will be asked why can they
compare just by counting the number of jellybeans instead of looking at fractions.
Then the students are presented with a different-sized bag of jellybeans and asked
which bag has the most of one flavor relative to the size of the bag. Students are then
asked if fractions are necessary in this task. Students should discuss this question in
small groups and then present their thoughts to the class. The activity will culminate
with a group discussion of how the task of comparing flavors in different bags is
different from the initial task.
H, E, R
Address inaccurate prior
knowledge; reinforce
component skills
19. Least Common Multiple - The teacher will write two fractions on the board and
show students how to compare them using the LCD method. The teacher will explain
that the students do not need to use this strategy all of the time. After prompting
students to provide any guesses as to why they are learning this method, the teacher
will explain that it is helpful when the fractions cannot be solved using the previous
methods they have learned and that they will explore examples of its usefulness in the
future.
W, E
Build fluency
E, R
Facilitate integration
and transfer; assess the
task at hand and plan an
appropriate approach;
reflect on one's
approach
21. Fraction reflection - Students are asked to think about what their favorite part of
learning fractions is, as well as what part of learning fractions they view as the hardest.
E2, T
Reflecting on and
adjusting one's
approach
21a. The students will then tell their partner what they think the hardest part of
learning fraction comparison is. The partner will play the role of the teacher in this
instance and attempt to help work out the difficulties.
E2, T
Targeted feedback (peer
feedback)
17. Why Hate Fractions? Tell the class that you have a problem: your
niece/nephew/son/daughter hates fractions and thinks they are useless in life. Tell the
class that you need some help convincing him/her that fractions are, in fact, useful,
and that we see fractions everywhere. Have your students write him/her a letter to
explain how you can use fractions in life and/or why it's worth learning about
fractions.
20. LCD Applied - Give each pair of students two cookies or Fruit Rollups (or a similar
food that can be cut easily). Then have the kids cut each into a different number of
pieces (fourths and 5ths for example). Then ask them how they could change the
representation to compare 4/5 to 3/4 (as an example) in light of what they learned
about finding LCDs. Allow students to provide suggestions to the whole class, and if
they're headed in the right direction, allow them to go back to their pairs and alter
their representations as needed. If they seem confused, provide them with some
scaffolding: First ask how they might figure out an LCD for those two numbers. Then
ask them to remind you what role the denominator plays when we are cutting shapes
to represent fractions (answer: the total number of pieces). Finally, ask how they could
change their pieces of food to reflect that change in denominator.
22. You're the Fraction! - The teacher gives each student in a group a piece of paper
with a different fraction on it. The students will then adopt this fraction as their
"fraction identity." The students must use whatever methods they choose to organize
themselves from least to greatest.
T
Value and expectancies
(provide flexibility and
control)
23. Efficiency Timing - In partner pairs, the students will be asked to time themselves
solving fraction comparison problems using different methods. The students will be
asked to solve and time themselves solving each problem using a set of methods for
each (common denominator, using shaded regions, decimal conversion, finding
reference points, number line). The problem set will be designed such that different
problems should take more or less time depending on the methods used. The students
are then asked to make generalizations about which methods worked better or worse
for which problem types.
E2, T
Reflect on various
approaches; promote
metacognition
23a. The students will then form groups and discuss as a whole their thoughts on
the generalizations they made in their partner pairs, and will share which methods
they thought worked best for which problems.
E2, T
Targeted feedback (peer
feedback)
R, T
Facilitate transfer;
motivate students;
planning an appropriate
approach
24. Choose Your Food: Have a diverse selection of food available for students to
choose. Have two different amounts of each type of food available, and label each
food with the fraction of a whole it represents. Break food into a variety of sizes of
pieces so that students cannot simply look at the food to choose which is larger. For
example, Provide 1/2 a cookie in one whole piece and 2/3 of a cookie broken into
smaller pieces. Which food would the student prefer? How did s/he make that choice?
(Elaborate on strategies with the class, as a review of different methods learned) Let
students keep the food they choose, thus giving them an incentive to determine the
larger portion.
25. Using Multiple Methods - The teacher will present two fractions on the
board and ask students to take turns coming up to the board and explain or
show different ways to compare the two. After many different problems have
been solved on the board, the teacher will ask the class to have a discussion
about which strategies they thought worked best for which problems.
Reflecting on and
adjusting one's
E, R, E2, T approach