Time
Task
Day 3
Materials & Notes
Welcome Back & Objectives
Address 4 point evaluations , questions and concerns
30 min
Total
10 min
10 min
10 min
Context and Models
While most of this day will be focused on the use of number lines as models it is important to note there are many valuable
models that can be used while learning about fractions, however context will determine which model would be most
appropriate.
The three most commonly used models are

Region or Area Models- Helps students visualize parts of a whole.

Length Models- Helps students visualize iteration, counting and measurement

Set models- Helps build understanding of division and ratio concepts
Fraction Brainstorm
1. Have participants fold a paper into three columns
2. Label each column after a model
3. With a partner, participants will brainstorm as many possible questions or contexts that would support each model
4. Challenge the participants to think of things that have nothing to do with pizza or food.
Why number lines activity
The Institute of Education Sciences is the research arm of the US Department of Education. Their mission is to provide rigorous
and relevant evidence on which to ground education practice and policy and share this information broadly. By identifying
what works, what doesn’t and why they aim to improve educational outcomes for all students particularly those at risk.
In 2010 they released the practice guided titled Developing Effective Fractions Instruction for Kindergarten Through 8th
Grade, in which they recommend 5 practices to improve fraction instruction. Let’s look at recommendation #2.
Help students recognize that fractions are numbers and that they expand the number system beyond whole
numbers. Use number lines as a central representational tool in teaching this and other fraction concepts from the
early grades onward


Have participants read slide #7 silently trying to address the question Why number lines?
Then do a think-pair-share
Essentially teachers should discover that the number line teachers children the “place value” of fractions.
Children need to understand:
•
Fractions have magnitude or value
•
Fractions can be counted
•
Fractions can be ordered
•
There are an infinite amount of fractions between any two points.
These understandings are essential in order to compute with fractions.
PPT
Notebook
30 Min
Progressions Document Reading Activity: Block Party
We will begin the day with an activity reading the Fraction Progression Documents for 3rd and 4th Grade. Teachers
will be assigned to question the importance of understanding unit fractions as they read the progression document
on fractions. After reading the progressions teachers will participate in a Block Party.
1.
2.
3.
4.
5.
6.
NF Progression Doc
Block Party Quote Card cut up and distributed
Allow 10 minutes for participants to read Progression Doc.
Participants randomly select quote cards, and take 3 min to read and think about its meaning. (Use block
party quote card document)
Participants mingle around the room sharing, explaining and discussing quotes. 5 min a rotation. Or triads
and quads could be formed with longer discussion time allowed.
Repeat as many mingles as time allows.
Whole group share out.
Ultimately teachers should have found many reasons unit fractions are important namely:
“Unit fractions are the basic building blocks of fractions in the same sense the number 1 is the basic
building block of whole numbers; just as every whole number is obtained by combining a sufficient
number of 1’s, every fraction is obtained by combining a sufficient number of unit fractions.”
6 Min
Watch the fractions progression video on Unit Fractions:
30 Min
Graphic Organizer of Strategies (Unit Fractions)
Begin talking unit fraction strategies, use graphic organizer. Whole group will draw models on the back of the organizer then
teachers as small groups can complete, the core connections, progression connections, and vocab, whole group will come
back and finish DOK component.
Be sure to show multiple models of unit fractions:

Unit fractions on a number line

Unit fractions as a tape diagram

Unit fractions as a set model

Unit fractions as an area model.
Unit fractions on a number line
Partitioning number lines to represent unit fractions. Students should be able to count unit fractions on the number line in 3 rd
grade and by 4th grade express fractions as the sum of unit fractions. This can be shown with jumps on a number line.
Emphasize the counting nature of unit fractions include a model similar to this one in showing what we would call improper
fractions
http://www.illustrativemathematics.org/fractions
_progression
Graphic Organizer of Strategies
Document Camera
Unit fractions as a tape diagram
Partitioning tape diagrams into equal shares. 3rd graders count the unit fractions, 4th graders
add them.
Unit fractions compose non-unit fractions
Both grades explore non unit fractions, by building them out of unit fractions. This is the foundation for fraction addition and
subtraction. We will look at addition with non-like denominators later.
Non-unit fractions can decompose to unit fractions
Students can work backward and decompose a fraction into its unit parts. 3rd graders simply count the pieces and break
them apart while 4th graders can use this to aid in subtraction.
Using unit fractions to compare fractions
Unit fraction make comparisons very simple students can reason about the size of the fraction based on the denominator.
Students will understand that the comparisons are made of just one unit the denominator will determine the size of the unit.
25 Min
Graphic Organizer Partner Activity:
This activity is essential. This is the time in which participants will comb through their core and make correlations between the
strategies just discussed and core standards. With a partner participants will complete the graphic organizer specific to their
grade level. As a group discuss any sticky concerns.
Fraction Equivalence & Comparing Fractions:
A serious mistake teachers make when teaching fraction equivalence is to rush to the algorithms (multiply or divide the top
and bottom numbers by the same non-zero number) When students follow a procedure they do not understand they
have no means of assessing their results to see if they make sense. Secondly mastery of a poorly understood algorithm in the
short term is quickly lost. Students have no way to reconstruct a forgotten procedure. Both of these skills require patience,
repeated exposures and discussion, discussion, discussion. We want students to be able to reason with fractions first!
Core
NF Progression
Strategy Graphic Organizer
Cognitive Rigor Matrix
20 MIN
Progressions Document Reading Activity: Snow Ball Fight (20 Min)
Back to fraction progression doc, this short read is now paying attention to fraction equivalence.
1.
2.
3.
4.
5.
6.
Fraction Progression Doc
Blank Notebook paper
Skim the progression doc for info about fraction equivalence. (5 min)
Teachers should find one statement or model to write down, and one question both pertaining to fraction
equivalence. (5 min)
Teachers crumble their notes and toss across the room.
Teachers pick up one note and take back to partners or triads to discuss. (5 min)
Whole class share out. (5 min)
Teachers should ultimately find:
“Fraction equivalence is founded in reasoning and experimentation with number lines and tape
diagrams. Students will need numerous exposures and opportunities to discover and explore equivalence
in order to form a solid conceptual understanding.”
12 Min
Watch the fractions progression video on Equivalency:
30 min
Patterns will lead to discovery of multiplication algorithm
Students will begin to recognize that equivalent fractions represent the same point on a number line, the same measurement
in a tape diagram, and the same area in an area model. It is important in 3rd grade to recognize 1 can be represented as a
fraction. 3rd graders will discover this by counting unit fractions and modeling in various contexts. 4th graders can extend this
understanding to recognize patterns with fraction equivalence, and in turn experimenting with multiplication.
http://www.illustrativemathematics.org/fractions
_progression
Graphic Organizer of Strategies
Doc Camera

20 Min
If time allows have the participants partner and complete the back side of the graphic organizer.
Van de Walle Comparison Strategies Activity
Present Task of fraction Comparisons below. Teachers must not use drawings, common denominators or cross multiplication.
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

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More of the same-sized parts (same denominators) Questions B & G
Same number of parts but different sizes (same numerator) Questions A, D, & H
More than, less than a benchmark fraction (1/2 or 1) Questions A, D, F, G, H
Closeness 9/10 is greater than 7/8 because each fraction is 1 fractional part away from a whole, however 1/10 is
smaller than 1/8 so 9/10 is closer to 1. Questions C, E, I, J, K, L
None of these strategies should be explicitly taught as rules. Exposure to contextual problems that allow students to reason
and apply number sense will develop these discoveries.
Blank Note book paper
Project or Write problem on board
Addition/Subtraction of Fractions:
30 Min
Miscellaneous fraction addition tasks
Discussion Questions
Task Sorting and Discussion Activity:
Table groups will be given a variety of tasks related to adding or subtracting fractions to review individually. After solving
and reviewing their tasks group members will discuss the following with their groups. Each person or pair of people will have
a different task.

Can this be modeled with a number line? Is there a better model?

How could a student use their understanding of unit fractions to solve this?

Can this be seen as composed of unit fractions or decomposed to unit fractions?

Does the student need to convert to common units?
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Discuss with a partner the task: What’s the math? How could it be improved? What is the relation to addition and
subtraction of fractions? What is the core connection? What support/Vocabulary/background would your
students need?
http://www.illustrativemathematics.org/illustrations/874 Cynthia’s Perfect Punch
http://www.illustrativemathematics.org/illustrations/856 Plastic Building Blocks
http://www.illustrativemathematics.org/illustrations/835 Writing a mixed nummber
http://www.illustrativemathematics.org/illustrations/968 Peaches
http://www.illustrativemathematics.org/illustrations/837 22 Seventeenths
IF other tasks are needed participants can pull from illustrative mathematics or core academy sites to supplement.
7 Min
Watch the fractions progression video on Addition:
http://www.illustrativemathematics.org/fractions
_progression
30 Min
Graphic Organizer of Strategies:
Graphic Organizer of Strategies
Doc Camera
Begin talking addition and subtraction strategies using graphic organizer. Whole group will draw models on the back of the
organizer then teachers as small groups can complete, the core connections, progression connections, and vocab, whole
group will come back and finish DOK component.
Fraction addition and subtraction
Addition and subtraction with like denominators will be a breeze if the teacher
has taken the time to solidify the idea of iterating unit fractions and fraction
equivalence. 4th graders should be able to simply compose a non- unit fraction
from unit fractions or decompose non-unit fractions. Students decompose and
compose fractions into unit fractions in order to add or subtract.
Be sure to include models.
Mixed Numbers
Mixed numbers with like denominators should be decomposed into a sum of unit fractions 2 ¾ is the same as 4/4 +
4/4 + ¾ or 11/4
Students will use their understanding of one flexibly to decompose
mixed numbers prior to adding or subtracting
The inverse of this, improper fractions, works equally well.
Multiplying Fractions by a Whole # and Decimal Fractions:
30 Min
Document Reading Activity:
7 Min
Watch the fractions progression video on Multiplication:
30 Min
Graphic Organizer of Strategies:
25 min
Marilyn Burns Fraction Multiplication article
Notebooks
Follow 4 A protocol
1. Have Participants read article (15 min) then respond with the 4 A’s
http://mathsolutions.com/documents/0-941355-64-0_L.pdf
o
What assumptions does the author of the text hold?
o
What do you agree with?
o
What do you want to argue?
o
What do you want to aspire to?
2. Table groups round robin discuss each A one at a time (10 min)
3. Whole group share out. (5 min)
http://www.illustrativemathematics.org/fractions
_progression
Begin multiplication and decimal fractions using graphic organizer. Whole group will draw models on the back of the
organizer then teachers as small groups can complete, the core connections, progression connections, and vocab, whole
group will come back and finish DOK component.
Multiplication
1. Begin with simple contextual tasks.
Context adds meaning. Problems do not need to be elaborate.
2. Connect the meaning of fraction multiplication with whole number computation.
Begin with a whole number multiplication and review the idea of repeated addition or groups of. 3 x 5=
5+5+5 so 3 x ½ = ½ + ½ + ½
3. Let estimation and informal methods play a big role in development of strategies.
We want students to be able to reason with fractions and mentally compute with them just as they had
with whole numbers, students should be reason about the approximate size of their answer and model it.
4. Explore the operation using models.
Strategy Graphic Organizer
Graphic Organizer Partner Activity:
Core
Progression Documents
Cognitive Rigor Matrix
Partners will then complete the front of the graphic organizer, finding the core connections, progression connections,
vocabulary etc.
Multiplying a whole number by fraction clip
https://www.teachingchannel.org/videos/multipl
ying-fractions-by-whole-numbers-lesson