1. A Story of Units www.EngageNY.org
December NTI - Fractions - Grades 3-5
TIME ALLOTTED FOR THIS SLIDE:
7 Minutes
MATERIALS NEEDED:
PPT Slides
Participant Packet (Modified Problem Set)
Post-It notes (Presenter only)
Personal White Boards, Markers, Eraser (Optional)
Pattern Blocks (Optional - Presenter Only)
Fraction Bars (Optional - Presenter Only)
Fraction Circles (Optional - Presenter Only)
A Story of Units
Fractions
Grades 3-5
One arithmetic for whole numbers and fractions.
Slide 1: Title Page
7 minutes
3 minutes:
Ask participants to find a partner not at their tables. Once they are in pairs, ask partner A to share their memories of learning fractions in school. After 1 minute, give a signal for the other partner to do likewise, and then return to their original seats, bidding farewell to their new friends.
4 minutes:
Call attention to the name of the PK-5 curriculum as “The Story of Units.” Ask participants to pair share as many units as they can think of.
Prompts: What would some measurement units be? Units in kindergarten? What units might be new in Grade 1? 2? 3? 4? 5? (See the last pages of the packet for a summation for your own reference, but do not direct the participants at this time.)
As more ideas come forward either from the prompts or from teachers’ prior knowledge, describe the units as “what we are counting” and the numbers as “adjectives describing units.” Think, “How many, of what.”
(Please see the back of the Modified Problem Set for a list of some units participants might state.)
**THIS SESSION IS DESIGNED TO BE 300 MINUTES.**

2. A Story of Units www.EngageNY.org
December NTI - Fractions - Grades 3-5
TIME ALLOTTED FOR THIS SLIDE:
10 Minutes
MATERIALS NEEDED:
Foundations of fractions – looking at prior learning
Equal Parts
Slide 2: Equal Parts
GRADES K, 1, 2 (See the standards pages in the back of the participants hand-out to read the G1-2 standards.)
3 minutes to discuss with partners.
3 minutes to discuss as a whole group.
Grade 3 starts the formal exploration of fractions.
What do students know before grade 3?
From Grade 1:
Rectangles, circles, etc.. (Grade K)
Halves, fourths, quarters, half of, fourth of, quarter of (1.G.3)
Partitioning into halves and quarters (1.G.3)
Half-Hour (1.MD.3)
From Grade 2:
Halves, thirds, fourths (2.G.3)
Partitioning into halves, thirds, and fourths (equal shares) and describing the shares (2.G.3)
Describing whole as 2 halves, 3 thirds, or 4 fourths (2.G.3)
Equal shares of the same whole need not have the same shape. (2.G.3)
If we were to partition this rectangle into equal parts, what might it look like? (Have participants partition this rectangle into quarters in 3 or 4 ways.)
What vocabulary might be difficult for students?
Halves, thirds (Why not “twoths” and threeths”?)
Quarter of (Why is a quarter of a whole 1 fourth, a quarter of a dollar 25 cents, and a quarter of an hour 15 minutes?)
(Click) Advance animation.
Are these equal parts?
Yes, each shows 1 fourth. (Equal shares need not be the same shape. 2.G.3)
Give participants time to look at the Participant Handouts for Grades 1 and 2. (Includes G1-M5 Topic Overview, G1-M5-L5; G2-M8 Topic Overview, G2-M8-L10, and the Standards.)
What prior learning do students need to be successful with each Problem Set?
G1-M5-L9 – half, quarter, equal parts, larger, smaller, identify shaded part as half or quarter, shapes
G2-M8-L10 – halves, thirds, fourths, partitioning into fractional parts, shading to show given fraction, 3 thirds and 4 fourths as a whole.
What prior learning will students be bringing to Grade 3 from these modules?
(Review discussion from 1st question on slide.)
How can understanding the prior learning help to close gaps in learning in G3, G4, and G5?
This is the prior learning we expect students to have coming into G3.
Close gaps by incorporating these concepts/vocabulary into lessons.

3. A Story of Units www.EngageNY.org
December NTI - Fractions - Grades 3-5
TIME ALLOTTED FOR THIS SLIDE:
20 Minutes
MATERIALS NEEDED:
Relating whole number units to fractional units
Slide 3: Duck and Rectangular Array
GRADE 3
3.NF.1: Understand 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.
Ask participants:
1 minute with partners
Describe the number of ducks in unit form. 1 duck
How would you describe the orange section of the rectangle? 1 eighth
(Model unit language on the document camera: “1 eighth.”)
3 minutes with whole group
How does unit form help students link what they know about whole numbers to fractions?
Just like we can talk about ducks, dogs, cats, ones, tens, hundreds, we can also talk about halves, thirds, fourths, etc….
The difference is just the units. Fractions work in the same way as whole numbers.
Since early grades, students have been able to add like units. If we have like units, we can add or subtract.
Participant Handout: (Note: Fractions with denominators of 2, 3, 4, 6, & 8 for 3rd Grade.)
Grade 3, Module Overview Lesson Objectives Chart
Identify places where prior learning from G1 and G2 is required.
Topic A: Equal parts, identifying and describing shares
Topic B: Partitioning, equal parts, identifying fractional parts, number bonds (part/whole relationship)
Topic C: Compare (larger/smaller), fractional parts in a whole
Topic D: Number lines, comparison (<,>,=) using number line
Topic E: Number lines, equal parts
Topic F: Number lines, comparison
Identify the sequence of learning. How does it move from simple to complex?
Topic A: Similar to Grade 2 work – identifying fractions (concrete, pictorial, abstract); extends beyond halves, thirds, fourths
Topic B: Introduces ‘unit fraction’ - identifying and building
Topic C: Unit fraction – comparing – part/whole relationship
Topic D: Extends to number line – representing fractions (placement and comparison)
Topic E: Still number line, but fractional equivalency  Leads to work with other models (number bond, fraction strips)
Topic F: Comparing fractions – different strategies
Where do you see repetition?
Interwoven throughout – “one step back, two steps forward”.
Grade 3, Module 5, Lessons 1 and 5 Problem Set.
Why do both lessons stress the unit fraction? Why is the unit fraction important?
Thinking of fractions in unit form ( 1 fourth, 1 third, etc.) leads to introduction of unit fraction 1/b; relates fractions back to whole numbers and to the idea of the unit.
Unit fractions tie fractions back to whole numbers and are the basic building blocks of whole numbers.
In kindergarten students learned to decompose 5 into “hidden partners.” We are doing the same thing now with fractions: 3/8 = 2/8 + 1/8, etc.
How does this sequence between both lessons support concrete to pictorial to abstract understanding?
Lesson 1 uses concrete models (fraction strips and beakers) to show fractional parts; familiar halves, thirds, and fourths.
Lesson 5 uses pictorial models; students determine the fractional part numerically.
We do many lessons before we introduce fraction notation.
What concrete experiences must students experience to understand the unit fraction? Compare those experiences to the experiences Grades PK-2 students have with units of ones and tens.
Idea of unit; manipulation of concrete units (Pattern blocks)
Concrete experience with whole numbers – similar – units  unit form

4. A Story of Units www.EngageNY.org
December NTI - Fractions - Grades 3-5
TIME ALLOTTED FOR THIS SLIDE:
5 Minutes
MATERIALS NEEDED: 10 10 10 10 10
100
Relates what we know about decomposing and renaming of whole numbers to decomposition and renaming of fractions. 10
Slide 4: Renaming 320 as 32 tens (from Grade 2 learning)
Using unit form, what do you see on the left?
3 hundreds, 2 tens
Say that in standard form.
320
What did we do going from left to right?
We renamed 320 as 32 tens.
Why might we do this?
To express an equivalent amount or to rename so that we have like units. Once we have like units, we can add or subtract.
How is the term “renaming” like “equivalent?”
Expressing the same amount in a different way.
Describe other examples of how units have been manipulated.
Place value chart  ones, tens, hundreds decomposed for other units or composed to other units through addition, subtraction, multiplication, and division. 10

5. A Story of Units www.EngageNY.org
December NTI - Fractions - Grades 3-5
TIME ALLOTTED FOR THIS SLIDE:
10 Minutes
MATERIALS NEEDED:
Using models to relate prior learning to fractional units (focusing on unit fractions)
Slide 5: Multiple Models
3.NF.2: Understand a fraction as a number on the number line; represent fractions on a number line diagram.
Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line.
Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line.
Ask participants to discuss the multiple models students use to explore and understand fractions.
How are these models used in earlier grades?
Unit form – to build a foundation for understanding numbers, place value, and the four operations (G2) Before grade 2 students use the “Say Ten Way.”
Number bonds – to show part/whole relationships (introduced in Kindergarten)
Tape diagram (here, fraction strip) – To pictorially represent a given quantity (G1)
Number Path – Now the path is divided into fractional parts. (Kindergarten)
Number line – In G2 students make their own rulers, which is the introduction to the number line. Students use it to show the relationship between numbers; addition and subtraction of numbers (G2)
Which of these models might be the most sophisticated?
The number line is used in fractions in a new way. Until now we have only shown whole numbers on the number line. The other models support this new complexity of the number line.
Have participants complete Grade 3, Module 5, Lessons 14 & 16.
Partners discuss, then whole group discusses:
What is the complexity between L14 and L16?
L14 decomposes 1 into parts using number bonds, fraction strips, and number lines. L16 decomposes whole numbers greater than one into parts using a number line.
Why do we want students to think about fractions beyond 1?
Fractions represent how whole numbers can be decomposed into smaller parts. Just as we can decompose, for example, a given number of hundreds and tens into smaller units, we can decompose whole numbers (ones units) into smaller parts as well  eventually, this relates not only to fractions greater than one but also to decimal numbers.
If we only examined the distance between zero and 1, students would not see that any place on the number line can be named. Perhaps, students could surmise that fractions are only numbers between zero and one.

6. A Story of Units www.EngageNY.org
December NTI - Fractions - Grades 3-5
TIME ALLOTTED FOR THIS SLIDE:
7 Minutes
MATERIALS NEEDED:
Models are again used here this time to show equivalence. Concrete and pictorial experiences help lead to understanding.
Slide 6: ¾ = 6/8
3.NF.3: Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.
Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line.
Recognize and generate simple equivalent fractions, e.g., ½ = 2/4, 4/6 = 2/3. Explain why the fractions are equivalent, e.g., by using a visual fraction model.
Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: Express 3 in the form of 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point of a number line diagram.
Have participants complete G3-M5, Lesson 22. (In the CD of this lesson, students use fraction strips and the number line, along with area models to show/prove equivalency. Here, on the Problem Set, work is more abstract.)
Obviously, this lesson explores equivalent fractions. Analyze these problems.
What adds to the complexity of these problems?
Support of equivalency using other models has been taken away.
Although equivalent amounts are shaded, they don’t appear to be the same.
What concrete experiences might students need to be exposed with prior to this Problem Set?
Pattern blocks – composing different shapes
Paper unit squares – manipulating size and shape of them – to ‘build’ what is seen here and to prove equivalency. If students struggle to see the equivalence pictorially, they can cut paper squares to prove equivalency.

7. A Story of Units www.EngageNY.org
December NTI - Fractions - Grades 3-5
TIME ALLOTTED FOR THIS SLIDE:
10 Minutes
MATERIALS NEEDED:
Having a firm grasp of equivalence, we are armed with strategies to look at fractions that are not equivalent and to compare them.
Slide 7: 4 as the Number of units/numerator
3.NF.3.d: Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, <, and justify the conclusions, e.g., by using a visual fraction model.
How does this G2-M2 visual help students to understand the size of units? (Relate centimeters to meters, ones to tens, fifths to thirds.)
Units represent different things. Here, the quantity is ‘4’ of each. The question is ‘4’ of what unit. 4 centimeters means something different than 4 meters, etc.. To compare, we must look at the unit and also at the quantity. We need to think, “How many of what?” If the “of what” is not identical, than the “How many” is insignificant in comparing quantities.
Give participants time to complete G3-M5, Lesson 28.
Then ask:
If we know that the units are of different sizes, but that the quantity of each is the same, we can use the quantity to help us compare (e.g., 4 meters > 4 cm). In other words, the how many is the same, but the of what is different.
How can we build on this understanding?
With fractions, we can use the numerators to compare fractions (e.g., thirds are greater than fifths, therefore 2 thirds are greater than 2 fifths, assuming that the two fractions refer to the same whole). How can understanding this help us compare 2/3 and 4/5? Write these 2 fractions under the document camera.
If we know that 2/3 is equivalent to 4/6, we are then comparing 4/6 to 4/5. Sixths is a smaller unit than fifths. Therefore, 4/6 < 4/5  2/3 < 4/5. (The visual for this is on the problem set of G3-M5-L28.)
What other ways could you compare 2/3 and 4/5 using mental math?
We can think of the units of thirds and fifths. 2/3 is 1 third less than 1. 4/5 is 1 fifth less than 1. Since thirds are greater than fifths, 2/3 will be further from 1 and, therefore, be less than 4/5.
Although not a Grade 3 standard, one could find a common denominator: 2/3 = 10/15 and 4/5 = 12/15; therefore, 4/5 > 2/3.

8. A Story of Units www.EngageNY.org
December NTI - Fractions - Grades 3-5
TIME ALLOTTED FOR THIS SLIDE:
5 Minutes
Grade 4 begins with decomposition of unit fractions using part/whole relationships, something with which students are familiar.
Slide 8: Decomposing a group of Ducks & Chickens
GRADE 4
Since kindergarten, students are learning that we can decompose any number into parts. The concept of part and whole starts early, and we continue to explore this throughout the Story of Units.
How does this prior learning support work with fractions?
Just as we can decompose whole numbers into smaller parts, we can also decompose fractions into smaller parts.

9. A Story of Units www.EngageNY.org
December NTI - Fractions - Grades 3-5
TIME ALLOTTED FOR THIS SLIDE:
10 Minutes
MATERIALS NEEDED:
Unit fractions are represented as the sum of smaller unit fractions. Grade 4 starts with decomposition of fractions building from G3’s knowledge of equivalence.
Slide 9: Decomposing ½
4.NF.3: Understand a fraction a/b with a>1 as a sum of fractions 1/b.
a. Understand addition and subtraction of fractions as joining and separating parts referring to the same whole.
Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions, e.g., by using a visual fraction model. Examples: 3/8 = 1/8 + 1/8 + 1/8; 3/8 = 2/8 + 1/8.
4.NF.4: Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.
a. Understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to represent 5/4 as the product 5 x (1/4), recording the conclusion by the equation 5/4 = 5 x (1/4).
How is this number bond like the one with the ducks and chickens?
We are looking at the part/whole relationship.
How is it different?
The units are fractions instead of whole numbers. Instead of representing whole numbers as two or more parts of a whole, we express a fraction less than 1 as two or more smaller parts.
Have participants study the G4-M5 Module Overview. Reminder that Grade 4 is limited to denominators of 2, 3, 4, 5, 6, 8, 10, and 12.
What prior understanding do G4 students need to be successful in this module?
Topic A: Decomposition, unit fractions, equivalence, tape diagrams, area model, part/whole relationships
Topic B: Area model, number lines, whole number multiplication and division, equivalence
Topic C: Representation of fractions on a number line, finding equivalent fractions
Topic D: Whole number addition and subtraction
Topic E: Decomposition of fractions, use of visual models, line plots
Topic F: Decomposition, whole number addition and subtraction
Topic G: Whole number multiplication, associative and distributive property, multiplicative comparison, line plots
Topic H: Patterns, whole number addition
How does the first half of the module support the second half of the module?
Students use what they learned in the first half of the module to build their understanding of fractions in the second half of the module. They apply their understanding of fractions less than 1 (first half) to fractions greater than 1 (second half).
Have participants complete G4-M5, Lessons 4 & 6.
Why use a tape diagram and an area model to decompose fractions? What multiple models do we use at lower grades to decompose whole numbers?
It allows us to visually see the decomposition and how the shaded part stays the same; it’s the size of the units and the quantity of the unit that is changing.
At lower grades, we use 5- and 10-group cards, Hide-Zero cards, place value disks and charts, Number paths, Rekenrek, number bonds, and tape diagrams.
How does the sophistication of recording increase? What does this prepare students for?
L4 – The problems increase in complexity. The first problems are shown in their entirely and then the other problems are scaffolded, requiring the students to do more and more work. It moves from decomposition of unit fractions to decomposition of non-unit fractions.
L6 – We decompose horizontally to show fraction equivalence. This prepares students to find equivalent fractions in a more abstract manner, recognizing that, for example, when we find the equivalence of 2/3 and 4/6, we are doubling the quantity of units and also making the size of the units half the size (thus doubling).

10. A Story of Units www.EngageNY.org
December NTI - Fractions - Grades 3-5
TIME ALLOTTED FOR THIS SLIDE:
20 Minutes
MATERIALS NEEDED:
Here, we see how the increased complexity of what we just examined comes into play as we compose to larger units.
Slide 10: 10/12
4.NF.1: Explain why a fraction a/b is equivalent to a fraction (n x a)/n x b) by using visual fraction models, with attention to how the number and the size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.
Read the Teach Using Feedback Protocol on the last page in the packet.
Let participants know they will be expected to teach this strategy to a peer in moments.
SUPER MODEL: (Refer to Grade 4, Module 5, Lesson 10.)
Ask participants to draw an area model that represents 10/12.
If we want to compose an equivalent fraction with smaller units, what can we do?
We can make groups of 2. If we do, we don’t have any remaining groups.
Ask participants to show this on the area model.
There should be 5 groups of 2 units shaded. There should be a sixth, unshaded group.
Write the equivalent fraction.
5/6.
Let’s consider the unit fractions of 1/12 and 1/6. What do you notice about the denominators?
6 is a factor of 12.
What do you notice about the numerators of 5 and 10?
5 is a factor of 10.
List the factors of 10 and 12.
10: 1, 2, 5, 10
12: 1, 2, 3, 4, 6, 12
Are there any common factors?
Yes, 1 and 2.
How does 2 relate to the area model you drew?
We made equal groups of 2.
Model how this can be written as a division problem.
DELIBERATE PRACTICE: G4-M5-L10 Problem Set 1(b) (9/12) and 1(c ) (6/10)
Review the protocol. Find a partner.
Facilitator uses a stopwatch to monitor the time and lead the group.
Remind Partner A not to respond to Partner B’s feedback. This is not time to discuss. Accept their feedback and your possible errors, looking forward to your improvement.
Have the participants complete G4-M5, Lesson 10.
What is the complexity in Problem 2?
When completing 2(b), we could compose to 4/6 or 2/3.
2(a) and 2(b) have the same fractional amount shaded in each. We can look at the relationship of the (a) and (b).

11. A Story of Units www.EngageNY.org
December NTI - Fractions - Grades 3-5
TIME ALLOTTED FOR THIS SLIDE:
16 Minutes
Armed with the knowledge of expressing a fraction as the sum of units fractions and of composing and decomposing to express equivalent fractions, students are ready to add and subtract with mixed numbers and fractions. Here, we examine prior learning that will help students be successful.
Slide 11: & 400 – 30
4.NF.1: Explain why a fraction a/b is equivalent to a fraction (n x a)/n x b) by using visual fraction models, with attention to how the number and the size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.
4.NF.3: Understand a fraction a/b with a>1 as a sum of fractions 1/b.
Ask participants to answer this question: Using mental math how could you solve the problem ?
( ) + (5 + 4) =
( )
( )
( )
Other ways are also possible.
In pairs, ask participants to answer this question. Then discuss it as a whole group.
What is the big idea that students must know in order to add these two numbers?
We can add like units; ones with ones and tens with tens.
We can add on one unit at a time; add the ten and then add the ones.
We can decompose either (or both) of the numbers in order to add more efficiently.
Model: using number bonds. This is how we help young students visualize the decomposition that eventually leads to mental math.
Model on the document camera Make a Ten using Decompose 8 as
Model How would you add using mental math and the Make the Next Ten strategy? Show using number bonds and the equation below.
(37 + 3) + 43 = 83 (or equivalent).
(CLICK) Advance the animation and repeat the questions.
We can add to make the next ten and then add the remainder of the second addend to find the total.
Model on the document camera 400 – 30, and model 52 – 26, 17 – 9.
400 – 30
Take from 1 hundred using number bonds: 100 – 30= 70; = 370
Unit form: 40 tens – 3 tens = 37 tens = 370
52 – 26
Take from the next ten using number bonds: 30 – 26 = 4; = 26
52 – 26 = 56 – 30 = 26 (Compensation – added 4 to the whole & the part or the minuend & subtrahend)
17 - 9
Take from 10: 10 – 9 =1; = 8
Compensation: 17 – 9 = 17 – = 8
Ask participants to complete G4-M5, Lesson 22.
How are the strategies Take from Ten, Take from the Next Ten, or Take from a Hundred like the strategy practiced in Lesson 22?
1(c-d) – We are taking from 1. Isn’t this like what they learned in earlier grades?
3(a-b) – We are taking from 1.
We make an easier problem by decomposing the minuend.

12. A Story of Units www.EngageNY.org
December NTI - Fractions - Grades 3-5
TIME ALLOTTED FOR THIS SLIDE:
7 Minutes
Here, we examine other strategies for adding and subtracting with whole numbers.
Slide 12: Number Path (G1-M1) and Number Line (G2-M3)
4.NF.3b: Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions, e.g., by using a visual fraction model. Examples: 3/8 = 1/8 + 1/8 + 1/8; 3/8 = 1/8 + 2/8; 2 1/8 = /8 = 8/8 + 8/8 + 1/8.
Discuss at tables what this model is and how it could be used.
It’s a number path.
Write 2 number sentences to match. Draw a number bond to support the part-whole relationship.
6 + 3 = 9
9 – 6 = 3
CLICK.
It’s a number line.
We can use it to show how we can add two numbers, focusing on the units.
We can use it to show how we can solve a subtraction problem by adding on to find the difference, again focusing on units.
We can use it to decompose a given number.
= 900
900 – 776 = 124
Have participants complete G4-M5-L24.
Pair-Share:
How would you describe what you did in this Problem Set?
We renamed a fraction as a mixed number by pulling out whole numbers.
How is this different than the way we used to simplify fractions?
We can visually see that they are the same as opposed to “dividing the numerator by the denominator and expressing the remainder as a fraction” which is the way that we likely learned.
How could this same idea be expressed as a number bond?
1(a) shows the solution using a number bond to express 11/3 as 9/3 + 2/3. 9/3 = /3 = 3 2/3.
How can whole number work with number line models (including the arrow way) support upcoming work with fractions?
We can use the same model to rename a “fraction greater than 1” into a mixed number. (This is the answer participants should say after studying G4-M5-L24.)
We can use the same strategies that we used with whole numbers to add and subtract with fractions.

13. A Story of Units www.EngageNY.org
December NTI - Fractions - Grades 3-5
TIME ALLOTTED FOR THIS SLIDE:
10 Minutes
MATERIALS NEEDED:
Representations of the arrow way. When subtracting fractions, we use these strategies.
Slide 13: ; 84 – 21; 84 – 19
4.NF.3c: Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction.
What’s going on here?
These are Grade 2 examples of the arrow way to add and subtract.
Adding to make the next ten; adding on a unit of ten; subtracting a unit of ten; subtracting one from a ten.
What model is similar to the arrow way?
The open number line, or the number line.
Give participants time to discuss what they see.
How would you generalize this strategy?
Making a simpler problem by adding and subtracting the ones and tens in two separate steps (units of ten and units of one) or by subtracting an easier number and then adding back the difference.
What do students need to know to be able to do these strategies?
Expressing a number as the sum of its parts (units)
Adding tens, adding ones.
Subtracting tens, subtracting ones.
Have participants analyze G4-M5, Lessons 30 and 33.
How is the Arrow Way used for fractions?
In the same way as whole numbers. We show how we can make the next whole, add to the next whole and then add on, subtract by adding on, or subtract by decomposing, or simply adding or subtracting like units.
How does the sequence scaffold in L30?
In Problem 1, students are adding like units with no regrouping 1(a) and then adding like units, making the next one (1b).
In Problem 2, students are thinking about how much more they need to get to the next one.
In Problem 3, students are decomposing the second addend in order to make the next one and then are adding on.

14. A Story of Units www.EngageNY.org
December NTI - Fractions - Grades 3-5
TIME ALLOTTED FOR THIS SLIDE:
20 Minutes
MATERIALS NEEDED:
In G5, the complexity increases….. 3 2
Slide 14: Rectangular Array & Area Model: Using the area model to Add and Subtract Fractions with unrelated units (G5-M3).
5.NF.1: Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators. For example, 2/3 + 5/4 = 8/ /12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.)
What model precedes the rectangular array?
The array is used in kindergarten to help students organize objects in order to decide, “How many?” Ten frames are one example of arrays used in many grade levels.
The rectangular array is used in Grade 2, Module 6 to introduce multiplication and division to students.
Review the Teach Using Feedback Protocol on the last page in the packet (used earlier).
Let participants know they will be expected to teach the modeled strategy to a peer.
Remind them to switch being Partner A and B so each person gets a turn to play “teacher”.
SUPER MODEL Round 1, G5-M3-L3:
Use area models to solve #1a: ½ + 1/3 and ½ + 2/3.
DELIBERATE PRACTICE Round 1: G5-M3-L3 Problem Set #1e ¾ + 1/5.
Review the protocol. Find a partner.
Facilitator uses a stopwatch to monitor the time and lead the group.
Remind Partner A not to respond to Partner B’s feedback. This is not time to discuss. Accept their feedback and your possible errors, looking forward to your improvement.
SUPER MODEL Round 2: (G5, M3, Lesson 12).
Use area models to solve #1a: 3 1/5 – 2 ¼ .
DELIBERATE PRACTICE Round 2: G5-M5-L12 Problem Set #1b (4 2/5 – 3 ¾ ).
Upon completion, briefly examine the Module Overview for G5-M3 in the Participant Handout. Look for comparisons in the sequence of learning from G4-M5 and how G4-M5 sets students up for success in G5-M3.
G5 -Topic A – Equivalent Fractions – All the standards in Topic A are Grade 4 standards.
G5 -Topic B – Making Like Units Pictorially -
G5 -Topic C – Making Like Units Numerically –
G5 -Topic D – Further Applications –
G4 – Fraction equivalence (Topics A,B,&E) sets students up for TA and TB of G5;
G4 – Finding common units (Topic C) – sets students up for finding common units (equivalence in G5);
G4 – Fraction addition and subtraction (Topics D&F);
All Topics – Relate to G5 – Further Applications
NOTE: Although the sequence jumped from G4 to G5 fraction addition and subtraction, on the next slide we will return to G4 fraction multiplication and continue to G5 fraction multiplication and division.
RECTANGULAR ARRAY
AREA MODEL

15. A Story of Units www.EngageNY.org
December NTI - Fractions - Grades 3-5
TIME ALLOTTED FOR THIS SLIDE:
10 Minutes
6 cm
We now switch gears to multiplication and division. Multiplication of fractions is introduced in G4.
Slide 15: 6 cm
4.NF.4: Apply & extend previous understanding of multiplication to multiply a fraction by a whole number.
5 minutes
3 minutes:
The purpose of this component is to clarify the meaning of a/b as a multiple of 1/b and to also show the relationship between measurement units and fractional units in regard to their behavior with the operations. (SEE G4-M5-Lesson 35 for further explanation of using unit form to multiply a whole number and a fraction.) Ask the participants:
True or false?
6 cm = 6 x 1 cm
Ask the participants to apply this same thinking to 6/5. (6/5 = 6 x 1/5)
6 fifths = (6 x 1) fifth = 6 x (1 fifth) = 6 x 1/5
Have participants review G4-M5, Lesson 35.
2 minutes
Debrief experience and be sure to bring out the easing of the students into fractional notation and what a stumbling block that can be without using unit form first. Call out the use of the associative property.
This is the associative property used to decompose a number into its unit parts. 6 x 1 cm is really showing the repeated addition of 1 cm + 1 cm + 1 cm + 1 cm + 1 cm + 1 cm. In turn 6 x 1/5 is showing the decomposition of 6/5 into its fractional units: 1/5 + 1/5 + 1/5 + 1/5 + 1/5 + 1/5.
We know this is true from early on in our counting experiences: 3 crackers is composed of 1 cracker + 1 cracker + 1 cracker. And in Grade 3 we see that as 3 crackers = 3 x 1 cracker.
Units change, but the mathematics, or the arithmetic, remains constant and true.

16. A Story of Units www.EngageNY.org
December NTI - Fractions - Grades 3-5
TIME ALLOTTED FOR THIS SLIDE:
15 Minutes
MATERIALS NEEDED:
We can use the associative property and what we know about multiplication to explain how to multiply a whole number by a fraction.
3 x 2 cm
3 x 2
Slide 16: 3 x 2; 3 x 2 cm
4.NF.4: Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.
The purpose of this sequence of 3 problems is to show the relationship between whole number multiplication with various units, multiplication of measurement units, and multiplication of fractional units.
Ask participants to consider the meaning of the factors 3 and 2 with their partner.
3 groups of 2
3 twos
3 lengths of 2 cm or 3 times as much a 2 cm
Share out whole group about their meaning, labeling the Grade 3 significance for now as the first factor indicating the number of groups, the second factor meaning the size of the group. What is the unit? (twos)
Ask participants to tell their partner a number sentence for the array of ducks.
3 x 2 = 6
Share out whole group: 3 times 2 ducks. What units are we using now? (ducks)
In the Participant Handout, follow the directions for making a tape diagram to solve for 3 x 2 cm. (Problem 1)
Share out whole group the significance of equal groups in the tape diagram. What is the unit? (centimeters)
In the Participant Handout, follow the directions for making a tape diagram to solve for 3 x 2 fifths. (Problem 2)
Share out whole group about how the previous examples shed light on successfully answering a whole number times a fraction problem. How is using unit form advantageous with fractions? Can you rename 6/5 as 1 1/5 using a number bond?
Using unit form helps us to relate what we know about whole numbers to fractions.
6/5 = 5/5 + 1/5; 5/5 = 1; 1 + 1/5 = 1 1/5
Model how the associative property works to solve this problem using unit or numerical form:
3 x 2 fifths = 3 x (2 x 1 fifth) = (3 x 2) x 1 fifth = 6 x 1 fifth or (6 x 1) fifth = 6/5
3 x 2/5 = 3 x (2 x 1/5) = (3 x 2) x 1/5= 6 x 1/5 or (6 x 1)/5 = 6/5
(Perhaps have participants try one on their own.)

17. A Story of Units www.EngageNY.org
December NTI - Fractions - Grades 3-5
TIME ALLOTTED FOR THIS SLIDE:
20 Minutes
MATERIALS NEEDED:
Here, we revisit the distributive property as it applies to mixed units in order to recognize its application to fractions.
Slide 17: Array of 3 x 6, 3 x 42; 3 x 4m 60cm
4.NF.4: Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.
4.NF.4c: Solve word problems involving multiplication of a fraction by a whole numbers, e.g., by using visual fraction models and equations to represent the problem.
Write a multiplication problem for the array. (3 x 6)
Use the distributive property, guided by the shading of the array, to write 2 multiplication problems that equal 3 x 6.
[(3 x 5) + (3 x 1)]
Use unit form to describe the array.
(3 fives + 3 ones)
Solve for 3 x 42 using the distributive property, separating the tens and ones. Write this as a multiplication equation and in unit form.
[(3 x 40) + (3 x 2)] = 126
OR [(3 x 4 tens) + (3 x 2 ones)] = 12 tens + 6 ones = 126
What other times have we used the distributive property to solve multiplication problems?
We also use the distributive property when we divide.
The models we use to show this are the Rekenrek with basic facts, the area models and number bonds.
3 minutes:
In the Participant Handout, have teachers draw a tape diagram representing 3 x 4 m 60 cm (Problem 3). (See Lessons 37 and 38 of G4-M5.)
Model after they do, showing 3 x 4 meter 60 cm tapes first not distributed to be adjacent and then redistributed to be adjacent.
4 minutes
Ask them, would this same model apply to multiply 3 x 4 3/5?
In the Participant Handout, have teachers draw a tape diagram representing 3 x 4 3/5 (Problem 4).
As you see is necessary, release, guide or show the work with the model and numerically. 3 x (4 + 3/5) and its equivalence to 3 x x 3/5 including the bond of 9/5 as 5/5 and 4/5.
4 minutes: Deliberate Practice
Re-read protocol for Deliberate Practice and prepare to practice Problem 1 from G4 Lesson minute prep time, 1 minute teach time, 30 seconds feedback, 1 minute teach, 30 seconds feedback.
5 minutes:
Conclude by having participants solve the problems in the Participant Handout from G4 Lesson 39.
Whole group shares out Problem Set analysis and experience.
NOTE: We will be returning to this idea in grade 5, when we solve a fraction times a whole number by using division.
3 x 42
3 x 4 m 60 cm

18. A Story of Units www.EngageNY.org
December NTI - Fractions - Grades 3-5
6 ÷ 2
TIME ALLOTTED FOR THIS SLIDE:
18 Minutes
MATERIALS NEEDED:
Let’s revisit the two types of division. We will see how this relates to work with fractions.
Slide 18: 6 ÷ 2
10 minutes
Have participants discuss the meaning of the divisor and of the unknown.
Review the distinction between measurement and partitive division first with 6 divided by 2 and then with 4 divided by ½ as pictured below.
Construct the work collaboratively and interactively with the participants, using both direct, guided and partner sharing processes to clarify the content. Different groups will need different kinds of support. Have participants generate word problems corresponding to each type.
Have participants review the Module Overview for G5-M4 in the participants’ packet.
Group Size Unknown
Partitive Division
“Dealing Cards”
Number of Groups Unknown
Measurement Division
“Repeated Subtraction”

19. A Story of Units www.EngageNY.org
December NTI - Fractions - Grades 3-5
TIME ALLOTTED FOR THIS SLIDE:
17 Minutes
MATERIALS NEEDED:
3 ÷ 2
In Grade 5, students begin to see a fraction as division and division as a fraction.
Slide 19: 3 Divided by 2
5.NF.3: Interpret a fraction as division of the numerator by the denominator (a/b = a divided by b).
Ask participants to draw a picture of this fraction problem. Hopefully there will be two interpretations.
Model on the document camera. Show 3 same colored post-it notes to represent the dividend. Under the post-its draw 2 circles to show the number of groups.
Think of this as 3 crackers divided by two people. What do we know? What is unknown?
We know the group size, but not the size of the group. (Mention to participants that we will return to this idea of group size unknown.)
Slide one full post-it note into each group. What do we do with the other piece?
We cut it in two equal pieces, and place a piece in each group.
What is the answer? What is the size of the group?
1 ½
Show 3 different colored post-it notes to represent the whole. Again, draw the 2 groups.
We need to know how much in each group. This time, however, we should consider that the post-it notes are 3 crackers of different flavors. Both people would like to taste each cracker. How does this change our picture?
Each piece is divided into 2. The answer is 3/2 rather than 1 ½.
Show this by dividing each post-it and moving it into the group.
Have participants practice 2 divided by 3, and 3 divided by 4.
If we do enough of these problems, what will students notice?
That we can interpret a division problem as a fraction.
Model unit form on the document camera:
Let’s go back to 3 divided by 2. What’s my divisor? 2.
Let’s state the whole (the dividend) as halves. How many halves in 3?
6 halves.
What is 6 halves divided by 2? How many is 6 divided by 2? What’s the unit?
6 divided by 2 is 3, and the unit is halves. Our answer is 3 halves.
Model the standard algorithm for 3 divided by 2.
Have participants complete G5-M4, Lessons 2 & 3 in the packet.

20. A Story of Units www.EngageNY.org
December NTI - Fractions - Grades 3-5
TIME ALLOTTED FOR THIS SLIDE:
10 Minutes
MATERIALS NEEDED:
Seeing a division problem as a fraction helps us to solve multiplication of a fraction by a whole number.
Slide 20: 1/3 of 18
5.NF.4: Apply and extend previous understanding of multiplication to multiply a fraction or whole number by a fraction.
a. Interpret the product of (a/b ) x q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a x q divided by b.
Ask participants to draw a tape diagram to match this problem.
When did we see a fraction times a whole number before now?
In Grade 4.
How did we interpret a fraction times a whole number?
We used the multiplication symbol, and we used repeated addition. So this problem would have been interpreted as 1/3 + 1/3 + 1/3 …… (eighteen times). We would have used the associative property to think “1 times 18 thirds.”
Model on the tape diagram on the document camera. Write: 3 units = 18; 1 unit = 18 divided by 3 = 18/3 or 6.
How does this thinking evolve from grade 4?
In grade 4 we thought, “1 times 18 thirds.” Now we think, “1 times 18 divided by 3 or 18/3.”
Have participants complete G5-M4-L7 in the packet.
How does Lesson 7 build on Lesson 3?
Lesson 3 taught us to think of a division problem as a fraction. Now we can use this learning to solve multiplication of a fraction.
L3 teaches us that 18÷3 can be interpreted as a fraction, 18/3. So when I think about solving 1/3 of 18, I can interpret the multiplication problem as 1 times 18/3, which is 18/3, the fraction, or the division problem. So now I can easily rename the fraction greater than 1 as 6. It’s showing how multiplication and division of fractions is related.

21. A Story of Units www.EngageNY.org
December NTI - Fractions - Grades 3-5
TIME ALLOTTED FOR THIS SLIDE:
15 Minutes
MATERIALS NEEDED:
2 x 20
Let’s take what we know and extend that learning….
Slide 21: 2 x 20
5.NF.4: Apply and extend previous understanding of multiplication to multiply a fraction or whole number by a fraction.
a. Interpret the product of (a/b ) x q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a x q divided by b.
3 minutes
In Grade 3, this means 2 units of 20 or 2 twenties.
In Grade 4, this can mean 20 times as much as 2.
… twenty times.
We can use multiplicative comparison language or think of the first factor as the unit, the second factor being the number of units.
As shown in G5-M4-L7:  
F: What does 2/5 x 20 mean in Grade 4 fraction work?
P: 2/5 + 2/5 + 2/5 …. Twenty times.
F: In Grade 5, this can now also mean 2 fifths of 20.
2/5 of 20 is represented as 2/5 x 20.
Model the problem.   
4 minutes
What if the unit is a fraction?
1 half of 2 fifths, for example.
Super Model: Use the document camera to model this problem using the area model. Explain to the participants that we are working in the first grid of the area model.
Also model this in unit language. ½ of 2 fifths. How many? 1. Of what? Fifths.
Do other examples as necessary such as 1/3 x 3/4 and 2/3 x ¾.
1/3 of ¾ Think 1 third of 3 is 1. What’s the unit? Fourths. What’s the answer? 1 fourth or 1/4.
2/3 of ¾ Think: 2 thirds of 3 is 2. What’s the unit? Fourths. What’s the answer? 2 fourths or ½.
Often when we solve in unit form, the answer is in a more simple form that if we did the multiplication.
5 minutes
Have participants deliberately practice G5-M4-L14, using examples (a) and (b).

22. A Story of Units www.EngageNY.org
December NTI - Fractions - Grades 3-5
TIME ALLOTTED FOR THIS SLIDE:
15 Minutes
MATERIALS NEEDED:
In Grade 5, students extend their knowledge of division…
1 ÷ 2 = ___
Slide 22: 1 ÷ 2
5.NF.7: Apply and extend previous understanding of division to divide unit fractions by whole numbers and whole numbers by unit fractions.
In pairs, have participants write a story problem that matches 1 divided by 2. Share some examples, and as you do, ask participants to label the stories as either missing group size ( or missing number of groups.
Consider 3 divided by 1/3.
Write a story problem that matches this problem. Determine if the story problem is partitive or measurement. Solve by using a tape diagram or an area model.
I have 3 apples. If I want to give a third of an apple to each friend, how many friends will get a piece? (Measurement – Number of Groups Unknown)
I apple = 3 thirds. Three apples equals 9 thirds. The answer is 9 friends.
I have 3 apples. This is a third of what I need to make applesauce. How many apples do I need? (Partitive - Group Size Unknown)
I need 9 apples.
Model 3 divided by 1/3 using a tape diagram.
Consider 1/3 divided by 3.
Write a story problem that matches this problem. Determine if the story problem is partitive or measurement. Draw a tape diagram or area model to solve.
I have a third of a pan of lasagna. I want to share it equally among 3 friends. How much (of the whole pan) will each friend get? (Partitive - Group size Unknown)
1 third divided by 3 is 1 ninth.
I can also think in unit form. 1 third is the same as 3 ninths. 3 ninths divided by 3 is 1 ninth.
I have a third of a pan of lasagna. I need 3 pans for a big family dinner. How much of the whole amount do I have? (Measurement – Number of Groups unknown)
I have 1 ninth of what I need.
Model 1/3 divided by 3 using a tape diagram.
5 minutes:
Have participants do G5, M4, L25 & 26.
3 minutes
Analyze Problem Set and experience.

23. A Story of Units www.EngageNY.org
December NTI - Fractions - Grades 3-5
TIME ALLOTTED FOR THIS SLIDE:
15 Minutes
MATERIALS NEEDED:
In G5, the complexity increases…..
3 1/2 2
Slide 23: Rectangular Array and Area Model Revisited
5.NF.5: Interpret a fraction as scaling (resizing).
What if our area model now has lengths that weren’t whole numbers? Can we find the area? Ask participants to find the area of a rectangle that is 2 ¾units long by 1 ½ units wide. Draw a model and show your work.
After participants have had an opportunity to work, Super model this problem on the document camera.
Participants should Deliberately Practice G5-M5-L11, Problems 1(b) and 1(c).
RECTANGULAR ARRAY
AREA MODEL

24. A Story of Units www.EngageNY.org
December NTI - Fractions - Grades 3-5
TIME ALLOTTED FOR THIS SLIDE:
5 Minutes
MATERIALS NEEDED:
How did we apply and extend previous understandings of whole numbers to operations with fractions?
Slide 24
Have groups discuss at their tables and then share out whole group.
Discussion may include, but is not limited to, the following:
In Grade 3, when formal introduction to fraction starts, we extend previous understandings by using unit language to describe fractions before we introduce fraction notation.
We can add and subtract with fractions by adding or subtracting like units just like with whole numbers.
The use of models with whole numbers extends to fractions: number path, number bond, number line, tape diagrams, array, and area model.
Understanding of number lines and comparison of whole numbers helps build an understanding of comparison of fractions.
Understanding of part/whole relationships extends to fractions.
We came to understand whole numbers and their operations using concrete, pictorial, and abstract representations. We can do the same with fractions.
We can decompose fractions just as we can decompose whole numbers by breaking them down into smaller parts.
We can rename fractions just as we can rename whole numbers.
Solution strategies of addition (Make a 10, Make the Next Ten, Make a Simpler Problem) and subtraction (Take from 10, Take from the Next Ten, Take from 100) with whole numbers builds a foundation for adding and subtracting with fractions (Make the Next Whole, Add to the Next Whole and then Add More, Take from 1).
We can use the number line and ‘arrow way’ to add and subtract with fractions just as we can use them with whole numbers.
Our understanding of multiplication and division of whole numbers as repeated addition and repeated subtraction extends to fractions.
The associative, commutative, and distributive properties apply to fractions just as they do to whole numbers.
We use our understanding of factors and multiples when finding equivalent fractions.
We can think of division of fractions as partitive division and measurement division just as we do when dividing whole numbers.

25. A Story of Units www.EngageNY.org
December NTI - Fractions - Grades 3-5
TIME ALLOTTED FOR THIS SLIDE:
3 Minutes
MATERIALS NEEDED:
Slide 25: Story of Units Curriculum Overview Map
2 minutes
Close the session by having participants notice the flow from whole number (yellow) to fraction operation modules (pink).
Thank them and encourage them to use ONE ARITHMETIC for both whole numbers, measurement units, and fractions and thus empower students to do the same.