Elementary School Math for Middle School Teachers
Sequence of Sessions
Overarching Objectives of this March 2015 Network Team Institute

Participants will be able to identify, practice, and use best instructional moves and scaffolds for chosen common core standards.
High-Level Purpose of this Session

Participants will deepen their understanding of foundations laid in grades 2-5 that support student success in middle school mathematics.
Related Learning Experiences



This session is part of a series supporting implementation of A Story of Ratios and A Story of Functions.
Sessions on Crafting Teaching Sequences for Interventions will also develop teachers’ ability to customize the curriculum to meet the specific needs of
their students.
Module Focus sessions also support implementation of the curriculum by closely examining each module in A Story of Ratios and
A Story of Functions.
Key Points



Participants will analyze the introduction, foundation and use of the Distributive Property, grades 2 through middle school.
Participants will analyze the algorithms we use for all four operations, and how these algorithms are linked to manipulatives and models.
Participants will analyze the two interpretations of division: partitive and measurement.
Session Outcomes
What do we want participants to be able to do as a result of this
session?
 Participants will be prepared to implement the modules and to make
appropriate instructional choices to meet the needs of their students while
maintaining the balance of rigor that is built into the curriculum.
How will we know that they are able to do this?
Participants will be able to articulate the key points listed above and will
demonstrate the ability to develop a plan for implementing the curriculum to
meet the specific needs of their students.
Session Overview
Section
Time
Overview
Prepared Resources
•
Introduction
1 min
Introduces the session objectives
regarding the Distributive
Property, the long division
algorithm, and partitive and
measurement division.
•
•
The Distributive
Property
90 min
Explores the Distributive Property
and its mathematical foundations.
•
•
Partitive and
Measurement
Division
45 min
Explores partitive and
measurement division and their
mathematical foundations.
•
•
The Long Division
Algorithm
45 min
Explores the long division
algorithm and its mathematical
foundations.
•
Elementary School Math
for Middle School
Teachers PPT
Elementary School Math
for Middle School
Teachers Facilitator
Guide
Facilitator Preparation
Review Elementary School Math
for Middle School Teachers PPT
and Facilitator Guide
Elementary School Math
for Middle School
Teachers PPT
Elementary School Math
for Middle School
Teachers Facilitator
Guide
Review Elementary School Math
for Middle School Teachers PPT
and Facilitator Guide
Elementary School Math
for Middle School
Teachers PPT
Elementary School Math
for Middle School
Teachers Facilitator
Guide
Review Elementary School Math
for Middle School Teachers PPT
and Facilitator Guide
Elementary School Math
for Middle School
Teachers PPT
Elementary School Math
for Middle School
Teachers Facilitator
Guide
Review Elementary School Math
for Middle School Teachers PPT
and Facilitator Guide
Session Roadmap
Section: Introduction
Time: 1 minute
In this section, you will begin to examine the session objectives
Materials used include:
regarding the Distributive Property, the long division algorithm, and
 Elementary School Math for Middle School Teachers PPT
partitive and measurement division.
 Elementary School Math for Middle School Teachers
Facilitator Guide
 Elementary School Math for Middle School Teachers
Participant Handout
Time Slide # Slide #/ Pic of Slide
Script/ Activity directions
1 min
Welcome! Introduce the presentation team.
MATERIALS FOR PRESENTERS:
• Facilitators’ Guide – This power point has minimal notes. The
facilitators’ guide is meant to be studied, and then used with this
slide set.
• Linker cubes – 20 of two colors
• Copy of G4, M3, L22 concept development for reference, if needed.
• Square tiles
• Place value disks
• Large Rekenrek (100 beads)
1.
MATERIALS FOR PARTICIPANTS:
• Template Grade 2, Module 6, Lesson 14, page 66 (three 2 x 4
rectangular arrays)
• Template Grade 3, Module 4, Lesson 9, page 4.C.14 (a 10 x 10 grid)
• Copy of Grade 4, Module 5, Lesson 37, Concept Development
• Scissors
• Rulers
• Place Value Disks
• Square Tiles
• Personal white boards, markers, erasers
GROUP
Section: Distributive Property
Time: 90 minutes
In this section, you will explore the Distributive Property and its
mathematical foundations.
Materials used include:
 Elementary School Math for Middle School Teachers PPT
 Elementary School Math for Middle School Teachers
Facilitator Guide
 Elementary School Math for Middle School Teachers
Participant Handout
Time Slide # Slide #/ Pic of Slide
2.
Script/ Activity directions
Participant Packet Pages 1 and 2
The focus of the first part of this session is a study of THE DISTRIBUTIVE
PROPERTY.
Start with the middle school assessment.
Ask the participants to complete the Grade 6, Mid-Module Assessment.
Then, they should analyze the student work.
• What misconceptions are evident?
• What gaps are evident?
3.
Participant Packet Page 3
(This demonstration should be deliberately practiced before the session.)
Display 7 linker cubes showing 5 of one color, and 2 of another color.
“How could you show me the same quantity using your fingers?”
• Participants will show 5 + 2.
“What property did we just model?”
• The Distributive Property.
• In the lower grades we call this Decomposing and Break Apart and
Distribute.
Refer back to the linker cubes. “What if the value of each cube is now 3?
Write an equation that shows this distribution.”
• (5 x 3) + (2 x 3) = 15 + 6 = 21
Switch to the document camera.
GROUP
“We do such activities starting in Grade 3 with a variety of models. One of
the most important models is the Number Bond.” In this demonstration,
use the words Part, Whole, Total.
Show the bonds that represent what we just did:
• 5+2=7
• 5 threes + 2 threes = 7 threes Help the participants see that unit form
is a very common way that we express quantities throughout The
Story of Units. Add a short drill. Deliberate practice. Give them
examples.
• (5 x 3) + (2 x 3) = 21
Uncover the connection to associated ratios. Connect this to the language of
parts and wholes.
Let’s rearrange the linker cubes to determine a comparison. (Rearrange
cubes to show the ratio 5:2) This is a ratio A:B. Note that not much has
changed here for middle school students. We can represent this
relationship with a number bond to clearly see the ratios associated with
5:2. (Display number bond and discuss and label the part as “A” and also as
“Part” and place the first number in the ratio (5) in the circle. Discuss the
second part of the number bond. Label the second part as “Part” and now as
“B” and place the second number in the ratio (2) in the circle.) We can
determine from the linker cubes that the total is 7 units. Let’s place the total
of 7 in the top circle, and now what might we label this portion of the
number bond? We know we can label it the total, but what ratio notation is
represented, here? We can label this portion or total “A+B.”
Using the number bond, we can determine the associated ratios B:A, A:A+B,
B:A+B where B:A is another part to part ratio, A:A+B is a part to total (or
part to whole) ratio and B:A+B is a part to total (or part to whole). Using
your whiteboards, show the associated ratio of B:A. Show the associated
ratio of A:A+B. Show the associated ratio of B:A+B.
Just as the number bond in primary grades allows us to use unit form, we
can also apply this concept to ratios. And we do. Let’s look again at the
ratio 5:2 using the number cubes first. We use these cubes at the middle
school level to represent 5 units to 2 units. We build strategically the
concept of equivalent ratios using the cubes first, and then representing
them with a tape diagram. Regardless of the representation, we can see the
connection to the primary grades with the unit form. What if each of the
cubes represented 3 units? (Draw the tape diagram and label one unit “3”)
If each unit represents 3, what is the equivalent ratio to 5:2? 5 threes and 2
threes. The ratio is 15:6. Can we build associated ratios from here? What
would be the associated ratio B:A? 6:15. Let’s turn and talk and see where
you think this concept will lead us:
The ratio B:A has a value of B/A, which can also be represented as y/x and
leads into scale factor, constant of proportionality, similarity and dilations,
slope, rate of change, etc.
Scenario: A student is confused in a ratio word problem. How could we use
number bonds, unit form to help him/her?
4.
Participant Packet Pages 4
Briefly show participants the sequence we use to help kindergarteners
count objects: arrange the objects in a line, in an array, in a circle, in a
scattered configuration using a “counting path.” The array allows students
to begin to subitize rather than count all. You subitized when you saw 7 as 5
and 2. Subitizing leads to counting on, which bridges on to reasoning in
parts and wholes.
Do the Grade 2, Module 6, Lesson 6 page in the participants’ packet.
Ask:
• What are the parts (or units) in these arrays? (Participants should
see the columns and rows.)
• What vocabulary is used in this lesson? Row and columns. Eventually
this will lead to student understanding of each of the factors in a
multiplication problem.
• What does the number bond for this graphic look like?
• Using objects, such as plastic bears, allows the students to concretely
move the objects in order to count them. In Grade 2, students start to
see that placing objects into equal groups allows them to add the
same number over and over. They skip count or use repeated
addition to find the total number in the group. In Grade 3, students
will start to multiply.
Analyze:
•
5.
Where do we find the roots of the distributive property?
Participant Packet Pages 5 and 6
PURPOSE OF THIS SLIDE: RECTANGULAR ARRAYS: Concrete
We shift to using square tiles, which allows students to make rectangular
arrays. This is the precursor to the area model.
Do Grade 2, Module 6, Lesson 13 page in participants’ package.
Do the Grade 2, Module 6, Lesson 15 page in the participants’ packet.
6.
Participant Packet Pages 7, 8 and 9
PURPOSE OF THIS SLIDE: Introduction to the Distributive Property
Do the Grade 3, Module 1, Lessons 9 & 10 pages in the participants’ packet.
Note on this page the use of unit form.
Afterwards, show how we can use the Rekenrek (or number rack) to
decompose 7 x 5 into (5 x 5) + (2 x 5).
Ask participants how they responded to unit form questions on the top of
page 9.
Have participants complete Grade 3, Module 1, Lesson 18.
In Grade 3, module 2, students are learning basic facts, and they are
introduced to tape diagrams for multiplication and division. We use
fluencies to help students memorize facts.
7.
Participant Packet Pages 10 and 11
PURPOSE OF THIS SLIDE: Introduction to the Area Model
In grade 3, students tile rectangles with square tiles, both centimeter and
inch. In this transitional lesson, students are given rectangles. They use
rulers to draw the squares, and then they count the number of squares to
find the area. In Grade 3, students use the terms “length” and “width.”
Previously, in Grade 2, students use the term “sides.”
Do the Grade 3, Module 4, Lesson 4 page in the participants’ packet.
In the problem set the length units are changing to square units.
What is happening to the units?
8.
Participant Packet Page 12
PURPOSE OF THIS SLIDE: Analyze the rectangular array and reason about
the factors.
Have participants complete Grade 3, Module 4, Lesson 9 page in the
participants’ packet.
Think, and then Pair and Share:
How can we use the concept taught in L9 to solve 225 12 by using mental
math?
The example (225 x 12) is not a grade 3 problem. However, the grade 3
strategy taught in this lesson can be used to solve a harder problem.
One way to solve this using mental math is to think, “I’ll halve one factor and
double the other.”
• 225 x 12 = This is not mental math for me.
• 450 x 6 = I struggle to solve this mentally as well.
• 900 x 3 = 2700. Bingo! I can do this in my head.
This strategy has its roots in Grade 3.
9.
Participant Packet Pages 13 and 14
PURPOSE OF THIS SLIDE: Distributive Property: 2-digits x 1 digit and then
3-digits x 1-digit
Analyze Grade 3, Module 4, Lesson 10 and then Grade 3, Module 4,
Lesson 11 pages in the participants’ packet.
10.
Participant Packet Pages 15 and 16
PURPOSE OF THIS SLIDE: Multi-Digit Factors with the Area model &
Multiplication Algorithm
The CC standards require Grade 4 students to multiply up to four-digit x 1digit numbers, and then two-digit x 2-digit numbers. (4.NBT.5)
We, of course, start with one-digit multipliers as shown in Lesson 9.
By Topic H in this module, we are introducing 2-digits x 2-digits. To build to
the standard algorithm, we use the area model and partial products. We
carefully connect all three (area model  partial products  standard
algorithm).
With the document camera model how we multiply 30 x 35 and then 34 x
35.
This is shown on page G4, M3, L36 on page 3.H.28. Show the area model,
with unit form, the partial products algorithm and the standard algorithm.
This is where we need to discuss the advantages of “new groups below”
algorithm.
Then have participants analyze Grade 4, Module 3, Lesson 37. Ask them
to include unit form, which is not required on this worksheet.
Multiplication of Polynomials.
11.
Participant Packet Pages 17 and 18
PURPOSE OF THIS SLIDE: Multiplication of Decimals using the area model.
This may seem out-of –order because we are skipping to grade 5. However,
this relates directly to what was done in grade 4 with the area model.
Have participants do Grade 5, Module 1, Lesson 11 in the participants’
packet. How is this like what we did in Grades 3 and 4?
Complete Grade 5, Module 2, Lesson 10 in the participants’ packet. What
we see in Lesson 10 is similar to how we learned to multiply decimals years
ago. We first multiply, and then consider the unit. We ask, “How much?
What’s the unit?” “How much of what?”
12.
Participant Packet Pages 19 and 20
PURPOSE OF THIS SLIDE: find the product of a whole number times a
mixed number using the distributive property.
We step back to grade 4, to look at the distributive property with fractions.
Have participants analyze the Concept Development of Grade 4, Module
5, Lesson 37.
This shows 2 copies of 3 and 1fifth using a tape diagram:
3 + 1/5 + 3 + 1/5
Then the parts of the tape are rearranged to show:
3 + 3 + 1/5 + 1/5
After the analysis, have the participants complete Grade 4, Module 5,
Lesson 37, Problem Set.
13.
Participant Packet Pages 21 and 22
PURPOSE OF THE SLIDE: Use the area model to solve two binomials.
Have the participants analyze Algebra, Module 4, Lesson 1.
PURPOSE OF THIS SLIDE: Progression of the concept to middle school
standards.
Talk to your table partner. How do these models progress? Where do we
change from pictorial to representational?
When students get to the algorithm, there are still levels of abstraction. We
can multiply and get each partial products that shows in the area model, or
we can combine steps and have only as many partial products as there are
in one of the factors – the standard algorithm.
14.
Section: Partitive and Measurement Division
Participant Packet Pages 23 and 24
Time: 45 minutes
In this section, you will explore partitive and measurement division. Materials used include:
 Elementary School Math for Middle School Teachers PPT
 Elementary School Math for Middle School Teachers
Facilitator Guide
 Elementary School Math for Middle School Teachers
Participant Handout
Time Slide # Slide #/ Pic of Slide
15.
Script/ Activity directions
Participant Packet pages 25 -33
Purpose of this slide is to discuss Partitive & Measurement Interpretation
of Division
These questions are in Grade 3, Module 1, Lesson 4, Concept Development
on page 1.B.5
What does the 12 represent in the picture?
• The total or the whole.
What does the 3 represent?
• Number of groups
What does the 4 represent?
• Number in each group.
Write a division sentence to match this problem.
• 12 / 3 = 4
• This story problem is asking us to find the size of the group. This is
the PARTITIVE interpretation of division.
Advance the animation to show both equations. What is the difference
between these division sentences?
• 12/4 = 3 does not match this story problem. We’d have to rewrite it
to say, “Mr. Ziegler had 12 markers. He put four in a bag. How many
bags does he need?” In this example, the number of groups is
unknown, hence it is the MEASUREMENT interpretation of division.
How would you show both equations in a number bond?
What would these 2 problems look like drawn in tape diagrams?
Have participants complete page Grade 3, Module 1, Lesson 6 in the
packet. Ask them which problem is partitive and which story problem is
measurement?
After participants have an understanding of partitive & measurement, have
them complete the participant pages for Grade 2, Module 6
• Problem 1 is measurement.
• Problem 2 is partitive.
GROUP
Have participants look at Grade 2, Module 6, Lessons 18 & 20. How are
finding even / odd numbers related to partitive and measurement division?
Have them discuss this at their tables.
NOTE: I would love to see a connection here to the even and odd
numbers with divisibility in primary and in G6 which leads into a great
discussion about distributive property.
Finally, have participants complete Grade 3, Module 1, Lesson 12. This
lessons shows the bar models for both interpretations of division.
16.
Participant Packet page 34 – Segue into division algorithm
Section: Long Division Algorithm
Time: 45 minutes
In this section, you will explore the long division algorithm and its
foundations.
Materials used include:
 Elementary School Math for Middle School Teachers PPT
 Elementary School Math for Middle School Teachers
Facilitator Guide
 Elementary School Math for Middle School Teachers
Participant Handout
Time Slide # Slide #/ Pic of Slide
Script/ Activity directions
GROUP
17.
Participant Packet pages 35 and 36
The focus of the second part of this session to study the algorithms through
the grade levels.
PURPOSE OF THIS SLIDE: ADDITION - Compose larger units in order to add
using the chip model and linking it to the standard algorithm.
Model 26 + 35 using the place value disks as you record the work in the
algorithm. Participants can follow along as they read the Concept
Development of Grade 2, Module 4, Lesson 7.
The benefits to “new groups below.”
• We can easily see 12 tens.
• The students write the greater digit before the lesser digit. (In this
example, they write 1 then 2.)
• They add the one last; usually this is easier to do.
Ask the participants to use the disks as they add 26 + 15.
Show that we can also draw the disks, or use dots on a place value chart.
Model 44 + 26 using the chip model. Complete Grade 2, Module 4, Lesson 8
using the chip mode.
Finally, they analyze Grade 5, Module 1, Lesson 9 concept development.
This shows decimal subtraction.
18.
Participant Packet Pages 37, 38, 39
THE PURPOSE OF THIS SLIDE: Subtraction algorithm.
Model 35 – 9 using the disks. The teachers can follow along with their disks,
and by reading the cd for Grade 2, Module 4, Lesson 11.
In Grade 2, Module 4, Lesson 13 we move to the chip model, showing 2
digits minus 1 digit.
We make a big jump to Grade 5, Module 1, Lesson 10 and subtract
decimals.
19.
Participant Packet Page 40
PURPOSE OF THIS SLIDE: Multiply using place value disks, then use the
chip model and link to the partial products algorithm and standard
algorithm.
Model 4 x 54 using number disks. Have participants follow along with
disks. Link the chip model to the algorithms.
Participants can follow along by reading the concept development of Grade
4, Module 3, Lesson 7.
They should draw the chips to do 5 x 42.
20.
Participant Packet Pages 41 – 48
PURPOSE OF THIS SLIDE: Use place value disks divide.
Have participants use the place value disks to divide 36 by 3. They can
follow the concept development Grade 4, Module 3, Lesson 21
21.
Participant Packet Pages 49
PURPOSE OF THIS SLIDE: Using the Distributive and Associative Property
to divide.
Use the document camera to show 1.2 divided by 60. This is example 1b on
page Grade 5, Module 2, Lesson 24.
Give participants time to analysis Grade 5, Module 2, Lesson 24.
22.
PURPOSE OF THIS SLIDE: Progression of Models
Talk to your table partner. How do these models progress? Where does it
change from pictorial to representational? How can we show this in an
abstract way?
When students get to the algorithm, there are still levels of abstraction. We
can multiply and get each partial products that shows in the area model, or
we can combine steps and have only as many partial products as there are
in one of the factors – the standard algorithm.
Use the following icons in the script to indicate different learning modes.
Video
Reflect on a prompt
Active learning
Turnkey Materials Provided
•
•
•
Elementary School Math for Middle School Teachers PPT
Elementary School Math for Middle School Teachers Facilitator Guide
Elementary School Math for Middle School Teachers Participant Handout
Additional Suggested Resources
•
•
How to Implement A Story of Ratios
A Story of Ratios Year Long Curriculum Overview
Turn and talk