K-5 Principals: Math Instruction Across the Grade Band
Sequence of Sessions
Overarching Objectives of this October 2014 Network Team Institute

Participants will be able to identify, coach, and provide feedback to support best instructional moves and scaffolds for chosen common core standards.
High-Level Purpose of this Session


Participants will understand how to draw connections between the progression documents and the careful sequence of mathematical concepts that
develop within each grade, thereby enabling them to support teachers as they enact cross-grade coherence in their classrooms.
Participants will be prepared to lead implementation of the modules within their schools and to identify appropriate instructional choices made to meet
the needs of their students while maintaining the balance of rigor that is built into the curriculum.
Related Learning Experiences
●
This session revisits some of the texts used in the early years of NTI as a tool to monitor our own progress over the past two years, recognize what we
still want to work on, and to collaborate with colleagues to create concrete steps that may help us get there.
Key Points


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Knowing the progression across the grade band prepares teachers to more effectively scaffold by knowing what comes before and after a particular
concept on the “ladder” of complexity.
Knowing the progression across the grade provides insight into how each concepts contributes to students’ overall conceptual understanding.
Knowing the progression across the grade band illustrates the power of coherent/consistent teaching of models, strategies, and approaches.
We want to create a culture in our schools where assessment and date result in analysis that drives planning practices, and from there teaching.
Session Outcomes
What do we want participants to be able to do as a result of this
session?



Knowing the progression across the grade band prepares teachers to
more effectively scaffold by knowing what comes before and after a
particular concept on the “ladder” of complexity.
Knowing the progression across the grade provides insight into how each
concepts contributes to students’ overall conceptual understanding.
Knowing the progression across the grade band illustrates the power of
coherent/consistent teaching of models, strategies, and approaches.
How will we know that they are able to do this?

Participants will be able to articulate the key points listed above.

We want to create a culture in our schools where assessment and date
result in analysis that drives planning practices, and from there teaching.
Session Overview
Section
Time
Overview
93 min
 Math Instruction Across the
Recaps topics covered in the July
Grade Band PPT
2014 sessions and revisits some of
 Math Instruction Across the
the texts used in the early years of
Grade Band Facilitators Guide
NTI as a tool to monitor our own
 CCSS Instructional Practice
progress over the past two years.
Guide
 Review CCSS Instructional
Practice Guide
Math Learning within
the K-2 Progression, 13 min
Level 1, Counting All
Examines how addition and
subtraction are introduced and
developed with coherence in A
Story of Units, starting with the
Counting All computation method.
 Math Instruction Across the
Grade Band PPT
 Math Instruction Across the
Grade Band Facilitators Guide
 Review Standards: K.OA. A3,
2.OA.2, 2.NBT.5
 Also review: K.CC.A.1,
K.CC.A.3, K.CC.B.4 and
K.CC.B.5.
Math Learning within
the K-2 Progression, 42 min
Level 1, Counting On
Examines the Counting On
computation method used in the
primary years to solve addition
and subtraction problems.
 Math Instruction Across the
Grade Band PPT
 Math Instruction Across the
Grade Band Facilitators Guide
 Review Standards: 1.OA.C.6,
and 1.OA.A.2
 Review Grade 1 Module 1
Math Learning within
the K-2 Progression,
Foundations for
38 min
Understanding Ten:
10 Ones
Explores the Foundations for
Understanding Ten.
 Math Instruction Across the
Grade Band PPT
• Review K-5 NBT Progressions
 Math Instruction Across the
Grade Band Facilitators Guide
 Counting On Video
Math Learning within
the K-2 Progression,
72 min
Foundations Level 3,
Make Ten
Explores using Make Ten in
composition and decomposition
methods and as the foundation of
place value strategies.
 Math Instruction Across the
Grade Band PPT
 Math Instruction Across the
Grade Band Facilitators Guide
Introduction
Prepared Resources
Facilitator Preparation




Review Standard: K.OA. A4
Review Standard: 1.OA.C.6
Review Grade 1 Module 2
Also review: K.NBT.1,
K.OA.3, K.OA.4. 1.NBT.4
 Review Grade 1 Module 4
 Review Standards: 2.NBT.7
and 2.NBT.9
Math Learning within
the K-2 Progression,
11 min
Foundations Level 3,
Take from Ten
Math Learning within
the K-2 Progression,
Use Units of 1s, 10,
114 min
and 100s to Add and
Subtract
Math Learning within
the Grade 3-5
125 min
Progressions
Conclusion
81 min
Explores using Take from Ten in
composition and decomposition.
 Math Instruction Across the
Grade Band PPT
 Math Instruction Across the
Grade Band Facilitators Guide
 Math Instruction Across the
Examines using units of 1s, 10, and
Grade Band PPT
100s to add and subtract.
 Math Instruction Across the
Grade Band Facilitators Guide
 Review Standard: 1.OA.C.6
 Review Grade 1 Module 2
Topic







Review Standard: 1.NBT.4
Review Grade 1 Module 4
Review Grade 1 Module 6
Review Grade 2 Module 3
Review Standard: 2.NBT.3
Review Standard: 2.NBT.6
Review Grade 2 Module 4




Review Grade 2 Module 6
Review Grade 3 Module 1
Review Standard: 3.MD.7
Review Standards: 3.OA.4,
3.OA.6, and 3.OA.7
Review Standard: 4.NBT.6
Review Grade 5 Module 5
Review Standard: 3.NBT.3
Review Standards: 4.NBT.1
and 4.OA.1
Also review: 4.NBT.5
Review CCSS Instructional
Practice Guide
Examines the topics in Grades 3-5
that build upon the work in K-2 to
move students from addition and
subtraction on to multiplication
and division
 Math Instruction Across the
Grade Band PPT
 Math Instruction Across the
Grade Band Facilitators Guide
 CCSS Instructional Practice
Guide




Discusses opportunities to
scaffold, additional coaching tools,
and the importance of planning.
 Math Instruction Across the
Grade Band PPT
 Math Instruction Across the
 Review CCSS Instructional
Practice Guide


Grade Band Facilitators Guide
 CCSS Instructional Practice
Guide
Session Roadmap
Section: Introduction
Time: 93 minutes
In this section, you will recap topics covered in the July 2014
Materials used include:
sessions and revisit some of the texts used in the early years of NTI
 Math Instruction Across the Grade Band PPT
as a tool to monitor our own progress over the past two years.
 Math Instruction Across the Grade Band Facilitators Guide
 CCSS Instructional Practice Guide
 Graphic Organizer
Time Slide # Slide #/ Pic of Slide
Script/ Activity directions
0 min
This session is designed to be 6 ½ hours in length, including two 15-minute
breaks.
1.
Materials for this session:
• “Practice Perfect/Switch” professional reading excerpts
• “Coaching Tools/Core Actions” professional reading excerpt (CCSS
Instructional Practice Guide)
•
Problem Set excerpts (Grades 3-5)
•
Handouts from the K-2 and 3-5 Grade Band sessions (including
graphic organizers)
•
Place Value Disks (Thousands to Thousandths)
•
Personal White Boards
•
Plain white paper
•
Chart Paper/Markers
GROUP
1 min
2.
This session revisits some of the texts used in the early years of NTI as a
tool to monitor our own progress over the past two years, recognize what
we still want to work on, and to collaborate with colleagues to create
concrete steps that may help us get there.
2 min
3.
Let’s just refresh ourselves on where we left off together in July. Here are
some of the main areas for discussion at that time.
1 min
4.
You shared several ideas for supporting teachers at your sites, such as
those on the screen.
40
min
5.
Write on your own for 10 minutes.
Talk with a partner for 20 minutes.
Whole group: 2 shared successes and 2 challenges (10 minutes).
Whole group: What do you hope to leave with after these next two days?
20
min
6.
Write on your own for 10 minutes.
Talk with a partner for 10 minutes.
20
min
7.
15 minutes with foursome
Complete this section by creating a sticky: What’s one area you want to
flesh out or work through, when it comes to math, over the next two days?
Write that twice- once for you to hold on to, once for us to gather on a
chart. (2 minutes)
(Break)
1 min
8.
On a scale of 1 to 10, where do you feel your teachers are with PUFM and
math PUID?
On a scale of 1 to 10, where do you feel YOU are with PUFM and PUID?
3 min
9.
Give participants about 2-3 minutes to review the Instructional Practice
guide and to familiarize themselves with the Core Action Indicators. These
indicators help to ensure the lesson reflects the Shifts required by the CCLS
for Mathematics.
2 min
10.
Reference the Core Action (Instructional Practice Guide) handout. Indicate
that, throughout the session, participants should be thinking about what
evidence they might see of the Core Actions within lessons which
incorporate the mathematical concepts that are introduced.
1 min
11.
Have participants examine the graphic organizer :
The purpose of the graphic organizer is for participants to take notes about
the development of the concept at each grade level. This will help them
support teachers as they work with students who may have conceptual
gaps in math understanding and as participants work with teachers who
are trying to identify where their grade level work fits within the
progression of learning across grade levels.
2 min
12.
This is the curriculum map for A Story of Units. It shows the sequence of
five to eight modules for each grade, PK through Grade 5.
Points:
• We are about to focus on the development of addition and
subtraction in Kindergarten, 1st, and 2nd grades.
• The modules in yellow deal with number.
• Coherence means that learning is connected within and across the
grades.
• Each grade level plays an important role in the students’ larger
experience.
• Knowing the work across the grade band means we can be more
flexible in meeting the needs of students performing at varied levels.
From Kindergarten to Grade 5, we move from ‘frogs, bananas, and beans’ to
‘ones, tens, and hundreds’, to ‘threes, fours and sevens’, to ‘thirds, fourths
and sevenths’. ONE big idea, the unit.
Section: Math Learning within the K-2 Progression, Level Time: 13 minutes
1, Counting All
In this section, you will examine how addition and subtraction are
introduced and developed with coherence in A Story of Units,
starting with the Counting All computation method.
Materials used include:
 Math Instruction Across the Grade Band PPT
 Math Instruction Across the Grade Band Facilitators Guide
Time Slide # Slide #/ Pic of Slide
Script/ Activity directions
1 min
13.
Explain the progression of the strategies for composing/decomposing.
2 min
14.
Look at all that Kindergarten does!!!
GROUP
1 min
15.
During our presentation, we will be exploring how A Story of Units for Grade
1 unfolds, with coherence to key concepts in kindergarten and second grade,
in order to support our students into meeting their grade level standards in
addition and subtraction and to build a strong foundation towards their
math understanding and work in the later grades.
By the end of Grade 1, students will:
• Fluently add and subtract within 10, which is the only grade level
fluency standard
• Add and subtract within 20 using strategies such as counting on,
making ten, taking from ten, using relationship between addition and
subtraction (knowing that 8 + 4 = 12, one knows 12 – 8 = 4)
• Use addition and subtraction within 20 to solve word problems
• Add within 100
• Subtract multiples of 10 from multiples of 10 in the range of 10
through 90 such as 90 – 20 = 70.
1 min
16.
Let’s explore how the story continues in G2, at the end of which students
will:
• Fluently add and subtract within 20, and know from memory all
sums of two one-digit numbers (2.OA.2)
• Fluently add and subtract within 100 (2.NBT.5)
• Add and subtract within 1,000
1 min
17.
Summarize below:
Have participants discuss the standards that are addressed as foundations
for addition and subtraction.
(Standards: Number words in sequence- K.CC.A.1 Know number names and
the count sequence. , Number recognition- K.CC.A.3: Know number names
and the count sequence 1:1 correspondence- K.CC.B4: when counting
objects, say the number names in the standard order, pairing each object
with one and only one number name and each number name with one and
only one object Cardinality: K.CC.B4 Understand the relationship between
numbers and quantities, connect counting to cardinality. )
Have participants articulate each standard.
Counting All in Kindergarten:
1. Learning that groups of objects have a numerical value.
2. Learning how to represent that value with an abstract numeral.
Students learn that 1 object is 1 unit, 2 objects is 2 units. 1 frog, 2 frogs,
3 frogs. And here comes 1 more frog (another unit) Now there are 4
frogs! 1 unit, 1 count.
Accurate, efficient counting of numbers 1-10 is the focus of the first 43 days
of Kindergarten. A lot of time is spent here because this is the foundation of
everything that follows. Students need 4 key understandings (the number
core):
• the number words in sequence
• 1:1 correspondence (connecting one number word to each object)
• reading, writing, and understanding the written numeric symbols
• Shifting from counting to cardinality, recognizing that the last
number word said when counting tells the total. This is crucial
because it leads to counting on. (Model by counting participants at a
table.) “How many are at this table?” “1, 2, 3, 4.” (Repeat.) “How
many are at this table?” “1, 2, 3, 4.” This student thinks the count is
the answer. He doesn’t understand that “4” names the total. When a
student makes the shift, he’ll be able to answer “4” and then count on
from that number. (Model counting two tables.) “How many at these
two tables?” “Fouuur, 5, 6, 7, 8, 9.”
In this curriculum, students move along a pathway from concrete to
pictorial to abstract, as shown in the image.
2 min
18.
Summarize below and include indicator as noted:
Core Action 3: Indicator D
How could you elicit a turn and talk about these varied configurations?
(3 minutes to discuss and 3 minutes to share out)
Counting All fluency activities:
Hands Number Line to 3 (5 minutes)
Materials: (S) Left hand mat, bag of beans (painted red on one side)
T: How many hands do you see on your mat?
S: 1.
T: How many real hands do you have?
S: 2.
T: Put 1 of your real hands down on the mat so that it matches the picture of
the hand exactly. Make sure to line up all of your fingers.
T: Take 1 bean out of your bag and put it on the pinky fingernail. How many
fingers have a bean?
S: 1
T: Which finger is it?
S: Pinky.
T: Show me your real pinky f start counting with (demonstrate).
S: 1 (hold up the pinky finger of the left hand, palm out).
T: Put another bean on the very next finger. How many fingers have beans
on them now?
S: 2.
T: Show me which fingers have beans. Use your mat to help you. on fingers
from 1 to 2. Ready?
S: 1 (hold up the pinky finger of the left hand), 2 (pinky and ring finger,
palm out).
T: Put another bean on the very next finger. How many fingers have beans
on them now?
S: 3.
T: Show me which fingers have beans. Use your mat to help you. on fingers
from 1 to 3. Ready?
S: 1 (hold up the pinky finger of the left hand), 2 (pinky and ring finger,
palm out), 3 (pinky, ring finger, and middle finger, palm out).
T: Very good! See if you can do it without looking at the mat. Close it up
(show closed fist)ready?
S: 1, 2, 3 (show fingers).
Continue practicing so that students get more comfortable with this way of
finger counting.
2. Number glove – M1 L1 (counting all; counting left to right
intentionally)
T: Watch my number glove and count with me. Ready? (Begin with closed
fist, then show the pinky finger, followed by ring finger, and then middle
finger.)
S: 1, 2, 3.
T: Stay here at 3. Let’s count back down to 1. Ready? (Put down the middle
finger, then ring finger.)
S: 3, 2, 1.
Continue counting up and down a few more times.
T: You’re ready for something harder! This time we’ll count up and down,
like a wave. Watch my glove and you’ll know just what to do.
S: 1, 2, 3, 2, 1, 2, 1, 2, 1, 2, 3, 2, 3, 2, 3.....
Listen for hesitation as students
count, rather than count along with them.
3. Line up, Sprinkle, Circle – M1 L10 (counting all; in more difficult
configurations)
Accurate counting of objects in different configurations (linear, array,
circular and scattered); One-to one correspondence (Each unit gets one
count)
T: Put 5 beans in your cup. (Wait for students to do this.) Spill them onto
your mat and put them in a straight line. Touch and count.
S: 1, 2, 3, 4, 5.
T: How many beans?
S: 5!
T: Put them back in your cup. Spill them onto your mat and sprinkle them
around. Touch and count.
S: 1, 2, 3, 4, 5.
T: Are there still 5?
S: Yes!
T: Put them back in your cup. Spill them onto your mat and put them in a
circle. Touch and count.
T: Are there still 5?
S: Yes!
2 min
19.
Summarize below:
Have the participants articulate the standard addressed. (Standard:
K.OA.A3: Understand addition as putting together and adding to, and
understand subtraction as taking apart and taking from. Decompose
numbers less than 10 into pairs in more than one way (using numbers
or drawings) and record each decomposition as a drawing or an
equation. )
Have teachers analyze what the kindergarten student is experiencing,
noticing and learning when engaging with this lesson.
Note: If participants have already seen the key points, do not reiterate them.
Points (do not feel compelled to say everything!!!!):
• Experience of embedded numbers
• Writing of equation starting with total to record the decomposition.
• Movement towards a unit of five.
• Equality of five and five in the image to the right. Equal.
• Growth pattern in orange and yellow colors. Starting to see one
more.
3 min
20.
Lead participants through this fluency:
5-group count on from 5 - M1 L19 (counting on; 5 + n pattern)
Click to advance to the next set of 5-groups. (Horizontal and vertical
representations are shown to remind participants to vary the orientation)
T: (Showing dot card.) Raise your hand when you know how many dots?
(Wait for all hands to be raised, then signal). Ready?
S: 8!
T: This time, just count the dots on the top row. Raise your hand when you
know how many dots on top. (Wait for all hands to be raised, then signal).
Ready?
S: 5.
T: This time, just count the dots on the bottom row. Raise your hand when
you know how many dots on the bottom. (Wait for all hands to be raised,
then signal). Ready?
S: 3.
T: (Showing the 7 dot card.) Raise your hand when you know how many
dots? (Wait for all hands to be raised, then signal). Ready?
S: 7!
T: Top? (Wait for all hands to be raised, then signal). Ready?
S: 5.
T: Bottom? (Wait for all hands to be raised, then signal). Ready?
S: 2.
T: Count from 5. Ready?
S: 5, 6, 7.
Reducing the questions to as few words as possible (top, bottom) once
students understand the essential task, will allow students to complete a
greater volume of problems in a short time, and maintain an energetic pace.
It’s important to allow wait time after asking students how many in the top
and bottom rows. Some students may see 5, but others may need to count
each dot.
Section: Math Learning within the K-2 Progression, Level Time: 42 minutes
2, Counting On
In this section, you will examine the Counting On computation
Materials used include:
method used in the primary years to solve addition and subtraction
 Math Instruction Across the Grade Band PPT
problems.
 Math Instruction Across the Grade Band Facilitators Guide
Time Slide # Slide #/ Pic of Slide
Script/ Activity directions
2 min
21.
Level 2: Counting On
Poll the participants about their experiences with counting on prior to
using the modules or their experience with counting on. Facilitate a
discussion about counting on.
If no one shares: share an experience when students struggled with
counting on. (When they need to count each finger on their hand, no
matter how many times they have counted them and realized there
were 5)
1 min
22.
Standard: K.CC.B.4: Count to tell the number of objects. Have teachers
analyze the sequence from simple to complex. Have them consider the
activity from a student’s point of view. What is the student
experiencing and learning?
Points:
• The foundation of counting on begins in Module 1 with seeing
embedded numbers within a larger group.
• Students start to see embedded numbers as units to count on from.
GROUP
2 min
23.
Point out another Indicator here.
Points: (Do NOT feel compelled to say everything.)
• “More than” “less than” are introduced in Module 3, the
measurement module, following the use of “longer than/shorter
than”, “heavier than/lighter than,” and “more than/less than” with
liquid capacity. Here, students are seeing that the next number is one
more. PRACTICE restraint. “More than” is complex language. Wait for
it to be introduced in Module 3.
• “One more” will later become “plus one.”
• They are beginning to be able to see five as a unit.
• They find embedded numbers in M1 and "see 2" without counting!
• That supports them to see 2 as a unit.
• That in turn leads to an understanding of multiplication, or copying a
unit, repeating it.
2 min
24.
Summarize below:
Points:
• Number stairs show the pattern of 1 more which leads to plus one,
the first step in counting on. Do NOT use the word “than here.” (See
next bullet.)
• Students construct this stair again in Module 3, the measurement
module, as students transition from longer than/shorter than, heavier
than/lighter than, more than/less than (with capacity), more
than/less than (with quantity).
• The smaller number is the structure on which we build 1 more to
make the next number.
• Number stairs show a color change at 5 (benchmark fluency).
• This representation helps in understanding numbers 6-10 in relation
to 5.
• Leads into addition and subtraction and measurement: more than,
less than, plus 1, minus 1, etc.
2 min
25.
Summarize below; throughout all slides, consider elements from
Instructional Practice Guide that can be highlighted and share as
pertinent.
Points:
• Students certainly can add 9 and 6 using Level 1 and 2 strategies.
• Research suggests that HOW students learn to add to and subtract
from the teens matters, that this is the point where their place value
understanding is forming.
2 min
26.
Summarize below:
Standard: 1.OA.C.6: Add and subtract within 20. Add and subtract
within 20, demonstrating fluency for addition and subtraction within
10. Use strategies such as counting on, making a ten, decomposing a
number leading to a ten, using the relationship between addition and
subtraction and creating equivalent but easier or known sums.
The big idea of Grade 1 Module 1 is around bringing students from counting
all to counting on in order to add and subtract numbers within 10. As you’ll
recall from the Progressions Document, there are 3 Levels to the addition
and subtraction strategies that the students employ. In GK, students
practiced Level 1 counting strategies where each object was counted as 1
unit. They also worked on Level 2 counting strategies by seeing embedded
numbers within a larger group.
For example, (hold up or show under doc cam) in a tower of 7, 5 are blue
and 2 are orange. They see 5 as a unit and count on 2 more to get to 7.
Now, in G1, we work with students to independently use the Level 2
strategy of counting on as they continue to see 1 addend as a unit to which
they add on or count on from.
Students will develop this strategy by:
• using embedded numbers in 5-group formations as
well in varied configurations of dots
and use this strategy as the basis to:
• find decompositions of numbers 6 through 10 in
various combinations
• solve addend unknown/change unknown problems
and
• understand the meaning of subtraction as it relates to
addition.
Let’s look at each of these experiences to see how they support student
understanding and use of counting on.
4 min
27.
Summarize below
Standard: 1.OA.C.6: Add and subtract within 20. Add and subtract
within 20, demonstrating fluency for addition and subtraction within
10. Use strategies such as counting on, making a ten, decomposing a
number leading to a ten, using the relationship between addition and
subtraction and creating equivalent but easier or known sums.
Lead the participants through the mini lesson using the 5 group cards.
Ask them to consider how the lesson, both the context and the use of
the 5 group cards, helps students to count on.
Demonstrate Lesson 6 Concept Development.
T:
Partner A, show how many animals there are on land with your cards,
using the number side.
S:
(Show the numeral 5.)
T:
Partner B, show how many animals there are in the pond your cards,
using the dot side.
S:
(Show 3 dots.)
Note to the PRESENTER: SHOW THE CARDS UNDER THE DOCUMENT
CAMERA.
T/S: Let’s find the total. We can count on from the number 5, using the 3
dots. Fiiiiive, 6, 7, 8. (Count, while pointing to the dots.)
T:
Work with your partner to write a number sentence that matches our
animals on land and in the pond on your personal white board.
S:
(Write 5 + 3 = 8 or 8 = 5 + 3)
T:
How else can we sort these animals into 2 groups? (ducks/frogs,
adults/babies)
Note to the PRESENTER: SWITCH BACK TO THE POWERPOINT AND SHOW
PICTURE.
Have participants use the picture and come up with other ways to decompose
8.
Some students may not need the dots to help use the counting on strategy.
They may be able to look at, say 5 and 3, choose the greater number and
count on from that larger unit in their head or by tracking with their fingers.
Points:
• Students use concrete and pictorial situations to describe all of the
decompositions of 6, 7, 8, 9 and 10 into 2 smaller units.
• Students continue to see an embedded number which becomes an
addend from which they count on to find the total.
• Students use 5-group cards to help count on from different
embedded numbers to compose 8 and find all decompositions of 8.
• They then record their decompositions in number bonds and as
expressions to total 8.
2 min
28.
Summarize below, focusing on fluency overall rather than only specific
fluency activities. Use specifics as an example across all grade levels.
Standard: 1.OA.C.6: Add and subtract within 20. Add and subtract
within 20, demonstrating fluency for addition and subtraction within
10. Use strategies such as counting on, making a ten, decomposing a
number leading to a ten, using the relationship between addition and
subtraction and creating equivalent but easier or known sums.
USE NUMERAL DICE RATHER THAN DOT DICE IN TARGET PRACTICE.
Ask the participants why we chose to use numeral dice rather than dot
dice.
Ask them what they think the goal of these fluencies is?
After using 5-group cards to count on, the goal is for students move on to
using the abstract number rather than relying on pictorial representation.
Number Bond Dash and Target Practice are 2 fluency activities that provide
continued practice for students to become fluent in decomposing numbers
on an abstract level.
Please find an example of Number Bond Dash in your handout and insert it
into your personal white boards, if you haven’t done so already.
In this particular Number Bond Dash, students are to complete as many
number bonds for 8 as they can in 90 seconds. Students get many
opportunities to work on decomposing numbers 5 through 10 using
different versions of Number Bond Dashes and see themselves making
improvements as they move on through the module.
Let’s have everyone do the Number Bond Dash. You’ll get 10 seconds,
although the students will be getting 90. (Give 10 seconds for workshop
participants. During this time, set up the document camera for “Target
Practice”.)
You will also find a template for Target Practice in your handout. The
teacher determines the target number, for instance, 7, which students write
on top of their paper. The student rolls a die and writes the number as one
of the parts in the number bond and determines and fills in the other part.
For instance, if the target number is 7 and the student rolls 2, she writes 2 as
one part and figures out and writes 5 to complete 7. (Demonstrate on the
document camera and have the participants say what numbers to fill in for
the number bond parts as the participant rolls the die.)
1 min
29.
Summarize below:
1.OA.A.2: Represent an solve word problems involving addition and
subtraction
Be transparent about our design of the PD experience. Tell the participants
that we deliberated about whether to use the three card or 3 bears and
decided not to use the 3 bears (you might replace the card with the 3 bears
to clarify the point). Why do you think we made that decision?
1 min
30.
Summarize below:
Standard: 1.OA.C.6: Add and subtract within 20. Add and subtract
within 20, demonstrating fluency for addition and subtraction within
10. Use strategies such as counting on, making a ten, decomposing a
number leading to a ten, using the relationship between addition and
subtraction and creating equivalent but easier or known sums.
Later in the module, students learn the meaning of subtraction as it relates
to addition. Using the same lesson with the bears playing in the wood,
students can see that subtraction can also be used to solve the problem. In
the previous demonstration, we solved the mystery number by setting up an
equation with a missing addend: 3 + ___ = 5. Let’s see how this problem can
be solved using subtraction.
Demonstrate Lesson 25.
T: In our story from the previous slide, what does the 5 stand for? (create a
number bond of 5, with one part 3, and one missing part.)
S: The number of bears playing at the end.
T: What does 3 stand for?
S: The number of bears playing in the beginning.
T: How many bears came over to play?
S: 2 bears.
T: Many of you used addition to figure out how many bears came over to
play. What addition sentence did you use? We know the whole and one
part. To find a missing part, we can also subtract.
As students practice solving these add to with change unknown story
problems, they write the number bond to match the parts and the total from
the story and write both addition and subtraction number sentences.
Representing the story as a number bond is a powerful method as students
see the relationship between the parts and whole within the story and can
relate it to both addition and subtraction sentences.
9 min
31.
Summarize below:
Standard: 1.OA.C.6: Add and subtract within 20. Add and subtract
within 20, demonstrating fluency for addition and subtraction within
10. Use strategies such as counting on, making a ten, decomposing a
number leading to a ten, using the relationship between addition and
subtraction and creating equivalent but easier or known sums.
This PATH needs to be in the materials list.
In M1, the number path is used for counting on to find the unknown part in
an addition problem as well as its related subtraction problem. This model
is reminiscent of the Rekenrek beads and the use of 5 groups. When
counting on, students can start by circling the addend and hop forward one
number at a time as they count on until they reach the total.
(Demonstrate on document camera.) First, when adding 2 numbers, 4 and 2
to find the total, we would circle the 4, and hop two places to add 2 and land
on 6.
To solve the problem, (write) 4 + ___ = 6, students first ask themselves, “4
plus what number is the same as 6?” since the equal sign means “the same
as.”
(Demonstrate.) They circle the 4 and hop as they count on until they reach
the total of 6. They see that the unknown part, the missing addend, is 2.
(Continue demonstration under the document camera.) When solving a
subtraction problem such as (write) 6 – 4, students can either use related
addition method by starting with one part and hopping forward to the total
as we just did. Or students may start at the total, circling 6 and hop
backwards to take away the given part. To solve 6 – 4, students have the
option of starting at 4 and counting their hops to get to 6, or staring at 6 and
counting backwards 4 hops to get to 2.
(Switch back to the power point.) Use the number path and work with your
partner to solve the problems on the slide. (Give 60 – 90 seconds.)
7 + ___ = 8, 8 – 7 = ___
2 + ___ = 9, 9 – 2 = ___
8 + ___ = 9, 9 – 8 = ___
As you use the number path to solve , think about the following question:
When is counting on more efficient and when is counting back more
efficient? This is a discussion students have after having lots of experience
solving related addition and subtraction problems.
Core Action 2: Indicator D
At your table discuss some ways you can check for understanding during the
Concept Development. (Allow 3 minutes to discuss, 3 minutes to share out)
1 min
32.
Summarize below:
Standard: 1.OA.C.6: Add and subtract within 20. Add and subtract
within 20, demonstrating fluency for addition and subtraction within
10. Use strategies such as counting on, making a ten, decomposing a
number leading to a ten, using the relationship between addition and
subtraction and creating equivalent but easier or known sums.
As we just discussed, sometimes, it’s more efficient to count on when the
total and the part are close to each other. As the total gets larger and the
part gets smaller, it can be more efficient to count back, as in 2 + ___ = 9. You
would want to start from 9 and just count back 2 times.
13 min
33.
Summarize below:
Standard: 1.OA.C.6: Add and subtract within 20. Add and subtract
within 20, demonstrating fluency for addition and subtraction within
10. Use strategies such as counting on, making a ten, decomposing a
number leading to a ten, using the relationship between addition and
subtraction and creating equivalent but easier or known sums.
By the end of Module 1, students add and subtract within 10 at the abstract
level. As you can see from this sprint, which you’ll find in your handouts,
students are able to look at a number sentence, whether it be addition or
subtraction and put their knowledge to work in finding the unknown
number.
What skills or strategies have we shared until this point that you can
imagine students employing to solve abstractly?
*Partners to 10
*Relationship between addition and subtraction
*Counting on to find the missing addend
*Understanding number sentences written in different order
Give the sprint: Modeling the sprint experience in the classroom.
Section: Math Learning within the K-2 Progression,
Foundations for Understanding Ten: 10 Ones
Time: 38 minutes
In this section, you will explore the Foundations for Understanding
Ten.
Materials used include:
• Major Work of the Grade Band: Grades K-2 PPT
• Major Work of the Grade Band: Grades K-2 Facilitator Guide
• Counting On Video
Time Slide # Slide #/ Pic of Slide
Script/ Activity directions
2 min
Summarize below:
What are the foundations for understanding 10?
Why 10 ones rather than 1 ten.?
Facilitate a discussion about the foundations for place value.
1.
GROUP
2 min
2.
Summarize below:
Have participants analyze the 5-group cards, why they are important.
Points:
• Shows 5 as a unit in each of these numbers and reinforcing the 5 + n.
• Organizing a unit of ten in this way will be used as students add,
subtract, multiply and divide in a place value chart with large
numbers in later grades
• Different from a ten frame in that there is no grid and the fives are
separated more, to allow students to quickly discern the fives.
• Derived from important research by Karen Fuson
• In Kindergarten we use the 10-frame starting in M5 when teen
numbers, the 10 ones + n ones pattern is introduced.
2 min
3.
Summarize below:
Have student articulate the standard addressed in this activity.
(Standard: K.NBT.A.1: Compose and decompose numbers from 11-19
into ten ones and some further ones.)
Have participants analyze the teen number bonds. What experiences
in M5 might the students have before this?
What would a kindergarten student be experiencing, noticing, and
learning when making these bonds?
Point:
In the curriculum, the first lesson with the ten ones and some ones is done
at the concrete level entirely, not abstract level. The students see 15 beans
both in the whole and in the parts.
This then supports their understanding so deeply of the meaning of the digit
1 in the tens place, that it has a value of 10 ones. This is so important, for the
students to have many experiences of the value of that digit being ten ones.
They have been rote counting the Say Ten way, now they get into the
concept and return to the rote with a new understanding.
2 min
4.
Summarize below:
Standard: K.NBT.A.1: Compose and decompose numbers from 11-19
into ten ones and some further ones
What would a kindergarten student be experiencing, noticing and
learning when making these bonds?
Point:
Why are ten frames preferable to popsicle stick bundles of 10 in
kindergarten? (can be counted and recounted)
*Distinguish between K and 1st grade standards: *
K.NBT.1 “…ten ones and some further ones” (In the curriculum we say
“some more ones” or specify how many ones.)
1.NBT.2a “10 can be thought of as a bundle of ten ones—called a ‘ten.’”
10 min
5.
Summarize below:
Standard: K.NBT.A.1: Compose and decompose numbers from 11-19
into ten ones and some further ones
Module 5 Brings us to Ten Ones and Some Ones
VIDEO – watch a demonstration on of counting on from 10 the math way
and the regular way (about 3 minutes).
• Teen numbers are learned as “10 ones and some ones” (10 is the unit
used to count on from)
• Students practice saying teen numbers in English and the “Say 10
way”
4 min
6.
Summarize below:
Task:
• Indicate the characters that represent the numbers 1, 2, 3 and 10 on
the chart (not necessary to pronounce the words in Chinese).
• Ask participants to try and figure out how to write the numbers 11,
16 and 20. After wait time lower screen and reveal the answer.
• Ask the participants to try again and write the number 21 in Chinese
characters.
• Can you find the mistake in the last row? (36 is written as “three 6”
rather than 3 ten 6)
Reflection:
• We expect students to understand teen numbers when many of our
conventions are arbitrary.
• Guide participants to realize that the pattern, along with their
knowledge of place value supported their understanding and ability
to complete the task.
Relate to Module:
Refer to this quote in the Module Overview: “They ‘stand’ on the structure
of the 10 ones and use what they know of numbers 1-9.” We just practiced
MP. 7 “Look for and make use of structure.”
• This was our inspiration for counting the “Say Ten” way. In some
countries this is called “Counting the Math Way.”
• Page 5 of the K-5 NBT Progressions refers to this as the East Asian
way of counting.
2 min
7.
Summarize below:
Let’s look at how the 5-group progresses in first grade. The quick ten
developed out of the five groups as they are first in two rows, either
horizontally or vertically.
(CLICK) The formation transforms so the two rows are end to end, and then
again with a line drawn through the dots. The way you saw the 5-group row
or the way students build a stick of ten to show 10 ones can now be thought
of and represented as 1 unit of ten.
(CLICK) Finally, the dots are taken away, leaving only a quick ten. Once the
quick ten is in place, the idea of a unit of ten is firmly established. This one
mark, a simple vertical line, proportionally represents 10 ones. 1 ten = 10
ones.
Again, point out how the quick ten develops out of the five groups as they
are first side by side, then end to end and then drawn with a line through the
five group dots and then the dots are taken away.
14 min
8.
Summarize below:
Have participants articulate the standard that is addressed with these tools
and fluency activities. (Standard: 1.NBT.B.2b: The numbers from 11 to 19
are composed of a ten and one, two, three, four, five, six, seven, eight, or nine
ones.)
Here are some fluencies which will support students in decomposing
numbers into tens and ones.
Happy Counting: the Say Ten Way
(You may use the Rekenrek first. Then use your thumb to signal counting up
and down.)
Let’s count up and down by 1s using the Say Ten Way. (i.e. 9, 10, ten 1, ten
2, ten 3, pause, ten 2, ten 1, pause, ten 2, ten 3, ten 4, etc.)
Now, let’s count up and down by 1’s alternating the Say Ten Way with the
Regular Way. (i.e.,, ten 1, 12, ten 3, 14, pause, ten 3, 12, ten 1, pause, etc.)
Tens and Ones: (Need Rekenrek.)
T:
(Show 16, 6 on top, 10 in the second row, on the Rekenrek). How
many tens do you see?
S:
1.
T:
How many extra ones?
S:
6.
T:
Say the number the Say Ten way.
S:
Ten 6.
T:
Good. 1 ten plus 6 ones is?
S:
16.
Hide Zero Number Sentences:
When I show you a number using my Hide Zero Cards, I want you to say the
addition sentence starting with the tens as your first addend. For example,
if I show you 14. You would say, 10 + 4 = 14. (Continue with other teen
numbers.)
All of the hard work in achieving fluency in these skills will prepare students
for their Level 3 strategy work of Making Ten to add.
**Analyze Problem Set Pages 1 and 2. ** Explain how the PS is
scaffolded. What models are being utilized and why? (10 minutes)
Section: Math Learning within the K-2 Progression,
Foundations Level 3, Make Ten
Time: 72 minutes
In this section, you will explore using Make Ten in composition and Materials used include:
decomposition methods and as the foundation of place value
 Math Instruction Across the Grade Band PPT
strategies.
 Math Instruction Across the Grade Band Facilitators Guide
Time Slide # Slide #/ Pic of Slide
Script/ Activity directions
GROUP
4 min
9.
Summarize below:
Ask participants to show with fist to 5 their familiarity with the three
Levels described on the slide. Fist being unfamiliar and 5 being
complete understanding.
Depending on their familiarity discuss the slide and give examples of
each level.
2 min
10.
Summarize below:
Have participants articulate the standard addressed in Kindergarten
to Make Ten. (Standard: K.OA.A4: For any number 1 to 9 find the
number that make 10 when added to the given number. )
This is the number bond model. What are students experiencing when
they use it?
Possible points:
Have a parent night to introduce the number bonds!
Three key “make ten” ideas in Kindergarten:
1. The decompositions of the numbers to 10.
2. How much more a number needs to make ten.
3. The ten plus facts.
The partners of ten are foundational for the make 10 and the take from 10
strategies that will be learned in grade 1 and 2.
4 min
11.
Summarize below:
Articulate the standard addressed with the Level 3 strategy. (Standard:
1.OA.C.6: Add and subtract within 20. Add and subtract within 20,
demonstrating fluency for addition and subtraction within 10. Use
strategies such as counting on, making a ten, decomposing a number
leading to a ten, using the relationship between addition and
subtraction and creating equivalent but easier or known sums. )
Grade 1 Module 2 is a crucial time period as student learn to use Level 3
strategies where they convert a problem into an easier problem. Some
students may solve 9 + 6 in this way but many will revert to counting on to
actually calculate sums. Nevertheless!!!! It is important to persist in practice
because the students are gaining an understanding of using ten, making ten,
and relationships to ten.
Students are introduced to and are practicing the Make 10 strategy and
Take from 10 strategy in order to add and subtract numbers within 20.
(CLICK) Students at Level 1 count all, starting with the first dot all the way
to 15 dots. Here, each object is 1 unit.
(CLICK) Students at Level 2 understand that the first addend is a unit of 9
and therefore, count on from 9 to get to 15.
(CLICK) Finally, students at Level 3 see each addend as a unit, one of which
can be manipulated to compose a new unit, a ten. At this point, students are
reminded how easy it is for them to add when one of the units is 10. 9 needs
1 more to complete a new unit of 10. 1 is taken from 6 and 5 are now
remaining. We now have 2 new units, 10 and 5, which equal 15. This
strategy is called Making a Ten.
2 min
12.
Summarize below:
Have participants articulate the standard:
K.NBT.1: Compose and decompose numbers 11-19 into ten ones and
some further ones. Understand that these numbers are composed of
ten ones and one, two, three, four, five, six, seven, eight, or nine ones.
K.OA.3: Decompose numbers less than or equal to 10 into pairs in
more than one way, and record each decomposition by a drawing or
equation.
K.OA.4: For any number from 1 to 9, find the number that makes 10
when added to the given number.
Show the equation using the animation. Ask participants to share with
a partner what kindergarten skills were used to solve 9 + 5 by making
ten.
•
•
•
How much more a number needs to make ten
(Partners to ten)
Decompositions of all numbers within 10
The ten plus facts (10+n)
In G1 M1, students gain fluency with these skills preparing to form a unit of
ten in M2.
This is giving the foundation of place value strategies.
To solve 9 + 5, students:
• know the 10 + facts are easy, giving a motivation to
make ten.
• know 9 needs 1 more to make 10. (Partners to ten.)
• take 1 from 5 so must know how to decompose 5
• add 10 + 4.
We will explore the ways in which students continue to practice and
strengthen these skills towards mastery in Module 1 and apply them in the
later modules and grades as they work on Level 3 decomposition and
composition methods and place value strategies
2 min
13.
Summarize below:
Have participants articulate the standard and how it changes in Grade
1.
(Standard: 1.OA.C.6: Add and subtract within 20. Add and subtract
within 20, demonstrating fluency for addition and subtraction within
10. Use strategies such as counting on, making a ten, decomposing a
number leading to a ten, using the relationship between addition and
subtraction and creating equivalent but easier or known sums. )
Have teachers analyze the slide. What is the student experiencing,
noticing, and learning?
Points:
• Module 2 begins with 2 lessons that set students up for the Make Ten
strategy .
• Students solve problems with 3 addends where each problem
encourages students to use the associative and commutative
properties to first compose a unit of 10 with 2 of the addends, then
adding the 3rd addend using 10 plus facts.
• (CLICK) If the problem was 1 + 4 + 9, students can still make 10 with
1 and 9, using the associative property (CLICK) and add on the 4 to
get 14.
Knowing partners to 10, in this case, knowing that 9 needs 1 more to get to
10, and the ten plus fact, as in 10 + 4, helps students easily solve this
problem.
13 min
14.
Summarize below:
Standard: 1.OA.C.6: Add and subtract within 20. Add and subtract
within 20, demonstrating fluency for addition and subtraction within
10. Use strategies such as counting on, making a ten, decomposing a
number leading to a ten, using the relationship between addition and
subtraction and creating equivalent but easier or known sums.
How will knowing 9 + 1 + 4 help students prepare for the make 10 strategy?
Let’s take a look at its related problem, 9 + 5.
(Demonstrate under document camera.)
(Lay out 9 red cubes and 5 yellow cubes in 5-group.)
How many does 9 need to get to 10? (1)
Let’s take 1 from 5. (Take 1 yellow cube and add to the 9 to make 10.)
We just made 10. To show we made 10, which is a friendly number, let’s put
a frame around it.
Let’s look at our new piles. What new piles do you see? (10 and 4)
So, 9 + 5 is the same as? (10 and 4).
What is 10 + 4? (14)
What is 9 + 5? (14)
(Demonstrate.) Here is a pictorial representation of the Make 10 Strategy
using o’s and x’s.
(Demonstrate.) Here is the abstract representation of solving 9 + 5. (Use the
number bond to decompose 5 into 1 and 4 and emphasize the connection to
the cubes and the math drawing. You can see that the 5 yellow cubes and
the 5 x’s were broken apart into 1 and 4.)
As you can see, students compose a unit ten from the larger addend, the 9,
by decomposing the second addend, the 5, into two smaller units, 1 and 4.
9 + 5 is composed to new units, 10 and 4. 10 + 4 = 14, so 9 + 5 = 14.
**Analyze Problem Set Pages 3 and 4.**
What supports are in place for students? Explain how the complexity
builds as you work down the page.
2 min
15.
Summarize below:
Standard: 1.OA.C.6: Add and subtract within 20. Add and subtract
within 20, demonstrating fluency for addition and subtraction within
10. Use strategies such as counting on, making a ten, decomposing a
number leading to a ten, using the relationship between addition and
subtraction and creating equivalent but easier or known sums.
Earlier in the year, students used the number path to 10 to use related
addition sentence to subtract. In Lesson 19, students apply the same idea as
they use related addition sentence to subtract within 20 by using the
number path that’s extended to 20. The number path invites students to
count on. However, students are counting on with efficiency. This is a BIG
step forward and a bridge to making ten to subtract.
For example, if students were to count on to solve 13 – 8 using the number
path, they might do the following.
Starting from 8, they may count eiiight, 9, 10, 11, 12, 13 (use pointer and
gesture hopping one number at a time) and count the number of arrows
made.
But a more efficient way to count on is to make 10 first.
Start from 8
Get to 10 (Click to animate.) by adding 2.
Then get to 13 (Click to animate.) by adding 3 more.
Here, students must know their partners to 10 as well as knowing that a
teen number is of the form10 + n. Eventually, students move away from
having the actual number path and work on a problem such as this by using
just the arrows:
(Click to animate.)
+2 +3
8  10  13
3 min
16.
Summarize below:
Standard: 1.OA.C.6: Add and subtract within 20. Add and subtract
within 20, demonstrating fluency for addition and subtraction within
10. Use strategies such as counting on, making a ten, decomposing a
number leading to a ten, using the relationship between addition and
subtraction and creating equivalent but easier or known sums.
Ask the teachers to solve 9 + 3 and 8 + 3 using the concrete-pictorialabstract progression for. What is a G1 student experiencing, noticing
and learning when doing this?
Concrete: linking cubes with frame.
Pictorial: Os and Xs
Abstract: Numbers with Number Bonds and 2 equations.
For the last 2 problems, solve using just the abstract level. Ultimately, our
goal is to have students move away from using concrete and pictorial
representations and use just numbers to solve. (Give 2 minutes.)
Topic A progresses in a way that allows students to focus adding with 9 as
an addend first, then 8 as an addend, and generalize this new making ten
strategy to a new number, 7 and move on to solving a variety of problems
involving a mixture of 7, 8, 9 as addends.
What will students notice when 1 addend is a 9? 8? (when an addend is 9,
you take out 1 from the other addend. When an addend is 8, take out 2 from
the other addend.)
4 min
17.
Summarize below:
Standard: 1.OA.C.6: Add and subtract within 20. Add and subtract
within 20, demonstrating fluency for addition and subtraction within
10. Use strategies such as counting on, making a ten, decomposing a
number leading to a ten, using the relationship between addition and
subtraction and creating equivalent but easier or known sums.
Module 2 culminates with naming 10 as a unit of 1 ten. This is the very first
time students are introduced to this language of ten as a unit. Up until now,
they have used 10 as a friendly number but might see it as 10 individual
ones. Now they are introduced to ten as a unit so that these 10 linking cubes
become 1 ten (show a ten stick) and this Rekenrek row becomes not only 10
individual beads but 1 ten (push 10 across all at once.)
Students go back to their work with their Hide Zero Cards and their Magic
Counting Sticks, now rethinking their teen number as 1 ten and some more
ones, as in 17 is the same as 1 ten and 7 ones.
Let’s see how this understanding will help students solve addition and
subtraction problems within 20.
NOTE to presenter: Will need 2 volunteers to be Partner A and Partner B
as the presenter projects the hide zero cards as scripted.
T: Using your magic counting sticks, show me 10 ones.
S: (Wiggle all 10 fingers.)
T: Show me 1 ten. (Clasp 1 both hands.)
S: (Clasp both hands.)
T: (Project 14 with Hide Zero Cards.) With your partner show me 14 as a
ten and some ones.
S: (Partner A clasps hands, Partner B shows 4 fingers.)
T: How many tens are in 14?
S: 1 ten.
T: 14 is 1 ten and how many ones?
S: 4 ones.
T: Let’s add 2. (Project 2 with the Hide Zero Card and write 14 + 2.) How
will you do this? Will you add 2 to the ten or to the ones? (Split the Hide
Zero Cards into 10 and 4)
S: To the ones.
T: Add 2 more fingers.
S: (Partner B adds 2 more fingers.)
T: 4 and 2 is?
S: 6.
T: 10 and 6 is?
S: 16.
T: How many tens and ones make up 16?
S: 1 ten 6 ones.
Repeat with 15 + 3.
Let’s subtract. With your partner, show me 13 as a ten and some ones.
(Write 13 – 2.) Let’s take away 2. Can I take from the ones? (Yes.) Partner
B, take 2 from 3. What do you have? (1.) Partner A and B, put your fingers
together. What is 13 – 2? (11.)
With your partner show me 12 as a ten and some ones. Let’s take away 9.
Can I take from the ones? (No.) Can I take from the ten? (Yes.) Partner A,
unbundle your ten. (Show all fingers.)
Take away 9 all at once. (Show 1 finger.) What is 1 and 2? (3) So what is
12-9? (3)
Say the number sentence. (12 – 9 = 3.) Let’s try some more: 13 – 9, 15 – 8.
5 min
18.
Summarize below:
Have participants articulate the standard that is addressed in Module 4:
(1.NBT.4: Add within 100, including adding a two digit number and a one
digit number, and adding a two-digit number and a multiple of 10, using
concrete models or drawings and strategies based on place value properties
of operations, and or relationship between addition and subtraction, relate
the strategy to a written method and explain the reasoning used.
Understand that when adding two-digit numbers, one adds tens to tens or
ones to ones and sometimes is necessary to compose a ten. )
Later in Module 4, students apply their work with the make ten strategy
when adding to a two-digit number. Let’s look at 28 + 6.
(Point to the image with linking cubes.) Students may make a ten first. 28
needs 2 more to get to 30. Take 2 from 6 and we have 4 more yellow cubes.
28 and 2 make 30; 30 and 4 make 34.
(Talk through the quick ten and number bond/number sentences images.)
Again, using concrete, pictorial and abstract representations, students use
their knowledge of the make ten strategy as they think about making the
next ten by decomposing the addend with 1 digit, using the arrow way or
the number bond as shown on this slide. Students also represent their
thinking by writing 2 addition sentences to show how they made the next
ten first.
Students must be able to use written notation to show how they solved 28 +
6. With every concrete and pictorial representation, the teacher must model
the written notation and give students opportunities to do the same.
For students who know 8 + 6 mentally, they may choose to add the ones
first and then add the 20 to find the total.
IF TIME ALLOWS:
Now You Try! 28 + 7; 27 + 5
5 min
19.
Summarize below:
Have participants articulate the standard that is addressed in Module 4:
(1.NBT.4: Add within 100, including adding a two digit number and a one
digit number, and adding a two-digit number and a multiple of 10, using
concrete models or drawings and strategies based on place value properties
of operations, and or relationship between addition and subtraction, relate
the strategy to a written method and explain the reasoning used.
Understand that when adding two-digit numbers, one adds tens to tens or
ones to ones and sometimes is necessary to compose a ten. )
Module 4 culminates with students adding two pairs of two-digit numbers
in two distinct ways using ten.
To explore both of these methods, students will begin their work with their
partner and use their ten sticks (linking cubes). Let’s take a closer look at
“adding on the ten first” method.
(Demonstrate with linking cubes.) Partner A makes 19 with her ten sticks
and Partner shows 15 with his ten sticks.
One way to add is to add on the ten first. (Add a ten stick from 15 to 19.)
We’ve added 10 to 19 which equals 29.
We now add the remaining ones, the 5, to 29 and get 34.
Let’s record how we added on the ten first. (Demonstrate.)
Another method is to make the next ten first. (Start with ten sticks
representing 19 and 15.)
T: How many does 19 need to get to the next ten?
S: 1
T: Where should we take 1 from ?
S: 15.
T: (Take 1 from 15 and complete the ten stick with 9.)
T: 19 and 1 is?
S: 20.
T: What do we still have remaining from 15?
S: 14.
T: What is 20 and 14?
S: 34.
Let’s see how we can record how we used the making the next ten strategy.
(Demonstrate.)
(Go right to the next slide. There are 2 animated images, but you’re
already demonstrated under the doc cam.)
12 min
20.
Summarize below:
Module 1 develops and strengthens this critical understanding of partwhole thinking to ensure that students advance from the counting all and
counting on strategies of Kindergarten and Grade 1 to the Level 3 strategy of
making a ten. Confidence with Level 3 strategies will enable students to
work fluently with larger numbers within all four operations as they
progress through later grades.
Success with addition and subtraction requires a deep understanding of the
base ten place value system, the understanding of which, in turn, rests upon
the knowledge of the bonds that make 10, the bonds within 10, and the
understanding that teen numbers are composed of a ten and some ones.
For this reason, Module 1 is squarely focused on a review of sums and
differences to 20 to ensure mastery of the grade level required fluency that
supports the major work of the grade.
** Analyze Problem Set Pages 5 and 6**
How does page 5 compare to the first grade PS with this strategy? How
does page 5 prepare students for page 6? What must a student know to
be able to use this strategy fluently?
2 min
21.
Summarize below:
The fluency activities of Grade 2 are designed to provide each student with
enough practice to achieve mastery of the required fluencies (i.e., adding
and subtracting within 20 and within 100) by the end of the year.
This fluency is essential to the work of later modules and future grade
levels, where students must fluently recompose place value units to work
adeptly with the four operations.
Let’s try a few fluency activities!
Think 10 to Add 9
I say, “9 + 5.” You say, “10 + 4.” (Continue with all the 9 + facts.)
Related Facts Within 20
I say, “15 - 8.” You write, “8 + 7 = 15.” (Continue with various facts within
20.)
3 min
22.
Summarize below:
Have participants articulate the standard addressed while building towards
the algorithm: (2.NBT. 7: Add and subtract within 1000, using concrete
models or drawings and strategies based on place value, properties of
operations, and/or the relationship between addition and subtraction,
relate the strategy to a written method. Understand that in adding or
subtracting three-digit numbers, one adds or subtracts hundreds and
hundreds, tens and tens, ones and ones, and sometimes it is necessary to
compose or decompose tens or hundreds.)
In Module 4, students extend and learn new place value strategies. They
work within 200, building confidence with smaller numbers before their
work within 1,000 begins. The standard addition and subtraction
algorithms are introduced here.
Module 5 extends this work to numbers within 1,000. For this reason, we
will examine the work of these two modules side by side. Both Modules 4
and 5 open with a topic on place value strategies.
Look at the images and talk with a neighbor:
• What connections can you see reaching back to GK and G1? How
does the work shown on this slide relate to the work of GK, G1, and
the beginning of G2? (e.g., decomposition of numbers within 10, bonds
of 10, the structure of 10)
•
What strategies are being used? (make a ten, make a hundred)
•
What understandings are necessary for students to be able to use
number bonds to solve these two problems? (the decomposition of
numbers into smaller units, and place value units of 1, 10, and 100.)
5 min
23.
Summarize below:
Articulate the standard addressed with the simplifying strategy of
compensation. (2.NBT. 9: Explain why addition and subtraction strategies
work using place value and the properties of operations.)
Students work with another strategy that highlights place value and the ten
structure, compensation. Progressing from a concrete model to work with
small numbers and then larger numbers, students learn that they can add or
subtract the same amount to or from both numbers to create an equivalent
problem that involves no renaming.
***Model with linker cubes, then with the tape diagram.***
(CLICK) You try! 440 – 280. Then model using the tape diagram.
(On the document camera show 699 – 210.) What about 699 – 210?
Why wouldn’t you add 1 to 699? (You can solve it mentally because no
renaming is necessary.) This points out the importance of thinking about the
relationship of the numbers before using strategies. Also, it’s subtracting a
multiple of 100 from a number that’s easier, not subtracting from the
multiple of 100.
Topic A of both Module 4 and Module 5 culminates with students sharing
and critiquing strategies. Giving them ample time to share their work and
explain their thinking using properties of operations and place value
reasoning deepens their conceptual understanding of addition and
subtraction.
The remainder of both modules shifts to conceptual understanding of the
addition and subtraction algorithms, as students use manipulatives and
math drawings to represent the composition and decomposition of tens and
hundreds and relate them step by step to a written method.
4 min
24.
Summarize below:
Have participants articulate the standard addressed while building towards
the algorithm: (2.NBT. 7: Add and subtract within 1000, using concrete
models or drawings and strategies based on place value, properties of
operations, and/or the relationship between addition and subtraction,
relate the strategy to a written method. Understand that in adding or
subtracting three-digit numbers, one adds or subtracts hundreds and
hundreds, tens and tens, ones and ones, and sometimes it is necessary to
compose or decompose tens or hundreds. In Module 5 Topic A, Strategies
for Adding and Subtracting Within 1,000, knowledge of place value and the
make a ten strategy helps students solve problems such as 590 + 240.
Talk with a neighbor: What is it about 590 + 240 that poses a challenge for
some students? (90 + 40) (CLICK twice to advance the number bond and
arrow notation.)
In all of this work, notice how students continue to look for and make use of
structure, building on their knowledge of partners to ten and applying it to
make a hundred. This work relies on an understanding of the associative
property. Arrow notation allows them to record the change in the numbers
as they work with them.
(CLICK) Now you try! 280 + 640.
Section: Math Learning within the K-2 Progression,
Foundations Level 3, Take from Ten
Time: 11 minutes
In this section, you will explore using Take from Ten in composition Materials used include:
and decomposition.
 Math Instruction Across the Grade Band PPT
 Math Instruction Across the Grade Band Facilitators Guide
Time Slide # Slide #/ Pic of Slide
Script/ Activity directions
2 min
Summarize below:
25.
Ask participants to show with fist to 5 their familiarity with the three
Levels described on the slide. Fist being unfamiliar and 5 being
complete understanding.
Depending on their familiarity discuss the slide and give examples of
each level.
1 min
26.
Summarize below:
Standard: 1.OA.C.6: Add and subtract within 20. Add and subtract
within 20, demonstrating fluency for addition and subtraction within
10. Use strategies such as counting on, making a ten, decomposing a
number leading to a ten, using the relationship between addition and
subtraction and creating equivalent but easier or known sums.
Module 2 Topic B is devoted to introducing the Take from Ten strategy and
provides opportunities for students to explore and practice this Level 3
strategy.
GROUP
1 min
27.
Summarize below:
Standard: 1.OA.C.6: Add and subtract within 20. Add and subtract
within 20, demonstrating fluency for addition and subtraction within
10. Use strategies such as counting on, making a ten, decomposing a
number leading to a ten, using the relationship between addition and
subtraction and creating equivalent but easier or known sums.
Module 2 Topic B is devoted to introducing the Take from Ten strategy and
provides opportunities for students to explore and practice this Level 3
strategy.
2 min
28.
Summarize below:
Standard: 1.OA.C.6: Add and subtract within 20. Add and subtract
within 20, demonstrating fluency for addition and subtraction within
10. Use strategies such as counting on, making a ten, decomposing a
number leading to a ten, using the relationship between addition and
subtraction and creating equivalent but easier or known sums.
Read the Bailey Bunny problem to the participants and have them
represent the story with a math drawing and number bond.
Bailey Bunny has15 carrots. 10 are in a basket and 5 on a plate. She ate 9
carrots from the basket. How many carrots were left?
Here, the story problem explicitly asks the students to take away 9 from 10.
(CLICK) When solving 15 – 9, students can see that 10 – 9 = 1 and 1 + 5 = 6.
This allow students to begin using the take from ten strategy because the
teen number is already separated into 2 smaller units for them, a unit of 10
and some ones and asks them to take 9 from 10 .
Note: The 5-group row is introduced the lesson prior to this one. Efficiency
is a good justification for drawing the ten in a row since you can just cross it
off.
2 min
29.
Summarize below:
Have participants articulate the standard addressed when using Take
from Ten (Standard: 1.OA.C.6: Add and subtract within 20. Add and
subtract within 20, demonstrating fluency for addition and
subtraction within 10. Use strategies such as counting on, making a
ten, decomposing a number leading to a ten, using the relationship
between addition and subtraction and creating equivalent but easier
or known sums. )
Concrete:
Model 12-9 WITH LINKER CUBES FIRST. Have the participants practice
with the linker cubes.
Model 11-9 and 12 - 9 with the REKENREK.
Pictorial:
Model 12-9 the 5 Group row way (use the slide’s animation).
Abstract:
(Use the document camera to demonstrate written notation. Always map
the concrete and pictorial representation to the abstract.
Write the expresson.
Show 12 was decomposed as 10 and 2 with a number bond.
Write 10 – 9 = 1
Write 1 + 2 = 3.
Write the answer into the original expression to make a complete number
sentence.
Guide participants to solving 11 – 9 using the 5-group row template and
recording abstractly.
** Analyze Problem Set Pages 7 and 8. **
Why do you think take from ten was introduced with word problems
first? Explain how the complexity builds on page 9. What scaffolds
support students with this strategy?
3 min
30.
Summarize below:
Ask participants to solve 13 - and 15 – 8 at the concrete, pictorial and
abstract levels.
Concrete: Linker cubes
Pictorial: 5 group row
Abstract: Number bond with 2 equations
13 – 9 =
15 – 9 =
15 – 8 =
13 – 7 =
Section: Math Learning within the K-2 Progression, Use
Units of 1s, 10, and 100s to Add and Subtract
Time: 114 minutes
In this section, you will examine using units of 1s, 10, and 100s to
add and subtract.
Materials used include:
 Math Instruction Across the Grade Band PPT
 Math Instruction Across the Grade Band Facilitators Guide
Time Slide # Slide #/ Pic of Slide
Script/ Activity directions
GROUP
4 min
31.
Summarize below:
Ask participants to show with fist to 5 their familiarity with the three
Levels described on the slide. Fist being unfamiliar and 5 being
complete understanding.
Depending on their familiarity discuss the slide and give examples of
each level.
3 min
32.
Summarize below:
Quick ten drawings helps students mentally add not only 1 more and 1 less,
but also 10 more and 10 less.
(Using the document camera, demonstrate drawing 15 in quick tens and
ones. Then show 1 more than 15 in a quick ten drawing and how it looks
different when showing 10 more than 15.)
As students explore tens and ones through concrete objects (mainly through
the use of ten sticks, 10 linking cubes put together like towers representing
tens, as well as the individual cubes to represent ones) and make quick ten
drawings to represent 2 digit numbers and reinforce the concept that 2 digit
numbers are made of tens and ones, they are introduced to the place value
chart. As you can see on the slide, using quick ten drawings and the place
value chart side by side, students are able to solve +1, +10, -1 and -10
problems with efficiency.
2 min
33.
Summarize below:
Have participants articulate the standard that is addressed in Module 4:
(1.NBT.4: Add within 100, including adding a two digit number and a one
digit number, and adding a two-digit number and a multiple of 10, using
concrete models or drawings and strategies based on place value properties
of operations, and or relationship between addition and subtraction, relate
the strategy to a written method and explain the reasoning used.
Understand that when adding two-digit numbers, one adds tens to tens or
ones to ones and sometimes is necessary to compose a ten. )
2 min
34.
Summarize below:
Ask the participants why they think we chose this model. What do
students notice, experience, and learn when using this model?
Here, dimes and pennies are used to represent and reinforce the concept of
tens and ones. The dime is another representation of the unit of 10 and
serves the same purpose as the quick ten. However, the proportionality is
gone. When you draw a quick ten, it is proportional to drawing 10 ones,
whereas the dime isn’t. But the idea is the same: one larger unit represents
10 smaller units.
As you can see in the first image, the places on the place value chart has
been renamed from tens and ones to dimes and pennies. In the image
below, students are asked to cross out the appropriate coin to show 1 less
and 10 less, just like how they would cross out 1 one or 1 ten in a quick ten
drawing.
2 min
35.
Summarize below:
Early on in Module 4, students see that they can apply their knowledge of 2
+ 1 to adding units of ten. Just as 2 + 1 = 3, 2 tens + 1 ten = 3 tens and 20 +
10 = 30, students use this understanding to add and subtract tens.
Students realize that when adding or subtracting tens only the unit changes
(e.g., 2 bananas + 1 banana = 3 bananas, just as 2 tens + 1 ten = 3 tens)
They also notice that when adding and subtracting ten the ones digit stays
the same.
Later on, students move on to adding multiples of tens to any given number,
such as 12 + 30 = 32 and subtract multiples of tens from tens, such as 30 –
10 = 20.
12 min
36.
Summarize below:
Have participants articulate the standard that is addressed in Module 4:
(1.NBT.4: Add within 100, including adding a two digit number and a one
digit number, and adding a two-digit number and a multiple of 10, using
concrete models or drawings and strategies based on place value properties
of operations, and or relationship between addition and subtraction, relate
the strategy to a written method and explain the reasoning used.
Understand that when adding two-digit numbers, one adds tens to tens or
ones to ones and sometimes is necessary to compose a ten. )
New groups below
Give participants a chance to record this method in their packet.
Core Action 3: Indicator E
Discuss with your table how a student using precise mathematical language
would explain their work for this problem. How can you elicit such answers
if students struggle to do so? (3-4 minutes to discuss and 2-3 minutes to
share out.)
7 min
37.
Summarize below:
Now you try! Solve these addition problems using both strategies. You may
want to work with a partner and try using your ten sticks at first. However,
using the written notation to connect what you did with concrete materials
is crucial, and the goal for students is to move onto using abstract
representation. (Give 2 minutes.)
19 + 17
18 + 22 (Note how compensation to make 20 + 20.)
49 + 25 (Module 6)
In the final module, Module 6, of Grade 1, students extend their learning
from Module 4 and add and subtract within 100! As you can see, even
though we are using bigger numbers to add, the strategies used are exactly
the same as those from Module 4 and are deeply rooted in all the work of
the Make Ten Strategy from Module 2.
** Analyze Problem Set page 12. ** Explain the placement of d. Why have a
problem where there is no bundling?
4 min
38.
Summarize below:
Have participants articulate the standard addressed when students bundle
10 ones to make 1 ten. Students bundle 10 tens to make 1 hundred. (2.NBT.
1 Understand that the three digits of a three-digit number represent
amounts of hundreds, tens or ones. A. 100 can be thought of as a bundle of
ten tens- called a hundred. )
In Module 3, students gain extensive experience physically bundling units.
We use the term “bundling” because it so perfectly describes the action
being taken. The repeated bundling establishes the pattern that 10 of a
smaller unit equals 1 of the next largest unit, here, 10 ones make 1 ten and
10 tens make 1 hundred.
Students bundle popsicle sticks in units of tens and hundreds to create a
unit of 1000. Students will use these bundles for skip-counting throughout
this module and in later topics, as needed, to support counting up and down
by ones, tens, and hundreds.
Bundling can be called renaming, regrouping, changing, or exchanging. The
terms are related to the action being taken. So when working with money
or place value disks later on, students “change” or “exchange” 10 smaller
units for 1 larger unit, or 1 larger unit for 10 smaller units. When relating
place value disks to math drawings when working with the addition and
subtraction algorithm, students “rename” or “regroup.” For example, 4 tens
3 ones can be renamed or regrouped as 3 tens 13 ones.
Models move from proportional, where size indicates value, to nonproportional, where value depends on placement, either on a place value
chart or in a number.
Bundles are proportional models because their value can be determined by
their size, i.e., a bundle of 100 is visibly larger than a bundle of 10. Ample
practice with bundling provides the conceptual foundation students need to
advance in later lessons to non-proportional models such as place value
disks.
1 min
39.
Summarize below:
Just as in G1 students worked with pennies and dimes to represent tens and
ones, now students progress to the non-proportional model of dollar bills to
represent ones, tens, and hundreds when modeling numbers. Dollar bills
are a non-proportional model because the value is not proportionate to the
size of the bill.
Students skip-count bills by ones, tens, and hundreds, and learn the
equivalence of 10 ten dollar bills to a hundred dollar bill.
Note that money in Module 3 is used only as it relates to units of 1, 10, and
100.
1 min
40.
Summarize below:
Students transition to another abstract, non-proportional model, place value
disks, also known as number disks. This is a big advance, as this model will
be used through Grade 5 for modeling very large and very small numbers.
This advance represents the culmination of all the foundational work that
preceded this topic, and in turn, it lays the foundation for composition and
decomposition in the standard addition and subtraction algorithm in
Modules 4 and 5. Students are now ready to manipulate place value units:
10 ones for 1 ten, 10 tens for 1 hundred, and 10 hundreds for 1 thousand.
1 min
41.
Summarize below:
Why are place value disks more a more abstract representation of tens and
ones than quick tens or 5 groups?
3 min
42.
Summarize below:
Student work leading up to this point is rooted in experiential learning to
develop an understanding of base-ten structure and the meaning of place
value.
Let’s look at how this work supports students as they compose and
decompose tens and hundreds and relate it to the addition and subtraction
algorithm.
Core Action 3: Indicator F
How do these models support students in choosing and using
appropriate tools to solve? (3 minutes to discuss and 3 minutes to
share out)
3 min
43.
Summarize below:
Have participants articulate the standard addressed: 2.NBT.3: Read and
write numbers to 1000 using base ten numerals, number names and
expanded form.
This work on the place value chart enables students to read, write, and say
numbers in all forms. It strengthens student understanding of place value,
as students relate the value of each digit to its place on the chart.
Note that unit form counting differs from Say Ten counting. In unit form, all
units are stated, e.g., 5 hundreds 7 tens 6 ones. Say Ten counting is used for
counting two-digit numbers, and the ones unit is not named, e.g., 27 would
be 2 tens 7, not 2 tens 7 ones.
Look at the example for unit form, and the various ways the units can be
expressed. This shows once again how powerful the unit is because it can
be manipulated in many ways, which translates into helping students solve
problems.
15 min
44.
Summarize below:
Have participants articulate the standard addressed: (2.NBT. 6: Use Place
value understanding and properties of operations to add and subtract.)
Beginning at the concrete level, students start with their hands (Magic
Counting Sticks) and then use place value disks to represent the
composition of 10 ones as 1 ten and 10 tens as 1 hundred. Then they relate
their models to a written vertical method.
Next they use math drawings, first of number disks and then chip models, to
represent the composition of tens and hundreds. Again, relating the
drawings to the algorithm.
Let’s walk through that progression. Have participants draw a place value
chart on their whiteboards with no headings. Have them model on the
personal board but write the algorithm on paper.
Guide participants through 183 + 319 with disks. > Discuss why we write
horizontally, then vertically. Count as you model each addend.
For each place ask, “Did we make a new ___?”
> Relate each step to the
algorithm.
Now you try! 167 + 256 with disks
Guide participants through 304 + 298 with a labeled drawing . > Discuss
vertical (ten-frame structure) vs. horizontal arrays (5 group structure).
(Flexible thinking, space constraints.)
Labeled vs. unlabeled place value chart (Depends on whether the model shows
the value.)
Now you try! 238 + 277 with a labeled drawing
Now you try! 387 + 534 with the chip model . > Discuss new groups
below (write in usual order, proximity of digits suggests their origin, adding
from the top down +1).
Relate to number bonds.
** Analyze Problem Sets 13 and 14. ** What is the added complexity?
5 min
45.
Summarize below:
At the end of Module 4, students are exposed to an addition method that
relates to their understanding of expanded notation and the concept of
adding like units. First, they write the addends in expanded form
horizontally. This then transitions to the vertical form, in which they record
the totals below. The totals below method gives students the option of
adding from left to right or from right to left.
Students represent and solve problems using both the new groups below
and the totals below methods, they relate both methods to their math
drawings of chip models and number bonds, and they discuss the
differences and similarities between the two.
***Model 23 + 48 both ways and show the number bond of 11, 60, 71, and 23,
48, and 71.***
Now you try! 125 + 75 and show a number bond for both.
25 min
46.
Summarize below:
Students use the same models and again work from concrete to pictorial as
they explore the subtraction algorithm. The magnifying glass presents a
new element. It is a visual cue serving several purposes:
• It reminds students to set the problem up for subtraction.
• It reminds them that the minuend is the whole from which they are
taking a part.
• It short-circuits the habit of seeing numbers as columns of isolated,
unrelated digits.
• It prevents the common error of switching the top and bottom digits
when the digit on top is too small.
• It prevents the other common error of forgetting to show a change to
the digits when regrouping has occurred.
Guide participants through 364 – 58 with disks.
> Magnifying glass. Point
out the connection to students understanding unit form, e.g., 364 can be seen
as 3 H, 5 tens, 14 ones.
Guide participants through 316 – 127 with labeled disks. > Do we have
enough ones, tens, hundreds? Relate each step.
Now you try! 584 – 147 with the chip model. > Write a number bond.
Explain why the addition and subtraction methods work.
**Analyze Problem Set Page 15 **
What support is in place for parents? How does the complexity build as you
work down the page?
5 min
47.
Summarize below:
That said, we are also leading students towards conceptual understanding
of the standard algorithms for addition and subtraction. So let’s look at how
students learn to subtract from multiples of 100.
(CLICK to advance the 2-step method) Rename 1 hundred as 10 tens then
rename 1 ten as 10 ones.
(CLICK to advance the 1-step method) Rename 1 hundred as 9 tens and 10
ones
These methods are also used when subtracting from numbers with zero in
the tens place.
***Model both 1- and 2-step methods for 700 – 463.***
Note that we return to that all-important understanding and manipulation
of units – 100 can be grouped as 10 tens or as 9 tens 10 ones.
Now you try! Use the chip model to show both methods of solving 600 –
347.
15 min
48.
Summarize below:
Coming out of Modules 4 and 5, students have at their disposal a range of
strategies based on place value, properties of operations, and the
relationship between addition and subtraction. It’s important to note that
there is no right answer to which strategy is the best or most efficient for a
given problem type. Different students may find certain strategies easier
than others. This is the reason we encourage sharing, reasoning, and
critiquing solution strategies – to provide as many points of entry to the
content as possible so that all students have access.
Here are two strategies students might use to subtract from a multiple of
100.
(CLICK to advance Compensation.)
(CLICK to advance Add to solve.)
Look at how these strategies build upon previous learning: compensation
(what you do to one number you do to the other), benchmark numbers
(friendly numbers that allow you to skip count), expanded notation and
mixed order (1 + 40 + 100), partners to 10 (6 + 4, 60 + 40, 160 + 40).
Now you try! Use compensation to solve 800 – 543 (Subtract 1 to get 799 –
542). Use arrow notation and add to solve 400 – 278 (278 + 2  280 + 20
 300 + 100  400).
**Analyze Problem Set pages 16 and 17.** Solve 2a using 2 step and 2b
using 1 step method.
Why are students given a choice of solving vertically or mentally on page
17?
1 min
49.
Summarize below:
The use of coherent models and strategies across grade levels enables
teachers to support students at their current level of understanding and
help them advance to Level 3 strategies, where they make an equivalent but
easier problem. Remember in GK students worked to see the 5 group as a
unit. This work makes seeing 10 as a unit a natural transition.
Because of the consistent reliance on the ten-structure and the continuous
practice with identifying a unit of ten, making a unit of ten, and taking from
a unit of 10, students’ conceptual understanding will rapidly and solidly
develop. This will be even more true as students enter Grade 2 having used
A Story of Units in Kindergarten and Grade 1.
2 min
50.
Summarize below:
There are multiple means of filling gaps in student understanding. For
example, return to the concrete level; have students represent their thinking
on multiple models, such as on their fingers, on the Rekenrek, and with tenframe cards and counters; move from simple to complex so that students
discover patterns, e.g., to solve 85 + 6, start at 5 + 6, then 15 + 6, and so on.
A word about working at the concrete level. We don’t want students
lingering there very long. It’s important to move them along the concretepictorial-abstract path quickly in order to close gaps.
1 min
51.
Let’s wrap up some key points regarding addition and subtraction in the K-2
learning progression.
Section: Math Learning within the Grade 3-5 Progressions Time: 125 minutes
In this section, you will examine the topics in Grades 3-5 that build
upon the work in K-2 to move students from addition and
subtraction on to multiplication and division.
Time Slide # Slide #/ Pic of Slide
2 min
52.
Materials used include:
 Math Instruction Across the Grade Band PPT
 Math Instruction Across the Grade Band Facilitators Guide
Script/ Activity directions
GROUP
1 min
53.
This is the Curriculum Map for A Story of Units. It shows the overall
sequence of the (varied number (5-8)) modules at each grade level, PK-5.
CLICK. We will now focus on multiplication and division in Grades 3-5.
CLICK. Our work comes from the highlighted modules.
General Points:
The colors within the map show the Domains of the Common Core
Standards.
• Yellow – Numbers
• Blue – Geometry
• Green – Number and Geometry, Measurement
• Pink - Fractions
4 min
54.
Speak briefly of Grade 2 work that leads to up to the work with
multiplication: (Background Information follows.)
G2-M6-TA Equal Groups Sequence:
• Begin making equal groups from a given number of objects using
concrete materials.
• Move to pictorial representations of equal groups shown here (circle
a group of 5 stars, add 5 more, then add 5 more).
• Relate drawings to repeated addition and write number sentences.
• Find sums by adding on each time, or by using
doubles/simplifications. (Skip-counting as a strategy is taught in G3.)
• Draw abstract tape diagrams that show the number in each group as
a new unit.
• Begin understanding that any unit may be counted, e.g., 3 dogs, 3
tens, or even 3 fives.
G2-M6-TB Array Sequence:
• Familiarity with arrays goes back to Kindergarten (rekenrek, 10
frames, etc.)
• Organize equal groups into arrays (concrete, not shown).
• Understand that either a row or column is the unit being counted.
•
•
•
•
Compose arrays one row or column at a time, and write repeated
addition sentences.
Continue to find sums by adding on each time. (Skip-counting as a
strategy is taught in G3.)
Represent arrays pictorially, distinguish rows/columns by separating
equal groups horizontally/vertically (Shown here: 3 rows of 5 or 5
columns of 3.)
Next use tiles to make arrays, pushing them together as a
foundational step toward work with area in G3.
General Points:
• Concrete experiences in this module set students up to begin at the
pictorial level in G3-M1.
• G2 focuses on manipulation of place value units (1, 10, 100). G3
focuses on manipulation of numbers 1-10 as units. This module
bridges those understandings.
Computation Strategies Sequence:
Level 1:
• Level 1 strategy would be to count dots one by one to find the total.
• Hopefully mastered in G2. G3 spends almost no time here.
• Could be used for remediation.
Level 2 and 3 strategies require an understanding of the unit and how to
manipulate it.
Level 2:
• G3-M1 starts at Level 2.
• Level 2 strategy would be to count the rows as units.
Count: 1 three, 2 threes, 3 threes, 4 threes, 5 threes, 6 threes, 7 threes.
• Seeing rows as units, students working at this Level could also skipcount. (3, 6, 9, etc.)
• L2 skip-counting strategies are used throughout G3-M1 and G3-M3.
Level 3:
• These are properties of operations strategies in G3. (Commutative,
Distributive, Associative.)
• Level 3 strategy would be looking at groups as units.
For example, using the Distributive Property to simplify 7 threes by thinking
of it as 5 threes and 2 threes.
• Level 3 strategies are introduced halfway through M1 and grow
slowly over M1 and M3.
• A strong foundation in models (e.g. array) is necessary so students
can use them as tools to access the properties.
Now, we move into the models of Grades 3-5. Let’s look at the progression
from an array to the area model.
Module 4:
• Area is introduced.
• Relate area to understanding of multiplication.
Sequence of progression to the area model:
This image shows how the array evolves to become the area model as
students’ understanding of area grows over the course of the module.
• Concrete/Pictorial materials:
• Columns and rows
• Grid with discrete objects in each box:
• Students often don’t perceive the space in each box as the unit
at first and mistakenly count lines.
• The object in each box is a scaffold to avoid that.
• Grid with no objects:
• Students use inch and centimeter tiles to cover shapes.
• Understand the importance of no gaps or overlaps for area.
• Students make use of grid paper as well.
• Empty square:
• This is what we call the area model.
• A critical module because it’s where the area model is
established.
• Folks with pacing problems, please do not gloss over/skip M4
because the model is used extensively in G4 and G5 as a model
for multiplication.
Standards Connection:
3.MD.7: Relate area to multiplication.
General Points:
3 min
55.
The models on the screen show the sequence of complexity in the first
lessons on multiplication.
Ask participants to analyze the sequence. You might ask them to notice how it
compares to Grade 2, or compare the way that each model shows 4, 3, and 12.
(Grade 2 – Repeated Addition; Grade 3 – Multiplication)
Teaching Sequence:
• Work with equal groups pictures, relate to repeated addition and
multiplication sentences.
• Re-discover that an array is a more efficient organization of an equal
groups picture.
• Formalize understanding that the array shows 1 equal group as 1
row. (Columns come halfway through the module.)
• Distinguish between number of groups and size of groups (meaning
of factors) as they count rows and how many in 1 row to write
multiplication facts.
• Array is a critical model for exploring multiplication, division, and the
arithmetic properties, so it’s highly emphasized.
• Arrays are in M1 & M3. They become an “old friend” that makes new
concepts/connections between concepts accessible.
• Number bond is familiar from earlier grades, but the part-whole
relationships it shows are harder to identify in terms of
multiplication. That’s what makes it the most abstract of these
models. (Possibly point out how 4 is shown.)
CLICK. Throughout these initial lessons, students use both unit form and
standard form to compose number sentences that represent the models.
(Unit form – the concept and/or the term – may be new to participants. If
necessary, practice counting by units of 3 with these models.)
3 min
56.
The models on the screen show the sequence of complexity in the first
lessons on division which immediately follow the sequence on the previous
slide. Multiplication and division are taught together in M1 & M3 so
students perceive the relationship between them from the beginning.
Ways this relationship is made evident:
• Understand division as an unknown factor problem: unknown in
division is size of groups or number of groups.
(Notice title of slide – “composing” units to divide.)
• Use the same models for division as for multiplication: equal groups
pictures, arrays (not shown), number bonds, and count-bys to
understand the meaning of the unknown in division as size of groups
or number of groups.
• Emphasis on conceptually understanding division and learning to
interpret problems by writing division expressions.
Review with participants:
• In each example on the screen, groups can also be referred to as
units: 7 groups of 2, or 7 units of 2, which is the same as 7 twos.
General Points:
5 min
57.
Core Action 1: Indicator B – The lesson intentionally relates new
concepts to students’ prior skills and knowledge.
What prior skills and knowledge are necessary? Look at the models as
well as the mathematical concepts. (3 minutes to discuss and 3 minutes
to share out.)
Distributive Property
• Introduced through the array (familiar friend)
• First with multiplication and then with division
• Learn this property as a strategy for approaching unknown
problems.
Ask participants to do the problem with you on their boards. Guide them
through the following steps:
1. Draw an 8 by 4 array (CLICK to show). How many fours does
the array show? (8 fours.)
2. Break apart 8 fours into 2 smaller parts: make number bonds to
show pairs with a sum of 8. (1/7, 2/6, 5/3, 4/4.)
3. Go with 5/3, add the unit “fours” to 8, 5, and 3 in the number
bond. CLICK to show bond.
4. Draw a line to show array breaking apart to make 5 fours and 3
fours. Shade 5 fours. CLICK.
5. Write 5 fours and 3 fours as equations next to the parts of the
array they represent. CLICK.
6. CLICK. Tell a partner how these equations show your work
with the array.
Background for presenter:
• G3 refers to the Distributive Property as the “break apart and
distribute strategy” with students.
• Focus on using five facts as a place to break apart arrays. Fives facts
are “easy.” Also, the Progressions call out the 5 + n pattern as a
strategy for multiplying.
• Although 5 + n is the focus, the efficiency of using doubles (4/4 in this
case) is also practiced.
• Number bond is just another perspective on the same idea.
Call attention to the final equation (5 + 3) x 4. This notation is not emphasized
in these lessons, but is shown to lay a foundation for the Associative Property.
General Points:
2 min
58.
Background for presenter:
• Area model is used for both multiplication and division.
• Given side lengths, students multiply to find the area.
• Given area and one side length, students find the other side length
either by solving an unknown factor multiplication equation or a
division equation.
• G3 area work is purely conceptual: the “formula” for area is not
formally taught until G4.
Standards Connection:
• 3.OA.4: Find the unknown number in a multiplication or division
equation.
• 3.OA.6: Understand division as an unknown factor problem.
General Points:
3 min
59.
Ask participants to study the work on screen. Which property is evidenced and
how?
• Distributive Property is evidenced. (CLICK to confirm Distributive
Property/show slide title.)
• Side length 8 is decomposed into 5 and 3.
• Smaller side lengths are used to find the areas of the smaller
rectangles.
• Areas of smaller rectangles are added to find the total area of the large
rectangle.
Application of this Level 3 strategy prepares students for work they do in G4
with multiplication and the area model.
Standards Connection:
3.OA.7: Fluently multiply and divide within 100.
General Points:
8 min
60.
MODEL. Guide participants in drawing an area model for 3 x 424.
In Grade 4, the area model is seen again; this time to support students as
they solve 1-digit by 3-digit multiplication.
Teaching sequence:
• Decompose units: 424 = 400 + 20 + 4
• Draw a rectangle with a width of 3 and a length of 400 + 20 + 4
• Multiply each composite rectangle. Focus on unit language (e.g. 3
times 4 hundreds is 12 hundreds)
• That multiplication is the distributive property in action
(3x400)+(3x20)+(3x4)
• Each partial product in the composite rectangles can be related to the
partial products when solved using a vertical problem.
CLICK. Image of problem just modeled displayed.
CLICK. Participants should complete Problem 7 on the Problem Set:
7 x 534
8 min
61.
Conceptually more abstract to process, the area model is also used for
division. Here, given the dividend and the divisor, students build an area
model in order to determine the quotient.
1,344 is the area and 6 is the width. Students solve for the length by thinking
6 times how many thousands is about 1 thousand? Zero thousands. So 6
times how many hundreds is about 13 hundreds? 6 times 2 hundreds is 12
hundreds, which is almost 13 hundreds.
(CLICK)
A new area model drawn below shows how one might decompose the area,
showing the 12 hundreds and a remaining 144. Then we can ask 6 times
how many tens is about 14 tens? 2 tens. That leaves a remaining 24 ones. 6
times 4 ones is 24 ones.
A number bond represents the 3 parts to the area model, and the number
sentence below shows the distributive property as we divided up the total
area into smaller parts.
Remember how this is similar to how we decomposed 1 division problem
into 2 division problems in Grade 3 using the distributive property, such as
48 ÷ 3 as 30÷3 and 18÷3.
Remember the area model breaks down for remainders, so the remaining
part of an area would get tacked on as 1 square unit to the end of the model,
as we showed a few slides back.
(Solve 1,214 ÷ 4 with participants.)
Standards Connection:
4.NBT.6: Find whole number quotients and remainders with up to 4digit dividends and 1-digit divisors.
General Points:
1 min
62.
Students record the 2 partial products under the problem.
Focus is on products of multiples of 10.
Connecting to the written notation is the step prior to solving for 4 partial
products. Students are already comfortable recording an area model and a
written notation for a single-digit times 2-digit number.
General Points:
3 min
63.
Background for presenter:
• To multiply a 2-digit by 2-digit number, students use unit language
(which supports place value) and the area model while recording
alongside the steps for the algorithm.
To solve the area model as it relates to the step of the algorithm, solve right
to left, top to bottom in the area model.
To start in the algorithm, we would start in the ones: 3 times 1 one. CLICK.
Record the partial product. Notice the distributive property at work,
recorded below the area model.
3 times 3 tens is 9 tens. CLICK. Record the partial product.
2 tens times 1 is 2 tens. CLICK. Record the partial product.
2 tens times 3 tens is 6 hundreds. CLICK. Record the partial product.
Add all partial products. CLICK.
General Points:
3 min
64.
Soon we start to see the multiplication of, for example, 43 and 67 as 3 times 67 and 40 times 67, moving us to the
recording of 2 partial products, more formally as the steps of the algorithm.
Multi-digit multiplication and use of the area model continues in G5 as
students multiply up to 4-digit numbers by 3-digit numbers, shown here.
Participants analyze and explain to a neighbor what they understand about
the area model on the left and its connection to the algorithm on the right.
Note: Though this is a 4-digit by 3-digit problem, the area model is a 3-by-3
model due to the zero in the tens place of the first factor.
General Points:
3 min
65.
In Grade 5, the area model comes back as students multiply a whole number
times a decimal. They see that the process is the same; it’s the units that are
different.
Again, notice how the area model, another old friend from previous grades,
can be used to support learning with decimal multiplication.
Give participants time to analyze and make sense of this area model.
Encourage participants to recognize and appreciate the unit language shown.
4 x 4 is always 16. The unit being counted here is tenths, so 4 x 4 tenths is
16 tenths, etc.
General Points:
5 min
66.
Background for presenter:
•
After a little practice students learn to draw a solid line and a dashed line at the same time.
We only do it as two separate steps initially.
•
The second shading should only shade the portion that we are finding. Some other models
shade 1/3 of the whole first, then shade ¼ of the whole next (and then name the portion that
is double-shaded). But we are ONLY finding ¼ of 1/3, so we only shade that portion.
First, we multiply unit fractions by unit fractions. We then increase the
complexity by multiplying unit fractions by non-unit fractions and finally
multiply non-unit fraction by non-unit fractions.
Core Action 2: Indicator A – The teacher makes the mathematics of the
lesson explicit by using explanations, representations, and/or
examples.
How does this representation help make the concepts explicit as opposed
to just teaching a short-cut? (3 minutes to discuss and 3 minutes to
share out)
Switch to Document Camera: Let’s multiply 2/3 x 2/3 together. (Model with
participants.)
CLICK: to show completed solution.
CLICK: Participants should complete Problem 17 in their Problem Set.
Solve /𝟑 × 𝟑/𝟒 = ____. Draw an area model to show your thinking.
Background for presenter:
We spend 3 lessons using models to show a fraction times a fraction.
Students will easily recognize the pattern that is happening: we can just
multiply the numerators and the denominators. Until students can articulate
why 1 third times 1 third results in 1 ninth, have them continue to use the
models to solve. However once they are able to articulate a clear
understanding of what is happening, the models are no longer necessary for
that student.
3 min
67.
In Module 5, students use a ruler to find the side lengths of a rectangle. Here
we have a rectangle measuring 2 1/2 in x 1 1/4 in.
CLICK:
Take a moment to analyze the work shown here to find the area.
Share with you neighbors what the work is showing.  Students are using the
Distributive Property to solve.
How is this work with the Distributive Property similar and different from the
work they’ve done previously?  It’s the same, except now we’re multiplying
with fractions and whole numbers.
General Points:
6 min
68.
The area model can also be used to show fractions as division. Here, we see
two ways to show 3 divided by 2. What is different about how these two
problems are solved?
Now put 3 crackers on your board. Talk with a neighbor about what you’ll do
to share these 3 crackers with 2 people.  (Participants share.)  Work with
a neighbor to show how you would share these crackers with 2 people. 
(Participants work.)  Write a division sentence that matches your work. 
(Participants work.)
CLICK: to show image. (Some, if not all, participants will have modeled the
sharing in this manner.)
Invite participants to share about what happened to the pattern that we saw
earlier.  Though 3 ÷ 2 does equal 3/2, the pattern (for the time being) has
disappeared because most people showed that if you have 3 crackers shared
by 2 people, each person can get 1 whole cracker and only the 3rd cracker
needs to be halved.
Ask participants to consider the following scenario: What if the 3 crackers
were each a different flavor, and each person wanted to taste each flavor?
Would this method of sharing still work? Tell your neighbor why or why not.
 This method would NOT work. Each cracker would need to be halved before
sharing.
Invite participants to share about what this image shows.
Would this method for sharing work if each cracker were a different flavor?
Explain. (Call on participants to share their ideas.)
Invite participants to share about what they notice is happening again in the
division sentence. 
The pattern returns! 3 ÷ 2 = 3/2. The numerator again shows the number of
crackers (or the whole) and the denominator shows the number of groups.
General Points:
3 min
69.
Familiar place value disks are used to model multiplication beginning in
Grade 3. Students use concrete disks to form arrays. They then represent
the disks pictorially on a place value chart.
Toward preparing students for Grade 4, they explore place value as a
strategy for multiplying with multiples of 10. They begin with concrete
experience.
Provide participants with place value disks. *This may be their first experience
with them. If so, review the tool.*
“We do” using the document camera (record equations and unit language):
•
•
•
•
•
•
Model 2 x 3 with place value disks as 2 rows of 3. Write ‘2 x 3 ones.’
Our array shows 2 x 3 ones, true? (True.)
What is the total value of 2 x 3 ones? (6 ones.) Record ‘2 x 3 ones = 6
ones’.
Let’s change our units from ones to tens (change disks).
What happens to our equations? (unit changes from ones to tens, and
total value changes from 6 to 60.)
We multiplied 2 x 3 ones by ten, so now we’ve got 2 x 30 = 60, or 2 x
3 tens = 60.
CLICK. This work is transferred to the place value chart. Students
pictorially model 2 × 3 in the ones place. Then they locate the same basic
fact in the tens column. They see that when multiplied by 10, the digit shifts
one place value to the left.
Standards Connection:
3.NBT.3: Multiply one-digit whole numbers by multiples of 10.
General Points:
4 min
70.
In Grade 4, we see similar modeling as we multiply by multiples of 10.
Background for presenter:
• Connection to the place value disk and chip model from Grade 3 with
multiplying by multiples of 10. (e.g. 3 tens times 2)
• Grade 4 extends multiplication into the hundreds and thousands to
multiply by 4 digit numbers.
• Grade 4 extends multiplication past multiplying number of groups or
number in each group, thinking about multiplicative comparison.
• “Times as many as” and multiplicative comparison set the stage for
Grade 5 scaling and Grade 6 proportional reasoning.
MODEL. 10 times as much as 1 one.
MODEL. Have participants follow along: 10 times as much as 3 ones.
CLICK. 3 tens is 10 times as much as 3 ones.
CLICK. To show multiplication of 100, first we break 100 into 10 times 10.
CLICK. Tell your partner a statement using multiplicative language for this
image. (300 is 100 times as much as 3 ones.)
Participants draw to show 1,000 x 3 = 3,000 and write a statement using
multiplicative comparison language. (3 thousands is 1,000 times as much as 3
ones.)
CLICK. Using place value disks, this image shows 3 times 4 ones is 12 ones.
CLICK. It is recorded here numerically and in unit form.
CLICK. Tell your partner the multiplication sentence for this image. (3 times
4 tens is 12 tens)
CLICK. Now we are multiplying by multiples of 10. (4 tens)
CLICK. Record the numerical multiplication sentence in unit form that
represents this image. Show me your boards.
CLICK. Check your work.
Standards Connection:
(CLICK for star)
• 4.NBT.1: A digit in the ones place represents 10 times what it
represents in the place to its right.
• 4.OA.1: Interpret a multiplication equation as comparison.
2 min
71.
Eventually use of the place value chart for multiplying multiples of 10
becomes more abstract. Here, in G5, we again see a place value chart used
to show multiplication by a multiple of ten. Notice what is different 
Numbers are used in place of disks and we now see the additional place
values of tenths and hundredths.  Students see that as we multiply by
multiples of ten, the digits shift one place to the left.
In G5, students continue to notice place value patterns and see how
adjacent place value units are related.
Encourage participants to analyze this slide and notice that exponents are an
added complexity.
• Here students are introduced to exponential notation where they
learn to recognize that multiplying by 10 to the 3rd power is
equivalent to multiplying 10 x 10 x 10.
• Each multiplication by 10 results in the movement of the DIGIT, one
place to the left.
General Points:
This work very closely mirrors the math on the place value chart from
grades 3 and 4.
2 min
72.
Using a similar model for division, students can see that the digits shift one
place to the left.
Students also use exponential notation when showing division by a power of
ten.
Background for presenter:
Previously, students learned that when moving to the left on the place value
chart, the units become 10 times greater than those to the right. Here,
students recognize that places to the right on the place value chart are 1/10
the size of the place to the immediate left.
• It is very important for students to realize, and for teachers to
reiterate, that the digits are doing the shifting.
• As you can see, the digit 7 in the hundreds place shifts 3 places to the
•
right when divided by 10 to the 3rd power.
The DECIMAL POINT DOES NOT SHIFT! The decimal is ALWAYS
between the ones place and the tenths place. It does not move.
Encourage participants to see the additional complexity on this slide. (In G5,
the only added complexity on the place value chart is the addition of a new
place value called, the “thousandths” place.)
General Points:
8 min
73.
Moving back to G4 and using place value disks, students multiply a 1-digit
number by a 2-digit number.
Background for presenter:
• Students use the distributive property, place value disks, partial
products, standard algorithm, and area model to multiple 2, 3, and 4
digit numbers times a single digit.
• Start concretely using place value disks.
• Move pictorially to the chip model.
• Align the steps of the algorithm directly to the work occurring
in the place value chart.
• Fluency with the standard algorithm for multiplication is not
expected until Grade 5 – 5.NBT.5.
• Introduced to the algorithm, supported by place value
strategies as stated in 4.NBT.4, to prepare for Grade 5
multiplication.
• Not assessed in Grade 4 on using the algorithm for
multiplication.
Switch to the document camera. MODEL 4 x 23 using place value disks and a
place value chart with ones and tens drawn on a personal white board.
• Display 2 tens 3 ones.
• Display 4 copies of 2 tens 3 ones to show multiplication times 4.
• Set up 4 x 23 vertically.
• Point to problem. 4 times 3 ones is 12 ones. Record 12 ones as a
partial product.
•
•
•
•
•
•
Point to the ones column in the place value chart. 4 times 3 ones
is 12 ones. Record 12 ones under the ones column.
Point to problem. 4 times 2 tens is 8 tens. Record 8 tens (80
ones) as a partial product.
Point to the tens column in the place value chart. 4 times 2 tens
is 8 tens. Record 8 tens under the ones column.
Regroup 10 ones as 1 ten.
Rename the number of units in the place value chart as 9 tens 2
ones.
Add the partial products as 92.
CLICK. Participants should analyze 4 x 34 and notice such items as:
• 16 ones required regrouping of 10 ones as 1 ten.
• The tens now have 13 tens. 10 tens are renamed as 1 hundred.
• 12 tens 16 ones is the same as 1 hundred 3 tens 6 ones.
CLICK. Participants should complete Problem 6 on the Problem Set: 3 x 424
General Points:
3 min
74.
Core Action 3: Indicator E – The teacher connects and develops
students’ informal language to precise mathematical language
appropriate to their grade.
What precise mathematical language would you expect to hear? (3
minutes to discuss and 3 minutes to share out)
The same model as on the previous slide is shown again here. This time,
showing decimal multiplication. Notice how the modeling is the same. It’s
the units that have changed.
After students multiply decimals by powers of 10, we move to the
multiplication of multi-digit decimal numbers by one-digit whole
numbers. As always, though the complexity of content may be changing,
the models remain the same.
In this example, we’re using the chip model and a place value chart to show
the multiplication and connecting it to the standard algorithm.
Invite participants to analyze the image and make sense of what is happening.
Participants should share with their neighbor what they understand about the
work shown.
Encourage participants to identify how this model would be different if it were
showing 423 x 4. (The ONLY difference would be the names of the places in the
PV chart–proof that multiplying decimals is just like multiplying whole
numbers.)
General Points:
10 min
75.
The division algorithm is introduced and supported alongside the place
value chart model in G4-M3-L17.
Switch to document camera.
Model. 4÷3 using place value disks.
Note the remaining 1 one cannot be further subdivided.
Model. Have participants following along with 42 ÷ 3 using place value disks.
Note the remaining 1 ten can be further subdivided; rename 1 ten as 10 ones.
Model. Have participants following along with 44 ÷ 3 using the chip model.
Note the remaining 1 ten can be further subdivided; rename 1 ten as 10 ones.
Note the remaining 2 ones cannot be further subdivided.
CLICK. Participants should analyze the 3 problems they just solved here on the
slide as the chip model and notice such items as:
• 4 is crossed off one by one, where 42 and 44 are crossed off
quickly, showing level of understanding for the model and facts.
• Remainders in the ones are circled.
• Each group of subdivided disks represents the quotient.
• Tens can be further subdivided because 1 ten is the same as 10
•
ones. G4 students do not yet know 1 one is equal to 10 tenths.
The steps of the algorithm (divide, multiple, subtract, bring
down) are better explained and modeled using place value
understanding instead of memorizing the 4 steps or acronyms
or songs.
CLICK. Students are asked to check their work using multiplication and
addition.
CLICK. Relating the part/whole relationship of the quotient and remainder
to the dividend can be shown using a number bond.
(If needed, allow participants to try 58 ÷ 3 or 1,534 ÷ 4.)
General Points:
5 min
76.
In G5, the same model, as was on the previous slide, is used here to show
how to divide a decimal dividend using a place value chart and the
standard algorithm.
Participants take a moment to analyze this work and make sense of what is
happening in this model. (The initial complexity of this problem is that the
ones must be changed for 10 tenths before sharing/dividing.)
Have participants discuss with a neighbor how they might use unit language
and place value reasoning to solve this division problem using both the chip
model and the written method.
Switch to Document Camera.
Model. 1.401 ÷ 3
(If necessary, check G5-M1-L14 CD to model appropriate language. It begins
like this.)
• Let’s begin by dividing or sharing our largest units. Can we share 1
one with 3 groups?  No, we must change 1 one for 10 tenths.
(Draw change using chip model.)
• 10 tenths plus the original 4 tenths in our whole makes… 14 tenths.
•
•
•
•
Can we share 14 tenths with 3 groups? How many tenths in each
group?  4 tenths. (Record 4 tenths in both groups using chip model
and record 4 tenths in algorithm.)
When we shared 4 tenths with 3 groups, how many tenths were
shared?  12 tenths. (Record in algorithm.)
How many tenths remain?  2 tenths.
Can we share 2 tenths with 3 groups?  No, we must change 2 tenths
for 20 hundredths, etc.
CLICK. Participants should complete Problem 13 of the Problem Set:
Use the place value chart and algorithm to solve 3.445 ÷ 5 = _______.
General Points:
5 min
77.
Work with each of the models readies students for solving using the
standard algorithm.
Switch to document camera. Model 6 x 612 using the algorithm. Record
regroupings “on the line”.
CLICK. Participants should analyze 5 x 2,374 and notice such items as:
• 5 times 4 ones is 20 ones. 20 ones is the same as 2 tens 0 ones. So
we record 2 tens “on the line” and a 0 in the ones column below
the line. The 2 is recorded before the 0 because this is the
number twenty, not the number “zero-two”. The proximity of
the 2 and 0 remind us of the number 20.
• Using unit language to multiply supports the place value of the
algorithm.
• Renaming 37 tens as 3 hundreds 7 tens helps in knowing where
to place the 3 and the 7.
CLICK. Participants should complete Problem 8 on the Problem Set:
4 x 8,618
General Points:
•
This is the standard algorithm for multiplication used in a Story of
Units, introduced in G4-M3-L9.
Standards Connection:
(CLICK for star) 4.NBT.5: Multiply a whole number up to 4 digits by
one-digit whole number.
5 min
78.
Switch to document camera.
Model. 46 x 63 as participants follow along.
CLICK. Display of problem just solved.
(Solve 48 x 74 with participants if needed.)
CLICK. Participants should complete Problem 10 on the Problem Set:
84 x 73 using an area model, partial products, and the standard
algorithm.
Standards Connection:
4.NBT.5: Multiply two 2-digit numbers.
General Points:
5 min
79.
Student are now able to complete the standard algorithm for division. As
they complete each step, they visualize the previous work done with
division (i.e., area model, place value disks).
Explain (model if time) each step of the process, being sure to use unit
language.
5 min
80.
Dividing by 2-digit divisors is new to Grade 5 students. Students round
the divisor in order to know where to put the first digit in the quotient of the
written method. Notice how unit language helps students estimate.
Use Document Camera to Model: (Use the vignette from M2-L22 as an
example of one way that unit language and place value reasoning could be
used to solve.)
• Let’s begin by sharing/dividing our largest units. Can we
divide 5 hundreds by 17?  Not without regrouping.
• We can rename/decompose 5 hundreds as/into 50 tens, plus
the 9 tens in our whole gives us 59 tens. We can divide 59
tens into 17 groups (or groups of 17). How can we estimate to
divide 59 tens by 17?  60 tens ÷ 20 = 3 tens.
• Record 3 tens in the quotient. 3 tens times 17 is?  51 tens.
• How many tens remain?  8 tens.
• Can we divide 8 tens by 17?  Not without regrouping.
• We need to decompose these 8 tens into 80 ones, and since
there are zero ones in our whole (point to 0 in dividend), we
have 80 ones divided by 17. How can we estimate to divide 80
by 17?  80 ones ÷ 20 = 4 ones.
• Record 4 ones in the quotient. What is 17 times 4 ones?  68
ones.
• How many ones remain?  12 ones.
• Can we make another group of 17?  No.
• What is our quotient?  34
• What is 34 units of 17, plus 12 ones?  590!
CLICK: To show solution.
CLICK: Participants should complete Problem 14 with a partner. (Partner A
uses unit language to explain each step of the algorithm in Problem (a).
Partner B does the same for Problem (b).)
PS shows 2 completed problems: (Directions) The following problems
have been completed for you. With a partner, take turns to explain
each step of the algorithm using unit language.
This problem should be modeled step-by-step so that participants can see how
we estimate, first using unit language. It is probably easier to calculate how
many groups of 20 are in 60 tens, compared with deciding how many groups
of 20 are in 600. Once we decide to start with 3 tens, the digit “3” is placed in
the quotient in the tens place.
5-10
min
81.
Give participants about 5-10 minutes to review the Instructional Practice
guide and familiarize themselves with the Core Action Indicators to ensure
the lesson reflects the shifts required by the CCSS for Mathematics.
Have participants highlight 1-2 indicators that they feel would be the most
important for each core action (all the indicators could be relevant).
Section: Conclusion
Time: 81 minutes
In this section, you will discuss opportunities to scaffold, additional Materials used include:
coaching tools, and the importance of planning.
 Math Instruction Across the Grade Band PPT
 Math Instruction Across the Grade Band Facilitators Guide
Time Slide # Slide #/ Pic of Slide
Script/ Activity directions
3 min
Multiple means of Representation is how we access information.
Information should be presented in a variety of modalities (visual, auditory,
kinesthetic) to address the needs of all types of learners.
82.
Multiple means of Action and Expression is how students demonstrate their
understanding. Students should be given a variety of ways to express their
learning.
Multiple means of Engagement is the motivation factor. Lessons need to
peak the interest of a variety of learners. Instruction should be
differentiated to meet the needs of all the students and keeping them
motivated and interested. This is best done through making lessons
challenging yet at the same time providing the necessary scaffolds.
5-7 min
83.
Let’s take a look at the video of a first grade classroom, using the
Instructional Practice Guide as a tool to examine evidence of effective
instructional strategies for students with disabilities. Also take into
consideration the Universal Design for Learning Scaffolds, and the work you
did earlier on Accommodations and Modifications.
Discussion:
Discussion:
Using both the Instructional Practice Guide and the Universal Design for
Learning Scaffolds, what evidence from the video highlights the indicators
that your group determined to be the most important for each Core Action.
GROUP
Were there missed opportunities to provide
scaffolds from an earlier grade.
What recommendations would you make to
the teacher to help meet the needs of students
who may be struggling or students that need to be challenged.
10 min
84.
Group conversation to address questions.
60 min
85.
Participants will self-group based on areas they would like to practice.
(Group will choose from various math content as well as particular areas on
the Instructional Practices Guide.)
Work group will spend 15 minutes in each round of practice for 3 rounds.
Participants will share highlights of their practice after rounds. (10 – 15
minutes)
86.
Either whole-group or partner work takes place to brainstorm strategies for
stickies based on conversation and learning of the past two days.
Participants will spend some time individually revising or refining their
strategic plans to support math instruction at their site.
87.
3 min
88.
(Click) after participants have read. Ask them to take a moment to synthesize
their thoughts about the day using the sentence starters. Next, have them
share a thought with a partner, and then ask participants to share out with
the whole group.
Use the following icons in the script to indicate different learning modes.
Video
Reflect on a prompt
Turnkey Materials Provided




Math Instruction Across the Grade Band PPT
Math Instruction Across the Grade Band Facilitators Guide
Counting On Video
CCSS Instructional Practice Guide
Additional Suggested Resources
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How to Implement A Story of Units
A Story of Units Year Long Curriculum Overview
A Story of Units CCLS Checklist
Active learning
Turn and talk