Introduction to Division: Two-Digit by One-Digit Problems
Pre-Assessment Data:
What Do Students Already Know: Multiplication facts up to 12, how to multiply four-digitnumbers by one-digit numbers using both the partial product and standard algorithm
methods, how to make a tape diagram for addition, subtraction, and multiplication, RDW
(read, write, draw) process, know how to identify KWFO (key words, facts, and operation)
in word problems, and how to solve multistep word problems. As far as division, some
students know and can demonstrate how to properly divide.
Formal/Informal Data Available: As far as formal data that is available on the students we
have their math interim scores, mid-module and end-of-module assessment scores, exit
ticket booklets, problem set booklets, and homework. The informal data available is usually
during the fluency, application, and concept development parts of the lesson where they
work on problems on their personal white boards. During this portion of the lesson they
solve problems and then hold up their white boards, I am then able to see who is getting
the concept and who may need more scaffolding. The whiteboards also help me see
common misconceptions amongst the students, which allows me to clarify and differentiate
if needed. On Tuesday I gave the students a “pre-test” with 100 division problems. They
were given 5 minutes to complete as much as they could. As far as the data taken from the
pre-test I concluded there is a range of background knowledge when it comes to division.
Surprisingly though, more students answered the problems correctly than I anticipated.
Objectives:
Content Language Objective: SWBAT explain, in writing and speaking, how to solve multidigit division problems with models using specific math vocabulary.
Group Objective: WWBAT solve and explain, in writing and speaking, a real world money
problem with 3 or more digits using division and models.
Key Vocabulary: Dividend, divisor, and quotient
Rationale & Standards:
Why is this important to us? Connection to students’ real lives: Someone give me an
example of where you would use division in the real world? How will this help you in the
future? Look at your partner and say “we divide all the time.”
We divide al the time, we can’t deny that. Whether it’s evenly sharing a pizza with
three friends or planning your money wisely to pay bills. We distribute, break apart, and
separate numbers by numbers all the time.
Relation to unit goals and essential questions: Multiply or divide to solve word problems
involving multiplicative comparison by using drawings and equations with a symbol for the
unknown number to represent the problem, distinguishing multiplicative comparison from
additive comparison 4.OA.A.2. Solve multistep word problems posed with whole numbers
and having whole number answers using the four operations, including problems in which
remainders must be interpreted 4.OA.A.3. Find whole-number quotients and remainders
with up to four-digit dividends and one-digit divisors, using strategies based on place
values, the properties of operations, and/or the relationship between multiplication and
division. Illustrate and explain the calculation by using equations, rectangular arrays,
and/or area models 4.NBT.B.6.
Assessment:
How will you measure students’ progress toward the objective?
1. Today we are going to start by making our individual tape diagrams, I want to see
you successfully fold and label a long piece of paper into twelve parts. To show me
you understand your tape diagram, you will fold the twelve-part tape by two, three,
and four to represent the dividend, divisor, and quotient in the equation. If I ask you
how many groups you have and the size of each group you should be able to identify
those numbers.
2. Quick division problems on white boards as a form of informal assessment.
3. In groups of three you will work with the large white boards and solve a real-world
money problem. This problem is designed to make you think about how people
manage their money by dividing it up into parts, delegating portions of their money
to be for savings, bills, groceries, internet/cable, and much, much more. On your
white boards I want to see…
a. A RDW process, show me you have read the problem by…
i. KW, F, O (on handout of problem)
b. Show me a drawing by…
i. A model, picture, or diagram (tape diagram, array, number bond, etc.)
c. The writing part of this problem will be shown by…
i. Work! I need to see how you solved the problem numerically!
ii. Check: How can you check your answer? Is your answer reasonable?
iii. Statement of the answer.
4. Individually, you will complete an exit ticket to assess your understanding of using
models to solve division.
Differentiation:
Will the data you gather allow you to differentiate future instruction? Yes, we tend to use
the same forms of gathering data; problem sets, white board work, and exit tickets from the
past help me understand students’ prior knowledge and ability. Since this is the
introduction to division lesson the data I collect during this time will be extremely
informative for how I will conduct further division lessons. It’s very important that the
class masters these basic principles of division before moving on to more complex
problems.
How will you use data to intentionally group students according to skills/readiness? During
the first activity where the students are folding tape diagrams I will be able to see how
students process the hands on, more visually appealing, explanation of division. Not all
students learn in abstract ways, therefore, the quick division work on their white boards
will register for those who like to see numbers in standard form. Once I complete these two
sections of the lesson I will be able to tell who may need more scaffolding and further
clarification. I will then assist those students or group of students. Prior to the lesson I will
have groups for the third part of the lesson already arranged. I will be making these groups
based off data and knowledge that I already have on the students’ ability. I intentionally
place a high, middle of the road, and low student in one group. This is where I hope further
scaffolding will occur from students to students. Sometimes my explanations don’t reach all
of the students and other students phrase or clarify the struggle child’s problem in a way
that I didn’t.
Student Choice: During the tape diagram session student’s can and should collaborate with
their tables. Once we move to the carpet for the division fact part I tend to direct students
to sit where they won’t be distracted. Some days they act accordingly to the expectations,
other days not so much. I like to start the day with trust and at least give them the
opportunity to prove me wrong. However, if they act out they know their assigned carpet
seats and will be told to move. For the real-world application problem I will create the
groups because I want to place specific students with each other to foster a beneficial
learning environment.
Lesson Components
Management
 Clip Chart: Individual, daily, behavior accountability system. Students start on silver
everyday, for positive behavior one can “clip up” to gold. For negative behavior, one
would “clip down” to stone, fool’s gold, or coal. Each level results in a specific
reward/consequence.
 Marble Jar: Whole class, ongoing, behavior accountability system. Whole group
positive or negative behavior results in addition or subtraction of marbles. When
the class collects 50 they receive a “party” of sorts.
 Chimes: We use chimes to signal when we want students’ attention.
Lesson Layout (Intro, Body, & Debrief)
1. Introduction: Review of Objectives, Rationale, and Assessment (10 MIN)
a. Good Morning! I hope everyone is ready to do some math. As I told you
Tuesday, today we are going to learn division. Can someone please read me
today’s PERSONAL objective in their best level 3 (presentation) voice?
b. Content Language Objective: SWBAT explain, in writing and speaking, how to
solve multi-digit division problems with models using specific math
vocabulary.
c. Can someone please read me today’s GROUP objective in their best level 3
(presentation) voice?
d. Group Objective: WWBAT solve and explain, in writing and speaking, a real
world money problem with 3 or more digit using division and models.
e. In today’s objective it says to explain your solution using specific math
vocabulary. Today’s key vocabulary is….
f. Key Vocabulary: Dividend, divisor, and quotient
g. What is a dividend? Divisor? Quotient? Division
h. Write on board: Dividend÷Divisor= Quotient & House Version
i. Circle the dividend, square the divisor, and triangle the quotient.
i. Rationale: So, why is division important to us? Someone give me an example
of where you would use division in the real world. How will this help you in
the future? Look at your partner and say, “we divide all the time.”
j.
Assessment: As far as how I will assess your understanding today, I will look
at your folded tape diagrams, white board work, group problem solution, and
your exit ticket.
k. Capiche? Any questions, comments, or concerns before we move on?
TRANSITION: Distribute tape diagrams. For this part of the lesson you will need a pen or
marker to write on your tape diagram. Take out a marker or pen and show me you are
ready by SILENTLY resting your head on your desk.
2. Folding Tape Diagram (20 MIN)
a. To start today’s lesson we are going to make our own tape diagrams (show
example). I want you to take this tape (show blank) and fold it into 12 equal
parts. I’m going to give you about 2 minutes to try and figure this out.
Collaborate with your table group! Before I say “go” show me with your
fingers what noise level should we be using? No more than a 2, maybe a 1.5….
Alright, go!
b. Who figured it out? It’s okay, it took me 5 different attempts, and then I had
to look it up on YouTube. What I want you to do is take your tape and
crumple it up into a ball. We made mistakes right? Is that ok? What do we do
with mistakes? Learn from them. Ok, everyone’s tape is in a ball? Throw it
away. Let’s start fresh.
c. Follow along as I demonstrate how to fold this tape into 12 equal parts. First,
fold the tape into half. When we fold it into half, what are we dividing 12 by?
Yes, 2. Write on board: Fold in ½ = divide by 2.
d. Next, fold it into thirds. When we fold the tape into thirds, what are we
dividing by? Yes, 3. Write on board: Fold in 1/3= divide by 3.
e. Last step is to fold it in half again. Similar to the first step, when we fold in
half, what are we dividing by? Yes, 2.
f. So, we folded it in half, then thirds, then in half again. 2x3x2=? 12. Woah, I
just multiplied those numbers and received 12. How does this show us the
relationship between multiplication and division?
g. Open up your tape diagram, can you see your 12 equal parts? Take a marker
are draw straight lines on your folds and label each box 1-12 (show
example).
h. Let’s go over division facts of 12 using our tape diagrams.
i. When you divide your tape by 2, what do you get? 6.
ii. What if you divide it by 3? 4.
iii. Divide it by 4? 3.
iv. We can play around with these tape diagrams later, but its time to
move on.
TRANSITION: Leave tape diagrams at your desk and take out your white board. You have
10 seconds to SILENTLY grab a marker and chose a smart seat on the carpet. Go!
3. Division Practice (20 MINS)
a. Okay, after looking over your pre-tests I noticed some common mistakes, so
we are going to practice on our white boards some division problems. No
need to show work, write your answer and hold it up.
i. A number divided by the same number will always equal what?
(7÷7=1) Yes, 1.
ii. A number divided by 1 will always equal? (7÷ 1= 7) Yes, it will always
equal the other number.
iii. As we know, division is the opposite of multiplication. So, the more
multiplication facts you have memorized the easier it will be to solve
these types of division problems.
b. Now we are going to move onto word problems where I want you to show
me the solution with a model or drawing. The three types of models we will
use are tape diagrams, arrays, and number bonds.
c. In this first word problem I want you to show me the solution with a tape
diagram. Remember with tape diagrams there is a whole, number of parts or
groups, and size of parts or groups.
d. For the second word problem I want you to show me the solution with an
array. What does an array look like? Yes, it’s a number of objects that follow a
specific pattern.
e. In the third problem I want you to show me the solution with a number bond.
Does anyone know what a number bond is?
f. Great, can you please give me a fist to five on your understanding of using
models to solve division problems? Ok, it’s okay to not feel 100% on this
because today’s the first time you are learning it. With more practice you will
master these different ways to draw diagrams of division.
g. Now we are going to move onto a multi-step word problem with 4 digit
numbers. You will be working in groups of 3 to solve this problem on the
large whiteboards.
TRANSITION: When I say, “go” you have 10 seconds to SILENTLY put your markers away,
sit at your desk, and put your white boards away. Show me you are ready by SILENTLY
putting your head on your desk. Go!
4. Group Problem Solving (15 MINS)
a. Listen carefully as I explain the directions and groups. As a group you are
going to solve this world problem together using the RDW process. The first
part of this process is read, so follow along as I read the problem aloud. Let’s
underline our KW/F/O. For the draw part your group will choose two of the
three models we just went over: tape diagram, array, or number bond. For
the write part I want to see your work, a statement of your answers, and how
you checked your answer.
b. Your groups are as follows…..
c. When I say, “go” I want you to get with your group, get a white board and
marker, and begin to solve. Make sure you are all collaborating with each
other. I am going to give you about 10 minutes to complete this and then we
will go over each group’s work as a class.
d. Any questions, comments, or concerns?
e. Someone repeat back to me the instructions and what I need to see on your
white boards. Yes, a diagram/model, written work, a statement of your
answers, and a check of your work. Capiche?
f. Wait, what noise level should you be using? Yes, 2, but a quiet 2.
g. Go!
TRANSITION: It’s time to go over each groups work as a class. Choose 1 person to take the
white board to the front of the front on the classroom. When I say, “go” I want you to
SILENTLY take a seat at the carpet WITH YOUR GROUP. Show me that you and your group
are ready by sitting silently and still, looking up at me. Go!
5. Review of Work (10 MINS)
a. Let’s look at each groups work. What similarities do you see? Differences?
(Drawings, answers, and check)
b. What’s the answer to the first part of the problem? Eric has $6,000 in his
savings.
c. What’s the answer to the second part of the problem? Eric has $2,000 for
bills, $2,000 for rent, and $2,000 for food.
d. What’s the answer to the third part of the problem? Eric has $1,000 for
dessert.
e. Do we see how different groups solved using different models?
6. Student Debrief and Closing (5 MINS)
a. Let’s look back at our objectives for today; we divided numbers all types of
sizes of digits. Give me a fist to five on how you feel about the PERSONAL
objective: dividing two-digit by one-digit numbers?
b. After some more practice with your group using models, give me a fist to five
on your understanding.
c. As far as the math vocabulary, rate your understanding of the words
dividend, divisor, and quotient.
d. Any final questions, comments or concerns before we do the exit ticket?
TRANSITION: When I say, “go” I want you to SILENTLY get up and go back to your desk.
Show me you are ready for the exit ticket by taking a pencil out and resting your head on
your desk. Go!
7. Exit Ticket (5 MINS)
a. Remind me again, how are exit tickets done? Independently.
b. So what noise level should you be using? 0
c. You will have 5 minutes to answer as many of the division problems as you
can.
d. Go!
Materials and Resources Needed
 2 tape diagrams
 Marker or pen
 Pencil
 Personal White-Boards
 Dry Erase Markers
 Large White-Boards
 Exit Tickets
 Chart Paper of the 3 Different Models
Extension of Learning: Homework
Division Practice: Solve two-digit by one-digit problems. Circle the dividend, square the
divisor, and triangle the quotient.
1÷1=
20÷1=
77÷1=
26÷26=
91÷91=
47÷47=
27÷3=
36÷9=
48÷8=
32÷4=
66÷2=
63÷7=
72÷9=
25÷5=
18÷6=
Division Practice: Solve two-digit by one-digit word problems using a model. Circle
the dividend, square the divisor, and triangle the quotient.
Maggie has 56 pieces of candy. She wants to divide her candy evenly amongst her 8 friends.
How many pieces of candy does each friend get? Solve using a tape diagram. (7 pieces of
candy)
Luke has 45 library books. He wants to divide the books amongst 5 classrooms. How many
books does each class get? Solve using a number bond. (9 books)
Miss Robbertz has 21 homework assignments to do. She decides to do 3 assignments each
day. How many days will it take Miss Robbertz to finish all of her homework? Solve using
an array. (7 days)
Group Problem: Solve a four-digit by one-digit word problem using the RDW process.
Eric has $12,000 in his bank account. He divides his money into 4 different amounts:
savings, bills, food, and rent.
He puts half of it into his savings account. How much money does he have in his savings
account? ($6,000)
With the remaining money, he divides it into three equal amounts to pay for his bills, food,
and rent. How much is each part worth? ($2,000 each)
Eric loves dessert so he decides to divide his “food money” into two equal parts: food and
dessert. How much money does Eric have for dessert? ($1,000)
Name:_________________________________
Date:__________________
Exit Ticket
Solve the following problem using a tape diagram.
40÷5=_______
Solve the following problem using an array.
16÷4=_______
Solve the following problem using a number bond.
72÷8=_______
Name:_________________________________
Date:__________________
Exit Ticket
Solve the following problem using a tape diagram.
40÷5=_______
Solve the following problem using an array.
16÷4=_______
Solve the following problem using a number bond.
72÷8=_______
Name:__________________________________
Date:__________________
Homework: Solving Division Using a Model
Solve the following problems using a tape
diagram.
Solve the following problems using an
array.
1. 54÷9=_______
1. 15÷3=_______
2. 48÷6=_______
2. 36÷6=_______
3. 30÷5=_______
3. 28÷4=_______
4. 18÷2=_______
4. 55÷5=_______
5. 88÷11=______
5. 42÷7=_______
Solve the following problems using a number bond.
1. 63÷9=______
6. 99÷9=_______
2. 121÷11=_______
7. 108÷12=________
3. 96÷8=_________
8. 132÷12=________
4. 84÷12=________
9. 100÷10=_______
5. 60÷5=_________
10. 1÷1=________