Repeated Addition, Arrays, and Equal Groups – Building a Foundation for
Multiplication
I.
II.
III.
Repeated Addition and Equal Groups: Students will learn about equal
groups and begin to build a foundation for multiplication and division. They will
learn how to solve repeated addition problems through problem solving, drawing
pictures, and making arrays. Students will extensively explore repeated addition
and will be able to write their own repeated addition stories. Students will also
be able to go from a picture or array to a repeated addition number sentence or
from a repeated addition number sentence to a picture or an array.
Essential Questions: Students will explore the essential questions, “how do we
solve repeated addition problems?” and “how do we work with equal groups to
solve repeated addition problems?”. The main idea of this unit is to have the
students explore repeated addition in many different ways, so that they build a
solid foundation for multiplication and division in the future. Students will work
with repeated addition in many different ways including word problems, skip
counting, number sentences, and arrays. The idea is for students to become
experts in repeated addition so that when they learn multiplication next year
they will have the knowledge they need to understand equal groupings.
Learning Goals:
A. Development of content understanding (key concepts and ideas): Students will
understand the concept of equal groupings. They will be able to solve
repeated addition number sentences and identify the array that goes with the
repeated addition number sentence, and vice versa. Students will also be able
to use problem solving to complete repeated addition word problems, and will
be able to write their own repeated addition word problems. In addition,
students will be able to talk about repeated addition and arrays in terms of
equal groups.
B. Development of habits of mind and work, including habits of independent or
collaborative thinking and doing typical of readers, writers, speakers,
creators, researchers, and thinkers in the discipline (ways of knowing):
Students will come to know about repeated addition through a variety of
independent work, group work, and class work. They will learn to explain
how to solve repeated addition problems and identify equal groups. Students
will engage in a variety of different activities, ranging from identifying arrays
in a book scavenger hunt, completing enVision workmats, creating their own
word problems, and playing repeated addition games. In addition, students
will explore repeated addition and arrays through problem solving. Before
introducing the topic, students will discover how to solve repeated addition
and create arrays through open ended word problems. They will work with
groups to explain their reasoning and analyze their solutions as a class.
C. Literacy development, including capabilities of proficient readers, writers, and
speakers: Although this is a math unit, students will be using their literacy
skills to demonstrate their understanding. Many lessons will include class
IV.
discussions which will allow the students to practice their speaking skills. In
addition, the two word problems students will be working on will be depicted
on a poster that they will solve in a group. Then, the groups will have to
present by explaining how they solved their problem. This will help them
practice writing in a group, as well as presenting in a group. It will also help
them to begin to explain, in words, how they are thinking about
mathematical concepts. Students will also begin to write their own word
problems with repeated addition. This will help students practice a different
kind of writing than they may be used to and make their writing skills more
versatile. They need to begin to learn how to use writing to explain their
mathematical thinking, as well as to create their own problems.
D. Development of the classroom as a learning community: Each lesson will
involve some sort of partner or group work. This will allow the students to get
to know each other as people and as mathematical thinkers. It will allow
them to talk in an informal way about their mathematical thinking, and will
give them more confidence when they present or share with the whole class.
Students will be asked to complete two open-ended word problems to explore
and discover the mathematical concepts of repeated addition and arrays.
These problems, although new to them, can be solved through a variety of
different methods. Students will be able to work on them independently for a
short period of time to get their ideas on paper, and then they will work in a
group before they present. Working in a group will allow them to talk
through their ideas and mathematical thinking and gain more confidence in
their mathematical abilities and problem solving skills. Students will be
encouraged to solve the problems in whatever way they are comfortable with,
and their different strategies will be represented on a chart, to include all
students and enforce the idea that there is more than one “right” way to solve
a problem. In addition, students will use their knowledge of arrays to work on
a quilt patch, which in the end will be put all together and hung up in the
classroom. This gives meaning to their work, and demonstrates the
importance of collaboration in learning. Students will also be starting math
centers in this unit, which they have not done before. This shows the
students that the teacher trusts them to be accountable for their behavior
and their learning while they are not in a small group with the teacher. In
addition, by teaching the students new concepts in small groups, it creates a
low-stakes environment and allows the students to feel more comfortable in
asking questions about the mathematical concepts.
Rationale:
A. Learning Goals: Students will begin to build a foundation for multiplication
and division by learning how to solve repeated addition number sentences.
Students will explore repeated addition using their problem solving skills to
solve an open ended word problem about M&Ms (something they love!), and
will use practice explaining their mathematical thinking by working in
groups and presenting their solution. Students will be asked to talk about
what they notice about the groups’ strategies, and they will also be asked to
discuss and identify the different problems and solutions in terms of equal
groups.
Students have already been encouraged to draw pictures to help them
solve repeated addition problems, and now they will have to use this picture
drawing practice to discover what arrays are. After they discover how to solve
the word problem (involving brownies and how we can get them into equal
pieces so that everyone in the class gets one), the will have to use this real
life problem to think about what their picture looks like, and come to the
conclusion that there are an equal number of brownies in each row.
Lastly, students will practice solving repeated addition word problems,
and use the strategy of “UNPACK” the question to help them. This strategy
can be used for any word problem, but is especially helpful for repeated
addition because they have to really think about what the question is asking
(underlining the word each, for example, is very helpful to understanding a
repeated addition word problem). In addition, this strategy will be helpful for
the students later on in math as well, because it is adaptable to all word
problems. Furthermore, students will learn how to write their own repeated
addition word problems, because if they can write their own problems, then
they can also solve them. The students have had a lot of trouble in the past
writing their own word problems, so I wanted to take the time in this unit to
go through how to write a word problem. If the students are comfortable
writing their own word problems and know how they should be worded and
what components go into it, then they will undoubtedly understand how to
solve a word problem.
B. Curriculum Standards:
CCSS.Math.Content.2.OA.C.4 Use addition to find the total number of
objects arranged in rectangular arrays with up to 5 rows and up to 5 columns;
write an equation to express the total as a sum of equal addends.
Students will create their own arrays, write number sentences from
arrays, solve repeated addition number sentences and word problems.
They will play several games to practice repeated addition and they
will practice skip counting to reinforce the idea of repeated addition.
Furthermore, they will make their own quilt patch with a decorated
array built from a number sentence.
C. Students Background and Readiness: Students have already learned how to
add three numbers together, and have had a lot of practice with word
problems. Although I have a lot of different learners in my classroom, they
will all be excited to solve the first introductory problem to repeated addition
because it involves M&Ms, and my students are very interested in anything
food related. They will be able to use the M&Ms as manipulatives, if they
need to, and if not they will get them as a snack after to make them very
interested. All students know how to use the strategy of drawing a picture to
help them solve a problem, so they should all be able to solve the open ended
word problems I give them. In addition, students have spent the last few
months working on different addition strategies, and they practice skip
counting every morning during morning meeting, so they already have
enough background knowledge to solve repeated addition number sentences.
D. Students Needs: I have a variety of students in my classroom, so I needed to
create a unit that would be applicable for all types of learners. The two word
problems I gave the students to discover arrays and repeated addition have
multiple entry points and may be solved in a variety of ways. Therefore, all
students, even students on IEPs and ELLs, should be able to solve them;
however, many of my students on IEPs and my ELLs will have the word
problems read to them to help with the literacy portion. Students will also
have a lot of time to work on these problems, and will work on them in
several different ways. First they will work individually so they can start to
think about the problem on their own. Then, they will work in their group
and explain how they solved the problem. Lastly, they will share with the
class. I decided to have these three steps so that students with learning
disabilities, or students who just take a longer time to process information
have the extra time to reread the question and get their ideas down before
they need to talk about them. In addition, students who might not have a lot
of confidence in math will be able to talk in a small group first before sharing
with the class. For the new concepts that are being taught, I will pull small
groups, and my students on IEPs will all be in the same group to assure that
they get the scaffolding that they need. In addition, all of the centers are
accessible to all students, and will be modeled to the whole class.
E. Research and Evidence-Based Best Practice Ideas: This unit involves learning
about repeated addition through many different methods. Before learning the
term “repeated addition” and “array”, students will discover how to solve
repeated addition problems and how to create an array through open ended
word problems (Zemelman, Daniels, & Hyde, 2005). By allowing students to
explore and discover these mathematical concepts, they are acting and
thinking like mathematicians and are active members of their learning
communities. This holds students accountable for their own actions, because
they are guiding their own learning. It holds the students to high standards,
and it lets them know that the teacher believes in their abilities and potential
(Smith, 2004). No matter what learning preference a student has, they will be
able to solve the open-ended problems given, because there are so many
different ways to think about solving them. In addition, because there are so
many ELLs and students on IEPs in my classroom, students will be read the
word problems, and will have a lot of time to talk about their mathematical
thinking in their groups. This will allow these students to talk about formal
academic thinking in an informal setting, making them more comfortable and
allowing them to think through their ideas before they discuss it with the
whole class or with the teacher. Furthermore, these students will benefit
from small group instruction, as well as math games, and other low stakes,
V.
informal assessments to help them practice their repeated addition
knowledge (Gibbons, 2009).
Assessments:
A. Explain your main assessments and how they are appropriate for your
learning goals: Students will be assessed many times throughout this math
unit, as to check for student understanding and misconceptions. This topic is
especially difficult for students to understand right away because it is the
basis of multiplication and division, which they have not been exposed to
before. Therefore, throughout the unit there are several places where student
misconceptions can be noticed. In both introductory activities (for repeated
addition and arrays), students will be given a worksheet with an open ended
word problem on it, on which they are asked to draw a picture and create a
number sentence. A variety of drawings or strategies would be accepted for
this worksheet; however, because this is the part that students are working
on individually, it will be helpful for the teacher to see what the students’
original thoughts are. Then, the students will share their mathematical
thinking with a group, and the group will present and explain to the class
how they solved the problem. In partners, and as a class, the students will
then discuss what they notice about the different strategies and the different
problems. The group work, discussions, and presentations will be an informal
assessment to see how students are thinking about these mathematical
concepts and how well they are able to explain their thinking in words.
In the math centers I have created for this unit, students will also be
assessed appropriately. All the centers will have some sort of a “worksheet”,
whether it be for the scavenger hunt, the coverall game, the quilt patches, or
the workmat, the students will be producing something in their center. This
is a way to make sure the students are working in their centers, and that
they are understanding the material, even if it is just with a dice game.
Furthermore, when students meet in small groups they will begin to work on
writing their own repeated addition word problems. This will allow the
teacher to talk with a small group of students about the different components
of a word problem and how to solve them and write them. Through discussion
with the students, and the students drafts and final products they will be
able to demonstrate their understanding of the material.
Finally, because this unit is based on a topic in the enVisions math
program, students will have to take several topic quizzes and a final unit
test. This will quantify their knowledge of repeated addition and arrays and
demonstrate whether or not they understood these mathematical concepts.
B. How will students know what to expect and the criteria for good work: At this
point in the year, students have had a lot of practice working in groups, and
have presented in groups more than once. They know what the expectation is
for working in a group, and understand that everyone must participate and
cooperate. Students also know the expectations for presenting, including
allowing everyone in the group a chance to speak, and speaking “loud and
proud”. I think the students will do very well with the open ended word
problem activities because they are used to this type of work. In addition,
although math centers are new for the kids, they do Daily 5 every day, and
most of them have been doing Daily 5 since kindergarten. They know that in
Daily 5 they must maximize their time on learning in the center, by working
the whole time. Therefore, I will tell the students that our new math centers
will have the same expectations as our Daily 5 centers. Students also know
that I will be collecting their work from every center, so they know a center
spent not maximizing their time on learning will not go unnoticed. In
addition, I will model each center for the students and clearly write out the
instructions for each center, so the students have a reminder of what they are
expected to be doing.
C. Culminating assignment and corresponding assessment criteria/rubric: The
students will have two final assessments. The first assessment will have two
parts, and is student centered; the students will write four of their own word
problems, and then switch with a partner to solve them. They must
demonstrate their solution through a drawing and a number sentence. In
addition, the students will have to take a unit test, which will assess their
understanding of repeated addition, arrays, and repeated addition word
problems.
VI.
Unit Calendar:
Lesson
Strategies
Lesson 1: What
Students will
is repeated
solve an open
addition?
ended word
problem based
on repeated
addition.
Students will
work
individually,
then in groups,
and then present
to the whole
class. We will
then discuss how
they discovered
repeated
addition and
equal groups.
Lesson 2:
Students will
Continuing to
engage in
work with
partner work
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
Materials
Differentiated
Word Problems
M&Ms
Construction
Paper
Markers
Chart Paper
Assessments
 Individual
worksheet
 Group poster
 Presentation
enVisions
interactive book
enVisions


Center
worksheets
enVisions quick
repeated
addition. How
do we UNPACK
word problems?
Lesson 3: What
is an array?
Lesson 4:
Applying our
knowledge of
arrays.
with whole class
word problem
examples.
Students will
then break into
different math
centers to work
on different
repeated
addition
activities.
Students will
work in small
groups with the
teacher to work
with word
problems.
Anchor chart,
whole group
discussion and
examples.
Students will
then work in
math centers to
practice repeated
addition and
arrays. Students
will also
continue to work
in small groups
with the teacher
to work on word
problems.
Students will
work on an openended word
problem
involving arrays.
Students will
work
individually,
then in a group,
and then will
present.
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workmat
Coverall game
(worksheet and
dice)
Skip counting
cards
Differentiated
repeated
addition
problems

Memory game
cards
Coverall game
Scavenger hunt
sheet
The Grapes of
Math
Amanda Bean’s
Amazing Dream
Differentiated
math problems
Workmats
Quilt patches
with number
sentence on back

Word problem
Brownies in a
rectangular pan
Posters
Markers

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


check quiz
Small group
work
Worksheets from
centers
Quilt patches
Small group
work
Individual
worksheets
Group posters
Presentations
Quick check quiz
Lesson 5:
Practicing
repeated
addition and
arrays.
Students will
continue to work
in centers to
practice repeated
addition and
arrays. In small
groups with the
teacher, students
will begin to
write their own
word problems.
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Lesson 6 (2
days): Word
problems and
review of
repeated
addition and
arrays.
Students will
complete their
word problems
(writing and
solving a
partner’s) and
quilt patches.
Students will
engage in a
review game for
their test.
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Word problem
graphic
organizers
Math notebooks
Coverall game
Memory game
Scavenger hunt
sheet
Amanda Bean’s
Amazing Dream
The Grapes of
Math
enVisions
workmat
enVision’s test
Center materials
(same as above –
for kids who
finish early)
Jeopardy
questions
Computer (to
type finish word
problems)
Math notebooks
Graphic
organizers

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

Center
worksheets
Small group
work
Word problems
enVisions test
Jeopardy game
VII. Resources:
A. How will you work to actively involve parents in their child’s academic activities and
performance, and communicate clearly with them?
All student enVision’s work will be sent home to their parents. In addition, students
will have homework every night that they will be encouraged to complete with their
parents.
B. Resources used:
 Van de Walle, J. A., Karp, K. S., & Bay-Williams, J. M. (2012). Elementary and
middle school mathematics: Teaching developmentally. Pearson.
 Zemelman, S., Daniels, H., & Hyde, A. (2005). Best practice, today's standards
for teaching and learning in America's schools. Heinemann.
 Common Core Standards (2012). Standards for mathematical practice. Retrieved
from http://www.corestandards.org/Math/Practice
 Neuschwander, C. & Burns, M. (1998). Amanda bean’s amazing dream.
Scholastic Press.
 Tang, G. (2004). The grapes of math. Scholastic Paperbacks.
C. Legal and ethical issues involved in Internet use and other resources: I made sure
that all of my resources were reliable and valid. All of the books I used I borrowed
from the curriculum lab in the education department at Clark University. In
addition, all of the ideas and inspiration I got for my unit came from scholarly math
textbooks and teaching and learning texts, also acquired through Clark University.
Furthermore, the mathematical concept I obtained from the enVision’s math
program, which is aligned with the Common Core standards.