Kathleen Sicula
CCLM Project 2
July 15, 2011
Project 2: CCSSM Interpretation Guide
Part 1: Standards
Grade 3
Domain: Number Operations in Base Ten
Cluster: Understand properties of multiplication and the relationship
between multiplication and division.
Standard 3.OA5: Apply properties of operations as strategies to multiply
and divide.
Standard 3.OA7: Fluently multiply and divide within 100, using strategies
such as the relationship between multiplication and
division or properties of operations. By the end of Grade
3, know from memory all products of two one-digit
numbers.
Part 2: Interpretation of the Standards
3.OA5 – Teacher Friendly Language
Students will apply the rules about how numbers work when multiplying and dividing.
These rules include:
Commutative Property of Multiplication – Order of factors does not affect
product. (4x5 = 5x4).
Associative Property of Multiplication: Grouping of factors does not affect
product. (4x5x2 can be thought of as 4x5= 20, 20x2 = 40 or
5x2 = 10, 10x4 = 40)
Distributive Property of Multiplication: Factors can be broken down into “friendly”
or workable numbers. (7x6 can be though of as 5x6=30, add two more groups of
six ~ 30+12 = 42)
3.OA5 – Classroom Practices
One way of developing an understanding of the commutative property of
multiplication is having students build arrays. For example, students may be given a
word problem that reads: There are 24 plants in a garden. Use counters to
represent the plants. Arrange the counters to show them in equal rows. (Everyday
DRAFT DOCUMENT, UNEDITED COPY. This material was developed for the Common Core Leadership
in Mathematics (CCLM) project at the University of Wisconsin-Milwaukee. (07.15.2011)
Math – Lesson 4.2) In building the models, students may have a variety of
solutions and should be encouraged to discuss their thinking. See Figure 1.
“Teacher Talk” around this work may include:
1. How many different ways can the garden be planted?
2. What do you notice about the arrays?
3. What does this show you about multiplication and division?
Building the models will lead the students to see that the factors 4x6, 6x4, 2x12,
12x2, 3x8, and 8x3 all are ways to make the product 24. Order of the factors may
affect the direction of the array (or garden) but does not affect the product.
They can build a fact family around the number models:
4x6=24, 6x4=24, 24/6=4, 24/4=6.
DRAFT DOCUMENT, UNEDITED COPY. This material was developed for the Common Core Leadership
in Mathematics (CCLM) project at the University of Wisconsin-Milwaukee. (07.15.2011)
One way of developing an understanding of the distributive property of
multiplication is for the teacher to ask students to take a single-digit
multiplication fact and break it down into more workable parts. Teachers should
start with facts that encourage easy composition or decomposition of numbers.
Examples of these kinds of facts might include 8x6, 9x3, or 4x7.
See Figure 2. “Teacher Talk” around these types of facts and solutions should help
students understand that it may be more efficient to solve a problem when working
from known facts or models.
One way of developing an understanding of the associative property of
multiplication is to have the students multiply single digit factors by multiples of
10. Examples of these kinds of facts might include 9x80, 6x30, 4x400. These
DRAFT DOCUMENT, UNEDITED COPY. This material was developed for the Common Core Leadership
in Mathematics (CCLM) project at the University of Wisconsin-Milwaukee. (07.15.2011)
kinds of problems also require a strong understanding of base-ten notation, which
is a critical area in grade 2. A strong understanding of place value is also important.
See Figure 3.
“Teacher Talk” around this work should help students understand that their
knowledge of base-ten can help them solve these types of facts.
It is important to note that students in third grade are not expected to be able to
name the properties of multiplication. However, they should have a strong
understanding how the properties work and be flexible in their ability to choose an
appropriate operation to solve a particular problem.
DRAFT DOCUMENT, UNEDITED COPY. This material was developed for the Common Core Leadership
in Mathematics (CCLM) project at the University of Wisconsin-Milwaukee. (07.15.2011)
3.OA7 – Teacher Friendly Language
Students will use strategies to accurately solve multiplication and division facts
within 100. Students will apply knowledge from working with manipulatives,
pictures, and arrays to fluently (3 – 4 seconds) determine answers.
3.0A7 - Classroom Practice
In order to build fact fluency in the classroom, teachers can use a variety of
strategies and activities. One of the most important is to continue to develop the
concept of “fact families.” Students should be able to transfer their knowledge of
addition and subtraction fact families to multiplication and division. When using
fact families, students make connections between facts. (4x3 = 12 helps a student
learn 3x4 = 12, 12/4 = 3, and 12/3 = 4) Games such as “Beat the Calculator” and
“Name that Number” (Everyday Math – Grade 3) also help build fluency. Activities
that develop strong comprehension of the commutative property of multiplication
will help students see the connection between multiplication and division facts.
Teachers can also help students build fact fluency by introducing “Fact Power.”
These are quick strategies and patterns students can use when multiplying by
specific factors.
Examples of “Fact Power”:
X2 – Double the other factor, Count by 2s
X4 – Think “Double Double” – 4x6 can be thought of as 6+6 = 12, 12+12 = 24
X5 – Count by 5s
Once students have these strategies and patterns in place, fact quizzes that
promote the use of Fact Power and allow 3 – 4 seconds per fact may be used to
maintain fact fluency.
Part 3: School Mathematics Textbook Program
Textbook Development for Standards 3.OA5 and 3.OA7
The Everyday Math program (EDM) begins to informally explore multiplication in
first grade through work with patterns and skip counting. Multiplication and
division concepts and strategies are introduced in second grade in one of the final
chapters. Instruction begins with lessons on multiples of equal groups. Models
such as counters, pictures and arrays are used to solve number stories involving
DRAFT DOCUMENT, UNEDITED COPY. This material was developed for the Common Core Leadership
in Mathematics (CCLM) project at the University of Wisconsin-Milwaukee. (07.15.2011)
subitizing. (The concept of subitizing, or grouping, must be understood before
students can apply the strategies.) Students are then formally introduced to
multiplication diagrams and number models. (Commutative Property of
Multiplication) Division is related to multiplication through equal-grouping and
equal-sharing number stories. Patterns of doubling and halving are explored.
(Distributive Property of Multiplication) Fact fluency is expected with the factors
0 and 1 and practiced with factors 2, 5, 10. Students begin to work with
multiplication and division fact families in second grade.
One of the critical areas in the CCSSM standards for third grade is “developing
understanding of multiplication and division and strategies for multiplication and
division within 100.” The EDM program strongly emphasizes multiplication concepts
and strategies in grade 3. Work with multiplication concepts begins in October
during Chapter 3 – Linear Measures and Area. Students explore areas of
rectangles and calculate areas using number models. (Commutative Property of
Multiplication) In Chapter 4 – Multiplication and Division, students build on
learning from second grade. The concept of equal groups is reviewed. Students
are given opportunities to use counters, arrays, area models, diagrams, and number
models to represent and solve multiplication and division stories. Fact fluency is
built through a review of fact families, use of the multiplication/division facts
table, and practice using fact triangles. Fact fluency is continued in Unit 7 –
Multiplication and Division when students learn square-number facts and review
multiplication and division patterns using a facts table. Students are expected to
be fluent solving facts with the factors 0, 1, 2, 5, 10. The associative and
distributive properties of multiplication are also introduced in this chapter as
students work with word problems and learn to write number models with
parentheses. Students also use the associative property of multiplication as they
learn to multiply and divide one digit numbers by multiples of 10, 100, and 1,000.
There is a strong emphasis on reasoning strategies and estimation when solving
multiplication word problems. Finally, in Unit 9 – Multiplication and Division,
students extend their thinking of multiplication strategies by learning the partialproducts and lattice algorithms. These algorithms use properties of multiplication
to solve problems involving multi-digit numbers. Students also build fact fluency by
finding factor pairs for given products. In Chapter 9 students also explore
computational strategies and their understanding of the connection between
multiplication and division while using counters to solve division problems that
involve remainders. Games such as Name that Number, Multiplication Bingo, Factor
DRAFT DOCUMENT, UNEDITED COPY. This material was developed for the Common Core Leadership
in Mathematics (CCLM) project at the University of Wisconsin-Milwaukee. (07.15.2011)
Bingo, Beat the Calculator, Baseball Multiplication and Roll to 100 are used
throughout the year to build and maintain fact fluency.
In fourth grade, EDM continues to build on multiplication concepts and the use of
properties as strategies that were introduced in third grade. One additional
model, the rate table, is used as a strategy to solve number stories. Students use
the distributive property of multiplication as they work with exponential notation.
The distributive property of multiplication is also used as students solve word
problems that require more than one step. Students work to use parentheses in
these number models. The partial-products and lattice algorithms are extended
and include multiplication of decimals. By the end of fourth grade, fact fluency is
expected using the factors 0 – 10. Teachers use the “50 Facts” test routine to
help students build and maintain fluency.
Conclusions
Overall, the EDM program is aligned with the mathematics of standard 3.OA5. All
three properties of multiplication are developed and used as strategies for solving
problems throughout third grade. Students are encouraged to use a variety of
models to help them visualize their learning. These models include manipulatives
(cubes), arrays, area models, base 10 blocks, number diagrams, and fact triangles.
Lessons are set up and develop math language using the model put forth in the
standards. Each lesson begins with a math message that helps introduce the
concept. The teacher leads the students in a discussion of strategy work.
(informal language) This allows the teacher to see the level of understanding and
leads him/her into a more formal discussion of the concept. (concept based
language) When appropriate, algorithms are taught as a method for solving
problems. Lessons include opportunities for individual practice, discussion, and
interaction between peers. Review work is provided through Math Boxes and
games. Each lesson includes ideas for differentiation. Two exploration activities
are included in each chapter to help students make generalizations about
strategies and develop concepts before learning formal algorithms and equations.
EDM expectations for fact fluency in third grade are not in alignment with the
CCSSM standard 3.OA7. This standard states that by the end of third grade,
students are expected to fluently multiply and divide within 100.
EDM limits fluency expectations in third grade to the x0, 1, 2, 5, 10 facts.
Although the remaining facts within 100 are not expected to be automatic in third
DRAFT DOCUMENT, UNEDITED COPY. This material was developed for the Common Core Leadership
in Mathematics (CCLM) project at the University of Wisconsin-Milwaukee. (07.15.2011)
grade, students must have strategies to compute the products. There is no
expectation for division fluency in EDM for third grade.
Suggestions
Teachers can help students achieve the expectations of standard 3.0A5 by
following the EDM program. The following suggestions are provided to help
teachers as they use the program to teach the mathematical properties and
multiplication/division: Set models, area models, and number lines should all be
used as ways to model multiplication facts. A list of math vocabulary is included at
the start of each lesson. It is important for the teacher to use this language when
teaching the concepts and expect students to use the language when explaining
answers or discussing thinking. Teachers will want to be sure to refer to the
Balanced Assessment page at the start of each chapter to find the portion of each
lesson that formatively assesses student progress. This information may be used
to guide the lesson and when making decisions on how to use the portion of the
lesson called Differentiation Options. The teacher might even choose to use some
of these differentiation options with the whole class. While there is much
opportunity for oral discussion of problem solving strategies, it is suggested that
more opportunities be provided for students to write constructed responses to
justify their reasoning when problem solving. Teachers may also want modify the
tests or include portions were students must show their work and construct
responses to defend their reasoning.
In order to meet the expectations of CCSSM standard 3.OA7, teachers will need
to make adjustments to their teaching of fact fluency. The following suggestions
are provided to help teachers as they teach fact fluency: Stronger emphasis on
fact shortcuts (fact power), pattern work with the multiplication facts table, drills
with fact triangles, and work with fact families will all help to make solving
multiplication facts more automatic.
Giving students time to play the fact fluency games is also extremely important in
building fluency. Creating weekly formative assessments that focus on fluency will
hold students accountable for learning their facts. When creating these
assessments, teachers should be mindful to group facts in a way that encourages
strategic thinking. For example, one fact test might only include x5 and x10 facts.
Students can use the “count by” strategy to solve these facts. Another fact text
might have the x2 and x4 facts. Students can use the “doubling” strategy to solve
these facts. Other assessments might include both multiplication and division
DRAFT DOCUMENT, UNEDITED COPY. This material was developed for the Common Core Leadership
in Mathematics (CCLM) project at the University of Wisconsin-Milwaukee. (07.15.2011)
facts for the same set of factors. These assessments would help the students
work on the connection between the multiplication and division facts. By using the
standards definition of fluent as being “within 3-4 seconds,” teachers should
expect students to be able to solve 20 problems correctly in approximately 70
seconds.
Part 4: “Check Point” Formative Assessment Tasks for Standards 3.OA5 and
3.OA7
The following pages contain a formative assessment for each standard. After each
assessment is an example of proficient student work, along with teacher
annotations explaining how the student’s reasoning is related to the mathematics in
the standard.
DRAFT DOCUMENT, UNEDITED COPY. This material was developed for the Common Core Leadership
in Mathematics (CCLM) project at the University of Wisconsin-Milwaukee. (07.15.2011)
Name_______________________________
Solve:
*Fill in the multiplication/division diagram.
*Use a strategy to solve the problem.
*Write the answer. Write a number model.
Mrs. Bay has 4 packs of colored pencils. Each box has 8 pencils. How many pencils
does she have?
Packs
Pencils per Pack
Pencils in all
Strategy Work:
Answer:______________
Number Model________________________
Suppose Mrs. Bay had only 7 pencils in a pack. How many pencils would she have
now? Explain your reasoning.
DRAFT DOCUMENT, UNEDITED COPY. This material was developed for the Common Core Leadership
in Mathematics (CCLM) project at the University of Wisconsin-Milwaukee. (07.15.2011)
DRAFT DOCUMENT, UNEDITED COPY. This material was developed for the Common Core Leadership
in Mathematics (CCLM) project at the University of Wisconsin-Milwaukee. (07.15.2011)
Name________________________________
Fact Fluency Assessment
Solve:
2x2 = _____ 6x4 = _____ 3x2 = _____ 4x7 = _____
4x4 = _____ 10x2 = ____ 8x4 = _____ 4x2 = _____
2x5 = _____ 3x4 = _____ 6x2 = _____ 5x4 = _____
4x9 = _____ 7x2 = _____ 10x4 = ____ 9x2 = _____
Circle a fact that you answered that you did not know
automatically. What strategy did you use to solve the fact
quickly?
DRAFT DOCUMENT, UNEDITED COPY. This material was developed for the Common Core Leadership
in Mathematics (CCLM) project at the University of Wisconsin-Milwaukee. (07.15.2011)
DRAFT DOCUMENT, UNEDITED COPY. This material was developed for the Common Core Leadership
in Mathematics (CCLM) project at the University of Wisconsin-Milwaukee. (07.15.2011)