3rd Grade Mathematics - Investigations
Unit 6: Multiplication and Division: Application and Fluency within 100
Teacher Resource Guide
2012 - 2013
In Grade 3, instructional time should focus on four critical areas:
1. Developing understanding of multiplication and division and strategies for multiplication and division within 100;
Students develop an understanding of the meanings of multiplication and division of whole numbers through activities and problems
involving equal-sized groups, arrays, and area models; multiplication is finding an unknown product, and division is finding an
unknown factor in these situations.
2. Developing understanding of fractions, especially unit fractions (fractions with a numerator of 1);
Students are able to use fractions to represent numbers equal to, less than, and greater than one. They solve problems that involve
comparing fractions by using visual fraction models and strategies based on noticing equal numerators or denominators.
3. Developing understanding of the structure of rectangular arrays and of area;
Students understand that rectangular arrays can be decomposed into identical rows or into identical columns. By decomposing
rectangles into rectangular arrays of squares, students connect area to multiplication, and justify using multiplication to determine
the area of a rectangle.
4. Describing and analyzing two-dimensional shapes;
Students compare and classify shapes by their sides and angles, and connect these with definitions of shapes. They also relate their
fraction work to geometry by expressing the area of part of a shape as a unit fraction of the whole.
3rd Grade
2012-2013
Page 1
3rd Grade Mathematics 2012 – 2013
Test By
1: Addition and Subtraction
7 weeks
8/27 – 10/12
October 12
2: Multiplication and Division:
Models within 100
5 weeks
10/15 – 11/16
November 16
3: Geometry/Measurement
4 weeks
11/19-12/21
December 21
4: Multiplication and Division:
Properties within 100
5 weeks
1/2 – 2/8
February 8
5: Fractions
8 weeks
2/11 – 4/12
April 12
6: Multiplication and Division:
Application & Fluency within 100
7 weeks
4/15 – 5/30
May 30
TRIMESTER 3
TRIMESTER 1
Time Frame
TRIMESTER 2
Unit
(Within 1,000)
3rd Grade
2012-2013
Page 2
Unit 6: Multiplication and Division: Application and Fluency within 100
Big Ideas
Essential Questions
Estimation is helpful in understating whether an answer is reasonable.
Multiplication and division are inverse operations.
STANDARDS
Identifier
April 15-May 30 (7 weeks)
Why do we estimate?
How are multiplication and division related?
Standards
Mathematical Practices
3.OA.3
Use multiplication and division within 100 to solve word problems in situations involving
equal groups, arrays, and measurement quantities,
1) Make sense of problems and persevere in
solving them.
3.OA.1
Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in 5
groups of 7 objects each.
2) Reason abstractly and quantitatively.
3.OA.2
Interpret whole-number quotients of whole numbers, e.g., interpret 56 ÷ 8 as the number of
objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of
shares when 56 objects are partitioned into equal shares of 8 objects each.
3) Construct viable arguments and critique the
reasoning of others.
3.NBT.3
Multiply one-digit whole numbers by multiples of 10 in the range 10–90 (e.g., 9 × 80, 5 × 60)
using strategies based on place value and properties of operations.
Fluently multiply and divide within 100, using strategies such as the relationship between
multiplication and division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or properties
of operations. By the end of Grade 3, know from memory all products of two one-digit
numbers.
4) Model with mathematics.
3.OA.4
Determine the unknown whole number in a multiplication or division equation relating three
whole numbers.
7) Look for and make use of structure.
3.OA.5
Identify arithmetic patterns (including patterns in the addition table or multiplication table), and
explain them using properties of operations. For example, observe that 4 times a number is
always even, and explain why 4 times a number can be decomposed into two equal addends.
8) Look for and express regularity in
repeated reasoning.
3.OA.6
Understand division as an unknown-factor problem.
3.OA.9
Identify arithmetic patterns (including patterns in the addition table or multiplication table), and
explain them using properties of operations.
Solve two-step word problems using the four operations. Represent these problems using
equations with a letter standing for the unknown quantity. Assess the reasonableness of
answers using mental computation and estimation strategies including rounding.3
3.OA.7
3.OA.8
3rd Grade
2012-2013
5) Use appropriate tools strategically.
6) Attend to precision.
Page 3
Unit 6: Multiplication and Division: Application and Fluency within 100
Identifier
Bloom’s
3.OA.3
Use multiplication and division within 100 to solve word problems in situations involving
equal groups, arrays, and measurement quantities,
3.OA.1
Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in 5
groups of 7 objects each.
3.OA.2
Interpret whole-number quotients of whole numbers, e.g., interpret 56 ÷ 8 as the number of
objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of
shares when 56 objects are partitioned into equal shares of 8 objects each.
3.NBT.3
Multiply one-digit whole numbers by multiples of 10 in the range 10–90 (e.g., 9 × 80, 5 × 60)
using strategies based on place value and properties of operations.
Fluently multiply and divide within 100, using strategies such as the relationship between
multiplication and division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or properties
of operations. By the end of Grade 3, know from memory all products of two one-digit
numbers.
3.OA.7
STANDARDS
Standards
3.OA.4
Determine the unknown whole number in a multiplication or division equation relating three
whole numbers.
3.OA.5
Identify arithmetic patterns (including patterns in the addition table or multiplication table), and
explain them using properties of operations. For example, observe that 4 times a number is
always even, and explain why 4 times a number can be decomposed into two equal addends.
3.OA.6
Understand division as an unknown-factor problem.
3.OA.9
Identify arithmetic patterns (including patterns in the addition table or multiplication table), and
explain them using properties of operations.
Solve two-step word problems using the four operations. Represent these problems using
equations with a letter standing for the unknown quantity. Assess the reasonableness of
answers using mental computation and estimation strategies including rounding.3
3.OA.8
3rd Grade
April 15-May 30 (7 weeks)
2012-2013
Skills
Concepts
Apply (3)
Solve (mult & div
word problems
w/in 100)
multiplication
division
equal groups
arrays
measurement
quantities
Apply (3)
Multiply & divide
(fluently w/in 100)
factor
product
Apply (3)
Solve (two-step
word problems
using the four
operations)
equation
mental
computation
estimation
rounding
Understand (2)
Represent (with
equations)
Evaluate (5)
Assess
(reasonableness)
Page 4
Unit 6: Multiplication and Division: Application and Fluency within 100
April 15-May 30 (7 weeks)
Instructional Strategies for ALL STUDENTS
The final unit will focus on the most difficult facts along with multiplication and division in context (word problems).
Critical Reading for Teachers Before Instruction Teaching Student-Centered Mathematics Grades K-3, Van de Walle & Lovin, Pearson, 2006, p. 77-79 & p. 82-85 (Building resource, SPED)
Children’s Mathematics, Carpenter, Heinneman, 1999 (CGI year 1 text)
Teaching Multiplication and Division Simultaneously – Multiplication and division are taught separately in most traditional programs, with multiplication preceding division. It is
important, however, to combine multiplication and division soon after multiplication has been introduced in order to help students see how they are related (Van de Walle,
2006).
Real-world context – For students to reach the level of rigor intended for the operations of addition and subtraction in the new Iowa Core, they must develop understanding of
the operations within real-world contexts. A lesson built around word problems focuses on how students solve the problem. They may use words, pictures, and numbers to
explain how they solved the problem and why they think they are correct. Allow students to use physical materials or drawings. Someone else should be able to understand how
they solved the problem when looking at their paper.
Multiplication and Division problem types – There are four structures for multiplication and division problems: Equal Groups, Comparison, Partitive (How many in each group?),
and Measurement (How many groups?). See page 9 of this guide for further explanation of the problem types. Students need regular opportunities to solve all of the different
types of problems in order to reach the level of rigor described in the Iowa Core.
Use of models to build conceptual understanding of multiplication – Drawings, counters, unifix cubes, and number lines are typically used to represent multiplication concepts.
It is essential for students to understand the relationship between addition and multiplication. (See examples of models below.) To make clear the connection to addition, early
multiplication work should include writing an addition sentence and a multiplication sentence. It is not necessary to write the products, but rather write one sentence that
expresses both concepts at once, for example, 3 + 3 + 3 + 3 = 3 x 4.
3rd Grade
2012-2013
Page 5
Unit 6: Multiplication and Division: Application and Fluency within 100
April 15-May 30 (7 weeks)
Routines/Meaningful Distributed Practice
Distributed Practice that is Meaningful and Purposeful
Practice is essential to learn mathematics. However, to be effective in improving student achievement, practice must be meaningful,
purposeful, and distributed.



Meaningful: Builds on and extends understanding
Purposeful: Links to curriculum goals and targets an identified need based on multiple data sources
Distributed: Consists of short periods of systematic practice distributed over a long period of time
Routines are an excellent way to achieve the mandate of Meaningful Distributed Practice outlined in the Iowa Core Curriculum. The skills
presented during routines do not necessarily reinforce the lesson concept for that day. Routines may be used to address a need for small
increments of exposure to a skill or review of skills already taught. Routine activities may be repeated several days in a row, allowing for a
build-up of conceptual understanding, or can be visited and re-visited over a period of time. Routines can be inserted as the schedule allows;
in short intervals throughout the day or as a lesson opener or closer. Selection of the routine should be made based on informal teacher
observation and formative assessments.
Concepts taught through Meaningful Distributed Practice during Unit 6:
Skill
Required: These concepts align to the supporting standards in this unit
Apply properties of operations for multiplication and division
Understand division as an unknown-factor problem
Multiply by multiples of 10
Additional: These concepts are optional, based on student need
Addition and Subtraction
Fraction concepts
Tell and write time
Graphing
Other skills students need to develop based on teacher observation and formative assessments.
3rd Grade
2012-2013
Standard
3.OA.5
3.OA.6
3.NBT.3
3.NBT.2
3.NF
3.MD.1
3.MD.3
Page 6
Unit 6: Multiplication and Division: Application and Fluency within 100
April 15-May 30 (7 weeks)
Investigations Resources for Unit 6- Multiplication and Division within 100 (Fluency)
Instructional Plan
Standards
Addressed
Resource
Multiplication and division fact
strategies from Math Resource
Binder
3.AO.3
3.AO.7
3.OA.8
3.0A1
3.0A.2
3.0A.4
3.0A.5
3.0A.6
3.0A.9
Additional Focus needed on:
 Multiply one-digit whole numbers by multiples of 10
in the range 10-90 (e.g., 9 x 80, 5 x 60) using strategies
based on place value and properties of operations
3rd Grade
CGI problems
2012-2013
3.NBT.3
Page 7
Unit 6: Multiplication and Division: Application and Fluency within 100
April 15-May 30 (7 weeks)
Lesson Progression
Lessons
Story Problem Bank
Teacher Directions
Every week you should do at least 2 but prefer 3 multi-step story problems. (Through
the use of the multi-step story problems the students will get practice in all more than
one operation.) And then one multiplication story problem and one division story
problems each week. This will allow students the ability to differentiate between the
different types of story problems.
These activities are for conversation and practicing basic facts.
Multiplication Shapes
How Many Points
These activities focus on multiplication facts of 6.
Marching Ants & Guitar Strings
There’s Always Another Way – 6’s
Making Snowmen
These activities focus on the relationship between 2, 4, and 8.
A Closer Look at our Snowmen Facts
Thunder Cake (literature connection with
This activity focus on facts of 7
Thunder Cake)
There’s Always Another Way – 9’s
Math Facts Column – 9’s
These activities focus on facts of 9.
Plates of Meatballs (literature connection with
Cloudy with a Chance of Meatballs)
These activities are for student practices to help them become more fluent at multiplication & Division
Capture – 6’s
These games practice facts of 6.
Ratio Race
Missing Numbers – 7’s
This game practice fact of 7.
Corners Spinners – 8’s
Easy Eights
Missing Numbers – 8’s
These games practice facts of 8’s.
Spaces -8’s
Crazy Eight
Corners spinners – 9’s
These games practice facts of 9’s.
Write to Divide – 9’s
Spinning Facts – 6, 7, 8, 9
This game practices facts 6, 7, 8, 9
3rd Grade
2012-2013
Standards
Addressed
3.OA.3
3.OA.8
3.OA.3
3.OA.7
3.OA.3
3.OA.7
3.OA.3
3.OA.7
3.OA.3
3.OA.7
3.OA.7
3.OA.7
3.OA.7
3.OA.7
3.OA.7
Page 8
April 15-May 30 (7 weeks)
Unit 6: Multiplication and Division: Application and Fluency within 100
Table 2.Common multiplication and division situations.1
Iowa Core Mathematics, p. 93; www.iowacorecurriculum.iowa.gov
Unknown Product
Equal
Groups
Arrays,4
Area5
3×6=?
There are 3 bags with 6
plums in each bag. How
many plums are there in all?
Group Size Unknown
("How many in each group?"
Division)
3 × ? = 18, and 18 ÷ 3 = ?
If 18 plums are shared equally
into 3 bags, then how many
plums will be in each bag?
? × 6 = 18, and 18 ÷ 6 = ?
If 18 plums are to be packed 6 to a
bag, then how many bags are
needed?
Measurement example. You
need 3 lengths of string,
each 6 inches long. How
much string will you need
altogether?
There are 3 rows of apples
with 6 apples in each row.
How many apples are there?
Measurement example. You
have 18 inches of string, which
you will cut into 3 equal pieces.
How long will each piece of
string be?
If 18 apples are arranged into 3
equal rows, how many apples
will be in each row?
Measurement example. You have
18 inches of string, which you will
cut into pieces that are 6 inches
long. How many pieces of string
will you have?
If 18 apples are arranged into
equal rows of 6 apples, how many
rows will there be?
Area example. What is the
area of a 3 cm by 6 cm
rectangle?
Area example. A rectangle has
area 18 square centimeters. If
one side is 3 cm long, how long
is a side next to it?
A red hat costs $18 and that is
3 times as much as a blue hat
costs. How much does a blue
hat cost?
Area example. A rectangle has
area 18 square centimeters. If one
side is 6 cm long, how long
is a side next to it?
A red hat costs $18 and a blue hat
costs $6. How many times as much
does the red hat cost as the blue
hat?
Measurement example. A
rubber band is stretched to be
18 cm long and that is 3 times
as long as it was at first. How
long was the rubber band at
first?
a × ? = p, and p ÷ a = ?
Measurement example. A rubber
band was 6 cm long at first. Now it
is stretched to be 18 cm long. How
many times as long is the rubber
band now as it was at first?
A blue hat costs $6. A red
hat blue hat. How much
does the red hat cost?
Compare
General
Measurement example. A
rubber band is 6 cm long.
How long will the rubber
band be when it is stretched
to be 3 times as long?
a×b=?
Number of Groups Unknown
("How many groups?" Division)
? × b = p, and p ÷ b = ?
The first examples in each cell are examples of discrete things. These are easier for students and should be given
before the measurement examples.
4
The language in the array examples shows the easiest form of array problems. A harder form is to use the terms
rows and columns: The apples in the grocery window are in 3 rows and 6 columns. How many apples are in
there? Both forms are valuable.
5
Area involves arrays of squares that have been pushed together so that there are no gaps or overlaps, so array
problems include these especially important measurement situations.
3rd Grade
2012-2013
Page 9
Unit 6: Multiplication and Division: Application and Fluency within 100
April 15-May 30 (7 weeks)
Story Problem Bank
Multi-step story problems
In the third grade Mrs. Smith’s class has 25 students, Mrs. Jones has 27 students and Mr. Perez has 28 students. If the tables in the lunchroom
can seat 10 students per table, how many tables will the third graders need to eat lunch? (joining & division)
At family night we had 800 tickets, we sold 755 tickets. We put the rest of the tickets into envelopes with 5 in each envelope. How many
envelopes do we need? (Separating & division)
We went to the movies last weekend. 3 adults and 4 children attended the movies. It cost $8 for each adult and $6 for each child. We also each
got a popcorn and pop combo that cost$7 per combo. How much money did we spend at the movies? (Multiplication & addition)
I made 8 pans of chocolate cookies; on each pan I can put 8 cookies at once. I also made 9 pans of peanut butter cookies and I can put 8 cookies
on each pan. I also made 7 pans of ginger snaps, were I can also put 8 cookies on each pan. How many cookies did I make all together?
(Multiplication and addition)
I need 185 stickers. I have 7 sheets of smiley faces, with 9 smiley faces on each sheet. I also have 8 sheets of butterflies, with 9 on each sheet.
How many more stickers do I still need?
My cell phone plan allows for 950 text messages. I went over and used 958 text messages. Each extra text message cost 5 cents. How much will
my extra texts cost?
I counted 42 tires in the parking lot. In the parking lot there are cars that have 4 tires and motorcycles that each has 2 tires. How many cars and
motorcycles are in the parking lot? (See how many different answers you can find.)
At the park I counted 56 legs. At the park there are humans who have 2 legs and dogs that have 4 legs. How many humans and dogs are in the
park? (See how many different answers you can find.)
I made 445 cookies. The third graders ate 396 cookies. With the left over cookies I put them in baggies with 7 cookies in each baggie. How many
baggies do I need?
I bought 56 markers. They came in packages of 8. Each package cost $3. How much money did I spend on markers?
The lunch cooks made 9 pans of pizza. Each pizza is cut up into a 5 by 4 array. The students ate 146 pieces of pizza. How many pieces of pizza
are left?
3rd Grade
2012-2013
Page 10
Unit 6: Multiplication and Division: Application and Fluency within 100
April 15-May 30 (7 weeks)
We were having breadsticks for lunch today. We had 435 breadsticks. The second and third graders ate 399 breadsticks. One serving of
breadsticks is 3 breadsticks. How many servings of breadsticks are left?
I have 4 boxes of tennis balls. In each box there are 5 tennis ball containers. In each container there are 3 tennis balls. How many tennis balls are
in all 4 boxes?
Mr. Nick sharpened 145 pencils. Mrs. Stone sharpened 179 pencils. They need 489 pencils sharpened, how many more do they need to
sharpen?
We took 3 buses to the Iowa Energy game. The first bus had 76 people on board. The second bus had 69 people on board and the third bus had
39 people. How many people went on the buses?
Miss House’s class earned a pizza party. Each pizza was cut into 8 slices and 6 pizzas were ordered. If each of the 20 student ate 2 slices and Miss
House ate 2 slices, how much pizza was leftover?
Farmer Max has 4 farms. Each farm has 6 animals. Each helper takes care of 8 animals. How many helpers did Farmer Max have?
Jamie and Johnnie are going to Home Depot. Johnnie spent $124 on paint. Jamie spent his money on flowers. The total bill comes to $214.
How much money did Jamie spend?
Mr. Larry totaled 724 points in four games of bowling. The highest possible score is 300. In game one he scored 50 points. In game two he had
the highest score possible. In game four he scored less than in game three. What are the possible scores for game three and four?
Paulina had 300 stuffed animals. She donated 258 of them. She organized her remaining stuffed animals into 7 boxes. How many stuffed animals
are in each box?
Daina and her sister made cupcakes for a bake sale. Daina made 259 cupcakes. Her sister made 459 cupcakes. They sold 662 cupcakes. They put
their remaining cupcakes into 8 boxes. How many cupcakes are in each box?
I bought 6 boxes of Go-Gurts. In each box is 8 go-gurts. If I need to share with a class of 36, do I have enough? Will I have extras? Or will I need
to get more?
3rd Grade
2012-2013
Page 11
Unit 6: Multiplication and Division: Application and Fluency within 100
April 15-May 30 (7 weeks)
Multiplication
I bought 6 packs of candy bars. In each package there are 6 candy bars. How many candy bars did I buy?
Mr. Norris’ classroom has 4 rows of 6 desks in each row. How many desks are in the classroom?
Mrs. Jones has 7 baskets. In each basket she has 9 eggs in each basket. How many eggs does she have all together?
I was building shapes with toothpicks and marshmallows. How many toothpicks do I need to make 6 hexagons?
I made cookies and packaged them into baggies. Each baggie holds 6 cookies, I used 9 baggies. How many cookies did I make?
I bought 4 packages of glue sticks. In each package was 6 glue sticks. How many glues sticks did I buy?
I bought 8 packages of pens. In each package there are 8 pens. How many pens do I have in all?
Division
I bought some 6-packs of pop. I bought 42 cans of pop. How many 6-packs did I buy?
Mrs. Clark has 72 eggs. She wants to evenly split them up between 9 baskets. How many eggs will she put in each basket?
I bought 56 markers. They came in packages of 8. How many packages did I buy?
Pencils come in boxes of 8. The third grade classrooms have altogether 64 students in the classroom. How many boxes of pencils will they need
so that each student has one new pencil?
I want to share my stickers with my friends. I have 36 stickers. If I share them with my 4 friends, how many will each friend get?
I need 30 juice boxes for a birthday party. They come in packages of 6. How many packages do I need to buy?
I need 45 notebooks. They come in packages of 5. How many packages of notebooks do I need to buy?
3rd Grade
2012-2013
Page 12