Unit 6: Multiplication and Division: Application and Fluency within 100
3rd Grade Mathematics - InvestigationsUnit 6: Multiplication and Division: Application and Fluency within 100Teacher Resource Guide2012 - 2013
In Grade 3, instructional time should focus on four critical areas: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.
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.
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.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 Mathematics 2012 2013UnitTime FrameTest ByTRIMESTER 11: Addition and Subtraction (Within 1,000)7 weeks8/27 10/12October 122: Multiplication and Division: Models within 1005 weeks10/15 11/16November 16TRIMESTER 23: Geometry/Measurement4 weeks11/19-12/21December 214: Multiplication and Division: Properties within 1005 weeks1/2 2/8February 85: Fractions8 weeks2/11 4/12April 12TRIMESTER 36: Multiplication and Division: Application & Fluency within 1007 weeks4/15 5/30May 30
Big IdeasEssential QuestionsEstimation is helpful in understating whether an answer is reasonable.Why do we estimate?Multiplication and division are inverse operations.How are multiplication and division related?
IdentifierStandardsMathematical PracticesSTANDARDS3.OA.33.OA.13.OA.23.NBT.3Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, Interpret products of whole numbers, e.g., interpret 5 7 as the total number of objects in 5 groups of 7 objects each.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.Multiply one-digit whole numbers by multiples of 10 in the range 1090 (e.g., 9 80, 5 60) using strategies based on place value and properties of operations.1) Make sense of problems and persevere in solving them.2) Reason abstractly and quantitatively.3) Construct viable arguments and critique the reasoning of others.4) Model with mathematics.5) Use appropriate tools strategically.6) Attend to precision.7) Look for and make use of structure. 8) Look for and express regularity in repeated reasoning.3.OA.73.OA.43.OA.53.OA.63.OA.9Fluently 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.Determine the unknown whole number in a multiplication or division equation relating three whole numbers.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.Understand division as an unknown-factor problem.Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations.3.OA.8Solve 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
IdentifierStandardsBloomsSkillsConceptsSTANDARDS3.OA.33.OA.13.OA.23.NBT.3Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, Interpret products of whole numbers, e.g., interpret 5 7 as the total number of objects in 5 groups of 7 objects each.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.Multiply one-digit whole numbers by multiples of 10 in the range 1090 (e.g., 9 80, 5 60) using strategies based on place value and properties of operations.Apply (3)Solve (mult & div word problems w/in 100)multiplicationdivisionequal groupsarraysmeasurement quantities3.OA.73.OA.43.OA.53.OA.63.OA.9Fluently 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.Determine the unknown whole number in a multiplication or division equation relating three whole numbers.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.Understand division as an unknown-factor problem.Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations.Apply (3)Multiply & divide (fluently w/in 100)factorproduct3.OA.8Solve 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.3Apply (3)Understand (2)Evaluate (5)Solve (two-step word problems using the four operations)Represent (with equations)Assess(reasonableness)equationmental computation estimationrounding
Instructional Strategies for ALL STUDENTSThe 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)Childrens 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.
Routines/Meaningful Distributed PracticeDistributed Practice that is Meaningful and PurposefulPractice 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 understandingPurposeful: Links to curriculum goals and targets an identified need based on multiple data sourcesDistributed: Consists of short periods of systematic practice distributed over a long period of timeRoutines 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:SkillStandardRequired: These concepts align to the supporting standards in this unitApply properties of operations for multiplication and division3.OA.5Understand division as an unknown-factor problem3.OA.6Multiply by multiples of 103.NBT.3Additional: These concepts are optional, based on student needAddition and Subtraction3.NBT.2Fraction concepts 3.NFTell and write time3.MD.1Graphing3.MD.3Other skills students need to develop based on teacher observation and formative assessments.
Investigations Resources for Unit 6- Multiplication and Division within 100 (Fluency)Instructional PlanResourceStandardsAddressed Multiplication and division fact strategies from Math Resource Binder3.AO.33.AO.73.OA.83.0A13.0A.23.0A.43.0A.53.0A.63.0A.9Additional 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 operationsCGI problems 3.NBT.3
Lesson ProgressionLessonsTeacher DirectionsStandardsAddressedStory Problem Bank 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. 3.OA.33.OA.8These activities are for conversation and practicing basic facts. Multiplication ShapesThese activities focus on multiplication facts of 6. 3.OA.33.OA.7How Many PointsMarching Ants & Guitar StringsTheres Always Another Way 6sMaking SnowmenThese activities focus on the relationship between 2, 4, and 8. 3.OA.33.OA.7A Closer Look at our Snowmen FactsThunder Cake (literature connection with Thunder Cake)This activity focus on facts of 73.OA.33.OA.7Theres Always Another Way 9sThese activities focus on facts of 9. 3.OA.33.OA.7Math Facts Column 9sPlates 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 & DivisionCapture 6sThese games practice facts of 6. 3.OA.7Ratio RaceMissing Numbers 7sThis game practice fact of 7. 3.OA.7Corners Spinners 8sThese games practice facts of 8s. 3.OA.7Easy EightsMissing Numbers 8sSpaces -8sCrazy EightCorners spinners 9sThese games practice facts of 9s. 3.OA.7Write to Divide 9sSpinning Facts 6, 7, 8, 9This game practices facts 6, 7, 8, 93.OA.7
Table 2.Common multiplication and division situations.[footnoteRef:1]Iowa Core Mathematics, p. 93; www.iowacorecurriculum.iowa.gov
Unknown ProductGroup Size Unknown("How many in each group?"Division)Number of Groups Unknown("How many groups?" Division)3 6 = ?3 ? = 18, and 18 3 = ?? 6 = 18, and 18 6 = ?EqualGroupsThere are 3 bags with 6 plums in each bag. How many plums are there in all? Measurement example. You need 3 lengths of string, each 6 inches long. How much string will you need altogether?If 18 plums are shared equally into 3 bags, then how many plums will be in each bag?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 plums are to be packed 6 to a bag, then how many bags are needed?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?Arrays,4Area5There are 3 rows of apples with 6 apples in each row. How many apples are there?Area example. What is the area of a 3 cm by 6 cm rectangle?If 18 apples are arranged into 3 equal rows, how many apples will be in each row?Area example. A rectangle has area 18 square centimeters. If one side is 3 cm long, how longis a side next to it?If 18 apples are arranged into equal rows of 6 apples, how many rows will there be?Area example. A rectangle has area 18 square centimeters. If one side is 6 cm long, how longis a side next to it?CompareA blue hat costs $6. A red hat blue hat. How much does the red hat cost?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 red hat costs $18 and that is 3 times as much as a blue hat costs. How much does a blue hat cost?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 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 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?Generala b = ?a ? = p, and p a = ?? 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.4The 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.5Area 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.
Story Problem Bank
Multi-step story problems
In the third grade Mrs. Smiths 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?
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 Houses 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?
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?
