Project 2: CCSSM Interpretation Guide Kathleen Sicula CCLM Project 2 July 15, 2011 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
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.
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
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.
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
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
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
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
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.
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.
Name Solve: Fact Fluency Assessment 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?