Mary Butkus
CCLM^2 Project , Summer 2012
DRAFT DOCUMENT. This material was developed as part of the
Leadership for the Common Core in Mathematics (CCLM^2) project at the University of Wisconsin-Milwaukee.
Grade: 4th Grade Domain: Operations and Algebraic Thinking Cluster: Use the four operations with whole number to solve problems. 1. Interpret multiplication equation as a comparison e.g., interpret 35=5x7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations. 2. Multiply or divide to solve word problems involving multiplicative comparison e.g., by using drawing and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison. (See table 2 in the glossary of the CCSSM materials.) Standard: 4.OA.1 and 4.OA.2 Explanation and Examples of the Standard What does the What concept-­‐based language might students and teachers standard really use? mean students will What representation, diagram, contexts and strategies support be understanding? the understanding? Interpret multiplication Students will be A multiplicative comparison is a situation in which one quantity equation as a comparison given opportunities is multiplied by a specified number to get another quantity. e.g., interpret 35=5x7 as a to write and Student should be able to identify and verbalize which quantity statement that 35 is 5 times identify equations is being multiplies and which number tells how many times. as many as 7 and 7 times as and statement for many as 5. Represent verbal multiplication Example: 4x5 =20 statements of multiplicative comparisons. Mary is four years old. Her brother is 5 times older than her. comparisons as multiplication How old is Mary’s brother? equations. Multiply or divide to solve Students will When using multiplicative comparison from additive comparison word problems involving translate students should note that: multiplicative comparison comparative • Additive comparisons focus on the difference between e.g., by using drawing and situation into two quantities equations with a symbol for equations with an A simple way to remember this “How many more?” the unknown number to unknown to solve. • Multiplicative comparison focus on comparing two represent the problem, quantities by showing that one quantity is a specified distinguishing multiplicative number of times larger or smaller than the other comparison from additive A simple way to remember this is “How many times as much? or comparison. (See table 2 in How many times as many?” the glossary of the CCSSM Examples: materials.) Unknown Product; 4x5=n Standard Mary ran 4 miles in a week. Tom ran 5 times as much. How many miles did Tom run? Groups Size Unknown; 20 /n=4 or 4xn=20 Carol ran 20 miles in a week. That is 5 times more than Mike. How many miles did Mike run? Number of Groups Unknown:20/5=n or 5xn=20 Mary ran 20 miles. That is 5 times more than Andy. How many times as much does Mary run as Andy has run? The terms students should use; multiplication/multiply, division/divide, equation, unknown, remainders reasonableness, mental computation. School Mathematic Text Book Program Standard How the mathematics is introduced and developed in the Everyday Math grade four programs? Interpret multiplication equation as a Develop fluency with multiplication comparison e.g., interpret 35=5x7 as a and division. Unit 3 and Unit 5 statement that 35 is 5 times as many Develop strategies for solving facts to as 7 and 7 times as many as 5. help student work toward instant Represent verbal statements of recall. 3.2, 3.3, 3.4 multiplicative comparisons as Develop the understanding of the multiplication equations. relationship between multiplication and division. 3.5 Develop the meaning of Open Sentences using an unknown variable. 3.9, 3.11 Introduce the basic principles of multiplication with multi digit numbers. 5.2 Multiply or divide to solve word Strategies for multiplying and divide problems involving multiplicative numbers. Units 3 and 5 comparison e.g., by using drawing and Introduce a simplified approach to equations with a symbol for the solving number stories 3.8 unknown number to represent the Develop the meaning of Open problem, distinguishing multiplicative Sentences using an unknown variable. comparison from additive 3.9, 3.11 comparison. (See table 2 in the Using diagrams when solving glossary of the CCSSM materials.) multiplication and division stories 6.1 Comparing number stories 8.8 How students develop understanding of this standard. Use rules to complete “What’s My Rule.” Make rectangular arrays to find factors of a number Playing “Name that Number” Practice multiplication fact with Fact Triangles and games Modeling multiplication with base -­‐10 blocks Students model square numbers with arrays Using true and false statements to show the understanding of an open sentence using a variable Guide for solving numbers stories which students discuss and apply as they solve problems. Students will first draw a picture or diagram to solve the problem. 3.8 Story Problems in math messages 3.8 Building Vocabulary for problem solving operations 3.8 Using diagrams to organize information when solving multiplication and division number stories. 6.1 Students will write numbers stories 6.1 Using true and false statements to show the understanding of an open sentence using a variable. 3.9, 3.11 Comparing Country areas 8.8 The terms students should use; number sentence, factors, products, multiples, dividend, division, quotient Skills Trace Grade 3: In the Everyday Math program, third grade students review multiplication and equal groups. They are provided opportunities to solve and write number stories involving equal groups. Students use physical models such as arrays, diagrams, base-­‐10 blocks, and number models to represent and solve multiplication number stories. Students discuss multiplication facts and the importance of fact strategies. They are introduced to number story diagrams to help them solve number stories. One of the three diagrams used is a comparison diagram. Students also solve number-­‐story problems using estimation and use estimates to check for the reasonableness of answers. Standard: Represent and solve problems involving multiplication and division: 3.OA.1 ,3.OA.2, 3.OA.3 Grade 5: In the Everyday Math program, fifth grade students review rectangular arrays and the use of multiplication number models to represent such arrays. Students are provided opportunities to factor numbers and develop automaticity with multiplication facts. Students do review the use of mathematical models to solve number stories; this includes practice with division number stories and interpreting the remainders. The fifth grade curriculum has several units on the study of fractions. This fraction work supports multiplication and division understanding. Pan-­‐balance models are used to develop linear equation of the unknown and to support multiplication and division concepts. Standard: Write and interpret numerical expressions: 5.OA.1. Conclusions: In the Everyday Math program, students are encouraged to invent their own method of solving problems. This approach requires student to focus on the meaning of the operation as they learn to think and use their common sense. Students are motivated because their understanding their own method as opposed to learning by rote. They become skilled at representing ideas with object words, pictures and symbols. The focus on the meaning of the operations supports 4OA.1. (35=5x7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5.) In Everyday Math (EM) students develop a variety of computational methods and flexibility to choose the procedure that is most appropriate in a given situation. EM emphasizes a variety of standard computational algorithms as well as student invented procedures. A lot of instructional time is spent on standard computational algorithms; teachers will need to help students also focus on the meaning of these operations. Secondly, the EM relies on practical application to teach story problem lessons. The program makes heavy use of work problem t o communicate the narrative in which the problem is set. Students learn to use these questions when problem solving “What do I know? “What do I want to find out?” “What do I need to do to get there?” and “Does my answer make sense?” EM does not cover all of CCSSM multiplication and division problem types with their story problems. Problem types are not clearly defined or organized with the CCSSM format, therefore, a teacher would need to search out and organize the different problem types throughout their manual. A teacher may also find the need to write out sample comparison problems types to support their students’ learning. A suggestion is to use the CCSSM format as a guide when teaching problem solving and develop problems to fill in the gaps for each problem solving lesson. Suggestions: The everyday math program has begun to provide crosswalks from the current curriculum and integrating the CCSSM. An example of this is in 4th grade lesson 3.1, “What My Rule?” they have provided guiding questions to align the lesson. “What do the numbers in in the column represent?” “What do the numbers in the out column represent?” These questions promote the thinking of how many groups and how many times more in the group. Secondly every day math curriculum has a variety of techniques to help student develop their “fact power” or basic facts. Students enjoy learning basic facts by playing games and reason through the games. It is suggested that this is an area for more classroom discussion on the meaning of multiplication and division reasoning in the games to support 4.OA.1. (35=5x7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5.) There are several 4th grade lessons that focus on problem solving with multiplication and division. They are not organized by problem types and some comparison problem types are missing. It is suggested that when lessons involving story problems arise, first the teacher identifies the problem type then fills in the gaps by writing the needed problem types to support their students’ learning.