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Rethinking Multiplication and Division
Gabr’l Stackhouse April 25, 2015
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Purpose To strengthen the teaching and learning of multiplication and division word problems in third, fourth, and fifth grade classrooms.
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Learning Goals To understand why students misapply addition and subtraction when solving multiplication and division word problems.
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Collaborative Norms Actively engage Openly share
Provide honest, constructive feedback Have fun
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Introductions Name Position Number of years as an educator
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I Know I Wonder Discuss in your groups how you teach students how to solve multiplication and division word problems. What do you know ? What do you wonder? Use a piece of chart in the middle of your table to create a T-chart and express your thoughts. I Know I Wonder
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Let’s Do Some Math Rock bands often stack their speakers in an array. One teen band has 24 speakers. They stack them at least 2 high, but no taller than 8 high. What are all the different arrays they can make?
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What would a student need to know and understand in order to solve the math problem?
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Let’s Do Some Math There are 415 biographies in the school library. If each shelf holds 27 books, how many shelves are completely filled? How many books are left?
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What would a student need to know and understand in order to solve the math problem?
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Multiplication and Division
Grade 3
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Common Core Grade 3 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. For equal-sized group situations, division can require finding the unknown number of groups or the unknown group size.
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Multiplication and Division
Grade 4
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Common Core Grade 4 Students use the four operations with whole numbers to solve problems. They develop fluency with efficient procedures for multiplying whole numbers; understand and explain why the procedures work based on place value and properties of operations; and use them to solve problems. Students apply their understanding of models for division, place value, properties of operations, and the relationship of division to multiplication as they develop, discuss, and use efficient, accurate, and generalizable procedures to find quotients involving multi-digit dividends.
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Multiplication and Division
Grade 5
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Common Core Grade 5 Write and interpret numerical expressions. Students develop understanding of why division procedures work based on the meaning of base-ten numerals and properties of operations. They finalize fluency with multi-digit addition, subtraction, multiplication, and division.
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Multiplication and Division
The teaching and learning of multiplication and division is more than knowing certain facts, being able to solve particular types of problems, and mastering relevant vocabulary. NCTM. (2011). Developing and essential understanding of multiplication and division: Grades 3-5. Reston, VA: National Council for Teachers of Mathematics.
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Multiplication as a Scalar Operation
In the multiplicative expression A x B, A can be defined as a scaling factor. The first factor will “scale” the second quantity It will resize the second factor proportionally according to the scale given by the first factor making a multiplicative change NCTM. (2011). Developing and essential understanding of multiplication and division: Grades 3-5. Reston, VA: National Council for Teachers of Mathematics.
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Representing Situations with Multiplication and Division
A situation that can be represented by multiplication has an element that represents the scalar and an element that represents the quantity to which the scalar applies. NCTM. (2011). Developing and essential understanding of multiplication and division: Grades 3-5. Reston, VA: National Council for Teachers of Mathematics.
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With a partner, use three different strategies to solve the problem.
Shirt Task If each of Stephen’s shirts has 4 buttons, then how many buttons are on 5 of his shirts? With a partner, use three different strategies to solve the problem.
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Reasoning about Situation and Representations
Each multiplicative expression developed in the context of a problem situation has on accompanying explanation, and different representations and ways of reasoning about a situation can lead to different expressions or equations. NCTM. (2011). Developing and essential understanding of multiplication and division: Grades 3-5. Reston, VA: National Council for Teachers of Mathematics.
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Reasoning about Situation and Representations
The expression we use to model a situation reflects our reasoning. Our explanation of our thinking guides our way of representing the situation and any expression or equation that we write. NCTM. (2011). Developing and essential understanding of multiplication and division: Grades 3-5. Reston, VA: National Council for Teachers of Mathematics.
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Representing Situations with Multiplication and Division
Students frequently choose an operation to solve a problem without trying to make sense of the choice. The look for “key words” or “phrases” or they just look at the numbers and choose an operation on the basis of their size. NCTM. (2011). Developing and essential understanding of multiplication and division: Grades 3-5. Reston, VA: National Council for Teachers of Mathematics.
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Representing Situations with Multiplication and Division Group Activity
In your groups, read through each of word problems in the Problem Type envelope. Sort each word problem on the Problem Type sorting mat.
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Representing Situations with Multiplication and Division
Which problem types should students master at the end of third, fourth, and fifth grade?
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3 3 3 3 3 3 4 4 4
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Representing Situations with Multiplication or Division
Students demonstrate many misunderstandings about when to multiply and when to divide in contextual situations. Problem solving is more complex than extracting numbers from given problems. Thinking is different from applying a “key word”. Fosnot, C., Dolk, M. (2001). Young mathematicians at work: constructing multiplication and division. Portsmouth, NH: Heinemann.
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Representing Situations with Multiplication or Division
The “key words” approach does not support students in the process of sense making Students need to analyze the full problem, taking every word and the contextual information into consideration to build their understanding of the problem’s meaning Fosnot, C., Dolk, M. (2001). Young mathematicians at work: constructing multiplication and division. Portsmouth, NH: Heinemann.
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Working with Context in Word Problems
The context of a problem situation and its interpretation can lead to different representations. Many children think the equal sign always comes before the answer. 2 x 6 = x 4 How might a child complete the number sentence?
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Working with Context in Word Problems
According to Fosnot and Dolk (2000), context is essential to the development of understanding and should be used from the start as a means of “construction” rather than as a culminating or extending activity for application at the end of a unit. Fosnot, C., Dolk, M. (2001). Young mathematicians at work: constructing multiplication and division. Portsmouth, NH: Heinemann.
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Working with Context in Word Problems
Presenting students with a situation provides the following: An opportunity for them to chart their own path to a solution An opportunity for them to construct mathematical ideas rather than apply an algorithm An opportunity for them to discuss and make sense of the situation rather than focusing on rules and procedures Fosnot, C., Dolk, M. (2001). Young mathematicians at work: constructing multiplication and division. Portsmouth, NH: Heinemann.
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Representing Situations with Multiplication and Division
Why is it important for students to make sense of situations that involve multiplication and division instead of looking for “key words?” NCTM. (2011). Developing and essential understanding of multiplication and division: Grades 3-5. Reston, VA: National Council for Teachers of Mathematics.
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Standards Group Activity
Read through the standards for third, fourth, and fifth grade. What is the major work for each grade level? How does the major work guide your instruction? Planning? Assessment? What are students expected to know, understand, and do with addition and subtraction word problems for each grade level?
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Session Evaluation Reflect on the session today.
What is your biggest “ah-ha?” What validated your current practice? What do you know you will work on moving forward? What support do you need?
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Resources Common Core Standards Writing Team (2015, April 4). Progressions for the common core state standards in mathematics: K-5, Number and operations in base ten. Retrieved from: Fosnot, C., Dolk, M. (2001). Young mathematicians at work: constructing multiplication and division. Portsmouth, NH: Heinemann. NCTM. (2015, March 31). Position Statements. Retrieved from: Statements/ NCTM. (2000). Principles and standards for school mathematics. Reston, VA: National Council for Teachers of Mathematics. NCTM. (2011). Developing and essential understanding of multiplication and division. Reston, VA: National Council for Teachers of Mathematics.
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Contact Information Gabr’l Stackhouse Math Consultant Creative Minds Consulting Phone:
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Thank you for your participation
“I hear and I forget. I see and I remember. I do and I understand.” ~ Chinese Proverb ~
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