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MATHEMATICS ACHIEVEMENT ACADEMY, GRADE 3
Day 2: Multiplication and Division
Purpose and Outcomes
• What is the purpose of the mathematics academies?
• What are the outcomes of the mathematics academies?
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Materials
Texas Gatewayhttp://www.texasgateway.org/
• Mathematics TEKS: Supporting Information
• Vertical Alignment Charts
Participation Norms
• Be fully present.
• Minimize distractions.
• Minimize “air time.”
• Take a chance.
• Celebrate accomplishments.
Discourse Norms
• Listen.
• Be involved.
• Contribute ideas.
• Participate by asking questions.
• Develop understanding, if not at the beginning, by the end.
Krusi, 2009
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Mathematics Norms
• Look for patterns in order to make generalizations.
• Make connections among models, representations, and generalizations.
• Communicate using academic vocabulary.
• Use mistakes as opportunities to support new learning about mathematics.
Yackel & Cobb, 1996
Learning Progression: Addition and Subtraction
Learning Progression: Multiplication and Division
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Grade 2: Multiplication and Division
• 2(6)(A) The student is expected to model, create, and describe contextual multiplication situations in which equivalent sets of concrete objects are joined.
• 2(6)(B) The student is expected to model, create, and describe contextual division situations in which a set of concrete objects is separated into equivalent sets.
Grade 3 STAAR®: Multiplication and Division
STAAR® Grade 3 Mathematics 2015 Release Questions
Modeling Whole Number Multiplication
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• Modeling Multiplication
• Check Point: Modeling Understanding of Multiplication
Think about the following:• Mathematics• Thinking• Vocabulary
o Academico Contextual
Modeling Whole Number Multiplication
Debrief: Modeling Whole Number Multiplication
• What mathematics are we expecting students to do in these activities?
• Describe the thinking that is expected of students in these activities.
• What mathematics vocabulary are we expecting students to use?
• What contextual vocabulary are we expecting students to use?
Using Anchor Charts
Concept Process Exemplar
A record of meaningful information and ideas about a concept or vocabulary of the mathematics being studied that may grow over multiple lessons
A record of complex mathematics skills broken down into a sequence of steps and may be extended to include additional solution strategies
An annotated example created by the teacher, student or teacher and students together that can be used as a guide
Example Example Example
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Concept
A record of meaningful information and ideas about a concept or vocabulary of the mathematics being studied that may grow over multiple lessons
Example
Anchor Charts:Created with students as a summary of learning
Using Anchor Charts
Modeling Whole Number Division
• Modeling Division
• Check Point: Modeling Understanding of Whole Number Division
Think about the following:• Mathematics• Thinking• Vocabulary
o Academico Contextual
Modeling Whole Number Division
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Debrief: Modeling Whole Number Multiplication and Division
• What mathematics are we expecting students to do in these activities?
• Describe the thinking that is expected of students in these activities.
• What mathematics vocabulary are we expecting students to use?
• What contextual vocabulary are we expecting students to use?
Situation 1Ann has 12 pieces of candy. She eats 2 pieces of candy every day. How many days will Ann have candy to eat?
Situation 2Bill has 12 pieces of candy. He shares the candy equally with a friend. How many pieces of candy will each person receive?
12 2 6 Quotative Division
Division by Grouping
Types of Division
Situation 1Ann has 12 pieces of candy. She eats 2 pieces of candy every day. How many days will Ann have candy to eat?
12 2 6 Quotative Division
Division by Grouping
12 2 6 Partitive Division
Division by Sharing
Situation 2Bill has 12 pieces of candy. He shares the candy equally with a friend. How many pieces of candy will each person receive?
Types of Division
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TEKS Connection
3(4)(H) The student is expected to determine the number of objects in each group when a set of objects is partitioned into equal shares or a set of objects is shared equally.
3(4)(K) The student is expected to solve one-step and two-step problems involving multiplication and division within 100 using strategies based on objects; pictorial models, including arrays, area models, and equal groups; properties of operations; or recall of facts.
Representing Multiplication and Division Problems
• Form groups of four by locating people around the room with cards that match yours.
• A matched group will consists of:o A word problemo An equationo A strip diagramo A number line or an array
Representing Multiplication and Division Problems: Card Match
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• How does the strip diagram represent the problem situation?
• How does the array or number line represent the problem situation?
• How does the equation represent the problem situation?
• How does the equation represent the strip diagram?
• How does the equation represent the array or number line?
Discussion Questions
• What models did you use to represent the problem situation? Why? I used _______ and _______ because _______.
• How does your model represent the problem situation?My _______ model represents the problem situation by _______.
• What other representation can be used to model the problem situation? I think we can use _______ to model the problem situation because _______.
Representations for Multiplication and Division
Select the Operation: Multiplication or Division
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• Represent the problem situation using at least two different models.
• Locate other groups with the same problem situation as your group.
• Hang your posters together, and look for similarities and differences among your representations.
Two-Step Problems
Check Point: Representing Multiplication and Division Problems
Check Point: Representing Multiplication and Division Problems
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• What representations included in this set of activities were different from the first two building blocks?
• How do these representations help students solidify their understanding of multiplication and division?
Debrief: Representing Multiplication and Division Problems
Concept
A record of meaningful information and ideas about a concept or vocabulary of the mathematics being studied that may grow over multiple lessons
Example
Using Anchor Charts
Solving Multiplication and Division Problems Using Models
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• Use the counters and grid paper to model and solve the problems individually.
• Share your models and solutions.
• Record one participant at your table explaining how they used the counters and grid paper to model and solve one of the problems.
Cafeteria Counters
• How did you use the counters and/or grid paper to model the situation?
• How did you use the counters and/or grid paper to determine the solution?
Cafeteria Counters
1. The Flower Farm 2. Petunia Palace3. The Pumpkin Patch 4. The Zucchini Garden
Solving Problems with Area Models
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Petunia Palace and Zucchini Gardeno How are the models alike? How are they different?o How did you decide how to decompose the factors?
The Flower Farm and The Pumpkin Patcho How are Aaron’s model and Carter’s model alike? How are they
different?o How do all four models help solve the problem?
Debrief: Solving Problems with Area Models
The Flower Farm planted 8 rows of bluebonnets with 12 bluebonnet plants in each row. How many bluebonnet plants were planted?
Farmer Brown planted 96 total pumpkins in his pumpkin patch. He planted 8 rows of pumpkins. How many pumpkins were planted in each row?
Debrief: Solving Problems with Area Models
Check Point: Solving Problems with Area Models
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Solving Multiplication and Division Problems Using Strategies and Algorithms
• Multiplying with the Associative Property• Multiplying with the Distributive Property• Dividing with the Distributive Property
Solving Multiplication and Division Problems Using Strategies and Algorithms
Debrief: Solving Multiplication and Division Problems Using Strategies and Algorithms
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48 750 7 350
2 7 14
35014
336 Fifty groups
of seven minus two groups
of seven makes 48 groups of
seven.
93 3
90 3 390 3 30
3 3 130 1 31
Debrief: Solving Multiplication and Division Problems Using Strategies and Algorithms
Check Point: Solving Multiplication and Division Problems Using Strategies and Algorithms
Debriefing Mental Strategies
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Solving Multiplication and Division Problems
How does the use of mental strategies connect to the standard algorithm?
Problem A Problem B
Partial Products Bridging
237
21 7 3140 7 20161
20 3
7140 21
7 20 7 3
140 21 161
Debriefing Mental Strategies
237
21 7 3140 7 20161
223
7161
How does the use of mental strategies connect to the standard algorithm?
Problem A Problem B Problem C
20 3
140 21 161
140 21 7 20 7 3
7
Partial Products Bridging Standard Algorithm
Debriefing Mental Strategies
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Making Connections to the Standard Algorithm
Solving Multiplication and Division Problems
Round Robin: Solving Problems
Making Connections to the Standard Algorithm
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• What connections do you see between the strategies your group used to solve the problems?
• Were there any answers that you disagreed with? Why?
• Did any of your group members think differently than you about how to solve a problem? Was the solution still the same?
Debrief: Making Connections to the Standard Algorithm
Check Point: Solving Multiplication and Division Problems
Fluency and Automaticity of Facts
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• Sort the cards into a strategy continuum.
• Why did you place the cards in the order that you did?
• Place your hand over one of the middle cards. What are the implications of that strategy being missed with students?
Debrief: Solving Multiplication and Division Problems Using Strategies and Algorithms
• Multiplication Facts: Find Someone Who . . .
• Exploring Facts
• Double the Fun
• Determining the Unknown
Fluency and Automaticity of Facts
× Description Examples
× 2× 10× 5× 1× 0
Students build on their understanding of skip‐counting and doubling for 2, 10, and 5. Multiplying by 1 is quick to memorize, but saved until students explore multiplication with groups greater than 1. Multiplying by 0 is also quick to commit to memory. Concrete thinkers may have difficulty visualizing groups of 0.
2 3 6 10 9 90
5 4 20 1 8 8
0 7 0
Debrief: Multiplication and Division Facts
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× Description Examples
×3 Students build on their understanding of multiplying by 2 and adding 1 more group.
3 4 12
2 4 1 4 12
Debrief: Multiplication and Division Facts
× Description Examples
×4 Multiplying by 4 can be thought of as doubling a double. 4 6 24
2 2 6 24
2 6 12
2 12 24
Debrief: Multiplication and Division Facts
× Description Examples
×6 Multiplying by 6 can be thought of as doubling a multiple of 3. 6 7 42
2 3 7 42
3 7 21
2 21 42
Debrief: Multiplication and Division Facts
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× Description Examples
×9 Students build on their understanding of multiplying by 10 and subtracting 1 group.
9 8 7210 1 8 72
10 8 8080 8 72
Debrief: Multiplication and Division Facts
× Description Examples
×8 Multiplying by 8 can be thought of as doubling a 4.
×7Students use number sense and mental strategies to explore decomposing 7.
8 6 48
2 4 6 48
4 6 24
2 24 48
7 8 56
5 8 40
2 8 16
Debrief: Multiplication and Division Facts
5 x 5
2 x 5
7 x 5
5
5 25
52 10
5
5
2
25
10
5
7 35
Number Sense Practice
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8 x 5
8 x 6
6 x 8
40
5
8
5
8 40
1
88
6
48
8
6 48
35
5
5
7
1
Number Sense Practice
3 x 8
3 x 7
3
8
246
8
483
8
‐ 3
7
3 21
Number Sense Practice
Multiplication and Division Facts
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Supporting Diverse Learners:Students with Special Needs
Expert Groups: Memory Challenges
0 1 2
Short‐termmemory
Workingmemory
Long‐term memory
Blended Groups: Memory Challenges
0 1 2
Short‐termmemory
Workingmemory
Long‐term memory
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Concepts
Mental Strategies, Procedures, and Algorithms
Applications
Supporting Diverse Learners: Students with Special Needs
Participation Norms
• Be fully present.
• Minimize distractions.
• Minimize “air time.”
• Take a chance.
• Celebrate accomplishments.
Discourse Norms
• Listen.
• Be involved.
• Contribute ideas.
• Participate by asking questions.
• Develop understanding, if not at the beginning, by the end.
Krusi, 2009
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Mathematics Norms
• Look for patterns in order to make generalizations.
• Make connections among models, representations, and algorithms.
• Communicate using academic vocabulary.
• Use mistakes as opportunities to support new learnings about mathematics.
Yackel & Cobb, 1996
• Anchor Chart• Learning Progression• Partitive Division (Division by Sharing)• Quotative Division (Division by Grouping)• Represent
• Array• Equation• Number Line• Number Sentence• Pictorial Model• Strip Diagram
• Standard Algorithm• Strategies/Mental Strategies/Solution Strategies
• Place Value• Properties of Operations• Relationship Between Multiplication and Division
Academic Vocabulary
Exit Slip
Learning ProgressionWhole Number Addition
and SubtractionDiverse Learners
What confirming/new ideas did you hear today?
How can you move new and intriguing ideas to action?