1
Day 1: Addition and Subtraction
MATHEMATICS ACHIEVEMENT ACADEMY, GRADE 2
• What is the purpose of the mathematics academies?
• What are the outcomes of the mathematics academies?
Purpose and Outcomes
2
Texas Gatewayhttp://www.texasgateway.org/
• Mathematics TEKS: Supporting Information
• Vertical Alignment Charts
Materials
0
1
2
Rotating Trios
• Be fully present.
• Minimize distractions.
• Minimize “air time.”
• Take a chance.
• Celebrate accomplishments.
Participation Norms
3
• Listen.
• Be involved.
• Contribute ideas.
• Participate by asking questions.
• Develop understanding, if not at the beginning, by the end.
Krusi, 2009
Discourse 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
Mathematics Norms
Learning Progression
A learning progression is a sequenced set of subskills and bodies of enabling knowledge that, it is believed, students must master en route to mastering a more remote curricular aim.
In other words, it is composed of step-by-step building blocks students are presumed to need in order to successfully attain a more distant, designated instructional outcome.
Popham, 2008
4
• Create a graphic organizer to show a learning progression from kindergarten to fourth grade for the concepts of addition and subtraction.
• Trade your poster with another table group.
o How are the big ideas of the other group’s poster similar to your group’s poster?
o How are they different?
Learning Progression
Where would you place the grade-level student expectations on your graphic organizer?
Learning Progression
Learning Progression
5
Learning Progression
How can a learning progression support planning for focused, targeted, and systematic instruction?
Learning Progression
• 2(4)(C) The student is expected to solve one-step and multi-step word problems involving addition and subtraction within 1,000 using a variety of strategies based on place value, including algorithms.
• 2(7)(C) The student is expected to represent and solve addition and subtraction word problems where unknowns may be any one of the terms in the problem.
Whole Number Addition and Subtraction
6
Balanced Pre-Assessment
The Role of Pre-Assessment
Where do we start with this year’s students?
What gaps do my students have?
Which adjustments are needed for the whole group?
Which adjustments are needed for a small group?
MSTAR Math Academy, 2010
Target Knowledge and
Skills
Foundational Knowledge and
Skills
Bridging Knowledge and Skills
Connections Across the Knowledge Representations
Balanced Pre-Assessment
7
• What are the broad and deep ideas that students must master to be successful in grade 2 mathematics?
• What are the concepts and procedures from grade 1 that students must have mastered?
• What are the foundational knowledge and skills that students may use to build towards mastery?
MSTAR Math Academy, 2010
Target Knowledge and Skills
Foundational Knowledge and Skills
Bridging Knowledge and Skills• What are the bridging knowledge
and skills that students may use to connect foundational understandings to target understandings?
• What are the target knowledge and skills mathematics that students must master?
Guiding Questions for Balanced Pre-Assessment
Guiding Questions for Examining Student Work
Models of Mathematics• What models do we see students using the most often?• With what models are students most successful?• What models are we not seeing students use?• With what models are students least successful?
Mathematical Processes and Procedures• What processes or procedures are students
using the most often?• With what processes or procedures are
students most successful?• What misconceptions are present in this work?• What steps are students taking most often?
Sorting Student Work
Where do we build from?
What do we need to develop?
What gaps do we need to address?
MSTAR Math Academy, 2010
Target Knowledge and Skills
Foundational Knowledge and Skills
Bridging Knowledge and Skills
Connections Across Knowledge and Representations
Instructional Decisions
8
Where do we build from?
What do we need to develop?
What gaps do we need to address?
Whole Class
Small Group
Pre-Assessment Instructional Decisions
Best Practices: Guiding Questions for BalancedPre-Assessment and Examining Student Work
1. Determining Sums Using Mental Strategies2. Determining Differences Using Mental Strategies3. Compensation with Addition4. Compensation with Subtraction5. Compatible Numbers and Making Tens
Computation of Addition and Subtraction
9
1. Determining Sums Using Mental Strategies2. Determining Differences Using Mental Strategies3. Compensation with Addition4. Compensation with Subtraction5. Compatible Numbers and Making Ten6. Solving Addition Problems Using Mental Strategies7. Solving Subtraction Problems Using Mental Strategies
Computation of Addition and Subtraction
2(4)(B) The student is expected to add up to four two-digit numbers and subtract two-digit numbers using mental strategies and algorithms based on knowledge of place value and properties of operations.
Computation of Addition and Subtraction
Whole Number Addition and Subtraction Based on Place Value and Algorithms
1. Determining Sums Using Mental Strategies2. Determining Differences Using Mental Strategies3. Compensation with Addition4. Compensation with Subtraction5. Compatible Numbers and Making Tens
10
1. Determining Sums Using Mental Strategies2. Determining Differences Using Mental Strategies
• Decomposing numbers and using expanded form
• Using place value knowledge and allowing students to think about joining or separating tens and joining or separating ones
Whole Number Addition and Subtraction Based on Place Value and Algorithms
3. Compensation with Addition4. Compensation with Subtraction
• The relative position on a number line – shifting numbers on a number lineNumbers are chosen to allow the use of compatible numbers for an equivalent expression.
• The preservation of a number• The visual distance between numbers
Whole Number Addition and Subtraction Based on Place Value and Algorithms
5. Compatible Numbers and Making Tens
• Making tens or multiples of ten
• Building on students prior knowledge of combining tens and building on understanding of place value
Whole Number Addition and Subtraction Based on Place Value and Algorithms
11
1. Determining Sums Using Mental Strategies2. Determining Differences Using Mental Strategies3. Compensation with Addition4. Compensation with Subtraction5. Compatible Numbers and Making Tens6. Solving Addition Problems Using Mental Strategies7. Solving Subtraction Problems Using Mental Strategies
Whole Number Addition and Subtraction Based on Place Value and Algorithms
Using Anchor Charts
Using Anchor Charts
Anchor Charts:Created with students as a summary of learning
12
• Why is it important for an anchor chart to be developed by the class rather than presented to the class?
• How and when might students refer to an anchor chart?
Using Anchor Charts
Check Point: Computation of Addition and Subtraction
Debriefing Mental Strategies
13
32 47 3247
79
30 40 2 7
70 9
79
How does the use of mental strategies connect to the standard algorithm?
Debriefing Mental Strategies
96 45 9645
51
90 40 6 5
50 151
How does the use of mental strategies connect to the standard algorithm?
Debriefing Mental Strategies
32 4930 40 2 9
70 1170 1110
8
0
1
1
80 1
13249
81
How does the use of mental strategies connect to the standard algorithm?
Debriefing Mental Strategies
14
100 38
100 30 8
100 0 0 30 0 8
How does the use of mental strategies connect to the standard algorithm?
10038
Debriefing Mental Strategies
How does the use of mental strategies connect to the standard algorithm?
0 10010
100 38
100 30 8
100 0 0 30 0
0 300
8
0 0 8
100
1 0 038
Debriefing Mental Strategies
How does the use of mental strategies connect to the standard algorithm?
1090
10 00
100 38
100 30 8
100 0 0 30 0 8
100 0 00 30 8
910 10 0
01 03 8
Debriefing Mental Strategies
15
How does the use of mental strategies connect to the standard algorithm?
910 100
1 0
2
03 86
900 100 10
100 38
100 30 8
100 0 0 30 0 8
100 0 0 30 0 8
60 262
Debriefing Mental Strategies
How does the use of mental strategies connect to the standard algorithm?
(100 1)(38 1)
993762
910 100
1 0 03 8
6 2
Debriefing Mental Strategies
1. Representation Card Match2. Fruit Stand3. Representations for Addition and Subtraction 4. Ice Cream Purchases
Represent and Solve One-step Problems
16
1. Representation Card Match2. Fruit Stand3. Representations for Addition and Subtraction 4. Ice Cream Purchases
Represent and Solve One-step Problems
1. Ice Cream Purchases2. Fruit Stand3. Representation Card Match4. Representations for Addition and Subtraction
Possible Sequence of Activities
Check Point: Represent and Solve Problems
17
All students must use the same model.
Agree or Disagree?
Students should choose the model that makes most sense.
Agree or Disagree?
All models work for all problems.
Agree or Disagree?
18
Solve One-Step Problems
Solve One-Step Problems: Carousel
• How are the two solution strategies similar?• How are they different?
Make notes on the poster to indicate similarities and differences.
• If you were asked to solve this problem, would your solution strategy look like one on the poster, or do you have thoughts about a third way to solve this problem?
Solve One-Step Problems: Carousel
19
Solve One-Step Problems: Speed Learning
• What representations did you find yourself using? Why?
• What mental strategies did you find yourself using? Why?
Solve One-Step Problems: Speed Learning
Solve One-Step Problems
20
• Be fully present.
• Minimize distractions.
• Minimize “air time.”
• Take a chance.
• Celebrate accomplishments.
Participation Norms
• Listen.
• Be involved.
• Contribute ideas.
• Participate by asking questions.
• Develop understanding, if not at the beginning, by the end.
Krusi, 2009
Discourse 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 learning about mathematics.
Mathematics Norms
21
AlgorithmAnchor ChartBalanced Pre-AssessmentFoundational–Bridging–Target Knowledge and SkillsLearning ProgressionRepresent
Equations Number Lines Pictorial Models
Strategies/Mental Strategies/Solution Strategies Place Value Properties of Operations Relationship Between Addition and Subtraction
Academic Vocabulary
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?
Exit Slip