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cell: 917-494-1606. phone: 212-766-2120. Agenda. Creating our Learning Community & Norms Setting Personal Goals and Identifying Themes Visualization through Quick Images’ Math Routines that Develop Thinking Exploring Addition and Mental Math Lunch Games. Eight CCSS Math Practices.

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slide1

cell: 917-494-1606

phone: 212-766-2120

agenda
Agenda
  • Creating our Learning Community & Norms
  • Setting Personal Goals and Identifying Themes
  • Visualization through Quick Images’
  • Math Routines that Develop Thinking
  • Exploring Addition and Mental Math
  • Lunch
  • Games
eight ccss math practices
Eight CCSS Math Practices
  • Make sense of problems and persevere in solving them.
  • Reason abstractly and quantitatively.
  • Construct viable arguments and critique the reasoning of others.
  • Model with mathematics.
  • Use appropriate tools strategically.
  • Attend to precision.
  • Look for and make use of structure.
  • Look for and express regularity in repeated reasoning.
construct viable arguments and critique the reasoning of others
Construct Viable Arguments and Critique the Reasoning of Others
  • Understand and use stated assumptions, definitions and previously established results
  • Make and explore conjectures using a logical progression of statements
  • Analyze arguments using cases, counterexamples, data, contexts
  • Compare effectiveness of two plausible arguments
  • Distinguish correct from flawed reasoning
model with mathematics
Model with Mathematics
  • Apply mathematics to solve everyday, society, and workplace problems
  • Make assumptions and approximations to simplify a complicated situation and revise as needed
  • Map quantitative relationships in situations using diagrams, charts, flow-charts, tables, etc.
  • Analyze quantitative relationships to draw conclusions
  • Interpret results in relation to the context determining whether answer makes sense or model needs revision
developing addition strategies
Developing Addition Strategies

Today’s Session Will Explore …

  • What does it mean to have computational fluency?
  • How do we develop computational fluency in our students?
developing addition strategies1
Developing Addition Strategies

What is in our addition toolbox?

developing addition strategies2
Developing Addition Strategies

39 + 68

Share your strategies in a small group.

Are your strategies the same? Different?

developing addition strategies3
Developing Addition Strategies

39 + 68

What mathematical ideas make these strategies work?

developing addition strategies4
Developing Addition Strategies

Did you use the standard algorithm?

39

+ 68

slide12

Developing Addition Strategies

What mathematical ideas make the standard addition algorithm work?

developing addition strategies6
Developing Addition Strategies

What mathematical ideas make the standard addition algorithm work?

  • Knowing that place determines value
  • Equivalence
  • Associativity
  • Commutativity
  • Unitizing
developing addition strategies7
Developing Addition Strategies

1

39

+ 68

107

(8+9) + (60 + 30) =

17 + 60 + 30 = 10 + 7 + 60 + 30 =

90 + 10 + 7 = 107

developing addition strategies8
Developing Addition Strategies

Other possible strategies for adding 39 + 68?

  • Partial sums (splitting)
  • Creating an equivalent problem (compensation)
  • Keeping one number whole and adding in parts
  • Using the ten structure of the number system to
    • Move to landmarks
    • Take landmark jumps
developing addition strategies9
Developing Addition Strategies

Other possible strategies for adding 39 + 68?

Partial sums (splitting)

(30 + 9) + (60 + 8) =

(30 + 60) + (9 + 8) =

90 + 17 = (90 + 10) + 7 =

100 + 7 = 107

developing addition strategies10
Developing Addition Strategies

Other possible strategies for adding 39 + 68?

Compensation (creating an equivalent problem)

39 + 68 =

(37 + 2) + 68 =

37 + (2 + 68) =

37 + 70 =

37 + 70 = 107

39 + 68 =

39 + (67 +1) =

39 + (1 + 67) =

(39 + 1) + 67 =

40 + 67 = 107

developing addition strategies11
Developing Addition Strategies

Other possible strategies for adding 39 + 68?

Keeping one number whole and adding in parts:

39 + 68 =

68 + (30 + 9) = (68 + 30) + 9

98 + 9 = 107

developing addition strategies12
Developing Addition Strategies

Other possible strategies for adding 39 + 68?

Making or using landmark jumps:

39 + 68 = 68 + 39

68 + (20 + 10 + 9) =

(68 + 20) + 10 + 9 =

(88 + 10) + 9 =

98 + (10 – 1) = 107

developing addition strategies13
Developing Addition Strategies

Other possible strategies for adding 39 + 68?

Moving to the nearest landmark:

39 + 68 =

39 + 1 + 67 =

40 + 67 = 40 + (60 + 7)

(40 + 60) + 7 =

100 + 7 = 107

developing addition strategies14
Developing Addition Strategies

It’s important to notice that many of the big ideas underlying these addition strategies are the same:

Big ideas:

  • Equivalence 39 + 68 = 40 + 67
  • Commutative property: a + (b + c) = a + (c + b)
  • Associative property of addition: a + (c + b) = (a + c) + b
  • Place determines value (the “3” in 39 is 30 or 3 tens)
developing addition strategies15
Developing Addition Strategies

All of these strategies can be used algorithmically.

The key is not just to have alternative mental-math strategies, but to know when to use them.

developing addition strategies17
Developing Addition Strategies

One of the hallmarks of number sense is flexible strategy use.

What does this mean?

In computation, it means looking to the numbers to pick the most efficient strategy.

developing addition strategies18
Developing Addition Strategies

HOW TO WE DEVELOP FLEXIBLE STRATEGY USE IN OUR STUDENTS?

developing addition strategies19
Developing Addition Strategies

One way to help students develop important number relationships is through computational mini-lessons.

developing addition strategies20
Developing Addition Strategies
  • Guided Mini-lessons
  • “Strings”
developing addition strategies21
Developing Addition Strategies
  • Strings are a series of interconnected bare number problems which teachers design and modify ad hoc in order to help students invent and/or use efficient mental-math computation strategies.
developing addition strategies22
Developing Addition Strategies

What mental–math strategy might a teacher using this mini-lesson be developing?

43 + 20

62 + 30

62 + 39

54 + 48

developing addition strategies24
Developing Addition Strategies

To successfully use mental-math mini-lessons, one must consider

  • The role of the student
  • The role of the teacher
  • The role (and power) of mathematical models
the role of the student
The Role of the Student

Students are expected to

  • Find their own solutions to the problem
  • Share their thinking publicly
  • Listen to and make sense of the strategies of others
  • Find and pose questions when they don’t understand or they need clarification
  • Try on new strategies
  • Practice until strategies become automatic
the role of student discourse
The Role of Student Discourse

Why is talk so important to the development of computational strategies?

1. Each time a strategy is discussed, students gain additional insights through other children’s explanations.

2. Over time, both through listening and questioning, students eventually make sense of the strategy and begin to feel comfortable with the strategy.

  • The students then may attempt to use the strategy in some situations.
  • Over time and with use, the strategy then becomes integrated into students’ mental-math repertoires and is used regularly when needed.
the role of the teacher
The Role of the Teacher
  • Encouraging students to make sense of situations;
  • Providing time for students to question each other’s thinking and strategies;
  • Connecting different strategies to help students understand each other’s thinking;
  • Highlighting efficient strategies;
  • Using questions such as:
    • How did you get your answer?
    • Can you explain it another way?
    • Did anyone do it the same way?
    • Can you put in your own words _______’s thinking?
the role of mathematical models
The Role of Mathematical Models
  • Models become important tools to represent student thinking.
  • Models become important tools to connect and juxtapose student strategies.
  • Models become important tools for students to think with.
developing addition strategies25
Developing Addition Strategies

Addition Strings

What addition strategies are the strings on the worksheet designed to develop?

developing addition strategies26
Developing Addition Strategies

What strategy is the string trying to develop?

54 + 20

55 + 19

42 + 40

44 + 38

66 +30

69 +27

89 +73

developing addition strategies27
Developing Addition Strategies

Choose one of the strings from the handout with a partner to analyze for:

(1) potential student strategies and struggles;

(2) how to model or represent their thinking;

(3) what questions to pose to students given their respective strategies or struggles.

games
Games
  • Why use games to teach math?
  • What do teachers need to do to ensure students are actually attending to the math in the game?
  • How can games be used to differentiate?
  • What kind of record keeping system would help us keep track of student learning when playing games?
  • How might we organize games for the greatest student autonomy and make sure they work with “just right” games as determined by informal assessment?
games2
Games
  • Choose a game and play it with a partner or in a small group.
  • Use the games analysis sheet to name the mathematics in the game, compare with games you already use,consider various ways to extend the game or make it less challenging.
  • Use the Games Continuum sheet to help you place the game on the continuum and begin building your games library.
reflections
Reflections
  • What are you questions and takeaways from today’s session?
  • Thank you!