UCDMP Saturday Series The Vision of the Common Core: Changing Beliefs, Transforming Practice Saturday, November 3, 2012

Math Tasks & Problem Solving (SMP #1) Basic Facts, Place Value, & Computation Parents and the CaCCSS Paul Giganti, Jr. Lunch SMP #6 Alternative and Invented Algorithms Multiplication Division Reflections and Feedback Agenda

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SMP #1 Make sense of problems and persevere in solving them

What is the intent of SMP #1? The goal of SMP#1 is for students to become successful problem solvers – Word problems Problem solving or non-routine problems

SMP #1 3 Key Parts to Learning Mathematics Skills – essentially the tools of mathematics Concepts – the ideas of mathematics we need to understand before we can do mathematics Problem Solving – the ability to apply the mathematics we know in different situations

SMP #1 What is a “problem solving”? ̶ Problem solving is knowing what to do when you don’t immediately know what to do (Lyle Fisher) ̶ Requires us to combine skills and concepts in order to deal with specific mathematical situations – problems.

Problem Solving Strategies Talk at your tables: What are some of the key problem solving strategies?

Problem Solving Strategies Guess and check (Guess and test) Look for a pattern Make a drawing or model Act it out Work backwards Simplify the problem Eliminate possibilities Make a systemic list Write and equations

Why Teach Problem Solving? Parent Involvement and Awareness Paul Gigante, Jr. CMC Math Festival Program CMC – California Mathematics Council

Fact Fluency Institute of Educational Sciences Practice Guide “Assisting Students Struggling with Mathematics: Response to Intervention for Elementary and Middle Schools” Recommends approximately 10 minutes per day building fact fluency

Fact Fluency The intent IS NOT to administer basic fact tests! Teachers need to build basic fact strategy lessons for conceptual development, which builds fluency.

Fact Fluency Fact fluency must be based on an understanding of operations and thinking strategies. Students must  Connect facts to those they know  Use mathematics properties to make associations  Construct visual representations to develop conceptual understanding.

Investigations have continually shown that an emphasis on teaching for meaning has positive effects on student learning, including Better initial learning Greater retention Increased likelihood that the ideas will be used on new situations. Also found in studies in high-poverty areas - Handbook of Research on Improving Student Achievement Why Teach for Understanding?

Math Facts Direct modeling / Counting all Counting on / Counting back / Skip Counting Invented algorithms  Composing / Decomposing  Mental strategies Automaticity

Developing Addition What do students need to understand about addition? What are some of the models used to develop addition concepts and written strategies

3 + 2    

4 + 3  

Tens Facts   = 10

Tens Facts   = 10

Tens Facts   = 10

Tens Facts   = 10

7 + 5      

8 + 6        

Addition – Make ten

Addition – Make ten

Addition –

Make tens

Addition –

8 ones + 6 ones = 14 ones 14 ones = 1 ten + 4 ones tens + 1 ten = 3 tens 3

Addition:

Addition – Plan to make tens

Addition – Plan to make tens

Addition:

…adds tens and tens, ones and ones…

Addition: … and sometimes it is necessary to compose a ten

Addition – =

Addition – Add Tens, Add Ones, and Combine = = = 84 This can also be done as add ones, add tens, and combine

Addition –

Addition – Expanded Form

Vertical vs Horizontal Why do students needs to be given addition (and subtraction) problems both of these ways? =

Decimal Addition How does the emphasis on place value support students as they begin to add decimals? How do we teach decimals so that we support students in overtly making the connection between whole number addition and decimal addition?

Decimal Addition What concrete models support decimal addition and subtraction? What representations support decimal addition and subtraction?

SMP #6 Refers to the need for teachers and students to communicate precisely and correctly Mathematics vocabulary Symbols (equal sign) Specifying units Clear and concise explanations Conceptual understanding Procedural understanding

SMP #6 Refers to the need for teachers and students to communicate precisely and correctly Level of precision needed  Is an estimate sufficient? How close an estimate?  How precise a measurement?

Teacher’s Role Model Appropriate Use of Mathematics Vocabulary, Symbols, and Explanation Teachers need to attend to precision as they are talking and teaching mathematics so that students do not learn unintended and inaccurate mathematics.

Students frequently copy teachers when they are not precise with language and definitions. For example: “A rectangle has 2 long sides and 2 short sides.” – This “definition” then excludes a square from being a special case of a rectangle. When solving by making tens, the student says “I did and then I added the 5. The teacher needs to be careful to write = 10 and = 15 rather than use a run-on equal sign (8 + 2 = = 15).

Or when using the distributive property to solve 4 x 28, the student might say “I multiplied 4 x 25 to get 100 and then I added 4 x 3 which is 12 to get 112.” The teacher needs to be careful NOT to record 4 x 25 = (4 x 3) = 112 which, even though it seems to represent the student’s verbal explanation, implies that 4 x 25 = 112. Instead show this as a series of steps: 4 x 25 = x 3 = = 112

Teacher’s Role Provide Opportunities for Students to Share Their Thinking  Ask students to explain and justify their solutions and mathematical ideas.  As students are explaining/ or justifying, check to see that they have solved the problem correctly and accurately and that they are using language to describe their process precisely.

For example - Back to 4 x 28. Suppose the student says “I multiplied 4 x 8 to get 32. I put down the 2 and carried the 3. Then I multiplied 4 x 2 to get 8 and added the 3 to get 11. The answer is 112.” The student is not using proper place value language in their explanation. You can now ask the student (and the class) questions to develop the proper language. “When you multiplied 4 x 2 to get 8, were you actually multiplying 4 x 2 to get a product of 8?” Focus on the precision of the language to get students to state “I multiplied 4 x 2 tens to get 8 tens” for example.

Teacher’s Role Prepare Students for Further Study Be careful to use language that supports the mathematics that comes later in the curriculum. Teachers often inadvertently set students up for confusion as they encounter mathematics in later grades by not being precise in the language they use or statements they make.

For example: “You can’t subtract a big number from a small number” or “Multiplication makes the number bigger and division makes the number smaller” or The alligator eats the larger number. These statements create issues for students as they go on to learn integers, decimals and fractions.

5 – 2 

   

– 2

– 2

Subtraction: 13 – 6 Decompose with tens 13 – 6 = 13 – 3 = – 3 = 7 33

Subtraction: 15 – 7 Decompose with tens 15 – 7 = 15 – 5 = – 2 = 8 52

Subtraction: 73 – 46 Take Away Tens, Then Ones 73 – 46 = 73 – 40 = – 6 =

Subtraction: 73 – 46 Take Away Tens, Then Ones 73 – 46 = 73 – 40 = – 3 = – 3 =

Subtraction: 73 – 46 Regrouping and Ten Facts 73 – – 6 = = 7 6 – 4 = 2

Subtraction: 42 – 29 Regrouping and Ten Facts 42 – – 9 = = 3 3 – 2 = 1

Subtraction: 57 –  34 (50 + 7)  (30 + 4) = 23 Do I have enough to be able to subtract?

Subtraction: 52 –  34 (50 + 2)  (30 + 4) ( )  (30 + 4) = 18 Do I have enough to be able to subtract?

Subtraction 300 – 87 Constant Differences Suppose I slide the line down 1 space? – 86 =

Subtraction: 73 – 46 Constant Differences 73 – = 77 = 77 = 50

Subtraction: 73 – 46 Regrouping by Adding Ten 73 –

What about decimals? How can you help students make sense of: 7.56 – – – 3.6

Multiplication What does 3 x 2 mean? Repeated addition groups of 2

Multiplication 3 rows of 2 This is called an “array” or an “area model”

Advantages of Arrays as a Model Models the language of multiplication 4 groups of 6 or 4 rows of 6 or

Advantages of Arrays as a Model Students can clearly see the difference between (the sides of the array) and the (the area of the array) 7 units 4 units 28 squares factors product

Advantages of Arrays Commutative Property of Multiplication 4 x 6 = 6 x 4

Advantages of Arrays Associative Property of Multiplication (4 x 3) x 2 = 4 x (3 x 2)

Advantages of Arrays Distributive Property 3(5 + 2) = 3 x x 2

Advantages of Arrays They can be used to support students in learning facts by breaking problem into smaller, known problems  For example, 7 x = = 56 +

Using Arrays to Multiply 23 x 4 4 rows of 20 4 rows of 3 = 80 =

Using Arrays to Multiply 23 x 4 4 rows of 3 4 rows of 20 = 12 =

Using Arrays to Multiply Use Base 10 blocks and an area model to solve the following: 21 x 13

Multiplying and Arrays 21 x 13

31 x 14 =

Partial Products 31 x (10  30) (10  1) (4  30) (4  1)

Partial Products 3 1 x (4  1) (4  30) (10  1) (10  30)

Pictorial Representation 84 x  80 4,  4 7  80 7 

Pictorial Representation  30 2,  7 4  30 4  x 94

Pictorial Representation 347 x ,000 2, ,

Decimals 0.4 x

Fractions 

Remember 21 x 13

Algebra (2x + 1)(x + 3)

Connecting Multiplication and Division

Division What does 6  2 mean?  Repeated subtraction group 2 groups 3 groups

Measurement Division I have 21¢ to buy candies with. If each gumdrop costs 3¢, how many gumdrops can I buy?

Fair Share Division Mr. Gomez has 12 cupcakes. He wants to put the cupcakes into 4 boxes so that there’s the same number in each box. How many cupcakes can go in each box?

Difference in counting? Measurement 3 for you, 3 for you, 3 for you And so on Like measuring out an amount Fair Share 1 for you, 1 for you, 1 for you, 1 for you 2 for you, 2 for you, 2 for you, 2 for you And so on Like dealing cards

Measurement Division What does 6  2 mean?  6 split into groups of 2

You know  The total amount of objects  The number of objects in each group You’re trying to find  The number of groups Counting  1, 2, 3, 4, ….  1, 2, 3, 4, ….. Measurement Division (Quotative)

Fair Share Division What does 6  2 mean?  6 split evenly into 2 groups

You know  The total amount of objects  The number of groups You’re trying to find  The number of objects in each group Counting  1 for you, 1 for you, 1 for you, 1 for you,….  2 for you, 2 for you, 2 for you, 2 for you, etc. Fair Share Division (Partitive)

Models for Division Repeated subtraction Groups  Finding the number in each group  Finding the number of groups Arrays – finding the missing side

Using Arrays 4 3 ? ?

Using Arrays 5 ? ? 3 6

Using Arrays ? )

Repeated Subtraction 23 ÷ 3  1 group for you 23 – 3 = 20 left  1 group for you20 – 3 = 17 left  1 group for you17 – 3 = 14 left  1 group for you14 – 3 = 11 left  1 group for you11 – 3 = 8 left  1 group for you8 – 3 = 5 left  1 group for you5 – 3 = 2 left 7 groups of 3 with 2 leftover

Repeated Subtraction 35 ÷ 4  Should I put 1, 2, or 3 in each group?  How many cubes did I give away?  How many cubes are left?

47 ÷ 6 6 Groups 6 ) Can I put at least 3 in each group?

47 ÷ 6 6 Groups 6 ) groups of 3 uses ___ pieces. 3 –18 29 How many pieces are left? Can I put 3 more in each group? 18

47 ÷ 6 6 Groups 6 ) groups of 3 uses ___ pieces –18 29 –18 11 How many pieces are left? Can I put 3 more in each group? 18

47 ÷ 6 6 Groups 6 ) How many more can I put in each group? –18 29 –18 11 – 6 5 How many pieces are left? Can I put any more in each group? 6 6 groups of 1 uses ___ pieces.

47 ÷ 6 6 Groups of 6 ) –18 29 –18 11 – R5 We have __ in each groups with ___ left 5 7

Expanded Multiplication Table Groups of 6 1’s 10’s100’s groups

338 ÷ 7 7 Groups 7 ) I can make 7 groups of at least Can I make 7 groups of at least 100? 7 groups of 1 ten is 7 tens or ___ 7 groups of 3 tens is __ tens or ___ 7 groups of 2 tens is __ tens or ___ 7 groups of 5 tens is __ tens or ___ 1’s 10’s So the answer is between 10 and 100

338 ÷ 7 7 Groups 7 ) 338 – ’s 10’s 7 groups of 3 tens uses ___ pieces. How many pieces are left? Can I put any more tens in each group?

338 ÷ 7 7 Groups 7 ) 338 – – ’s 10’s 7 groups of 1 ten uses ___ pieces. How many pieces are left? Can I put any more tens in each group?

338 ÷ 7 7 Groups 7 ) 338 – – – ’s 10’s 7 groups of 5 uses ___ pieces. How many pieces are left? Can I put any more ones in each group? How many ones can I put in each group?

338 ÷ 7 7 Groups 7 ) 338 – – – – ’s 10’s 7 groups of 3 uses ___ pieces. How many pieces are left? Can I put any more ones in each group?

338 ÷ 7 7 Groups 7 ) 338 – – – – ’s 10’s We have ___ in each group with ___ left R2

Try a couple! 858 ÷ 4 3,793 ÷ 9