Second Grade CCSS–M, and Daily Math Vacaville USD August 27, 2013

AGENDA The CCSS-M: Math Practice Standards Daily Math Programs Subitizing Ten Frames Number Bonds Place Value Computation And other areas Addition and Subtraction Planning/Discussions

The Common Core State Standards – Mathematics

CCSS – M The CCSS in Mathematics have two sections: Standards for Mathematical CONTENT and Standards for Mathematical PRACTICE know The Standards for Mathematical Content are what students should know. do The Standards for Mathematical Practice are what students should do.  Mathematical “Habits of Mind”

Standards for Mathematical Practice

CCSS Mathematical Practices OVERARCHING HABITS OF MIND 1.Make sense of problems and persevere in solving them 6.Attend to precision REASONING AND EXPLAINING 2.Reason abstractly and quantitatively 3.Construct viable arguments and critique the reasoning of others MODELING AND USING TOOLS 4.Model with mathematics 5.Use appropriate tools strategically SEEING STRUCTURE AND GENERALIZING 7.Look for and make use of structure 8.Look for and express regularity in repeated reasoning

Reflection How are these practices similar to what you are already doing when you teach? How are they different? What concerns do you have with regards to the Standards for Mathematical Practice?

Standards for Mathematical Content

Are a balanced combination of procedure and understanding. Stress conceptual understanding of key concepts and ideas

Standards for Mathematical Content Continually return to organizing structures to structure ideas place value properties of operations These supply the basis for procedures and algorithms for base 10 and lead into procedures for fractions and algebra

“Understand” means that students can… Explain the concept with mathematical reasoning, including  Concrete illustrations  Mathematical representations  Example applications

Organization K-8 Domains  Larger groups of related standards. Standards from different domains may be closely related.

Domains K-5 Counting and Cardinality (Kindergarten only) Operations and Algebraic Thinking Number and Operations in Base Ten Number and Operations-Fractions (Starts in 3 rd Grade) Measurement and Data Geometry

Organization K-8 Clusters  Groups of related standards. Standards from different clusters may be closely related. Standards  Defines what students should understand and be able to do.  Numbered

A Daily Math Program

5 Big Ideas 1. From Kindergarten on, help children develop flexible ways of thinking about numbers by having them “break apart” numbers in multiple ways

5 Big Ideas 2.From their earliest days in school, children should regularly solve addition, subtraction, multiplication, and division problems.

5 Big Ideas 3. Problem solving of all types should be a central focus of instruction.

5 Big Ideas 4.Develop number sense and computational strategies by building on children’s ideas and insights.

5 Big Ideas 5.Teach place value and multi-digit computation throughout the year rather than as “chapters”.

Number Sense What is “number sense”? The ability to determine the number of objects in a small collection, to count, and to perform simple addition and subtraction, without instruction.

Visualize Numbers I am going to show you a slide for a few seconds Record the number of dots in Box A and in Box B READY?

Box A Box B

Record your answers  Box A  Box B 

Share On a scale of 1-5, how confident are you that your answer is correct?

SUBITIZING Ability to recognize the number of objects in a collection, without counting When the number exceeds this ability, counting becomes necessary

Box A Box B

Perceptual Subitizing Maximum of 5 objects Helps children  Separate collections of objects into single units  Connect each unit with only one number word  Develops the process of counting

Subitizing Let’s try again. Ready??

Box C Box D

Record your answers  Box C  Box D 

Share On a scale of 1-5, how confident are you that your answer is correct?

Box C Box D

Box C Box D

Conceptual Subitizing Allows children to know the number of a collection by recognizing a familiar pattern or arrangement Helps young children develop skills needed for counting Helps develop sense of number and quantity

Children who cannot conceptually subitize will have problems learning basic arithmetic processes

Practicing Subitizing Use cards or objects with dot patterns Groups should stand alone Simple forms like circles or squares Emphasize regular arrangements that include symmetry as well as random arrangements Have strong contrast with background Avoid elaborate manipulatives

How Many Dots?

What’s 1 more than

What’s 1 less than

Ten Frames and Dot Patterns

Ten Frames  

   

    

Base 10 Blocks

Base 10 Shorthand

Tens Facts   = 10

Tens Facts   = 10

Tens Facts   = 10

Learning Progression Concrete  Representational  Abstract

Part-Whole Relations

44444 Number Bonds

Number Bonds – 17 17

Number Bonds – 43 43

Number of the Day Number of the Day of School Counting Counting back Place Value  Straws  Base 10 Blocks  Hundred’s Chart Computation

Number of the Day Today is the 9 th day of school  What is 1 more than 9?  What is one less than 9?  Find all the possible number bonds (using 2 numbers) that you can make with 9.

Number of the Day Today is the 78 th day of school  Write 78 in expanded form.  What is 1 more than 78? 1 less?  What is 10 more than 78? 10 less?  Find at least 3 number sentences for 78.  Use at least 3 numbers  Use at least 2 different operations

Random Number of the Day The number of the day is: 436  Who can read the number?  What digit is in the ten’s place? The hundred’s place?  Write the number in expanded form.  What is 1 more than 436? 1 less?  What is 10 more than 436? 10 less?  What is 100 more than 436? 100 less?  Find at least 3 number sentences for 436.

Random Number of the Day II Popsicle sticks What is the number? Write it in words. Where would it be located on the number line?  Hundred’s  Tens Counting  Start at number and count by 1’s; 2’s; 5’s; 10’s

My Number of the Day Is my number larger or smaller than your number?  How do you know?  Fill the number in so that each makes a true statement: ___ Write a number that is larger than the number of the day. Write a number that is smaller than the number of the day.

CCSS - NBT Understand place value. 1.Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases: a. 100 can be thought of as a bundle of ten tens — called a “hundred.” b. The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones).

CCSS - NBT Understand place value. 2.Count within 1000; skip-count by 2s, 5s, 10s, and 100s. CA 3. Read and write numbers to 1000 using base-ten numerals, number names, and expanded form. 4. Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons.

CCSS – NBT 8. Mentally add 10 or 100 to a given number 100–900, and mentally subtract 10 or 100 from a given number 100–900.

CCSS – MD 6. Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2,..., and represent whole-number sums and differences within 100 on a number line diagram.

Math Talk Students do better in classrooms where teachers use numbers as regular part of day

Reflection Where, in the course of a normal day, can you find places to talk about numbers OUTSIDE OF MATH TIME? Where do numbers occur in the everyday lives of your students?

Daily Math, continued Number of Day on Calendar Rote Counting Place Value with smaller numbers, i.e., 10 and ______ more Calendar Questions – Days of the week, months of the year, tomorrow and yesterday, how many Saturday’s have we had, looking at the columns of the calendar, etc.)

Daily Math, continued Number of Day on Calendar Addition Problems Number Bonds 1 more 1 less, 10 more 10 less Predicting

Daily Math, continued Word Problems All four operations ( +, -, x, ÷) Clear action problems verses passive problems All problem types appropriate to grade level (see chart)

CCSS – OA Represent and solve problems involving addition and subtraction. 1. Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.

Daily Math, continued Geometry Plane Shapes: Triangles, Quadrilaterals, Pentagons, Hexagons Solids: Cubes Be able to identify critical attributes Name shape based on critical attributes Continue to review shapes from K-1

CCSS – Geometry 1. Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. 5 Identify triangles, quadrilaterals, pentagons, hexagons, and cubes. Sizes are compared directly or visually, not compared by measuring. 5

Daily Math, continued Patterns Predict the next element in the pattern (shape, numeric, location, etc.) Identifying the repeating part

Daily Math, continued Graphs and Data At least once a month – related to things about the kids Graphs represent real people and real data Ask a wide variety of problems related to the graph including “What would happen if….” questions

CCSS – MD Represent and interpret data. 9. Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by making repeated measurements of the same object. Show the measurements by making a line plot, where the horizontal scale is marked off in whole-number units. 10. Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories. Solve simple put-together, take- apart, and compare problems using information presented in a bar graph.

Daily Math, continued Time Morning, afternoon, evening, am, pm Order of events To the nearest 5 minutes (depends on grade level)

Daily Math, continued Money Names of Coin Values of Coin Make 37  in at least 3 ways Write 84 cents in 2 different ways

CCSS – MD Work with time and money. 7. Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m. Know relationships of time (e.g., minutes in an hour, days in a month, weeks in a year). CA 8. Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and ¢ symbols appropriately. Example: If you have 2 dimes and 3 pennies, how many cents do you have?

Addition and Subtraction

CCSS – M Add and subtract within Fluently add and subtract within 20 using mental strategies. 2 By end of Grade 2, know from memory all sums of two one-digit numbers. 2 2 See standard 1.OA.6 for a list of mental strategies.

CCSS – M Use place value understanding and properties of operations to add and subtract. 5. Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction. 6. Add up to four two-digit numbers using strategies based on place value and properties of operations.

CCSS – M 7. Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.

CCSS – M 7.1Use estimation strategies to make reasonable estimates in problem solving. CA 9. Explain why addition and subtraction strategies work, using place value and the properties of operations. Explanations may be supported by drawings or objects.

Teaching for Understanding Telling students a procedure for solving computation problems and having them practice repeatedly rarely results in fluency Because we rarely talk about how and why the procedure works.

Teaching for Understanding Students do need to learn procedures for solving computation problems But emphasis (at earliest possible age) should be on why they are performing certain procedure

Research Students who learn rules before they learn concepts tend to score significantly lower than do students who learn concepts first Initial rote learning of a concept can create interference to later meaningful learning

Gretchen – 1 st Grade 70 – 23

Progression Concrete Pictorial or Visual or Representational Abstract  Invented Algorithms  Alternate Algorithms  Traditional Algorithms

Invented Procedures Allow students to invent and develop their own procedures based on what they already know

Fact Fluency Fact fluency must be based on understanding 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.

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

Addition

3 + 2 

4 + 3  

 

Domino Facts

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

Adding 2-digit numbers Miguel – 1 st Grade Connor – 1 st Grade How is the way these students solved the problems different from the way we typically teach addition?

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 – =

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

Addition –

Addition – Expanded Form

Addition – Add Tens, Add On Ones Add tens = 70 Add on ones = = 84 Be careful about run on equal signs!

Addition – Add On Tens, Then Ones Add on tens = 76 Add on ones = 84 Be careful about run on equal signs!

Addition – Add On Hundreds, Tens, and Ones = = = =

Addition – Compensate Add a nice number = 86 (Think: 38 is 2 less than 40) Compensate86 – 2 = 84

Addition Try at least 2 different strategies on each problem

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

Subtraction

1. Katie had 5 candy hearts. She gave 2 of them to Nick. How many hearts does Kate have left for herself? 2. Katie has 5 candy hearts. Nick has 2 candy hearts. How many more does Katie have?

5 – 2 

 

– 2

– 2

Subtraction How do you currently teach subtraction?  “Take-away”  “The distance from one number to the other” Additional Strategies

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

Developing Subtraction Connor – 1 st Grade 25 – 8 Connor – 1 st Grade 70 – 23

Subtraction: 43 – 6 Take Away Tens, Then Ones 43 – 6 = 43 – 3 = – 3 =

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: 53 – 38

Subtraction: 73 – 46 Regrouping and Ten Facts 73 – – 40 = 20

Subtraction: 42 – 29 Regrouping and Ten Facts 42 – – 20 = 10 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 –

Subtraction – Adding On 471 – 285 Start at 285Add 5 Now at 290Add 10(15) Now at 300Add 100(115) Now at 400Add 70(185) Now at 470Add 1(186) Now at 471 – DONE!

Subtraction Try at least 2 different strategies on each problem – – – – – 298

Subtraction Planning your strategy Not all problems are created equal! What strategy would be the most effective. NOT “one size fits all”