Second Grade CCSS–M, and Daily Math

Vacaville USDAugust 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

The Standards for Mathematical Content are what students should know.

The Standards for Mathematical Practice are what students should do. Mathematical “Habits of Mind”

Standards for Mathematical Practice

CCSS Mathematical Practices

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REASONING AND EXPLAINING2. Reason abstractly and quantitatively3. Construct viable arguments and

critique the reasoning of others

MODELING AND USING TOOLS4. Model with mathematics5. Use appropriate tools strategically

SEEING STRUCTURE AND GENERALIZING7. Look for and make use of structure8. 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

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 3rd 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

Ten Frames

Ten Frames

Base 10 Blocks

Base 10 Blocks

Base 10 Shorthand

Base 10 Shorthand

Tens Facts

7 + 3 = 10

Tens Facts

6 + 4 = 10

Tens Facts

8 + 2 = 10

Learning Progression

Concrete

Representational

Abstract

Part-Whole Relations

4 4 4 4 4

Number Bonds

Number Bonds – 17

1717

17

17

1717

17

17

Number Bonds – 43

4343

43

43

4343

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 9th 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 78th 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 numbersUse 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:___ < ___ and ___ > ___

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 - NBTUnderstand 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 - NBTUnderstand place value.2. Count within 1000; skip-count by 2s, 5s, 10s, and 100s. CA3. 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, continuedNumber 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, continuedNumber of Day on Calendar Addition Problems Number Bonds 1 more 1 less, 10 more 10 less Predicting

Daily Math, continuedWord 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, continuedGeometry 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.

Daily Math, continuedPatterns Predict the next element in the pattern

(shape, numeric, location, etc.) Identifying the repeating part

Daily Math, continuedGraphs 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, continuedTime Morning, afternoon, evening, am, pm Order of events To the nearest 5 minutes (depends on

grade level)

Daily Math, continuedMoney 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). CA8. 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 20.2. 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 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 7.1 Use estimation strategies to make reasonable estimates in problem solving. CA9. 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 – 1st Grade70 – 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

4 + 3

Domino Facts

Domino Facts

Tens Facts

7 + 3 = 10

7 + 5

8 + 6

Addition – 7 + 5 Make ten

7 + 5

3 2

210 +

12

Addition – 8 + 6 Make ten

8 + 6

2 4

410 +

14

Addition – 28 + 6

Addition – 28 + 6 Make tens

28 + 6

2 4

430 +

34

Addition – 28 + 6

Addition – 28 + 6

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

28+ 6

1

4

2 tens + 1 ten = 3 tens

3

Adding 2-digit numbers

Miguel – 1st Grade30 + 16

Connor – 1st Grade39 + 25

How is the way these students solved the problems different from the way we typically teach addition?

Addition: 28 + 34

Addition – 28 + 34 Plan to make tens

28 + 34

2 32

3230 +

62

Addition – 46 + 38 Plan to make tens

46 + 38

4 34

3450 +

84

Addition: 28 + 34

Addition: 28 + 34

…adds tens and tens, ones and ones…

Addition: 28 + 34

… and sometimes it is necessary to compose a ten

Addition: 28 + 34

Addition: 28 + 34

28 + 34

20 + 8 + 30 + 4

Addition – 28 + 34

50 12

= 62210

Addition – 46 + 38 Add Tens, Add Ones, and Combine

46 + 38

40 + 30 = 706 + 8 = 1470 + 14 = 84

This can also be done as add ones, add tens, and combine.

701484

Addition – 546 + 278

546 + 278

500 + 200 40 + 70 6 + 8

700110 14824

Addition – 546 + 278

Expanded Form

500 + 40 + 6 + 200 + 70 + 8

700 + 110 + 14 810 + 14

824

Addition – 46 + 38 Add Tens, Add On Ones

46 + 38Add tens 40 + 30 = 70

Add on ones 70 + 6 = 76

76 + 8 = 84

Be careful about run on equal signs!

Addition – 46 + 38 Add On Tens, Then Ones

46 + 38Add on tens 46 + 30 = 76

Add on ones 76 + 8 = 84

Be careful about run on equal signs!

Addition – 546 + 278

Add On Hundreds, Tens, and Ones546 + 278 = 546 + 200 = 746 + 70 = 816 + 8 =

746816824

Addition – 46 + 38 Compensate

46 + 38 Add a nice number 46 + 40 = 86

(Think: 38 is 2 less than 40)

Compensate 86 – 2 = 84

Addition

Try at least 2 different strategies on each problem1. 57 + 6 2. 48 + 37

3. 63 + 29 4. 254 + 378

5. 538 + 296

Vertical vs Horizontal Why do students need to be given

addition (and subtraction) problems both of these ways?

279 + 54 = 279+ 54

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

5 – 2

0 1 2 3 4 5 6 7 8 9 10 11

12

5 – 2

0 1 2 3 4 5 6 7 8 9 10 11

12

5 – 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 = 10

10 – 3 = 7

3 3

Subtraction: 15 – 7 Decompose with tens

15 – 7 =

15 – 5 = 10

10 – 2 = 8

5 2

Developing Subtraction

Connor – 1st Grade25 – 8

Connor – 1st Grade70 – 23

Subtraction: 43 – 6 Take Away Tens, Then Ones

43 – 6 =

43 – 3 = 40

40 – 3 = 37

33

Subtraction: 73 – 46 Take Away Tens, Then Ones

73 – 46 =

73 – 40 = 33

33 – 6 = 27

40 6

Subtraction: 73 – 46 Take Away Tens, Then Ones

73 – 46 =

73 – 40 = 33

33 – 3 = 30

30 – 3 = 27

40 633

Subtraction: 53 – 38

Subtraction: 53 – 38

Subtraction: 53 – 38

Subtraction: 53 – 38

Subtraction: 53 – 38

Subtraction: 53 – 38

Subtraction: 53 – 38

Subtraction: 73 – 46 Regrouping and Ten Facts

73

– 46

6

72

60 – 40 = 20

Subtraction: 42 – 29 Regrouping and Ten Facts

42

– 29

3

31

10 + 2- 9

30 – 20 = 10

1

Subtraction: 57 – 34 57 34

(50 + 7) (30 + 4)

20 3+ = 23

Do I have enough to be able to subtract?

Subtraction: 52 – 34

52 34 (50 + 2) (30 + 4)

(40 + 12) (30 + 4)

10 8+ = 18

Do I have enough to be able to subtract?

Subtraction 300 – 87 Constant Differences

0 87 300

Suppose I slide the line down 1 space?

299

86

299 – 86 =

Subtraction: 73 – 46 Constant Differences

73

– 4627

+ 4

+ 4 = 77

= 50

Subtraction: 73 – 46 Regrouping by Adding Ten

73

– 46

13

5

27

Subtraction – Adding On

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

Subtraction

Try at least 2 different strategies on each problem

1. 53 – 7 2. 58 – 36

3. 73 – 29 4. 554 – 327

5. 538 – 298

Subtraction

Planning your strategy Not all problems are created equal! What strategy would be the most

effective.

NOT “one size fits all”