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First Grade CCSS–M, and Daily Math
Vacaville USDSeptember 6, 2013
AGENDA The CCSS-M: Math Practice Standards Daily Math Programs
Subitizing Number of the Day Word Problems Model Drawing (steps) And other areas
Addition and Subtraction Facts Exploring Resources Report Cards and Assessments
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
OVE
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ion
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
Renee – Modeling Daily Math
Subitizing
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?
How Many Dots?
What’s My Number
What’s 1 more than
What’s 1 less than
Ten Frames and
Dot Patterns
Ten Frames
Ten Frames
Ten Frames
Ten Sticks
Base 10 Shorthand
Kindergarten – OA 4
For any number from 1 to 9, find the number that makes 10 when added to the given number, e.g., by using objects or drawings, and record the answer with a drawing or equation.
Tens Facts
7 + 3 = 10
Tens Facts
6 + 4 = 10
Tens Facts
8 + 2 = 10
Kindergarten – OA 3
Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings, and record each decomposition by a drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1).
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
Renee – Modeling Daily Math
Number of the Day of School Number of the Day on the Calendar Random Number of the Day
Number of the Day
Number of the Day of School Counting
Ones Tens Counting On
Counting back
Number of the Day
Place Value Straws Ten Frames Ten Sticks Hundred’s Chart
Computation
Number of the Day
Today is the 16th day of school Put in one more straw. How many ones do
we have? Do we need to make another bundle?
Count the number of ten’s bundles – 1 So, how many tens to make 16? How many ones? Let’s count the number of the day by
counting tens and counting on the ones – 10, 11, 12, 13, 14, 15, 16
Number of the Day
Today is the 16th day of school What is 1 more than 16? What is one less than 16? What if I put in another bundle of 10? Now
what would the number be? What if I took out a bundle of 10? What
would the new number be? Find at least 5 possible number bonds
(using 2 numbers) that you can make with 16.
Number of the Day
Today is the 78th day of school Put in one more straw. How many ones do
we have? Do we need to make another bundle?
Count the number of ten’s bundles Let’s count the number of the day by
counting tens and counting on the ones – 10, 20, 30, 40, 50, 60, 70, 71, 72, 73, 74, 75, 76, 77, 78
How many ten’s in 78? How many ones in 78?
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
Number of the DayNumber of Day on Calendar Rote Counting 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.)
Number of the DayNumber of Day on Calendar Addition Problems Number Bonds 1 more 1 less, 10 more 10 less Predicting
Daily Math, continuedPatterns Predict the next element in the pattern
(shape, numeric, location, etc.) Identifying the repeating part
Random Number of the Day
The number of the day is:
36 Who can read the number? What digit is in the ten’s place? The one’s
place? Write the number in expanded form. What is 1 more than 36? 1 less? What is 10 more than 36? 10 less? Find at least 3 number sentences for 36.
Random Number of the Day II
Popsicle sticks to generate random number of the day 5 tens, 9 ones
What is the number? Etc.
3 ones, 7 tens What is the number? Etc.
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 - NBTExtend the counting sequence.1. Count to 120, starting at any number
less than 120. In this range, read and write numerals and represent a number of objects with a written numeral.
CCSS - NBTUnderstand place value.2. Understand that the two digits of a
two-digit number represent amounts of tens and ones. Understand the following as special cases:a) 10 can be thought of as a bundle of
ten ones — called a “ten.”b) The numbers from 11 to 19 are
composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones.
CCSS - NBTUnderstand place value.
c) The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones).
3. Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <.
CCSS – NBT
Use place value understanding and properties of operations to add and subtract.5. Given a two-digit number, mentally
find 10 more or 10 less than the number, without having to count; explain the reasoning used.
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, continuedWord Problems All four operations ( +, -, x, ÷) Clear action problems verses passive
problems All problem types appropriate to grade
level (see chart)
Model Drawing Steps
1. Read the entire problem, “visualizing” the problem conceptually
2. Decide and write down (label) who and/or what the problem is about
3. Rewrite the question in sentence form leaving a space for the answer.
4. Draw unit bars of equal length that you’ll eventually adjust as you construct the visual image of the problem
Model Drawing Steps
5. Chunk the problem and adjust the unit bars to reflect the information in the problem
6. Determine exactly “what” you’re being asked to find and place a question mark in the place on the model drawing that reflects the “what”
Model Drawing Steps
7. Compute the problem to come up with an answer (show all work!)
8. Write the answer in a complete sentence that clearly states the solution
Pre-Model Drawing Steps
1. Read the entire problem, “visualizing” the problem conceptually
2. Decide and write down (label) who and/or what the problem is about
3. Rewrite the question in sentence form leaving a space for the answer.
Pre-Model Drawing Steps
4. Chunk the problem to determine what you know, what the action is, and what you are being asked to find
5. Compute the problem to come up with an answer (show all work!)
6. Write the answer in a complete sentence that clearly states the solution
CCSS – OA
Represent and solve problems involving addition and subtraction.1. Use addition and subtraction within 20
to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.
CCSS – OA
Represent and solve problems involving addition and subtraction.2. Solve word problems that call for
addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.
Modeling Daily Math
Renee and Pam Word Problems
Add To – Result Unknown
There are seven children playing at the park. Four more children come to the park. How many children are in the park now?
Taken From – Result Unknown
There are thirteen children playing at the park. Seven children goes home. How many children are in the park now?
Put Together/Take Apart – Total Unknown
At the park, 8 children are in the playground and 5 are by the pond. How many children are in the park?
Put Together/Take Apart – Both Addends Unknown
There are 16 children in the park. They are at either at the playground or by the pond. How many are at the playground and how many are by the pond?
Put Together/Take Apart – Both Addends Unknown
Rene has 14 bears. They are all red or blue. How many red bears and how many blue bears could she have?
Add To – Change Unknown
Renee has nine games. She got some more games for her birthday. Now she has 15 games. How many games did Renee get for her birthday?
Taken From – Change Unknown
Renee has 12 stuffed animals. Her new puppy got loose and chewed up some of them. Now she has only 9 stuffed animals. How many stuffed animals did the puppy chew up?
Multiplication
There are 3 boxes. Each box has 4 cookies in it. How many cookies are there in all?
Group Size Unknown
Renee has 15 cupcakes. She arranges them on three plates so that there is the same amount of cupcakes on each plate. How many cupcakes are on each plate?
Number of Groups Unknown
Renee bought 12 extra pencils to give to her best friends. If she gives each of her best friends 4 pencils, how many best friends does she have?
Renee – Modeling Daily Math
Shapes Time Money
Daily Math, continuedGeometry Plane Shapes: Rectangles, Squares,
Triangles, Trapezoids, Half-Circles, Quarter-Circles
Solids: Cubes, Right Rectangular Prisms, Right Circular Cones, Right Circular Cylinders
Be able to identify critical attributes Continue to review shapes from K
CCSS – Geometry
Reason with shapes and their attributes.1. Distinguish between defining
attributes (e.g., triangles are closed and three-sided) versus non-defining attributes (e.g., color, orientation, overall size); build and draw shapes to possess defining attributes.
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.4. Organize, represent, and interpret
data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another.
Daily Math, continuedTime Morning, afternoon, evening, am, pm Order of events To the nearest 5 minutes (depends on
grade level)
CCSS – MD Tell and write time.3. Tell and write time in hours and half-
hours using analog and digital clocks.
Daily Math, continuedMoney Names of Coin Values of Coin
Addition and Subtraction Facts
CCSS – OA Understand and apply properties of operations and the relationship between addition and subtraction.3. Apply properties of operations as
strategies to add and subtract.3 Examples: If 8 + 3 = 11 is known, then 3 + 8 = 11 is also known. (Commutative property of addition.) To add 2 + 6 + 4, the second two numbers can be added to make a ten, so 2 + 6 + 4 = 2 + 10 = 12. (Associative property of addition.)
CCSS – OA Understand and apply properties of operations and the relationship between addition and subtraction.4. Understand subtraction as an
unknown-addend problem. For example, subtract 10 – 8 by finding the number that makes 10 when added to 8.
CCSS – OA Add and subtract within 20.5. Relate counting to addition and
subtraction (e.g., by counting on 2 to add 2).
6. Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 – 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).
CCSS – OA Work with addition and subtraction equations.7. Understand the meaning of the equal
sign, and determine if equations involving addition and subtraction are true or false. For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 – 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2.
CCSS – OA 8. Determine the unknown whole number
in an addition or subtraction equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 + ? = 11, 5 = – 3, 6 + 6 = .
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
Try at least 2 different strategies on each problem1. 43 + 6 2. 68 + 7
Vertical vs Horizontal Why do students need to be given
addition (and subtraction) problems both of these ways?
79 + 5 = 79+ 5
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 – 20 Take Away Tens
43 – 20 =
40 – 20 = 20 so
43 – 20 = 23
40 + 3
Subtraction: 43 – 20 Count back
43 – 20 =
43, 33, 23
Subtraction: 43 – 20 Count up
43 – 20 =
20,
30, 40 2 tens
41, 42, 43 3 ones
23
Subtraction: 53 – 30
Subtraction: 53 – 30
Subtraction: 53 – 30
Subtraction: 53 – 30
Subtraction: 53 – 30
Subtraction: 53 – 7
40 + 3 == 46 + 3
Subtraction: 53 – 30
Subtraction: 53 – 30
Subtraction: 73 – 6 Regrouping and Ten Facts
73
– 6
6
76
60
Subtraction: 42 – 9 Regrouping and Ten Facts
42
– 9
3
33
10 + 2
30
1
Exploring Resources
Doc Locker
Illustrative Mathematicswww.illustrativemathematics.org