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# Common Core Curriculum Framework Training 3 - 5

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## Malissa Jacks

on 8 September 2014

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#### Transcript of Common Core Curriculum Framework Training 3 - 5

"And once I had a teacher who understood. He brought with him the beauty of mathematics. He made me create it for myself. He gave me nothing, and it was more than any other teacher has ever dared to give me." -Cochran
SEDOL Common Core Curriculum Framework Training
Number Sense
Instructional Strategies
Line Models
a/b
1/b
Limit denominators to 2, 3, 4, 6, and 8 (3rd grade)
Limit denominators to 2, 3, 4, 6, 8, 10, 12, and 100 (4th grade)
Area Models
Set Models
Order and Compare
Fractions and Decimals
p. 24
Models - not algorithm
Algorithm

more experience
with models
"benchmark fractions"
3/4 < 5/4
3NF.c includes representation of whole
numbers as fractions (1/1, 2/1, etc.)
Region/Area Models
Line Models
.13 < .2
.134 > .12
.301, .031, .003, .31
Base TEN Fractions
3/10 or 30/100
as .3 or .30
27/10?
ACTIVITY
Place Value
again... always
use visual models
misconception usually occurs with rounding down
400 + 20 + 7 = 427
3 x 100 + 4 x 10 + 7 x 1 + 3 x 1/10 +
9 x 1/100 + 2 x 1/100 = 347.392
Expanded Form
Factors and Multiples
(P. 22)
Strong Connection to....
Multiplication and division
Area
Fractions
Models SHOULD
be used to help students find
factors and multiples
Which numbers had the least amount of arrays?
What numbers have a factor of 2?
What numbers have factors that form a square?
What can you say about the factors for even numbers?
Do even numbers always have two even factors?
Multiplication and Division
(P. 34)
DO NOT...
Have students memorize facts (using drilling)
Teach multiplication and division separately
Teach in chunks of facts (e.g. 5s)
There are three levels of multiplication and division understanding, ALL build off of students' understanding of arrays and repeated addition developed in 2nd grade.
Level 2: Repeated counting on by a given number
Level 3: applying the associative and distributive property to compose and decompose (either into addition/subtraction or multiplication)
All of students’ understanding of multiplication and division situations, levels of representing and solving, and patterns need to culminate by the end of grade 3 in fluent multiplying and dividing.
Level 1: making and counting all quantities involved
Multiplication and Division
(P. 34)
K-2 standards
k-2
Strategies
Making a ten
Decomposing to a ten
Counting on
Related Facts
Using the relationship between addition and subtraction
Properties of Operations
Fluency is NOT automaticity
It does not come from memorization
It is the process of using strategies to solve problems more efficiently
This is something that cannot be assessed through a paper and pencil assessment. Needs to be assessed through an interview based assessment.
Strictly about understanding what multiplication and division mean and how the operations can be applied to real life situations.
Fluency in multiplication and division is much harder than addition and subtraction because there are no general strategies, just patterns and strategies dependent on specific numbers.

Example, when multiplying by 5, the product will always end is a 5 or 0.
(essentially students at this stage are just laying the foundations for standards 2.OA.3 & 2.OA.4)
This is easier for division than multiplication
Easier for multiplication than division
Example:
7 x 6 =
(6 + 1) x 6 =
6 x 6 + 1 x 6 =
36 + 6 = 42
Meaning, at the end of grade 3 all students should be at level 3 or beyond (have facts memorized).
Other key points about teaching multiplication and division.
Multiplication and division must be taught at the same time, NOT as separate units.
Models (especially area/array) are EXTREMELY important in developing understanding of multiplication and division, especially for future fractional concepts.
Present multiplication and division problems in real-life contexts
3 x 50 =
3 x 5 tens =
15 tens =
150
How do they know 15 tens is 150?
Skip counting by 50. 5 tens is 50, so 50, 100, 150.
Counting on by 5 tens. 5 tens is 50, 5 more tens is 100, 5 more tens is 150
Decomposing 15 tens. 15 tens is 10 tens and 5 tens. 10 tens is 100. 5 tens is 50. So, 15 tens is 100 and 50, or 150.
(relies on a strong place value understanding)
How do you move onto long division?
Think of how multi-digit multiplication was introduced.

If a student knows 56 / 8 then they can solve 560 / 8 and 5600 / 8
Our ultimate goal is fluent division...
Place value
Properties of operations
Relationship between multiplication and division
Round UP!
Rounding
Decimal Placement in Multiplication
DO NOT teach the rule of counting digits after a decimal to teach decimal placement.
If students understand place value well, they can reason about the placement of the decimal.
What is a tenth x a tenth?
Students can convert decimals to base-ten fractions
Estimation
Decimal Placement in Division
Connect to multiplication
Use place value reasoning
DO NOT teach students how to count digits after the decimal
Estimation
In mathematics, an algorithm is defined
by its steps and not by the way those
steps are recorded in writing. With this in
mind, minor variations in methods of recording
standard algorithms are acceptable.
remember the pizza's
this can't equal 3/4
3/3 = 1/3 + 1/3 + 1/3
3/3 = 2/3 + 1/3
Like Denominators including Mixed Numbers
this whole idea is built off of always representing
fractions >1 (5/3, 9/4,)
building off that strong k-2 number sense
17/6 - 5/6 or 7/5 +4/5
80% of students on NAEP:
4 1/5 as 21/5, but didn't know 4+1/5
the concept of a mixed number is defined only after
How do we develop students thinking vs. directly teaching
the algorithm?
Add and Subtract with unlike denominators including mixed numbers
Equivalency is the key prerequisite
Estimation should occur before computation
Informal methods should be supported
1/4 + 1/2
3/4 + 2/3
obviously visual fraction models ALWAYS are employed at
the beginning of any new concept so that students construct
their own meaning so that their invented approaches contribute
to the develpment of the algorithm
Nothing states they have to have LCM - they will discover it's easier
How is 5.NF.4 similar to the 4.NF.4a &4.NF.4b standards?
Lets try some problems
without
1 1
_ _
2 3
X
2 4
_ -
3 5
x
Use the manipulatives in front of you to solve the following problem...
Jim is making four pizzas to be shared equally among 5 people. How much pizza does each person get?
This is an example of an equal sharing problem.
How can we use a number line to solve the following problem?
Five people need to run a relay race of four miles, how far does each person have to run if they all run the same distance?
Hunter has 3/4 of a cake left to share among 4 visitors at his house. How much of the leftover cake does each person get?
Cynthia would like to bake cookies. She has one cup of sugar in the pantry. How many batches of cookies can she make using the one cup if each batch requires
A) 1/2 cup B) 1/3 cup C) 2/3 cup D) 3/4 cup
Grade 1 - hour and half hour
- analog and digital
- a.m. and p.m.
Now - nearest minute, solving time intervals (elapsed time)
11:30 12:30 1:30 1:45
The game begins at 11:30 a.m. If it lasts 2 hours and 15 minutes, when will it be over?
8:45 - 11:15
Algebraic Thinking
(P.40)
Do NOT teach students tricks to decode word problems!
This can lead to misconceptions about what operations to use to solve a problem.
This does not create true problem-solving sense.
For example,
Julie has 5 more apples than Lucy. Lucy has 4 apples. How many apples does Julie have?
Lucy has 4 fewer apples than Julie. Lucy has 5 apples. How many apples does Julie have?
What is an arithmetic pattern?
Patterns that change by the same rate, such as adding the same number.
For example, the series 2, 4, 6, 8, 10 is an arithmetic pattern that increases by 2 between each term.
What patterns do you see?
What patterns do you see?
Any sum of two even numbers is even.
Any sum of an even number and an odd number is odd.
The multiples of 4, 6, 8, and 10 are all even because they can all be decomposed into two equal groups.
The doubles (2 addends the same) in an addition table fall on a diagonal while the doubles (multiples of 2) in a multiplication table fall on horizontal and vertical lines.
The multiples of any number fall on a horizontal and a vertical line due to the commutative property.
All the multiples of 5 end in a 0 or 5 while all the multiples of 10 end with 0. Every other multiple of 5 is a multiple of 10.
Any sum of two odd numbers is even.
Why, on a multiplication chart, does the products in each row and column increase by the same amount?
There are 4 beans in a jar. Each day 3 beans are added. How many beans are in the jar for each of the first 5 days?
Today, both Sam and Terri have no fish. They both go fishing each day. Sam catches 2 fish each day. Terri catches 4 fish each day. How many fish do they have after each of the five days? Make a graph of the number of fish.
Compare 3 x 2 + 5 and 3 x (2 + 5)
Which equation is true?
15 - 7 – 2 = 10
15 - (7 – 2) = 10
Parenthesis
Exponents
Multiplication
and
Division
Order of Operations
(left to right)
and
Subtraction
(left to right)
Expression or Equation?
Operation Comprehension
(P. 43)
Your objective in the beginning isn't necessarily to develop an area formula but to apply students' developing concept of multiplication to the area of rectangles
(Van de Walle, 2010)
Area, Perimeter, and Volume
Today we are going to find the area of a shape
Focus Question:
See if you can find another way to find
the area without counting each tile
8x4 = 32 and 8x3 = 24 32+24 = 56
Think back to the activity we just did
“A rectangular garden has as an area of 80 square feet. It is 5 feet wide. How long is the garden?” Here,specifying the area and the width, creates an unknown factor problem
Helping students understand the properties of operations for multiplication and division
MODELS!
Associative property
Commutative property
Distributive property
"What amount would be added to a quantity?"
"What factor would multiply one quantity?"
Consider two diving boards. One is 40 feet high and the other is 8 feet high.
In first and second grade students would compare these two values in an additive sense.
One is 32 feet higher than the other.
In fourth grade students start comparing the two value multiplicatively.
One is 5 times higher than the other.
How should you correctly interpret the statement "times more than"?
One person take three counters, and give another person three times more than you.

How many counters did you give your partner?
Ideally, the wording should be changed to "times as much"
We need to expose students to both interpretations, but let them know that typically the accepted interpretation would be just to use multiplication (e.g., 3 x 3)
Lines and Angles
(P.50)
Initially, an informal definition of a right angle, acute angle, and obtuse angle should be given.
1/360 degrees
An entire rotation is 360 degrees
Have students practice identifying a variety of different sized angles
"packing" volume is more difficult than iterating a unit to measure length and measuring by tiling.
eventually students learn to conceptalize a layer as a unit that itself is composed of units of rows, each row composed of individual cubes ....
Meaning must be attached to a unit
(Fractions)
Multiplication and Divison
(Fractions)
p. 34
Primary Objectives
3. how to utilize SEDOL's curricular tools to address Common Core State Standards
2. how to read SEDOL's Scope and Sequence
1. how to read SEDOL's Curriculum Framework
Understand:
Secondary Objectives
2. Introduce the instructional routine for the math block
1. Increase your math content knowledge so that you can analyze student development.
What are the Common Core Standards?

Designed with focus, coherence, and rigor in mind.
Narrows the scope of content in each grade
Domain
Cluster
Standard
8 Math Practice Standards
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning
HOW
Math should be taught
Time and Money
P.45
Measurement
P.47
(p. 19 )
P. 28
P. 31
p. 56
P. 31
Be careful of cases involving 0 in division
Represent and Interpret Data
P. 53
Measure the length of your shoe in inches.

What length(s) were the most common?
How many more people had lengths of 7 than 6 1/4?
How many students had measurements larger than 7?
What is the total shoe length for the students who measured six and one-half?
What is the total number of inches for shoes measuring six and one-half and seven and one-half?
If we added all the shoe lengths above seven and one-fourth together what would we get?
What would be the total length of both shoes for ALL the people who had shoe lengths of six and one-half?
Instructional Strategies(purple box)
Key Idea: A fraction does not say anything about the size of the whole or the size of the parts. A fraction tells us ONLY about the RELATIONSHIP between the part and whole.
Use a variety of fractional models
Fractional parts are the building block for all fraction concepts
Much practice with dividing shapes and representing on a number line to build this type of number sense (MP6)
Promote reasoning with benchmark fractions: 0, 1/2, 1/1
Misconception
Misconception:
After several examples that 8/12 = 2/3, one would make an argument that each was divided or multiplied by 4
(
Through reasoning
- they haven't learned about fraction multiplication)
Again they don't know about fraction multiplication, but are beginning to discover algorithm
4/5 or 4/9 (same numerator)
4/7 or 5/7 (same denominator)
3/8 or 4/10 (different numerator & denominator)
4/6 or 7/12
WHich is Greater
NOT expected to find common multiple - will discover using models
so, let's test our number sense using benchmark fractions
5/3 or 7/4 (remember we are always going to work with
fractions greater than 1)
Always
taught simultaneously - this is just
a new way to write the number
1. Using the meaning of a decimal as a fraction generalizes to work with decimals in grade 5 that have more than two digits (thousandths).
2 Arguments
2. Visual fraction models don't generalize well to 5th grade.
That's why it's so necessary to to build the conceptual knowledge and view decimals as fractions when first introducing.
see if this activity is easier now
Calculators
Shape Reasoning
P. 62
Coordinate Graphing
P. 60
The PARCC Assessment
The Partnership for Assessment of Readiness for College and Careers
http://www.parcconline.org/parcc-assessment
" The Mathematical Practices will be taken seriously in curriculum and teaching if, and only if, they are taken seriously in testing. It can be expected, then, that the developers of the CCSS, and the States that collaborated in calling for the development of the CCSS, will work with the developers of assessments to ensure that the Mathematical Practices are taken seriously in testing.

Math "IPF"
Primary Objectives
Secondary Objectives
1. Increase your math content knowledge so that you can analyze student development.
2. Introduce the instructional routine for the math block
TRY A NUMBER TALK!
Closing Thoughts
8+6=(8+2=10+4=14)
13-4=(13-3=10-1=9)
6+7=(6+6=12+1+13)
8+4=12, one knows 12-8=4
8+3=11, 3+8=11(commutative)
& 2+6+4=2+10=12(associative)
"Nice Numbers"
2. laying a stronger
foundation
1.
3. questioning
1/8 + 4/5 =
2/5 + 1/2 =
This will eliminate one of the biggest misconceptions of
2/5 + 1/2 = 3/7 because 3/7 isn't even a half
1 2 3
find end (given start & elapsed)
find start (given end & elapsed)
Chapter 2 - p.13
5292 divided by 42
If one were to take an AngLeg and rotate it all the way around, what would be formed?
same perimeter different area:
6x2
4x4
1x12
4x3
2x6
same area different perimeter
same perimeter different area:
12 x 1
4 x 3
6 x 2
6 x 2
4 x 4
same area different perimeter:
Usually this is based off of
place value understanding
1 2 3
1 2 3
The goal is for students to see unit fractions
as the basic building blocks of fractions, in the
same sense that the number 1 is the basic building
block of the whole numbers; just as every whole
number is obtained by combining a sufficient number
of 1s, every fraction is obtained by combining a
sufficient number of unit fractions.
build fraction language by
2. know what is meant by "equal parts"
1. specifying the whole
3. 5/3 is the quantity you get when combining 5
parts together when the whole is divided into 3 equal
parts
**no need to introduce "proper" or "improper" fractions
denominator is the unit we are counting
numerator is the number of those units we have
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