**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

Not until 4th grade

Order and Compare

Fractions and Decimals

p. 24

Models - not algorithm

Algorithm

develops in 4th grade after

more experience

with models

Reasoning about "size" and

"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

**Grades 3 - 5**

**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?

What about odd numbers?

Questions to ask students about factors and multiples

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.

Start with multiples of 10, 100, and 1000.

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

Add and Subtract

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

fraction addition has been defined.

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

using the traditional algorithm.

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

Grade 2 - 5 minutes

- 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?

Start with 3, add 5

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)

Addition

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

Additive comparison types asks:

"What amount would be added to a quantity?"

Multiplicative comparison types ask:

"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

Addition and Subtraction

(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

Addition and Subtraction

P. 31

Be careful of cases involving 0 in division

Represent and Interpret Data

P. 53

Measure the length of your shoe in inches.

Third grade questions:

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?

Fourth grade questions:

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?

Fifth grade questions:

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.

http://www.parcconline.org/samples/mathematics/grade-3-mathematics-fluency

http://www.parcconline.org/samples/mathematics/grade-3-mathematics-field

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