Transitioning to the Common Core State Standards – Mathematics

Transitioning to the Common Core State Standards – Mathematics
Pam Hutchison

AGENDA Party Flags Overview of CCSS-M Integers
Standards for Mathematical Practice Standards for Mathematical Content Integers Word Problems and Model Drawing Expressions and Equations Fractions

Expectations We are each responsible for our own learning and for the learning of the group. We respect each others learning styles and work together to make this time successful for everyone. We value the opinions and knowledge of all participants.

Erica is putting up lines of colored flags for a party.
The flags are all the same size and are spaced equally along the line. 1. Calculate the length of the sides of each flag, and the space between flags. Show all your work clearly. 2. How long will a line of n flags be? Write down a formula to show how long a line of n flags would be.

CaCCSS-M Find a partner Decide who is “A” and who is “B”
At the signal, “A” takes 30 seconds to talk Then at the signal, switch, “B” takes 30 seconds to talk. “What do you know about the CaCCSS-M?”

CaCCSS-M “What do you know about the CaCCSS-M?”
Using the fingers on one hand, please show me how much you know about the CaCCSS-M

National Math Advisory Panel Final Report
“This Panel, diverse in experience, expertise, and philosophy, agrees broadly that the delivery system in mathematics education—the system that translates mathematical knowledge into value and ability for the next generation — is broken and must be fixed.” (2008, p. xiii)

Common Core State Standards
Developed through Council of Chief State School Officers and National Governors Association

Common Core State Standards

How are the CCSS different?
The CCSS are reverse engineered from an analysis of what students need to be college and career ready. The design principles were focus and coherence. (No more mile-wide inch deep laundry lists of standards)

Real life applications and mathematical modeling are essential.

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

Mathematical Practice
Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning.

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

CCSS Mathematical Practices
Cut apart the Eight Standards for Mathematical Practice (SMPs) Look over each Tagxedo image and decide which image goes with which practice The more frequently a word is used, the larger the image Using the Standards for Mathematical Practice handout…did you get them right? Glue the Practice title to the appropriate image. What did you notice about the SMPs?

Reflection How are these practices similar to what you are already doing when you teach? How are they different? What do you need to do to make these a daily part of your classroom practice?

Supporting the SMP’s Summary
Questions to Develop Mathematical Thinking Common Core State Standards Flip Book Compiled from a variety of resources, including CCSS, Arizona DOE, Ohio DOE and North Carolina DOE flipbooks

Standards for Mathematical Content

Content Standards Are a balanced combination of procedure and understanding. Stressing conceptual understanding of key concepts and ideas

Content Standards Continually returning 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 Standards
Groups of related standards. Standards from different clusters may be closely related. Standards Defines what students should understand and be able to do. Numbered

Integers

Similarities and Differences
Look at the former standards related to integers

Now read the current CCSS standards related to integers How are they similar? How are they different?

Current CCSS standards related to integers 6th Grade NS 5 NS 6 NS 7 NS 8 How are these different from what you are currently teaching?

Integers – 6th Grade Denver, Colorado is called “The Mile High City” because its elevation is 5280 feet above sea level. Someone tells you that the elevation of Death Valley, California is −282 feet. Is Death Valley located above or below sea level? Explain. How many feet higher is Denver than Death Valley? What would your elevation be if you were standing near the ocean?

Integers – 6th Grade On the same winter morning, the temperature is -28℉ in Anchorage, Alaska and 65℉ in Miami, Florida. How many degrees warmer was it in Miami than in Anchorage on that morning?

Word Problems and Model Drawing

concrete – pictorial – abstract
Model Drawing A strategy used to help students understand and solve word problems Pictorial stage in the learning sequence of concrete – pictorial – abstract Develops visual-thinking capabilities and algebraic thinking.

Steps to Model Drawing Read the entire problem, “visualizing” the problem conceptually Decide and write down (label) who and/or what the problem is about Rewrite the question in sentence form leaving a space for the answer. Draw the unit bars that you’ll eventually adjust as you construct the visual image of the problem H

Steps to Model Drawing Chunk the problem, adjust the unit bars to reflect the information in the problem, and fill in the question mark. Correctly compute and solve the problem. Write the answer in the sentence and make sure the answer makes sense.

Missing Numbers 1 Mutt and Jeff both have money. Mutt has $34 more than Jeff. If Jeff has $72, how much money do they have altogether? H

Missing Numbers 2 Mary has 94 crayons. Ernie has 28 crayons less than Mary but 16 crayons more than Shauna. How many crayons does Shauna have?

Missing Numbers 3 Bill has 12 more than three times the number of baseball cards Chris has. Bill has 42 more cards than Chris. How many baseball cards does Chris have? How many baseball cards does Bill have?

Missing Numbers 4 Amy, Betty, and Carla have a total of 67 marbles. Amy has 4 more than Betty. Betty has three times as many as Carla. How many marbles does each person have?

Representation Getting students to focus on the relationships and NOT the numbers!

Practice with Model Drawing

Expressions and Equations Real World

Fruit Salad A fruit salad consists of blueberries, raspberries, grapes, and cherries. The fruit salad had a total of 280 pieces of fruit. There are twice as many raspberries as blueberries, three times as many grapes as cherries, four times as many cherries as raspberries. How many cherries are there in the fruit salad?

Firefighter Allocation
A town’s total allocation for firefighter’s wages in a new budget is $600,000. If wages are calculated at $40,000 per firefighter and benefits at $20,000 per firefighter, write an equation whose solution is the number of firefighters the town can employ if they spend their whole budget. Then solve the equation.

Log Ride A theme park has a log ride that can hold 12 people. They also have a weight limit of 1500 pounds per log for safety reasons. If the average adult weighs 150 pounds, the average child weighs 100 pounds, and the log itself weighs 200 pounds, the ride can operate safely if the inequality 150A + 100C  1500 is satisfied (A is the number of adults and C is the number of children in the log ride together). There are several groups of children of differing numbers waiting to ride. Group one has 4 children, group two has 3 children, group three has 9 children, group four has 6 children, and group five has 5 children. If 4 adults are already seated in the log, which groups of children can safely ride with them?

Fractions on a Number Line

So what is the definition of a fraction?

Definition of Fraction:
Start with a unit, 1, and split it into ___ equal pieces. Each piece represent 1/___ of the unit. We’re talking about ___ of those ___ equal pieces when we name the fraction __/__.

1 5 ● How many pieces are in the unit? Are all the pieces equal?
We write that as …… 1 ● 5

1 1 5 ● How far (from 0) to the point? We write that as ……
And we name the point …… ● 1 1 5

1 ● How many pieces are in the unit? Are all the pieces equal?
So each piece represents ● 1

1 ● How many pieces are there from 0 to the point?
So the name of the point is …. ● 1

4 Definition of Fraction:
When we name the point , we’re talking about a distance from 0 of ___ of those ___ pieces. 4

1 ● How many pieces are in the unit? Are all the pieces equal?
So the denominator is and each piece represents ● 1 5

1 The distance from 0 to the point is ___ of those pieces are shaded?
So the numerator is and the fraction represented is ● 1 3 3

1 The denominator is ● So each piece represents The numerator is
And the fraction is ● 1 6 5

Academic Vocabulary What is the meaning of denominator?
What about numerator? Definitions should be more than a location – the denominator is the bottom number They should be what the denominator is – the number of equal parts in one unit

1 Locate on the number line: ● Start with a unit, 1,
1 Start with a unit, 1, Split it into __ equal pieces. Each piece represents of the unit We want __ of those pieces So this point represents 7 2

1 6 8 Locate on the number line: ● Start with a unit, 1,
1 Start with a unit, 1, Split it into __ equal pieces. Each piece represents of the unit We’re want __ of those pieces So this points represents 8 6

1 1 2 3 3 | | | | | | | | | The denominator is …….
| | | | | | | | | The denominator is ……. The numerator is ……… Another way to name this point? 1 2 3 3 1 Page 83

1 1 2 5 3 2 3 | | | | | | | | | The denominator is ……
| | | | | | | | | The denominator is …… The numerator is ……… Another way to name this point? 1 2 5 3 1 2 3

2 1 2 7 3 1 3 | | | | | | | | | The denominator is …..
| | | | | | | | | The denominator is ….. The numerator is ……… Another way to name this point? 1 2 7 3 2 1 3

1 2 15 3 | | | | | | | | | Suppose the line was shaded to 5.
| | | | | | | | | Suppose the line was shaded to 5. How many parts would be shaded? So the numerator would be ……… 1 2 15 3

1 2 30 3 | | | | | | | | | Suppose the line was shaded to 10.
| | | | | | | | | Suppose the line was shaded to 10. How many parts would be shaded? So the numerator would be ……… 1 2 30 3

Rally Coach Partner A goes first Partner B: coach SWITCH
Name the point as a fraction and as a mixed number. Explain your thinking Partner B: coach SWITCH Partner B goes Partner A: coach Page 93-94

Rally Coach Partner A goes first Partner B: coach (as needed)
Locate the point on the number line Rename the point in a 2nd way (fraction or mixed number) Explain your thinking Partner B: coach (as needed) SWITCH ROLES What would you look for in a “good” explanation?

Rally Coach Partner A Partner B 6. 7. 8.

Connect to traditional
Change to a fraction. How could you have students develop a procedure for doing this without telling them “multiply the whole number by the denominator, then add the numerator”?

Connect to traditional
Change to a mixed number. Again, how could you do this without just telling students to divide?

Student Thinking Video Clips 1 – David (5th Grade) Two clips
First clip – 3 weeks after a conceptual lesson on mixed numbers and improper fractions Second clip – 3.5 weeks after a procedural lesson on mixed numbers and improper fractions

Student Thinking Video Clips 2 – Background
Exemplary teacher because of the way she normally engages her students in reasoning mathematically Asked to teach a lesson from a state-adopted textbook in which the focus is entirely procedural. Lesson was videotaped; then several students were interviewed and videotaped solving problems.

Student Thinking Video Clips 2 – Background, cont.
Five weeks later, the teacher taught the content again, only this time approaching it her way, and again we assessed and videotaped children.

Student Thinking Video Clips 2 – Rachel
First clip – After the procedural lesson on mixed numbers and improper fractions Second clip – 5 weeks later after a conceptual lesson on mixed numbers and improper fractions

Classroom Connections
Looking back at the 2 students we saw interviewed, what are the implications for instruction?

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

Equivalent Fractions

Equivalent fractions can be constructed by partitioning equal fractional parts of a whole into the same number of equal parts. The length of the whole does not change; it has only been partitioned into more equal sized pieces. Since the length being specified has not changed, the fractions that describe that length are equal.

CaCCSS Fractions are equivalent (equal) if they are the same size or they name the same point on the number line. (3.NF3a)

Equivalent Fractions ● 1 ●

Equivalent Fractions

Equivalent Fractions Start by splitting each part into:
2 equal parts 3 equal parts 4 equal parts Predict the fraction if you split each part into: 5 equal parts 8 equal parts 10 equal parts

Locate on the top number line.
● 1

Copy onto the bottom number line.
● 1 ●

Are the lengths equal? ● 1 ●

● 1 ●

So ● 1 ●

Fraction Families

Adding Fractions

So what about adding fractions?
What do we already know about adding? How do we add whole numbers on a number line?

| | | | | | | | | Add 3 + 4 Move a distance of 3 From that point, move a distance of 4 You end at 7 So = 7

ADD: 1 1

ADD: | | | | 1 | | | | | 1

ADD: + = | | | | | | | | | | | | | 1 | | | | | | | | | | | | | 1

Subtracting Fractions

Subtraction What do we know about subtracting so far? 579 – 34
5.79 – 3.4 Just like adding, we can only subtract things that are alike

/ / /3 / / / / / / / /6

Subtraction What’s the biggest issue that arises when we go to teach subtraction?

Subtract: | | | | | | | | | | | | 1 2 3 4 5 6

| | | | | | | | | | | | | | | | | | | | | | | | |
Subtract: | | | | | | | | | | | | | | | | | | | | | | | | | 1 2 3 4 5 6

Reflect How does the number line support students in developing/deepening their understanding of fraction operations? How does the use of a consistent model throughout the curriculum support all students?

Multiplying Decimals and Fractions

Using Arrays to Multiply
Use Base 10 blocks and an area model to solve the following: 21 x 13

Multiplying and Arrays
21 x 13

Decimals 0.4 0.4 x 0.6 0.6

Fractions 1 1

Fractions 2 3  3 5

Transitioning to the Common Core State Standards – Mathematics

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