Content Deepening6th Grade Math

September 16, 2013Jeanne SimpsonAMSTI Math Specialist

2

•Name•School•Classes you teach•Your favorite math topic to teach

Welcome

3

4

He who dares to teach

must never cease to

learn.John Cotton Dana

acos2010.wikispaces.com•Electronic version of

handouts•Links to web resources

6

Intervention

Assessment

Collaboration

Five Fundamental Areas Required for Successful Implementation of CCSS

7

Instruction•Deep conceptual understanding•Collaborative lesson design•Standards for Mathematical

Practice

Content• Fewer standards with greater

depth• Understanding, focus, and

coherence• Common and high-demand tasks

Intervention• Common required response to

intervention framework response

• Differentiated, targeted, and intensive response to student needs

• Student equity, access, and support

Assessment•PLC teaching-assessing-learning

cycle•In-class formative assessment

processes•Common assessment instruments

as formative learning opportunities

Collaboration

How do we teach?

8

SMP1 - Make sense of problems and persevere in solving them

SMP2 - Reason abstractly and quantitativelySMP3 - Construct viable arguments and critique the

reasoning of othersSMP4 - Model with mathematicsSMP5 - Use appropriate tools strategicallySMP6 - Attend to precisionSMP7 - Look for and make use of structureSMP8 - Look for and express regularity in repeated

reasoning

Standards for Mathematical Practice

9

•Capture the processes and proficiencies that we want our students to possess

•Not just the knowledge and skills but how our students use the knowledge and skills

•Describe habits of mind of the mathematically proficient student

•Carry across all grade levels, K-12

What Are The Practice Standards?

10

•√ I already do this.

• ! This sounds exciting!• ? I have questions.

Standards of Mathematical Practice

11

•Problem solving•Demanding tasks•Student understanding•Discussion of alternative

strategies•Extensive mathematics

discussion•Effective questioning•Student conjectures•Multiple representations

High-Leverage Strategies

12

Instruction• D

eep conceptual understanding

• Collaborative lesson design

• Standards for Mathematical Practice

Content• F

ewer standards with greater depth

• Understanding, focus, and coherence

• Common and high-demand tasks

Intervention• C

ommon required response to intervention framework response

• Differentiated, targeted, and intensive response to student needs

• Student equity, access, and support

Assessment• P

LC teaching-assessing-learning cycle

• In-class formative assessment processes

• Common assessment instruments as formative learning opportunities

Collaboration

What are we teaching?

Critical Focus AreasRatios and Proportional

RelationshipsConnect to whole number multiplication and division

Applying to problems

Standards 1-3

Number Systems

Dividing fractionsNegative numbersCoordinate plane

Standards 4-11

Expressions and Equations

Variables and expressionsSolve one-step equations

Standards 12-20

Statistics

Understanding different measures of center

Standards 25-29

Geometry – Standards 21-24

Recommend Emphases from PARCC Model Content Framework for Mathematics

15

Ratios and Proportional Relationships

Cluster Analysis Tool

Analysis Tool

Content Standard

Cluster

Which Standards

in the Cluster Are Familiar?

What’s New or

Challenging in These

Standards?

Which Standards

in the cluster Need

Unpacking or

Emphasizing?

How Is This Cluster

Connected to the Other 6-8 Domains

and Mathematical Practice?

Understand ratio concepts and use ratio reasoning to solve problems.

6.RP.1 Understand the concept of a ratio, and use ratio language to describe a ratio relationship between two quantities.

17

UNDERSTANDING•6.RP.1 – Understand the concept of a

ratio, and use ratio language to describe a relationship between two quantities.

•6.RP.2 – Understand the concept of a unit rate a/b associated with a ratio a:b with b 0, and use rate language in the context of a ratio relationship.

18

Kim and Bob ran equally fast around a track. Kim started first. When she had run 9 laps, Bob had run 3 laps.

When Bob had run 15 laps, how many laps had Kim run?

Explain your reasoning.

19

Solving ProportionsSolve Kanold, p. 94

• The traditional method of creating and solving proportions by using cross-multiplication is de-emphasized (in fact it is not mentioned in the CCSS) because it obscures the proportional relationship between quantities in a given problem situation.

• If two pounds of beans cost $5, how much will 15 pounds of beans cost?

20

Ratios and Proportional Relationships Progression, pages 6-7

• Although it is traditional to move students quickly to solving proportions by setting up an equation, the Standards do not require this method in Grade 6. There are a number of strategies for solving problems that involve ratios. As students become familiar with relationships among equivalent ratios, their strategies become increasingly abbreviated and efficient.

21

6.RP.3 Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.a. Make tables of equivalent ratios relating quantities with

whole-number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.

b. Solve unit rate problems including those involving unit pricing and constant speed.

c. Find a percent of a quantity as a rate per 100; solve problems involving finding the whole, given a part and the percent.

d. Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.

22

1. The ratio of free throws that Omar made to the ones he missed at practice yesterday was 7:3. If he attempted 90 free throw at practice, how many free throws did Omar make?

made 7

missed 3

Attempted 90

9 x 7 = 6390 ÷ 10 = 9

Omar made 63 free throws.

9

9 9 9

9 9 9 9 9 9

23

2. At FDR High School, the ratio of seniors who attend college to those who do not is 5:2. If 98 seniors do not attend college, how many do?

24

3. At Mesa Park High School, the ratio of students who have driver’s licenses to those who don’t is 8:3. If 144 students have driver’s licenses, how many students are enrolled at Mesa Park High School?

25

4. Of the black and blue pens that Mrs. White has in a drawer in her desk, 18 are black. The ratio of black pens to blue pens is 2:3. When Mrs. White removes 3 blue pens, what is the new ratio of black pens to blue pens?

26

GeometryUnpacking Standards

27

Unpacking the StandardsStep 1: Target a standardStep 2: Chunk the Main

CategoriesStep 3: Identify all standard

componentsStep 4: Identify the

Developmental Progression

Step 5: Identify Key Vocabulary

Step 6: Add Clarifying Information

“To increase student achievement by ensuring educators understand specifically what the new standards mean a student must know, understand, and be able to do. (Unpacking) may also be used to facilitate discussion among teachers and curriculum staff and to encourage coherence…(Unpacking), along with on-going professional development is one of many resources used to understand and teach the CCSS.”

-North Carolina Dept of Public Instruction

28

Why are we Unpacking Standards?

To understand what the standards are asking students to know, understand, and be able to do

To make time for professional discussion about the standards

To build upon and use common terminology when discussing the implementation of the standards

Unpacking is standards is not a substitute document for the Common Core Standards, it is a

record of the conversation of those who are involved in the process of digging into the

standards.

29

Step 1 – Target a Standard

•6.G.1 – Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles, or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.

302.G.3Partition circles and

rectangles into two, three, or four

equal shares

Partition circles and rectangles into two equal shares, using

the word halves, half of2.G.3

Partition

Builds on 1.G.3 Needed for 3.G.2

partition

Equal shares

rectangle HalvesHalf of

circlePartition a shape into fourths in different ways

Pattern Blocks Fraction

Bars/Circles

Describe

ThirdsThird of

FourthsFourth of

whole

Identicalwhole

2/2 = one whole

Partition circles and rectangles into three equal shares, using

the word thirds, third of2.G.3

Partition circles and rectangles into four equal shares, using

the word fourths, fourth of

2.G.3

Describe the whole as two halves, three thirds, four fourths

2.G.3

Recognize that equal shares of identical

wholes need not have the same shape

2.G.3

Recognize

The final product….

31

Example2.G.3 Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.

Step 2: Chunk the Main Categories

1.All Standard(s) in the cluster(s)

2.Identify Key Verbs

2.G.3Partition

circles and rectangles into two,

three, or four equal shares

Describe RecognizePartition

32

Step 3: Identify all standard components

Components from CCSS:Analyze nouns and verbs

What do students need to do?Include bullets, examples, footnotes, etc.

Take standard apart according to the verbs to separate skills within the standard

What do the students need to know?

lt blue

33

Example 2.G.3

Partition circles and rectangles into two, three, or four equal

shares

Partition RecognizeDescribe

Partition circles and rectangles into four equal shares, using

the word fourths, fourth of

2.G.3

Describe the whole as two halves, three thirds, four

fourths2.G.3

Recognize that equal shares of identical wholes need not have

the same shape2.G.3

Partition circles and rectangles into three equal shares, using

the word thirds, third of2.G.3

Partition circles and rectangles into two equal shares, using

the word halves, half of2.G.3

34

Step 4: Identify the Developmental Progression

Questions to consider when looking at the developmental progression of the standards…

•How would you utilize these chunks (blue) for scaffolding toward mastery of the entire standard?

• Where would you start when teaching this standard?

• What is the chunk that demonstrates the highest level of thinking?

35

Vertical AlignmentUsing the progression document(s) from Ohio Department of Education and CCSS Writing Team:Look to the grade level(s) below to see if the standard is introduced.

Look to the grade level(s) above to see if the standard is continued.

Code each standard on the poster with: builds on introduced needed for or mastered

and the grade level to which the standard aligns.

36

Introduced?Mastered?

Needed for?Builds on?

2.G.3Partition circles and rectangles into two, three, or four equal

shares

Partition RecognizeDescribe

Partition circles and rectangles into four equal shares, using

the word fourths, fourth of

2.G.3

Describe the whole as two halves, three thirds, four

fourths2.G.3

Recognize that equal shares of identical wholes need not have

the same shape2.G.3

Partition circles and rectangles into three equal shares, using

the word thirds, third of2.G.3

Partition circles and rectangles into two equal shares, using

the word halves, half of2.G.3

Builds on 1.G.3

Needed for 3.G.2

37

Step 5: Identify Key Vocabulary

Identify content vocabulary directly from the standard. Identify additional vocabulary students will need to know to meet the standard.

green

382.G.3Partition circles and

rectangles into two, three, or four

equal shares

Partition circles and rectangles into two equal shares, using

the word halves, half of2.G.3

Partition

Builds on 1.G.3 Needed for 3.G.2

partition

Equal shares

rectangle HalvesHalf of

circle

Describe

ThirdsThird of

FourthsFourth of

whole

Identicalwhole

Partition circles and rectangles into three equal shares, using

the word thirds, third of2.G.3

Partition circles and rectangles into four equal shares, using

the word fourths, fourth of

2.G.3

Describe the whole as two halves, three thirds, four fourths

2.G.3

Recognize that equal shares of identical

wholes need not have the same shape

2.G.3

RecognizePartition circles and rectangles into two equal

shares, using the word halves,

half of2.G.3

39

Step 6: Add Clarifying Information

Kid-friendly language to add clarity Clarifying pictures, words, or phrasesDefinitions, examplesSymbols, formulas, pictures, etc.

CAUTION: do not replace important vocabulary that is included in the standard.

yellow

402.G.3Partition circles and

rectangles into two, three, or four

equal shares

Partition circles and rectangles into two equal shares, using

the word halves, half of2.G.3

Partition

Builds on 1.G.3 Needed for 3.G.2

partition

Equal shares

rectangle HalvesHalf of

circlePartition a shape into fourths in different ways

Pattern Blocks Fraction

Bars/Circles

Describe

ThirdsThird of

FourthsFourth of

whole

Identicalwhole

2/2 = one whole

Partition circles and rectangles into three equal shares, using

the word thirds, third of2.G.3

Partition circles and rectangles into four equal shares, using

the word fourths, fourth of

2.G.3

Describe the whole as two halves, three thirds, four fourths

2.G.3

Recognize that equal shares of identical

wholes need not have the same shape

2.G.3

Recognize

Partition circles and rectangles into two equal

shares, using the word halves,

half of2.G.3

41

Transfer Unwrapping to Chart

42

Divide and conquer…•6.G.2 – Volume of right rectangular prism

•6.G.3 – Polygons in the coordinate plane

•6.G.4 – Nets and surface area

432.G.3Partition circles and

rectangles into two, three, or four

equal shares

Partition circles and rectangles into two equal shares, using

the word halves, half of2.G.3

Partition

Builds on 1.G.3 Needed for 3.G.2

partition

Equal shares

rectangle HalvesHalf of

circlePartition a shape into fourths in different ways

Pattern Blocks Fraction

Bars/Circles

Describe

ThirdsThird of

FourthsFourth of

whole

Identicalwhole

2/2 = one whole

Partition circles and rectangles into three equal shares, using

the word thirds, third of2.G.3

Partition circles and rectangles into four equal shares, using

the word fourths, fourth of

2.G.3

Describe the whole as two halves, three thirds, four fourths

2.G.3

Recognize that equal shares of identical

wholes need not have the same shape

2.G.3

Recognize

Partition circles and rectangles into two equal

shares, using the word halves,

half of2.G.3

Main Idea of Standard

Key Verbs

Take standard apart according to the verbs to separate skills within the standard.

Use all components of standard. Put in a logical sequence

Vertical alignment

Vocabulary

Clarifying information, student-friendly

44

The Number SystemVertical Alignment

45

Mathematics consists of pieces that make sense; they are not just independent manipulation/skills to be practiced and memorized – as perceived by many students.

These individual pieces progress through different grades (in organized structures we called “flows”) and can/should be unified together into a coherent whole.

Jason Zimba, Bill McCallum

Create a chart that lists…•What students need to know and be able

to do to demonstrate mastery of these standards

•Prerequisite skills that are needed for these standards

47

5th Operations and Algebraic Thinking•Evaluate expressions with ( ), [ ], { }•Write and interpret numerical expressions•Generate numerical patterns from rules•Form ordered pairs from patterns•Graph ordered pairs on the coordinate

plane

Prerequisite Skills•Order of operations•Whole number operations

48

Fluency•The word fluent is used in the Standards

to mean fast and accurate. Fluency in each grade involves a mixture of just knowing some answers from patterns (e.g., “adding 0 yields the same number”), and knowing some answers from the use of strategies.

• Progressions for the Common Core State Standards in Mathematics

49

Fluency•Fluent in the standards means “fast and

accurate.” It might also help to think of fluency as meaning more or less the same as when someone is said to be fluent in a foreign language. To be fluent is to flow; fluent isn’t halting, stumbling, or reversing oneself.

• Jason Zimba

50

Fluency ExpectationsGrade Required Fluency

K Add/subtract within 51 Add/subtract within 102 Add/subtract within 20

Add/subtract within 100 (pencil and paper)3 Multiply/divide within 100

Add/subtract within 10004 Add/subtract within 1,000,0005 Multi-digit multiplication6 Multi-digit division (6.NS.2)

Multi-digit decimal operations (6.NS.3) 7 Solve px + q = r, p(x + q) = r

51

Mathematical Fluency

52

Multiplication Strategies

53

Division Strategies

54

Unit Fractions

55

Fraction Multiplication in Grade 5

56

Fraction Division in Grade 5

57

Fraction Division in Grade 6•6.NS.1 – Interpret and compute quotients of

fractions, and solve word problems involving division of fractions, e.g., by using visual fraction models and equations to represent the problem.

•Examples: ▫Create a story context…▫Use a visual fraction model to show the quotient…▫Explain division using its relationship with

multiplication▫Sample problems

58

6NS - Understanding•6.NS.5 – Understand that positive and negative

numbers are used together to describe quantities having opposite directions or values…

•6.NS.6 – Understand a rational number as a point on the number line…

•6.NS.6b – Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane…

•6.NS.7 – Understand ordering and absolute value of rational numbers .

59

Expressions and EquationsProgression Documents

• K–6 Geometry• 6-8 Statistics and Probability• 6–7 Ratios and Proportional Relationships• 6–8 Expressions and Equations• 6-8 Number Systems

• These are the documents currently available. They are working on documents for the other domains (Functions, Geometry 7-8).

Progressions Documents

http://ime.math.arizona.edu/progressions/

61

Jigsaw•Read your assigned section

•Chart paper▫Summarize what you read.▫How can this document help you in your

classroom?▫What is said about the Math Practices?

•Be prepared to share

62

Understanding•6.EE.5 – Understand solving an equation

or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.

63

64

65

66

Statistics and ProbabilityResources

67

Illustrative Mathematics•Illustrative Mathematics provides

guidance to states, assessment consortia, testing companies, and curriculum developers by illustrating the range and types of mathematical work that students experience in a faithful implementation of the Common Core State Standards, and by publishing other tools that support implementation of the standards.http://www.illustrativemathematics.org/

68

Buttons: Statistical Questions•Zeke likes to collect

buttons and he keeps them in a jar. Zeke can empty the buttons out of the jar, so he can see all of his buttons at once.

•What statistical questions could we ask about these buttons?

69

North Carolina’s Lessons for LearningBake Sale Brownies (p. 3) - 6.RP.3, 6.EE.9• I can use knowledge of multiplication and division to scale recipes to make different

numbers of servings.• I can identify quantitative relationships between scaled recipes.• I can represent relationships between quantities by plotting points on the coordinate grid.Paper Clip Comparisons (p. 13) - 6.RP.1, 6.RP.2, 6.EE.6, 6.EE.9• I can represent equivalent ratios as a ratio table, as a graph, and on a double number line.• I can analyze ratio tables to identify qualitative patterns.• I can calculate unit rates by analyzing patterns in data.I'll Race You? (p. 19) - 6.EE.1, 6.NS.3• I can write an exponential expression as repeated multiplication.• I can evaluate an exponential expression with whole number exponents.Block Part-y (p. 22) - 6.G.2• I can find the volume of a right rectangular prism with fractional edge lengths.• I can apply my knowledge of the formula V = lwh to determine possible fractional

dimensions of right rectangular prisms when given the volume.How MAD Are You? A Deeper Look at Mean Absolute Deviation (p. 28) - 6.SP.5, 6.SP.2, 6.SP.3,

6..SP.4• I can use measures of center and measures of variability to summarize data sets in

context.• I can determine measures of center and variability of a data set and use the measures to

draw conclusions.• I can connect the measures of center and variability to the shape of the data distribution

within the given context.Shakespeare vs. Rowling (p. 34) - 6.SP.5, 6.SP.2, 6.SP.3, 6.SP.4)• I can collect data and display that data on tally charts, histograms, and box plots.• I can quantitatively analyze data to determine measures of center and variability.• I can use measures of center and variability of data sets to compare data sets.

70

Mathematics Assessment Project• Tools for formative and summative assessment that make

knowledge and reasoning visible, and help teachers to guide students in how to improve, and monitor their progress. These tools comprise:

• Classroom Challenges: lessons for formative assessment, some focused on developing math concepts, others on non-routine problem solving.

• Professional Development Modules: to help teachers with the new pedagogical challenges that formative assessment presents.

• Summative Assessment Task Collection: to illustrate the range of performance goals required by CCSSM.

• Prototype Summative Tests: designed to help teachers and students monitor their progress, these tests provide a model for examinations that may replace or complement current US tests.

http://map.mathshell.org/

Mean, Median, Mode, and RangeProjector Resources:

Computer Games: Ratings

P-71

Imagine rating a popular computer game.

You can give the game a score of between 1 and 6.

Mean, Median, Mode, and RangeProjector Resources:

Bar Chart from a Frequency Table

P-72

Mean score

Median score

Mode score

Range of scores

Mean, Median, Mode, and RangeProjector Resources:

Matching Cards1. Each time you match a pair of cards, explain

your thinking clearly and carefully.

2. Partners should either agree with the explanation or challenge it if it is unclear or incomplete.

3. Once agreed stick the cards onto the poster and write a justification next to the cards.

4. Some of the statistics tables have gaps in them and one of the bar charts is blank. You will need to complete these cards.

P-73

Mean, Median, Mode, and RangeProjector Resources:

Sharing Posters

1.One person from each group visit a different group and look carefully at their matched cards.

2.Check the cards and point out any cards you think are incorrect. You must give a reason why you think the card is incorrectly matched or completed, but do not make changes to the card.

3.Return to your original group, review your own matches and make any necessary changes using arrows to show if card needs to move.

P-74

75

Understanding

•6.SP.2 – Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.

76

Jeanne SimpsonUAHuntsville AMSTI

jeanne.simpson@uah.edu

acos2010@wikispaces.com

Contact Information

77

Feedback3 things I learned

2 things I liked

1 thing I want to know more about