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  1. Content Deepening6th Grade Math September 16, 2013 Jeanne Simpson AMSTI Math Specialist

  2. Welcome • Name • School • Classes you teach • Your favorite math topic to teach

  3. He who dares to teach must never cease to learn. John Cotton Dana

  4. acos2010.wikispaces.com • Electronic version of handouts • Links to web resources

  5. Five Fundamental Areas Required for Successful Implementation of CCSS

  6. How do we teach?

  7. Standards for Mathematical Practice SMP1 - Make sense of problems and persevere in solving them SMP2 - Reason abstractly and quantitatively SMP3 - Construct viable arguments and critique the reasoning of others SMP4 - Model with mathematics SMP5 - Use appropriate tools strategically SMP6 - Attend to precision SMP7 - Look for and make use of structure SMP8 - Look for and express regularity in repeated reasoning

  8. What Are The Practice Standards? • 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

  9. Standards of Mathematical Practice • √ I already do this. • ! This sounds exciting! • ? I have questions.

  10. High-Leverage Strategies • Problem solving • Demanding tasks • Student understanding • Discussion of alternative strategies • Extensive mathematics discussion • Effective questioning • Student conjectures • Multiple representations

  11. What are we teaching?

  12. Critical Focus Areas Ratios and Proportional Relationships Connect to whole number multiplication and division Applying to problems Standards 1-3 Number Systems Dividing fractions Negative numbers Coordinate plane Standards 4-11 Expressions and Equations Variables and expressions Solve one-step equations Standards 12-20 Statistics Understanding different measures of center Standards 25-29 Geometry – Standards 21-24

  13. Recommend Emphasesfrom PARCC Model Content Framework for Mathematics

  14. Ratios and Proportional Relationships Cluster Analysis Tool

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

  16. 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.

  17. 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.

  18. Solving Proportions Solve 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?

  19. 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.

  20. 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. • 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. • Solve unit rate problems including those involving unit pricing and constant speed. • Find a percent of a quantity as a rate per 100; solve problems involving finding the whole, given a part and the percent. • Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.

  21. 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 9 9 9 9 9 9 9 Attempted 90 missed 3 9 9 9 90 ÷ 10 = 9 9 x 7 = 63 Omar made 63 free throws.

  22. 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?

  23. 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?

  24. 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?

  25. Geometry Unpacking Standards

  26. Unpacking the Standards “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 Step 1: Target a standard Step 2: Chunk the Main Categories Step 3: Identify all standard components Step 4: Identify the Developmental Progression Step 5: Identify Key Vocabulary Step 6: Add Clarifying Information

  27. 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.

  28. 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.

  29. 2.G.3 Partition circles and rectangles into two, three, or four equal shares The final product…. Describe Recognize 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 Partition Partition circles and rectangles into three equal shares, using the word thirds, third of 2.G.3 Partition circles and rectangles into two equal shares, using the word halves, half of 2.G.3 Fourths Fourth of Identical whole Builds on 1.G.3 Needed for 3.G.2 Thirds Third of whole partition rectangle Halves Half of Partition a shape into fourths in different ways Pattern Blocks Fraction Bars/Circles 2/2 = one whole Equal shares circle

  30. Step 2: Chunk the Main Categories Example 2.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. • All Standard(s) in the cluster(s) • Identify Key Verbs 2.G.3 Partition circles and rectangles into two, three, or four equal shares Describe Recognize Partition

  31. lt blue 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?

  32. Example 2.G.3 Partition circles and rectangles into two, three, or four equal shares Partition Describe Recognize Partition circles and rectangles into two equal shares, using the word halves, half of 2.G.3 Partition circles and rectangles into three equal shares, using the word thirds, third of 2.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

  33. 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?

  34. Vertical Alignment Using 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.

  35. 2.G.3 Partition circles and rectangles into two, three, or four equal shares Partition Describe Recognize Recognize that equal shares of identical wholes need not have the same shape 2.G.3 Partition circles and rectangles into two equal shares, using the word halves, half of 2.G.3 Partition circles and rectangles into three equal shares, using the word thirds, third of 2.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 Introduced? Mastered? Needed for? Builds on? Builds on 1.G.3 Needed for 3.G.2

  36. 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

  37. 2.G.3 Partition circles and rectangles into two, three, or four equal shares Describe Recognize Partition circles and rectangles into two equal shares, using the word halves, half of 2.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 Partition Partition circles and rectangles into three equal shares, using the word thirds, third of 2.G.3 Partition circles and rectangles into two equal shares, using the word halves, half of 2.G.3 Fourths Fourth of Identical whole Builds on 1.G.3 Needed for 3.G.2 Thirds Third of whole partition rectangle Halves Half of Equal shares circle

  38. Step 6: Add Clarifying Information • Kid-friendly language to add clarity • Clarifying pictures, words, or phrases • Definitions, examples • Symbols, formulas, pictures, etc. CAUTION: do not replace important vocabulary that is included in the standard. yellow

  39. 2.G.3 Partition circles and rectangles into two, three, or four equal shares Describe Recognize 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 Partition circles and rectangles into two equal shares, using the word halves, half of 2.G.3 Partition Partition circles and rectangles into three equal shares, using the word thirds, third of 2.G.3 Partition circles and rectangles into two equal shares, using the word halves, half of 2.G.3 Fourths Fourth of Identical whole Builds on 1.G.3 Needed for 3.G.2 Thirds Third of whole partition rectangle Halves Half of Partition a shape into fourths in different ways Pattern Blocks Fraction Bars/Circles 2/2 = one whole Equal shares circle

  40. Transfer Unwrapping to Chart

  41. 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

  42. 2.G.3 Partition circles and rectangles into two, three, or four equal shares Main Idea of Standard Key Verbs Describe Recognize 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 Partition circles and rectangles into two equal shares, using the word halves, half of 2.G.3 Partition • Take standard apart according to the verbs to separate skills within the standard. • Use all components of standard. • Put in a logical sequence Partition circles and rectangles into three equal shares, using the word thirds, third of 2.G.3 Partition circles and rectangles into two equal shares, using the word halves, half of 2.G.3 Vertical alignment Fourths Fourth of Identical whole Vocabulary Builds on 1.G.3 Needed for 3.G.2 Thirds Third of whole partition rectangle Halves Half of Partition a shape into fourths in different ways Pattern Blocks Fraction Bars/Circles Clarifying information, student-friendly 2/2 = one whole Equal shares circle

  43. The Number System Vertical Alignment

  44. 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

  45. 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

  46. 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

  47. 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

  48. 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

  49. Fluency Expectations