2009 Mathematics Standards of Learning – Implementation Supported by Professional Development Second Grade Math Presentation Session #1 February Mathematics Standards of Learning – Implementation Supported by Professional Development Second Grade Math Presentation Session #1 February Parts taken from Michael Bolling’s (Mathematics Coordinator) – VSUP Presentation

December 9, Identify/ write/compare halves, thirds, fourths, sixths, eighths, tenths A sample of the progression of fractions. K.5 Identify halves and fourths New content 1.3 Identify/ write halves, thirds, fourths New content 5.2 a) Recognize equivalent fractions/decimals. B) compare and order fractions & decimals 3.3 c) compare fractions with like/unlike denominators 4.2 a) compare and order fractions /mixed numbers 6.2 a) compare/order fractions, decimals, and % 6.4 model multiplication and division of fractions 7.1 c) Compare and order fractions, decimals, percents, and scientific notation

December 9, Gr 2 - New Content Changes Leave outWhat’s New? Rectangular prism (was solid) Grouping objects by threes and fours 2.2b Write the ordinal numbers Id parts of sets/regions – halves, fourths, sixths, eighths, and tenths (not just the unit fractions) 2.8 Create and solve one and two step addition and subtraction problems Estimate and measure liquid volume ( was compare) Estimate and measure weight/mass in pounds, ounces, kilograms, and grams New Vocab. Draw line of symmetry Tell and write time to nearest five minutes (was quarter hour) Predict outcomes when an experiment is repeated Analyze data displayed in picture graphs, pictographs, and bar graphs Demonstrate an understanding of equality using equal signs and non- equal signs Measure length to determine perimeter of polygon Count square units/ cubes to determine area/volume Square pyramid Cylinder Cone Sorting 3D shapes Compare Unit Fractions (1/2, 1/3, ¼, 1/6, 1/8, 1/10)

December 9, 2010 K.5 Identify the parts of a set and/or a region that represents halves and fourths. Recognize that fractions represent parts of equal size of a whole

December 9, 2010 Grade 1: start taking about fair share. Write the fraction

December 9, K-1 Fractions of Sets Count students in halves, fourths, and thirds as often possible and let the students use the vocabulary words: halves – fourths - thirds Make sure they know the parts created are subsets of the whole.

December 9, 2010 Why is this not cut into equal parts? 7 How many equal parts do you see? Which is cut into fourths?

December 9, 2010 K – Halves and Fourths of a Set

December 9, Ask them to show you one-fourth, two-fourths, three-fourths, even four-fourths. K - Fourths of a Set (beyond the unit fractions) K - Fourths of a Set (beyond the unit fractions)

December 9, st grade adds - Thirds of a Set 4 4 4

December 9, 2010 Thirds ( 1 st grade) out of 12 New: Write the fraction

December 9, /6 th as FAIR SHARE K (halves and fourths) + 1 st (adds thirds) 2nd grade adds 1/6 th, 1/8 th, and 1/10 th + (4 th – add 12ths)

December 9, 2010 K (halves and fourths) + 1 st (adds thirds) Second grade adds sixths, eighths, and tenths + (4 th – add 12ths) One sixth One eighth

December 9, 2010 K (halves and fourths) + 1 st (adds thirds) Second grade adds sixths, eighths, and tenths. 2/6 2/8

December 9, 2010 K (halves and fourths) + 1 st (adds thirds) Second grade adds sixths, eighths, and tenths. Worksheets – If no conceptual understanding – they can’t do it One sixth One eighth

December 9, a,b - Identify parts of sets and/or regions that represent halves, thirds, fourths, sixths, eighths, and tenths. Write the fractions ( and not just unit fractions) Which model represents 2/3 of a set?

December 9, 2010 Gr 1 - Unit Fractions (1/2, 1/3, ¼,) Help them understand the size relationship between ¼, 1/3, and ½ of a given whole. Talk about: Which is greater? Which is less?

December 9, c - Compare Unit Fractions (1/2, 1/3, ¼, 1/6, 1/8, 1/10)

December 9, 2010 Assessing Higher-level Thinking Skills b) The student will represent probability as a number between 0 and 1, inclusive. Jennifer has 12 marbles. 1 Blue 3 Red 8 Green Where on the number line would you place an arrow to show the probability of choosing a green marble? 8/12 2/3

December 9, a) identity/ commutative properties for add/mult Equality and Properties – preparation for justifications 1.18 demonstrate equality using equal signs New from grade demonstrate an understanding of equality using = and ≠ New content 4.16 b) associative property for add/mult Newfrom grade distributive property of multiplication over addition New from grade a-c) investigate and identify property of +/X, multiplicative property of zero, inverse property for multiplication 7.16 a-e) apply properties with real numbers, comm/associative property of +/X, distributive, +/X identity, +/X inverse, X property of 0 Leading into students giving justifications to steps when solving equations and inequalities in MS and HS

December 9, 2010 Equations and Inequalities What does the equal sign mean?

December 9, 2010 EqualityEquality Connected to N&NS SOL 2.1c 1.18 The student will demonstrate an understanding of equality through the use of the equal sign The student will demonstrate an understanding of equality by recognizing that the symbol = in an equation indicates equivalent quantities and the symbol ≠ indicates that quantities are not equivalent. Connected to N&NS SOL 2.1c 1.18 The student will demonstrate an understanding of equality through the use of the equal sign The student will demonstrate an understanding of equality by recognizing that the symbol = in an equation indicates equivalent quantities and the symbol ≠ indicates that quantities are not equivalent = = AND THE ANSWER IS…….? Now students should think about options to balance the equation on the right side. List 5 options that would make the sentence balance? 8, 10-2, 1+7, 8, 10-2, 1+7, , Where are we headed?

December 9, 2010 SOL 1.18 demonstrate equality using an equal sign SOL 2.22 demonstrate understanding of equality and not equal signs Equal Sign = Not Equal Sign =

December 9, Inequalities 2009 SOL 3.20 (C.F. - Essential Understanding )

December 9, Gr 3 - Commutative Property of addition Equalities 2009 SOL 3.20 (C.F. - Essential Understanding ) Grade 1

December 9, Equalities SOL 2.22 and SOL 3.20 (use to prove properties) Equalities SOL 2.22 and SOL 3.20 (use to prove properties)

December 9, 2010 Equalities SOL 4.16a recognize/demonstrate meaning (thinking) of equality in an equation. 8 = = = 2 x 3 True or False? 7 x 4 = What will the students say? 27 How many different ways can you show 9 = 9?

December 9, Modeling One-step Linear Equations 2009 SOL 5.18c Using a cup and candy corn, construct a model for J = 6

December 9, Modeling One-step Linear Equations 2009 SOL 5.18c How many to balance ?

December 9, Modeling One-step Linear Equations 2009 SOL 5.18c Using your cups and candy corn, construct a model for J + 4 = 7

December 9, Modeling One-step Linear Equations 2009 SOL 5.18c J = 3 pieces of candy

December 9, What equation is modeled below? B + 2 = 9

December 9, 2010 Assessing Higher-level Thinking Skills 5.8 c) The student will model one-step linear equations in one variable, using addition and subtraction. = x = 1

December 9, 2010 How many apples would you need to replace the barrel ? Remember you must keep the equation equal Apples

December 9, Order of Operations 6.8 Order of Operations no { }, | | Only ( ) 7.13 evaluate algebraic expressions 7.3 operations with integers 8.1 simplify numerical expressions involving positive exponents, using rational numbers, order of operations, and properties to justify Alg1.1 represent verbal quantitative situations algebraically/evaluate expressions for given replacement values of variables New from grade 7 Expressions and Operations New content including (modeling)

December 9, Mean as Fair Share 6.15 Mean as Balance Point New content Statistics Alg1.9 Standard Deviation Alg1.9 – Standard deviation, mean absolute deviation, variance, dispersion, z-scores New content Alg2.11 Normal Distributions New content

December 9, 2010 Mean as Fair Share Average: ( ) / 3 items =7

December 9, 2010 Mean as Fair Share 777 Average: ( ) / 3 items =7

December 9, 2010 Mean as Balance Point It’s all about the total distance away from the “mean/average” Helps to create a foundation to understand “absolute value”

December 9, 2010 Statistics in Algebra One 40 How can you help? Help students become comfortable in collecting, displaying, and analyzing data. They should also be able to make logical predictions from the data.

December 9, 2010 The 2009 SOL and the new SOL Assessments 41 Increased rigorIncreased rigor Higher-level questionsHigher-level questions Technology enhanced itemsTechnology enhanced items Increased rigorIncreased rigor Higher-level questionsHigher-level questions Technology enhanced itemsTechnology enhanced items y The point U( - 6, - 3) is translated 3 units right. What are the coordinates of the resulting point, U′?

December 9, 2010 Assessing Higher-level Thinking Skills The student will estimate…area and perimeter. Don’t have to call it area but they could count the number of squares. Build and name two geometric shapes using 18 squares.

December 9, 2010 Higher Order Thinking Skills Connected to N&NS SOL The student will create and solve one-step story and picture problems using basic addition facts with sums or less and the corresponding subtraction facts. 2.8 The student will create and solve one- and two-step addition and subtraction problems, using data from simple tables, picture graphs, and bar graphs 3.4 The student will estimate solutions to and solve single-step and multistep problems involving the sum and difference of two whole numbers, each 9,999 or less, with or without regrouping. 43 the use of two or more operations; and operations can be different.

December 9, 2010 Emily is reading the latest Magic Maggie book. She reads 12 pages each day. After 7 days, Emily still has 20 pages left to read. How many pages are in Emily's book? 44 Zach had 64 ounces of soda. He poured 8 ounces into each of 5 glasses. How much soda was left over? Modeling to solve word problems Tamara had 3 pennies. She got 5 pennies for cleaning her room. Then she lost 2 pennies. How many pennies does she now have? Build skills to solve multi-step problems

December 9, 2010 Students need opportunities to solve various problem types through modeling, reasoning, and reflection to strengthen their mathematics understandings and use of concepts and skills. Students need opportunities to solve various problem types through modeling, reasoning, and reflection to strengthen their mathematics understandings and use of concepts and skills. Let the students struggle, take a risk at getting it wrong, explain why, re-think, re-do! Check out this site:

December 9, 2010 Assessing Higher-level Thinking Skills 5.4 The student will create and solve single-step and multistep practical problems involving addition, subtraction, multiplication, and division with and without remainders of whole numbers. 5.5 The student will a) find the sum, difference, product, and quotient of two numbers expressed as decimals through thousandths (divisors with only one nonzero digit); and a) find the sum, difference, product, and quotient of two numbers expressed as decimals through thousandths (divisors with only one nonzero digit); and b) create and solve single-step and multistep practical problems involving decimals. b) create and solve single-step and multistep practical problems involving decimals. 5.6 The student will solve single-step and multistep practical problems involving addition and subtraction with fractions and mixed numbers and express answers in simplest form. 5.4 The student will create and solve single-step and multistep practical problems involving addition, subtraction, multiplication, and division with and without remainders of whole numbers. 5.5 The student will a) find the sum, difference, product, and quotient of two numbers expressed as decimals through thousandths (divisors with only one nonzero digit); and a) find the sum, difference, product, and quotient of two numbers expressed as decimals through thousandths (divisors with only one nonzero digit); and b) create and solve single-step and multistep practical problems involving decimals. b) create and solve single-step and multistep practical problems involving decimals. 5.6 The student will solve single-step and multistep practical problems involving addition and subtraction with fractions and mixed numbers and express answers in simplest form b) Michael jogged 3.4 miles each day for 3 days. Jennifer jogged 4.2 miles each day for the same 3 days. What is the difference between the number of miles Jennifer jogged and the number of miles Michael jogged on these 3 days?

December 9, 2010 Assessing Higher-level Thinking Skills 47 Order of Operations evaluate, given x = b 1 st 2 nd 3 rd

December 9, 2010 A focus on content plus….  a balance between conceptual and procedural approaches.  include relevant and real world applications.  give students intentional vertical connections to other grade level content and practices.  reflection time – to answer “the why”, “what if”! A focus on content plus….  a balance between conceptual and procedural approaches.  include relevant and real world applications.  give students intentional vertical connections to other grade level content and practices.  reflection time – to answer “the why”, “what if”! 48 We Must Provide….

December 9, Technology Enhanced Items (TEI) Format of Questions: Fill in the blank Click and drag Hot-spots: Select one or more answer options, placing points on coordinate planes Creation of graphs Approximately ten practice questions for each mathematics test, Grades 3-8 and EOC addressing – February 2011 increased rigor for existing SOL items that address new SOL technology enhanced items

December 9, Mathematics Standards of Learning Implementation Timeline 2010 – 2011 Teach old and new SOL content Field Test items on new 2009 SOL – live test items on 2001 standards Grade 3 live test is still cumulative but field test items on new content is only from grade 3 content New 2009 SOL taught and fully assessed New Grade 3 assessment covers 2009 grade 3 content only Gr. 3-5 technology enhanced items are live spring 2013

December 9, Can be solved or explained in a variety of ways 2. Focus on conceptual aspects of mathematics 3. Have the potential to expose student understanding and misconceptions 5. Lend themselves to a scoring rubric (see the rubric included) PIVOTAL QUESTIONS They serve a vital and critical role in unveiling student understanding and misconceptions in ways that knowledge- recall questions do not allow.

December 9, Try to make some simple shifts in what you expect from students. That means….asking it differently! Here are some examples of how you might adjust a few typical elementary concepts.

December 9, How did you arrive at that answer? Why do you think that? What have you discovered? Have you thought of another way this could be done? Does that make sense? Does that always work? How could we prove that? Have we solved a problem similar to this one? Is that the only possible answer? Is your solution reasonable? Is there a real-life situation where this could be used? Where else would this strategy be useful? Do you see a pattern? Is there a general rule? What other questions does this bring up? What is the math in this problem? Have you tried making a guess? Would another recording method works as well or better? Give me another related problem. Is there another way to draw or explain that? How did you organize your information? Would it help to draw a picture? Incorporate Good Mathematical Questioning

December 9, Try to make some simple shifts in what you expect from students. That means….asking it differently! Find a rectangle in the classroom. What shape are the student desks? Instead ask: How do you know the chalk board is a rectangle? How do you know the student desks are not a square?

December 9, Try to make some simple shifts in what you expect from students. That means….asking it differently! What is the probability of drawing a red marble from bag one? Instead ask: If you close your eyes, reach into a bag, and remove 1 marble, which bag would give you a better chance of picking a blue marble? How could we prove that? Is there a real-life situation where this could be used? 75 red 25 blue 40 red 20 blue 100 red 25 blue 1 2 3

December 9,

December 9, 2010 ResourcesResources 57 Blueprints are currently available – effective in Formula sheets for 6-8 and EOC are currently available – effective Curriculum Framework – New Enhanced Scope and Sequence – coming soon : summer 2011 Will include differentiation strategies for all learners. Math Resource page Resource page Vocabulary Vertical Articulation Documents – handoutsVertical Articulation Documents – handouts Blueprints are currently available – effective in Formula sheets for 6-8 and EOC are currently available – effective Curriculum Framework – New Enhanced Scope and Sequence – coming soon : summer 2011 Will include differentiation strategies for all learners. Math Resource page Resource page Vocabulary Vertical Articulation Documents – handoutsVertical Articulation Documents – handouts