1. Kindergarten Math Presentation
2009 Mathematics Standards of Learning – Implementation Supported by Professional Development
Kindergarten Math Presentation
Session #1
February 2011
Parts taken from Michael Bolling’s (Mathematics Coordinator) – VSUP Presentation

2. VDOE K – 2 Professional Development Focus
October 21, 2010
VDOE K – 2 Professional Development Focus
Unpack Number and Number Sense
Fact Strategies
Inverse Relationships
Fractions
Equality
Problem Solving 2 2

3. A sample of the progression of fractions.
3.3 c) compare fractions with like/unlike denominators
4.2 a) compare and order fractions /mixed numbers
5.2 a) Recognize equivalent fractions/decimals.
B) compare and order fractions & decimals
K.5 Identify halves and fourths
New content
2.3 Identify/ write/compare halves, thirds, fourths, sixths, eighths, tenths
1.3 Identify/ write halves, thirds, fourths
New content
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

4. Number and Number Sense
October 21, 2010
Number and Number Sense
Standard: K.5
What do students have to do?
Look for verbs.
With what?
With what parameters? Which figures, numbers, shapes?
Vocabulary
Essential Understandings
Sharing
Halves
Fourths
Parts
Whole
Set
Region
Equal parts
Equal size
Equal area
Recognize
Parts of a whole
(sets and regions)
Given a region – id
Given a set - id
Identify fractions
Identify equal parts
Understand fraction concept (equal parts – sharing)
Sets
Regions
Concrete
Halves
Fourths 4 4

5. Equal Parts - SORTING
Equal Parts
Not equal

7. Recognize that fractions represent parts of equal size of a whole
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

8. start taking about fair share.
Grade 1:
start taking about fair share.
Write the fraction

9. Why is this not cut into equal parts?
How many equal parts do you see?
Which is cut into fourths?
Why is this not cut into equal parts?

10. Halves and Fourths of Sets
Worksheets – circle one half, circle one fourth
If no conceptual understanding – they can’t do it

11. Halves and Fourths of Sets
Count students in halves and fourths as often possible and let the students use the vocabulary words:
halves and fourths
Make sure they know the parts created are subsets of the whole.

12. Halves of a Set

13. Halves of a Set 6 6

14. Halves of a Set
Now they know to circle 6 triangles

15. Fourths of a Set 3 3 3 3

16. (beyond the unit fractions)
Fourths of a Set
(beyond the unit fractions) 3 3
Ask them to show you one-fourth, two-fourths, three-fourths, even four-fourths. 3 3

17. Equality and Properties – preparation for justifications
5.19 distributive property of multiplication over addition
New from grade 7
4.16 b) associative property for add/mult
Newfrom grade 7
3.20 a)
identity/
commutative
properties
for add/mult
1.18 demonstrate equality using equal signs
New from grade 3
2.22 demonstrate an understanding of equality using = and ≠
New content
6.19 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

18. Equations and Inequalities
Vertical Alignment
Modeling one step linear equalities
What does the equal sign mean?

19. Equality Where are we headed? 5 + 3 =
October 21, 2010
Equality
Where are we headed?
5 + 3 =
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, ,
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.

20. SOL 1.18 demonstrate equality using an equal sign
SOL 2.22 demonstrate understanding of equality and not equal signs

21. Equalities SOL 2.22 and SOL 3.20

22. Inequalities 2009 SOL 3.20 (C.F. - Essential Understanding )
Using the Commutative property, we can allow students more opportunities to explore equality. Representing this property in a variety of forms will provide them more opportunities to connect the characteristics of these properties to future mathematical endeavors. 2 3 4

23. Equalities 2009 SOL 3.20 (C.F. - Essential Understanding )
Commutative Property of addition
Using the Commutative property, we can allow students more opportunities to explore equality. Representing this property in a variety of forms will provide them more opportunities to connect the characteristics of these properties to future mathematical endeavors. 3 4

24. Equalities/Properties 2009 SOL 4.16a
K.6
Bag of 10 dual color chips (beans), spill on the table, and sort the two colors. Count and write the number of each color. Continue until all the facts have been created. Will see the inverse when you have all 10 fact.
Will build a foundation for 3+7 = 7+3 and = 4 +6
The finished list will show all the fact families for adding to the sum of 10
Red Blue
Identity Property of Addition
8 + 0 = 8
Commutative Property of Addition
4 + 3 = 3 + 4
Identity Property of Multiplication
8 x 1 = 8
Commutative Property of Multiplication
2 x 5 = 5 x 2
Show the blue number balances…participants write on the smart pals….Robin makes a model, Kim make a model…Who can come up with a third?

25. What will the students say?
Equalities SOL 4.16a recognize/demonstrate meaning (thinking) of equality in an equation.
What will the students say?
True or False?
8 = 1 + 7
2 + 3 = 2 x 3
3 + 5 = 5 + 3
Will students agree that 8 = 1 + 7? Many times they think that it has to say = 8. Use the balance to prove that they are equal. Continue with the other equations. Discuss commutative property? How many different ways can you show 9 = 9? Can you show ?
7 x 4 =
9 = 9

26. Linear Equations 2009 SOL 5.18 a-bP Y
Investigate and describe concept of a variable.
Write open sentences using a variable W Z
Again, this is an equation using multiplication. This would NOT be tested in fifth grade, but could be used to differentiate for those students who are ready for that discussion. What is the value for B here? How might you write this equation? 26 26

27. Modeling One-step Linear Equations 2009 SOL 5.18c
Using a cup and candy corn, construct a model for
J = 6
Again, this is an equation using multiplication. This would NOT be tested in fifth grade, but could be used to differentiate for those students who are ready for that discussion. What is the value for B here? How might you write this equation?

28. Modeling One-step Linear Equations 2009 SOL 5.18c
How many to balance?
Students can discover what the unknown variable is here without having to write an equation.

29. Modeling One-step Linear Equations 2009 SOL 5.18c
Using your cups and candy corn, construct a model for
J + 4 = 7
Again, this is an equation using multiplication. This would NOT be tested in fifth grade, but could be used to differentiate for those students who are ready for that discussion. What is the value for B here? How might you write this equation?

30. Modeling One-step Linear Equations 2009 SOL 5.18c
Now you try. How might you write this equation? What is the value for J here?

31. What equation is modeled below?
How might you write this equation? What is the value for B here?

32. 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

33. K.6 Model add/sub of whole numbers up to 10
Bears in a cave
Partners
Count out a total number of bears 10 or below
Partner 1 turns their back.
Partner 2 hides some bears under the cup.
Partner 1 turns back around and determines the number of bears missing

34. 7.13 evaluate algebraic expressions
Expressions and Operations
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
7.13 evaluate algebraic expressions
6.8 Order of Operations
no { }, | |
Only ( )
5.7 Order of Operations
7.3 operations with integers
New from grade 7
New from grade 7
New content including (modeling)

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

36. Mean as Fair Share
Average: ( ) / 3 items = 7 10 8 3

37. Mean as Fair Share
Average: ( ) / 3 items = 7 7 7 7

38. Mean as Balance Point 3
Sigma ( xi – u ) 7 38 38

39. Statistics in Algebra One
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. 39

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

41. Assessing Higher-level Thinking Skills
3.9 The student will estimate…area and perimeter.

42. Higher Order Thinking Skills
October 21, 2010
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.
the use of two or more
operations; and operations can be different. 42 42

43. Modeling to solve word problems
October 21, 2010
Build skills to solve multi-step problems
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?
Zach had 64 ounces of soda. He poured 8 ounces into each of 5 glasses. How much soda was left over?
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? 43 43

44. October 21, 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.
Let the students struggle, take a risk at getting it wrong, explain why, re-think, re-do!
Check out this site: 44

45. 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 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.
5.5 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?

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

47. Assessing Higher-level Thinking Skills
3rd
Order of Operations
5.7
2nd
1st
6.8
, given x = -2
7.13b
evaluate

48. We Must Provide…. 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”, “ the what if”

49. 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
- 49 - 49

50. 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
- 50 -

51. They serve a vital and critical role in
PIVOTAL QUESTIONS
The following questions are aligned with the Standards of Learning.
They serve a vital and critical role in
unveiling student understanding and misconceptions in ways that knowledge-recall questions do not allow.

52. PIVOTAL QUESTIONS 1. 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)

53. 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.

54. Incorporate Good Mathematical Questioning
• 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?

55. 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?

56. 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 1
40 red
20 blue 2
100 red
25 blue 3

58. Resources
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
Vocabulary
Vertical Articulation Documents – handouts