2009 mathematics standards of learning – implementation supported by professional development
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2009 Mathematics Standards of Learning – Implementation Supported by Professional Development Third Grade Math Presentation Session #1 February 2011. Parts taken from Michael Bolling’s (Mathematics Coordinator) – VSUP Presentation. - PowerPoint PPT PresentationTRANSCRIPT
2009 Mathematics Standards of Learning – Implementation Supported by Professional
Development
Third Grade Math PresentationSession #1
February 2011
1Parts taken from Michael Bolling’s (Mathematics Coordinator) – VSUP Presentation
December 9, 2010
The 2009 SOL and the new SOL Assessments
2
• Increased rigor
• Higher-level questions
• Technology 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
3
Gr 3 - New Content Changes
Moved out of your grade to..
What’s New?
Angles
Vertices
Points
Lines
rays
Id examples of the
identity and commutative properties for addition
and multiplicati
on
Compare fractions using WORDS and SYMBOLS
(greater than, less than, equal to)
ESTIMATE and solve up to multistep problems
Estimate and measure length to
the nearest ½ inch (was inch)
New Vocab.Tell time to
nearest minute
Add/sub proper fractions with
like denominators of 12 or less
(was 10 or less)
Estimate and measure area and perimeter
Determine elapsed time
in one-hr. increments
over 12 hour periodRecall multiplication facts through
12’s (was 9’s)
Moved to grade 1 & 2
Demonstrate understanding of
equality sign
Moved to grade 4
Reading, writing, adding, subtracting
decimals
MODEL fractions including
mixed and WRITE
NUMBERS (was divide regions and
sets – moved down)
Represent multi. and div. using area, set, and NUMBER LINE MODELS
Count square to determine
area
December 9, 2010 4
2.3 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
numbers6.2 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, 2010
Grade 1:start taking about fair
share.
Write the fraction
K.5Identify 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
Why is this not cut into equal
parts?6
How many equal parts do you see?
Which is cut into fourths?
December 9, 2010
( K – halves and fourths1st grade - Thirds )
4
4
4
13
13
13
4 out of 12
New: 1st - Write the
fraction
Model one-third with 12 triangles?
December 9, 2010
2.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
FAIR SHARE
2
2
2
2
2
Third grade adds 1/12ths(previously students will learn ½, ¼, 1/3, 1/8, 1/10) 2
2
2
December 9, 2010
FAIR SHARE
2
2
2
2
2
Third grade adds 1/12ths(previously students will learn ½, ¼, 1/3, 1/8, 1/10) 2
2
2
December 9, 2010
Grades 1 & 2 – Unit Fractions (1/2, 1/3, ¼, 1/6, 1/8, 1/10)
13
1
0
14
12
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, 2010
3.3c – Compare Fractions using >, <, or = signs)(1/2, 1/3, ¼, 1/8, 1/10, 1/12)
13
12
14
10
18
110
4th grade will order the unit fractions (Number line)
112
December 9, 2010
Assessing Higher-level Thinking Skills
13
4.13 b) The student will represent probability as a number between 0 and 1, inclusive.
Jennifer has 12 marbles.
1 Blue3 Red8 Green
Where on the number line would you place an arrow to show the probability of choosing a green marble?
8/122/3
December 9, 2010 14
3.20 a) identity/
commutative properties
for add/mult
Equality and Properties – preparation for justifications
1.18 demonstrate
equality using equal
signs
New from grade 3
2.22 demonstrate
an understanding
of equality using = and ≠
New content
4.16 b) associative property for
add/mult
Newfrom grade 7
5.19 distributive property of
multiplication over addition
New from grade 76.19 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 0Leading 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
Equality
Connected to N&NS SOL 2.1c1.18 The student will demonstrate an understanding of equality through the use of the equal sign.
2.22 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.
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, 5 + 3 2+5+1, 3+10-5
16
Where are we headed?
December 9, 2010
SOL 1.18 demonstrate equality using an equal sign
http://illuminations.nctm.org/ActivityDetail.aspx?id=33
SOL 2.22 demonstrate understanding of equality and not equal signs
Equal Sign=
Not Equal Sign=
December 9, 2010 18
Inequalities 2009 SOL 3.20 (C.F. - Essential Understanding )
2 34 4
December 9, 2010 19
3 34 4
Equalities 2009 SOL 3.20 (C.F. - Essential Understanding )
Grade 1
The order with which you add the numbers
doesn’t change
anything. Both sides are
still equal.
Commutative
Property of
addition
December 9, 2010
Identity and Commutative Property of addition/multiplicationHow would you show this with a
balance scale? 2X3 = 3X2
• Add 2 groups of three on one side. • Add three groups of two on the
other side.
Both sides will be equal
December 9, 2010 21
EqualitiesSOL 2.22 and
SOL 3.20 (use to prove properties)
http://illuminations.nctm.org/ActivityDetail.aspx?id=26
December 9, 2010
Equalities/Properties2009 SOL 3.20
22
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
December 9, 2010
Equalities -2009 SOL 4.16a recognize/demonstrate meaning (thinking) of equality in an equation.
8 = 1 + 7
3 + 5 = 5 + 32 + 3 = 2 x 3
True or False?
7 x 4 = 4 + 4 + 4 + 4
What will the students say?
23
How many different ways can you show
9 = 9?
December 9, 2010 24
Modeling One-step Linear Equations2009 SOL 5.18c
Using a cup and candy corn, construct a model for
J = 6
December 9, 2010 25
Modeling One-step Linear Equations2009 SOL 5.18cHow
many to
balance?
December 9, 2010 26
Modeling One-step Linear Equations2009 SOL 5.18c
Using your cups and candy corn, construct a model for
J + 4 = 7
December 9, 2010 27
Modeling One-step Linear Equations2009 SOL 5.18c
J = 3
pieces of candy
December 9, 2010 28
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.
3 5x = x= 1
December 9, 2010
We can all continue concept of variable (Previous Grades)
?
December 9, 2010
How many apples would you need to replace the barrel ?Remember you must keep the equation equal.
31
3 Apples
December 9, 2010 32
5.7 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
New from grade 7
Expressions and Operations
New content including
(modeling)
December 9, 2010
Assessing Higher-level Thinking Skills
33
Order of Operations
40 16 2 (1 3)
340 2 2 (1 3)
3 3x x evaluate , given x = -2
5.7
6.8
7.13b
1st2nd
3rd
December 9, 2010
Assessing Higher-level Thinking Skills
34
6.20 The student will graph inequalities on a number line.
4
4
x
x
4
4
Students will need a solid conceptual
understanding of inequalities before going
to Middle School
December 9, 2010 35
5.16 Mean as
Fair Share
6.15 Mean as Balance
Point
New content 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
108
3
Average: (10 + 8 + 3) / 3 items = 7
December 9, 2010
Mean as Fair Share
7 7 7
Average: (10 + 8 + 3) / 3 items = 7
December 9, 2010
Mean as Balance Point
38
7
3 8 10
It’s all about the total distance away from the
“mean/average”
Helps to create a foundation to understand
“absolute value”
December 9, 2010
SOL 3.17 – Analyze and interpret information with up to 30 data points and up to 8 categories, by
writing one sentence.
• collect and organize data• Recollect and compare data• observe, measure, surveys,
experiments• Construct line plots, bar graphs and
picture graphs to represent the data• Read and Interpret the data in these
graphs
December 9, 2010
Statistics in Algebra One
Collect data, display and analyze data, understand the behavior of data sets, understand how data is spread about the mean, how is this used to inform decisions
40http://www.mathwire.com/
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
Ages of parents and grandparents
Talk about :
• how data is sometimes grouped in clusters
• how sometimes there are data points that lie far from the group of data.
Discuss what conclusions can be made?
December 9, 2010
Assessing Higher-level Thinking Skills
42
3.9d The student will estimate…area and perimeter.
Count the number of squares.Build and name geometric shapes
using 24 squares.
December 9, 2010
Higher Order Thinking Skills Connected to N&NS SOL 3.21.6 The student will create and solve one-step story and picture problems using basic addition facts with sums 10 18 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 solutionsto 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 moreoperations; 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 toread. 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. Let the students struggle, take a risk at getting it
wrong, explain why, re-think, re-do!
Check out this site:
http://www.mathwire.com/problemsolving/probsk12.html#k12number
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
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.
46
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?
December 9, 2010
Assessing Higher-level Thinking Skills
47
7.5 c) The student will describe how changing one measured attribute of a rectangular prism affects its volume and surface area.
Describe how the volume of the rectangular prism shown (height = 8 in.) would be affected if the height was increased by a scale factor of ½ or 2. V = h X 3 X 5
8 in.
5 in.
3 in.The volume would be cut in half or doubled accordingly.
V = (8)(15) - originalV = (4)(15) – height is halfV = (16)(15) – height is double
SA = 2(l*w)+ 2(w*h) + 2(l*h)
How you can help.
Necessary Background:
Give the students word problems to
solve. Then ask them what would
happen if one variable changed.
Example: If you ran 3 minutes each day at recess
for a total of 5 days. How many minutes would you
have run for the week?
Next: Ask how many total minutes would you have
run if on Tuesday you ran more than usual and ran
8 minutes.
December 9, 2010
Note: Blueprints Changes
- 48 -
• Some Reporting Categories Combined
• Watch the growing emphasis on the Statistics, Patterns, Functions, and Algebra Reporting Category shown on the next slides
December 9, 2010
December 9, 2010 50
December 9, 2010 51
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”!
52
We Must Provide….
December 9, 2010 - 53 -
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, 2010 - 54 -
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
2011-2012
• New 2009 SOL taught and fully assessed• New Grade 3 assessment covers 2009 grade 3 content only•
2012-2013• Gr. 3-5 technology enhanced items are live spring 2013
December 9, 2010 55
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)
PIVOTAL QUESTIONSThey serve a vital and critical role inunveiling student understanding and
misconceptions in ways that knowledge-recall questions do not allow.
December 9, 2010 56
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, 2010 57
• 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, 2010 58
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, 2010 59
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 red25 blue
40 red20 blue
100 red25 blue
1
2
3
December 9, 2010 60
December 9, 2010
Resources
61
• Blueprints are currently available – effective in 2011-2012 http://www.doe.virginia.gov/testing/sol/standards_docs/mathematics/review.shtml
• Formula sheets for 6-8 and EOC are currently available – effective 2011-2012 http://www.doe.virginia.gov/testing/test_administration/ancilliary_materials/2011-12/index.shtml
• Curriculum Framework – http://www.doe.virginia.gov/testing/sol/standards_docs/mathematics/review.shtml
• New Enhanced Scope and Sequence – coming soon : summer 2011 Will include differentiation strategies for all learners.
• Math Resource page http://www.doe.virginia.gov/instruction/mathematics/high/index.shtml
• Vocabulary http://www.doe.virginia.gov/testing/sol/standards_docs/mathematics/vocabulary/index.shtml
• Vertical Articulation Documents – handouts