2009 Mathematics Standards of Learning – Implementation Supported by Professional

Development

4th – 5th 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 5 - New Content Changes

Equilateral

Scalene

Isosceles

Volume

5.4 – Create and solve single step and multistep practical

problems for addition, subtractions, multiplication and. INCLUDE DIVISION with and

without remainders.

5.8a – Find perimeter, areas and VOLUME in standard units

Measure and classify angles (added straight angles)

Classify triangles - add - Equilateral, Scalene, and Isosceles Triangle

COMPARE and ORDER fractions

and decimals

5.16 – Describe mean as Fair Share (M, M, & M as measures of

center

New Emphasized

Vocab.

Model one-step linear equations in one variable using

addition and subtraction.

What’s new/ in focus?Identify and

describe prime and composite

numbers

Identify and describe the

characteristics of even and odd

numbers

Division with decimals –

computation only

Estimate and measure length,

weight/mass, area, and liquid

volume

Measure and draw angles and triangles.

Line graphs

Stem and Leaf plots 5.5b – Create and solve single

step and multistep practical problems involving decimals.

5.7 – Evaluate whole number expressions using order of

operations limited to parentheses, +, -, X and /

5.19 – Investigate and recognize distributive property of addition

over multiplication

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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

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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?

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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

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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

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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?

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3.3c – Compare Fractions using >, <, or = signs)(1/2, 1/3, ¼, 1/8, 1/10, 1/12)

13

12

14

10

18

110

4.2 Compare and order fractions/mixed numbers (Number line will be important!)

112

December 9, 2010

4th Grade - Comparing fractions to benchmarks

Gr. 3 will add and subtract

fractions with denominators of 12

or less. They will also learn X

facts through 12’s (was 9’s)

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Assessing Higher-level Thinking Skills

11

3.7 The student will add and subtract proper fractions having denominators of 12 or less.

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Assessing Higher-level Thinking Skills

12

4.3 d) The student will, given a model, write the decimal and fraction equivalents.

0.2

or

4.2 equivalent fractions

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Assessing Higher-level Thinking Skills

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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

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The Number line!

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Assessing Higher-level Thinking Skills

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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 based

upon the NUMBER

LINE!

December 9, 2010

Assessing Higher-level Thinking Skills

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8.5 b) The student will find the two consecutive whole numbers between which a square root lies.

Between which two square roots does 5 lie?

Between which two whole numbers does lie?Between 5 and 6 Between

SQR(16) and SQR (36)

SQR (16) (25) (36) (49)

4 5 6 7

SQR (16) (25) (36) (49)

4 5 6 7

How you can help.Necessary Background:

Students will need a thorough understanding and mastery of the number line and perfect squares.

(also numbers, fractions, and decimals)

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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

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Equations and Inequalities

What does the equal sign mean?

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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

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Where are we headed?

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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?

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How many different ways can you show

9 = 9?

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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=

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

2 34 4

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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

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3 34 4

Equalities - 2009

Gr 5 – Distributive Property of Multiplication over Addition

2 X ( 3 + 1 ) ( 2 X 3 ) + (2 X 1)

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Equalities (use to prove properties)

http://illuminations.nctm.org/ActivityDetail.aspx?id=26

December 9, 2010

Equalities/Properties2009 SOL

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Identity Property of Addition/Multiplication

8 + 0 = 8 8 x 1 = 8

Commutative Property of Addition/Multiplication

4 + 3 = 3 + 4 2 x 5 = 5 x 2

Gr. 4 – Associative property of addition and multiplication

Gr. 5 Distributive Property of

multiplication over addition

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Modeling One-step Linear Equations2009 SOL 5.18c

Using a cup and candy corn, construct a model for

J = 6

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Modeling One-step Linear Equations2009 SOL 5.18c

How many

pieces are

represented by

the jack-o-

lantern?

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Modeling One-step Linear Equations2009 SOL 5.18c

Using your cups and candy corn, construct a model for

J + 4 = 7

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Modeling One-step Linear Equations2009 SOL 5.18c

J = 3

pieces of candy

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What equation is modeled below?

B + 2 = 9

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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

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We can all continue concept of variable (Previous Grades)

?

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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)

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Assessing Higher-level Thinking Skills

37

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

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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

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Mean as Fair Share

108

3

Average: (10 + 8 + 3) / 3 items = 7

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Mean as Fair Share

7 7 7

Average: (10 + 8 + 3) / 3 items = 7

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Mean as Balance Point

41

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 4.14 – Collect, organize, display, interpret data from a variety of graphs

• 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

43http://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?

M T W T F Sat Sun

Number of students ordering lunch

Ask extended questions:

• Define the days where the number of students ordering lunch is increasing.

• Which days is the number of students ordering lunch decreasing?

• Where is the data not increasing?• Where is the data not decreasing?

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Assessing Higher-level Thinking Skills

45

3.6 The student will represent multiplication and division, using area, set, and number line models…

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Assessing Higher-level Thinking Skills

46

3.9d The student will estimate…area and perimeter.

The curriculum framework says the student will also measure to

determine area and perimeter.

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.

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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?

48

Grade 3Zach had 64 ounces of soda.

He poured 8 ounces into each of 5 glasses. How much soda was

left over?

K – 1 - 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?

4.4b & 4.5d – Solve single and multi-step practical problems involving add/sub/multi. of whole number and add/sub of fractions, and decimals

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.

50

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

51

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 only, you ran more than usual

and ran 8 minutes instead.

December 9, 2010

Assessing Higher-level Thinking Skills

52

8.11The student will solve practical area and perimeter problems involving composite plane figures.

Find the area of

the shaded region. A1

A2

Blue Area = A1+A2

A2 = Area of Square

Blue Area = A1 - A2

A1 = Area of Circle

December 9, 2010

Assessing Higher-level Thinking Skills

53

A.10 The student will compare and contrast multiple univariate data sets, using box-and-whisker plots.

Which class had the most students scoring higher than 83?

Class A has 36 students and Class B has 20 students. Which class has more students scoring above 83?

25% of 36 = 9

50% of 20 = 10

Longer (to 98)?

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Note: Blueprints Changes

- 54 -

• 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

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December 9, 2010 57

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”!

58

We Must Provide….

December 9, 2010 - 59 -

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 - 60 -

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 61

As the Virginia Department of Education revised the Standard of Learning assessments to

increase the level of rigor and include higher-level questions (M. Bolling, VDOE, 2010)

teachers will need to prepare students to respond well to these kinds of questions.

December 9, 2010 62

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 63

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 64

• 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 65

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 66

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 67

Before…….

Write 0.1 as a fraction.Or Reduce 2/20.

After…..

Write 0.1 with three different equivalent numbers.

Or

Which of the following are equivalent to 0.1?{ 1/10, 10%, .02, 2/20, 1%, 0.10}___________________________________________

December 9, 2010 68

Try asking:

Write two numbers that are greater than 95.867 by re-arranging only two digits for each new number created. Defend your answer choice.

Instead of:

Fill in the blank with > or <:

95.867 ____ 95.876

December 9, 2010 69

Before:

Identify each angle as a right angle, acute angle, or an obtuse angle?

After:

Draw a playground using at least five different angles that are between 45 degrees and 125 degrees. Name each resulting angle.

December 9, 2010 70

Before:

The temperature of the water in a swimming pool is 51°F. Since the freezing point of water is 32°F, how many degrees would the temperature of thewater have to drop to reach the freezing point?

After:

If the water in a swimming pool was 67 degrees at 8:00 am, and at 11:00 am that morning it had risen 8 more degrees, how many more degrees would the water need to rise in order to swim in water that was 80 degrees?

December 9, 2010 71

72

Content moved away from Gr. 4

Remove

id ordered pairs in the

first quadrant of coordinate

plane

Have students apply onlyVocabulary – flip, slide, turn

from grade 3 – compare fractions, id congruent and non-congruent shapes,

perimeter and area, analyze properties of 2-D and 3-D figures,

From grade 2 - Certain-likely-unlikely-impossible

Describe the path of shortest distance

between two points on a flat

surface.

Moving to 6th gradeEstimate conversions between metric and

US Customary Units (weight, length, volume) – still have to do it among metric

units and among US customary units.

December 9, 2010

73

Gr 4 - New Content Changes

Associative Property

Equivalent

Tons

Miles

4.2c – Id the division statement that represents a fraction

4.3d Given a model, write the decimal and fraction equivalents

4.2a – compare and order

fractions and mixed numbers

with like and unlike

denominators.

Use denominators of

12 or less by comparing to benchmarks (number line)

Equivalent fractions through

twelfths using regions, set models, and

measurement models (#line)

4.4d & 4.4d Solve single step and multi-step +, -, X, problems with whole numbers, fractions, and

decimals

4.9 – determine elapsed time in hours and minutes within a 12 hour

period

4.12a, b – define polygon. Identify polygons with 10 or fewer sides

4.13b – Represent probability as a

number between 0 and 1 inclusive

( need fractions on a number line)

New Emphasized

Vocab.

4.16 a/b (from 7.3a) recognize

and demonstrate the meaning of equality in an

equation. Investigate the

equality in + and X equations that

demonstrates the associative

property of + and X.

What’s new/ in focus?

December 9, 2010

Resources

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• 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