Algebraic Reasoning OSPI Math Coaching Workshop March 9, 2009 David Foster.

Algebraic Reasoning

OSPI Math Coaching Workshop

March 9, 2009

David Foster

California 44thIn 2007 NAEP, CA Ranked 44th out of 49 participating.

CA was 42nd in 2005.

NAEP 2007 8th grade

National Perspective

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

Washington

15th of 49

Jurisdiction Score EnrollmentNational 281 38%Massachusetts 298 45%Minnesota 292 35%North Dakota 292 21%Vermont 291 26%Kansas 290 39%New Jersey 289 40%South Dakota 288 30%Virginia 288 42%New Hampshire 288 30%Montana 287 24%Wyoming 287 32%Maine 286 29%Colorado 286 44%Pennsylvania 286 42%Texas 286 28%Maryland 286 52%Wisconsin 286 30%Iowa 285 27%DoDEA 285 40%Indiana 285 33%Washington 285 31%Ohio 285 35%North Carolina 284 33%Oregon 284 39%Nebraska 284 35%Idaho 284 37%Delaware 283 36%Connecticut 282 39%South Carolina 282 41%Utah 281 58%Missouri 281 33%Illinois 280 33%New York 280 21%Kentucky 279 34%Florida 277 42%Michigan 277 38%Arizona 276 32%Rhode Island 275 41%Georgia 275 49%Oklahoma 275 27%Tennessee 274 31%Arkansas 274 33%Louisiana 272 24%Nevada 271 34%California 270 59%West Virginia 270 33%Hawaii 269 28%New Mexico 268 34%Alabama 266 30%Mississippi 265 21%District of 248 51%

Comparison of 8th Grade Students by State

Percent Enrolled in College Prep Math Courses

(Advanced)

versus

Mean Score of Students on NAEP (8th grade 2007)

Comparison of 8th Graders by State Enrollment in CP Math versus Student

Achievement on NAEP

0%

10%

20%

30%

40%

50%

60%

70%

260 265 270 275 280 285 290 295 300

Mean Score on 8th Grade NAEP 2007

% of 8th Graders Enrolled in

CP Math Courses

Washington

National Average

MassachusettsCalifornia

QuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.

“We have made significant gains in enrolling students in Algebra I in eighth grade in recent years, surpassing other state in the U.S. But we must set our goal higher.”

Algebra for All or Algebra Forever

We have also made more significant gains in FAILING students in Algebra I in eighth grade in recent years, surpassing other state in the U.S.

Arnold Schwarzenegger July 8, 2008

California Algebra – How are we doing?

Algebra CST

Students Passing

Prof or Adv

Students Failing

Basic or Below

2003506,000 students

106K21% meet standards

400K79% of students failed

the test in 2003

2008747,000 students

47% increase

187K25% meet standards

560K 40% more students failed in 2008 than in 2003

In 2008 more students failed the Algebra CST than took it in 2003!

3 out of 4 Fail

Does Having Students Take Algebra More TimesIncrease District Performance?

R2 = 0.73

260

280

300

320

340

360

380

400

420

440

0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4

Average # of Years Students Take the Algebra CST

Algebra CST (Mean Scale Score)

Copyright Tucher

Basic

Proficient

“I see trouble with algebra.”

Big Ideas in Algebra

• Variable

• Equivalence

• Functions

• Representation

• Modeling

• Structure

• Generalization

Big Idea

• EQUALITY – Understanding that two quantities are the same is an extremely important attribute in mathematics. Using number properties to maintain equality is central to algebra. Equations are to mathematics what sentences are to language. If two quantities are not equal, then their relationship is an inequality. A set of properties solely for inequalities is also studied in algebra.

EqualityWhat number would you put in the box to make this a true number sentence?

+ 58 + 4 =

Response/Percent Responding

Grade 7 12 17 12 &17

1 and 2 5 58 13 8

3 and 4 9 49 25 10

5 and 6 2 76 21 2

From Falkner, Levi, and Carpenter, 1999

Big Idea

• STRUCTURE – Algebra is often referred to as generalized arithmetic. Our number system is made up of sets of numbers and operations. Basic properties (axioms) are defined as the underpinnings of a system. These are used to govern how numbers and operations function together. New rules (theorems) are developed and expand understanding of the system. Algebra is the study of that structure.

Fruit for Thought

Algebra Reasoning

Fruit for Thought Part 1 Each type of fruit represents a single digit (0 – 9). Each fruit represents the same number in every equation. Using what you know about the number system and properties, determine the value of each fruit.





+



=





• =






+ +







=

- =









+





= • =











•



=





÷ =









÷ =





Fruit for Thought Part 2 Each type of fruit represents a single digit (0 – 9). Each fruit represents the same number in every equation. Using what you know about the number system and properties, determine the value of each fruit. Match fruit to the digit. Provide reasons.





• =




- =

• =









+ =







•



=









+



=


+ + =











÷ =






QuickTime™ and aDV/DVCPRO - NTSC decompressor


FRUIT FOR THOUGHT VIDEO DISCUSSION QUESTIONS

What did the teacher do to help set class norms and expectations for participation and behavior? How did the teacher help the students in making their explanations clear to the rest of the class? What did the teacher do to foster algebraic thinking? What kinds of questions did she ask?

Big Idea

• VARIABLE – In mathematics quantities that may change is an important idea. In algebra literal symbols are used to represent mathematical constructs. Algebra makes use of variables as a primary construct in equations, inequalities, formulas, functions, identities, and properties.

Variable

Is h + m + n = h + p + n Always, Sometimes, or Never true?

Grade 6 Grade 7

Grade 8

Always True 9.0% 9.6% 8.9%

Sometimes True 27.0% 36.8% 45.2%

Never True 26.2% 18.4% 27.4%

No response 34.4% 30.7% 16.3%

Don’t know/Other 3.3% 4.4% 2.2%

Some Diff erent Uses of Literal Symbols f , y in 3f = 1y (3 f eet in 1 yard) π, e (irrational numbers), c (speed of light) x in 5x – 9 = 91 a,b in a + b = b + a x,y in y = 9x - 2 m,b in y = mx + b * in e * x = x Place the sense in which variable is used in the box to the left of the best match Abstract symbols Parameters Constants Unknowns Generalized numbers Varying quantities Labels

Big Idea

• FUNCTION – A function is a relationship between one set of objects that maps to another set of objects. This structure is fundamental to algebra and can be used to study many ideas in mathematics. Students develop early ideas of functions by examining, extending and predicting patterns. As part of developing early algebra proportional reasoning should be learned through a functional approach.

Banquet TablesYou are helping to plan a big reception for your sister’s wedding. The reception hall has banquet tables shaped as hexagons. Six people can sit around a table.

Banquet TablesYou have just found out the hall where you are holding the reception is long and narrow. There is not enough room to spread the tables out. Your brother has an idea, what happens if we push two tables together so that one of the sides from the first table is touching a side from the second table.

Banquet TablesWhat happens to the number people when you push two tables together?

Banquet TablesHow would you find how many people can sit around any given number of tables?

The Banquet Tables• Individual Investigation- Square Tables

– Show NUMERICALLY- set up an arithmetic t-table, include arithmetic number sentences for 5, 13, and 100 tables

– Show GEOMETRICALLY –draw a minimum of 4 tables and color code how you determine the number of seats

– Show VERBALLY – describe your visualization for the number of seats in words

– Show ALGEBRAICALLY – determine the algebraic expression that matches your visualization

– Show GRAPHICALLY – create an appropriate graph

Share with a Partner

• Share your representations with a partner

– Are they alike? – If they are alike, can you think of OTHER

strategies students would use to determine the number of seats?

– Are they different?

– If they are different, do you understand your partner’s representations?

Whole Group Process• Sharing of Different Representations

– Do you understand these different representations?

– What questions might you ask to better understand?

– Can you think of any OTHER representations students might come up with?

– Are they similar to shared representations?

The Banquet Table• Second Stage of the Investigation

– Use triangles, trapezoids, and hexagons as banquet table shapes

– Then, show the 5 Representations: NUMERICAL/ARITHMETIC

– GEOMETRICAL/CONTEXT – VERBAL – ALGEBRAIC – GRAPHICAL

GRAPHICAL • Graph all four equations onto ONE graph

– Talk with a partner about what you see– What are the similarities? – What are the differences?– Can you determine the number of people seated

at 25 tables/35 tables from your graph?• How?

– How do you know that this answer is correct?

REFLECTION Select just one to write on

• What did I learn today that I would like to use in my classroom?

– How and where in my curriculum would I introduce this “learning” in my classroom?

• What did I learn today that I would like to share with my colleagues?

– How and where in my curriculum would I be able to introduce this “learning” with my colleagues?

• I was pleased that I …because…

• Something that was meaningful to me was….because…

Big Idea

• REPRESENTATION – Algebra is often thought of as a language. But the language is not limited to symbolic notation. The ideas of mathematics can be represented by graphs, tables, arrays, diagrams, words, as well as, symbolic notation. Connecting the linkages between different representations provides insights and understanding to mathematics concepts. Moving comfortably between the representations promotes cognitive flexibility that one can use in solving problems and modeling situations.

“Where is the ten?”

Determine the perimeter of the arrangement of these algebra tiles.

1 ut

1 ut

1 ut

x ut

Algebra Tiles

QuickTime™ and aH.263 decompressor


Video Discussion: Where is the 10?• INDIVIDUAL THINK TIME

• What were the teacher’s expectations for her students?• What moves did the teacher make in order to help her

students reach her expectations?• Was she successful? Why or why not?• What can you take away from this video to help you in

your own classroom? • Pair/Share THEN Whole Group Share

Matching Multiple

Representations: Malcolm Swan from the

Shell Center, University of Nottingham• We will be looking at this task through a teacher lens and through a learner

lens.• In a classroom, this matching multiple representations task would be used

after there has been some preliminary work on how the order of operations is presented algebraically, e.g., “Multiply n by 3, then add 4.”

• Today, you are going to jump right into this task as a learner.• You will work with a partner; a group of 3 will work.

Matching Multiple Representations

• Each pair of you will be getting a baggie with 4 sets of cards, each a different color.

• First, you will be taking just two of those sets out and you will match these two sets.

– The two sets will be turned face UP so that each of you can clearly see the cards.– Each person will take turns selecting one card from one color and verbally

explaining to their partner what that card represents, which card it matches in the other color, and why the two cards are a match.

– You will notice that there are BLANK cards. Some cards are missing; you will need to make matching cards. Please do not write on the blank cards.


• Once you and your partner have matched the first two sets, you will take out the third set - Tables of Numbers.

• You will match this new set with the original two sets.• Again,

– The new set will be turned face UP so that each of you can clearly see the cards.– Each person will take turns selecting one card from Tables of Numbers and

verbally explaining to their partner what that card represents, which cards it matches from the first two sets, and why this is a match.

– There are some missing tables which you will need to complete.


• Once you and your partner have matched the first three sets, you will take out the fourth set – Areas of Shapes.

• You will match this new set with the original three sets.• Again,

– The new set will be turned face UP so that each of you can clearly see the cards.– Each person will take turns selecting one card from Areas of Shapes and verbally

explaining to their partner what that card represents, which cards it matches from the first two sets, and why this is a match.

– There are some missing area shapes which you will need to complete.


• Once you and your partner have matched ALL FOUR SETS, • Discuss what you think are:

– the educational objectives of this task– Questions you might pose to your students to help them generalize what they have

discovered by engaging in this task

• Share with another pair at your table• Whole group sharing• Malcolm Swan suggests that each pair creates and presents their own poster

with cards and explanations.

Big Idea

• GENERALIZATION – Moving from the specific to a generalization is a powerful process in mathematics. In early algebra generalizing about number is important. Algebra 1 uses basic generalizations from number theory. Generalization is often the outcome of making conjectures and justifying. Proof and justification form the foundation for higher mathematics.

Find Out What Your Students Know

1. y = 3x + 4 5. P = 2l + 2w 9. 3 + 2n = 8

2. 40 = 5x 6. 3x + 2y 10. a(b + c) = ab + bc

3. 3x + 2x = 5x 7. a + b = b + a 11. 5x – 2x

4. A = bh 8. 3y – 5x =17 12. d = rt

1) Sort the cards in any way that makes sense to you.2) Write a sentence or two that describes the symbol strings that you have grouped together.3) Write another symbol string for each sorted group.

Sorting Symbol Strings

I. 5x – 2x Incomplete Equations

3x + 2y No solutions

New: 4y + 3w

II. d = rt Functions and Formulas

y = 3x + 4 Equations with two or

P = 2l + 2w more variables

A = bh New: A = r2

Sorting Symbol StringsIII. a + b = b + a Identities

a(b + c) = ab + ac Equations that are

3x + 2x = 5x always true

New: 2(6x) = 12x

IV. 3y – 5x = 17 One-sided Equations

40 = 5x Equations with 3 + 2n = 8 variables on

only one side

New: 3x + 4 = 22

Sorting String SymbolsI. 5x – 2x 3x + 2y Expressions

No equals sign New: 6x - 2

II. d = rt A = bh Formulas P= 2l + 2w Equations used

for a purpose New: C=r2

III. a + b = b + a Identities a(b + c) = ab + ac Equations that 3x + 2x = 5x are always

true New: 2(6x) = 12x

Sorting Symbol Strings

IV. y = 3x + 4 Linear Equations

3y – 5x = 17 Two variable

equations

New: 2x + 2y = 6

V. 40 = 5x One Variable Equations

3 + 2n = 8 One variable

equations

New: 9x + 3 = 15

Benefits of Sorting Symbol Strings

• Assessing students’ recall of vocabulary related to symbol strings

• Identifying some student understandings about uses of variables

• Finding some student misunderstandings about symbol strings

• Raising questions for further discussion

Help Students Develop Symbol Sense

Before we can expect students to “represent and analyze mathematical situations and structures using algebraic symbols” (NCTM, 2000), we must help them “work meaningfully with variables and symbolic expressions .” (NCTM, 2000)

Big Idea

• MODELING – Mathematizing a situation transforms a problem into a schema that can be altered or simplified in order to better understand the situation or solve the problem. Problems can be modeled using algebraic representations. Once a problem is represented algebraically it can be manipulated, extended or changed using properties of mathematics.

HURDLE #3

Symbolic Translation

What’s the problem and the research?


3 feet = 1 yard

3f = yOr

3y = f


There are 9 students for each professor at a

university

9s = p9p = s

Algebraic Reasoning OSPI Math Coaching Workshop March 9, 2009 David Foster.

Documents

Transcript of Algebraic Reasoning OSPI Math Coaching Workshop March 9, 2009 David Foster.