quadratics: how a function-based approach can help

Quadratics: How a Function-Based

Approach Can HelpQuickTime™ and a

TIFF (Uncompressed) decompressorare needed to see this picture.

http://generative.edb.utexas.edu/materials/aisd09/aisd10.htm

Today

• FBA can help with distinguishing when/how to use rules for simplifying from rules for solving in a way that students can check themselves.

• Go over FBA activities that can be used “tomorrow” in class to introduce quadratics

• Make connection from solving with quadratics back to solving with linear functions.

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

Big Picture(FBA)



Aug

Feb

Feb

Today

But first some data about the kinds of algebra-related challenges the function-based approach can address

The survey was given to 80 Algebra 1 students at a local high school on November 24, 2009. All students were taught by the same teacher. 63 of the 80 students completed the survey.

Research from Prudie Cain

Question #1: Simplify 3x+2-2x-1

0%10%20%30%40%50%60%70%80%90%

100%

combine like terms apply the distributiveproperty

do the same thing toboth sides

cancel multiply both terms by it do the opposite substitute a number forx

Examine graphs using acalculator or computer

Examine tables using acalculator or computer

Examine graphs usingpencil/pen or paper


Definitely would use this strategyLikely to use this strategyUnlikely to use this strategyDefinitely would NOT use the strategyNo Response

Question #2: Simplify -7(x+2)+4

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

combine like terms apply thedistributive

property

do the same thingto both sides

cancel multiply both termsby it

do the opposite substitute anumber for x

Examine graphsusing a calculator

or computer

Examine tablesusing a calculator

or computer

Examine graphsusing pencil/pen or

paper


paper

Definitely would use this strategy

Likely to use this strategy

Unlikely to use this strategy

Definitely would NOT use the strategy

No Response

Question #3: Solve 2x-5=x+4

0%10%20%30%40%50%60%70%80%90%

100%




undo the operation(for example: undo

the division)

substitute a numberfor x

Examine graphsusing a calculator or

computer

Examine tablesusing a calculator or

computer


paper


paper


Question #4: Solve x-4-3x=3+2x+1

0%10%20%30%40%50%60%70%80%90%

100%




do the opposite substitute a numberfor x


computer


computer


paper


paper




Definitely wouldNOT use the strategyNo Response

Question #1: Simplify 3x+2-2x-1

0%10%20%30%40%50%60%70%80%90%

100%



cancel multiply both terms by it do the opposite substitute a number forx

Examine graphs using acalculator or computer

Examine tables using acalculator or computer




Goodnews

SizableFraction (~1/3) Select “Solving” Strategies

Most seem unaware of how calculator (FBA) might help

More likely to view paper version as useful

See thesestrategies asmeaningful

Question #2: Simplify -7(x+2)+4

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

combine like terms apply thedistributive

property

do the same thingto both sides


do the opposite substitute anumber for x

Examine graphsusing a calculator

or computer

Examine tablesusing a calculator

or computer


paper


paper




Definitely would NOT use the strategy

No Response

Goodnews

Still sizablefraction (~1/3) Select “Solving” Strategies

And FBA is seen as even less useful

Although seemingly more useful or acceptable if done with paper and pencil

Question #3: Solve 2x-5=x+4

0%10%20%30%40%50%60%70%80%90%

100%




undo the operation(for example: undo

the division)

substitute a numberfor x


computer


computer


paper


paper


Goodnews

And FBA is EVEN LESS USEFUL for solving than for simplifying

Still sizablefraction (~1/3) seems to select an inappropriate or

irrelevant strategy

Yet don’t seem to be as able to connect to potentially relevant sub-strategies

Although seemingly more appropriate if done with paper and pencil

Question #4: Solve x-4-3x=3+2x+1

0%10%20%30%40%50%60%70%80%90%

100%




do the opposite substitute a numberfor x


computer


computer


paper


paper




Definitely wouldNOT use the strategyNo Response

Goodnews

Still sizablefraction (~1/3) seems to select an inappropriate or

irrelevant strategy

But somehow it is more likely to be useful if you graph or create tables using pencil and paper

And FBA not seen as useful

Again, seemingly not connected to potentially relevant sub-strategies

Simplifying and Solving are Confused

Function-Based Approach

(1) Can Help (see images)

(2) Can Help in Ways that Will Help with Future Challenges

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


Today

• FBA can help with distinguishing when/how to use rules for simplifying from rules for solving in a way that students can check themselves.

• Go over FBA activities that can be used “tomorrow” in class to introduce quadratics

• Make connection from solving with quadratics back to solving with linear functions.



Big Picture(FBA)



Aug

Feb

Feb

Today

For lines, solving for xmeans getting to a machine that gives out what you put in, knowing what you get out and thereby knowing what you put in.

x

79.8568

x

?

3?

IN

OUT

(Must be 3)

79.8568

e.g., X = 3



unsolving_lines_00.ggb[1]


[3] QuickTime™ and aTIFF (Uncompressed) decompressor


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

2nd TRACECALC


GRAPH


unsolving_lines_02.ggb


ENTER ENTER

ENTER

2nd MODEQUIT

[1]

[2]

[3]

2nd PRGMDRAW

ENTER

ENTER

GeoGebra

Type3orx

Calculator



Double-click on object to change color or thickness (style)



Click to turn on

Enter each pair (side) in solving

Example of “unsolving”; Can do (almost) anything to both sides[e.g., multiply both sides by x]

Repeat [1] - [3]

or

(because after using 5:intersect the x is set to the x-value of the intersection)



Can turn off functions by driving over equal sign and pressing ENTER

If intersection happens off the screen use WINDOW



Each transformation changes lines but preserves solution

Unsolving … Do whatever you want to both sides.






• Check intersection (same x-value?)• Draw vertical line at this value• Put new pair of functions on the board

For quadratics, solving for xmeans getting to a place where what we know about roots helps us know the original intersection(s).

Thankfully, doing the same thing to both sides does the same sort of thing to the original functions (replaces the original pair with functions that are NOT equivalent but do have the same solution set).

(Use other activities to help understand the importance of roots)







[1]

[2]

[3]

solving_quads_00.ggb


Calculator:

Follow earlier steps and use 5:intersect and draw 4:Vertical





Draw vertical lines (& adjust appearances)



What Do We Notice?

Like for lines, can we do (almost) ANYTHING to both sides? [Unsolving] Try it. What happens?


Why is this good?: We know how to find roots.

solving_quads_03plus.ggb

Just like for lines, enter each pair (side) in solving

Use 5:Intersect to find x-values for drawing vertical lines


Where could we go from here?Assignment: Get parabola and line to intersect in second quadrant. [1] plot each pair of functions in solving [2] confirm that intersections with original functions have the same x values as each step [3] confirm that what comes out of the quadratic formula are these x values.




Challenge

handouts





[3] QuickTime™ and aTIFF (Uncompressed) decompressor



2nd TRACECALC


GRAPH




ENTER ENTER

ENTER

2nd MODEQUIT

[1]

[2]

[3]

2nd PRGMDRAW

ENTER

ENTER

GeoGebra

Type3orx

Calculator



Double-click on object to change color or thickness (style)



Click to turn on

Enter each pair (side) in solving

Example of “unsolving”; Can do (almost) anything to both sides[e.g., multiply both sides by x]

Repeat [1] - [3]

or

(because after using 5:intersect the x is set to the x-value of the intersection)



Can turn off functions by driving over equal sign and pressing ENTER

If intersection happens off the screen use WINDOW



Each transformation changes lines but preserves solution





[1]

[2]

[3]



Calculator:

Follow earlier steps and use 5:intersect and draw 4:Vertical




What Do We Notice?

Like for lines, can we do (almost) ANYTHING to both sides? [Unsolving] Try it. What happens?


Why is this good?: We know how to find roots.

solving_quads_03plus.ggb

Just like for lines, enter each pair (side) in solving

Use 5:Intersect to find x-values for drawing vertical lines


Where could we go from here?Assignment: Get parabola and line to intersect in second quadrant. [1] plot each pair of functions in solving [2] confirm that intersections with original functions have the same x values as each step [3] confirm that what comes out of the quadratic formula are these x values.




Challenge

quadratics: how a function-based approach can help

Documents

fba activities

boardfor quadratics

calculator fba

paperand fba

pencilgoodnewsand fba

strategiesand fba

imagescan help

linear functions