Sleepy Hollow High Sleepy Hollow High School Mathematics School Mathematics

and and the Shift to Common the Shift to Common

CoreCoreFaculty PresentationFaculty Presentation

October 16, 2013October 16, 2013Dan Larkin, Math ChairDan Larkin, Math Chair

So what is different?So what is different?

• Depth vs. BreadthDepth vs. Breadth• Overarching themes (through all Overarching themes (through all

levels of instruction)levels of instruction)• Read criticallyRead critically• Think criticallyThink critically• More complex solutionsMore complex solutions• Technical languageTechnical language• ApplicationsApplications

8 Standards for 8 Standards for Mathematical PracticeMathematical Practice

• Make Make sense of problemssense of problems and and perseverepersevere in solving themin solving them

• ReasonReason abstractly and quantitatively abstractly and quantitatively• Construct viable argumentsConstruct viable arguments and and

critiquecritique the reasoning of others the reasoning of others• ModelModel with mathematics with mathematics• Use Use appropriate toolsappropriate tools strategically strategically• Attend to Attend to precisionprecision• Look for and make use of Look for and make use of structurestructure• Look for an express regularity in repeated Look for an express regularity in repeated

reasoningreasoning

Pedagogical ShiftsPedagogical Shifts

1) Focus1) Focus

2) Coherence2) Coherence

3) Fluency3) Fluency

4) Deep Understanding4) Deep Understanding

5) Application5) Application

6) Dual Intensity6) Dual Intensity

FocusFocus

• Narrow Narrow • DeepenDeepen• Time/Energy in the classroomTime/Energy in the classroom

CoherenceCoherence

• Connect learning across gradesConnect learning across grades• Build new understandingsBuild new understandings

FluencyFluency

• Speed/AccuracySpeed/Accuracy• Class timeClass time• Homework timeHomework time• RepetitionRepetition

Deep UnderstandingDeep Understanding

• Ease of operationsEase of operations• More than the trickMore than the trick• Learn the mathLearn the math

ApplicationApplication

• Expected to useExpected to use• Choose appropriate concept even Choose appropriate concept even

when unpromptedwhen unprompted

Dual IntensityDual Intensity

• Practice vs. Understanding Practice vs. Understanding • BalanceBalance

What is the purpose of What is the purpose of mathematics?mathematics?

Take something complex…Take something complex…

and make it simpleand make it simple

Math illiteracy is not a Math illiteracy is not a jokejoke

• Would you ever joke about not Would you ever joke about not being able to read?being able to read?

• Would you ever joke about not being able to write?

A different mindsetA different mindset

• We need to move away from saying, We need to move away from saying, “It’s ok, I was never good at math”“It’s ok, I was never good at math”

• No one should take pride in not No one should take pride in not being able to do mathbeing able to do math

When am I ever going to When am I ever going to use this?use this?

• When you were very young, you When you were very young, you spent hours and hours playing with spent hours and hours playing with all kinds of toys, e.g., blocks that you all kinds of toys, e.g., blocks that you would pound through holes with the would pound through holes with the same shapes. same shapes.

Were you preparing to Were you preparing to pound blocks through pound blocks through

holes on a daily basis as holes on a daily basis as an adult? an adult? or or

Were you developing a fairly Were you developing a fairly general set of problem-general set of problem-solving skills that you solving skills that you would use someday on would use someday on

problems that you couldn't problems that you couldn't even begin to imagine at even begin to imagine at

that age?that age?

For instance:For instance:

• Suppose you're running a factory. There Suppose you're running a factory. There are a number of workers who do certain are a number of workers who do certain tasks, each task requiring a particular tool. tasks, each task requiring a particular tool. The workers, when needing a tool, stop The workers, when needing a tool, stop what they’re doing, walk over to the what they’re doing, walk over to the toolbox, sign out the tool, walk back to toolbox, sign out the tool, walk back to their station, use the tool, walk back to the their station, use the tool, walk back to the toolbox, and sign the tool back in. toolbox, and sign the tool back in.

This This exhausting processexhausting process happens happens

several times a day. several times a day.

A similar mathematical A similar mathematical exampleexample

• If you're familiar with factoring If you're familiar with factoring common monomials, you might think common monomials, you might think about this: about this:

Is it easier to computeIs it easier to compute 3*19 + 12*19 + 2*19 + 5*193*19 + 12*19 + 2*19 + 5*19or or (3 + 12 + 2 + 5)*19 ? (3 + 12 + 2 + 5)*19 ?

• It's obvious that it's easier to compute the It's obvious that it's easier to compute the latter, because it requires fewer steps. latter, because it requires fewer steps.

• You might consider, can the underlying You might consider, can the underlying idea be applied to what's happening in the idea be applied to what's happening in the factory? factory?

In fact, it can:In fact, it can:

• You could have each worker check You could have each worker check out the tools that she/he needs one out the tools that she/he needs one time, at the beginning of the shift, time, at the beginning of the shift, and check them back in one time, at and check them back in one time, at the end of the shift. It's really the the end of the shift. It's really the same kind of reasoning, except one same kind of reasoning, except one deals with mathematical deals with mathematical expressions, and the other deals expressions, and the other deals with manufacturing processes. with manufacturing processes.

Do you need to have studied Do you need to have studied monomial factoring in monomial factoring in

order to come up with this order to come up with this solution?solution?

• Not necessarily. But if you've spent Not necessarily. But if you've spent time in math classes, learning to time in math classes, learning to look for ways to look for ways to eliminate wasted eliminate wasted operationsoperations, you're much, much , you're much, much more likely to think to look for a more likely to think to look for a solution, eliminating waste in the solution, eliminating waste in the first place, which is the first step to first place, which is the first step to finding one. finding one.

Can you solve this?Can you solve this?

• 4x + 7 = 2 – (x + 10)4x + 7 = 2 – (x + 10)

Can you solve Can you solve and use the and use the

number/equality properties number/equality properties to describe how you to describe how you

converted your equation converted your equation line by line?line by line?• 4x + 7 = 2 – (x + 10)4x + 7 = 2 – (x + 10)

• 4x + 7 = 2 – (x + 10)4x + 7 = 2 – (x + 10)

4x + 7 = 2 – x 4x + 7 = 2 – x - 10- 104x + 7 = 2 – 10 - x4x + 7 = -8 - x

4x + x = -7 - 8

5x = -15x = -3

Distributive Property

Commutative Property of Addition

Combine like terms

Additive Property of Equality

Combine like terms

Multiplicative Property of Equality

Integrated Algebra vs. Algebra 1

• The cost of three notebooks and The cost of three notebooks and four pencils is $8.50. The cost of four pencils is $8.50. The cost of five notebooks and eight pencils five notebooks and eight pencils is $14.50. Determine the cost of is $14.50. Determine the cost of one notebook and the cost of one one notebook and the cost of one pencil. [Only an algebraic pencil. [Only an algebraic solution can receive full credit.]solution can receive full credit.]

• Next weekend Marnie wants to Next weekend Marnie wants to attend either carnival attend either carnival AA or or carnival carnival BB. Carnival . Carnival AA charges $6 charges $6 for admission and an additional for admission and an additional $1.50 per ride. Carnival $1.50 per ride. Carnival BB charges charges $2.50 for admission and an $2.50 for admission and an additional $2 per ride.additional $2 per ride.

• a) In function notation, write a) In function notation, write A(A(xx) to represent the total cost of ) to represent the total cost of attending carnival attending carnival A A and going on and going on x x rides. In function notation, write rides. In function notation, write B(B(xx) to represent the total cost of ) to represent the total cost of attending carnival attending carnival B B and going on and going on x x rides.rides.

• b) Determine the number of b) Determine the number of rides Marnie can go on such that rides Marnie can go on such that the total cost of attending each the total cost of attending each carnival is the same. [Use of the carnival is the same. [Use of the set of axes below is optional.]set of axes below is optional.]

• c) Marnie wants to go on five c) Marnie wants to go on five rides. Determine which carnival rides. Determine which carnival would have the lower total cost. would have the lower total cost. Justify your answer.Justify your answer.

PARCC Assessment

Examples