20
The stIntegrMetho The De In the proble which solve m examp A.A.6 variabl A.A.7 two va A.A.8 A.A.9 A.A.10 A.A.11 is requ intege A.A.22 A.A.23 A.A.24 A.A.25 equatio A.A.26 rategies of R ated Algebra od and Divide etective Met Detective Me em is exposed uses inverse most type of ple is illustrat- Analyze an le or linear in - Analyze an ariables - Analyze an - Analyze an 0 - Solve syst 1 Solve a sys uired Note: T rs. 2 - Solve all t 3 - Solve liter 4 - Solve line 5 - Solve equ ons in one va 6 - Solve alge 330 East 85th Rush Hourca curriculum. and Conque hod ethod, the prd and solved i operations to equations an ed below. nd solve verba nequality in ond solve verba nd solve verba nd solve verba tems of two li tem of one li he quadratic types of linear ral equations ear inequalitie ations involv riable. ebraic proport h Street, Suite C an be used to This docume r relate to sta oblem is defi n a back to fr o undo stepsd inequalitieal problems w ne variable al problems w al problems th al problems th inear equation near and one equation sho r equations in for a given va es in one varia ing fractional tions in one v C • New York, RUSH HOU address num nt cites speci andards unde ned, question ront way. Th, starting with s, including o whose solution whose solution hat involve qu hat involve ex ns in two vari e quadratic eq ould represen n one variable ariable able l expressions variable which NY 10028 • Te UR merous NY Sta fic examples er the Algebra ns are asked is supports th h the last one nes addresse n requires sol n requires sol uadratic equaxponential gro iables algebra quation in tw t a parabola e Note: Expre h result in l el: (212) 7170 ate Math Stan of ways in wa strand. based on cau he strategy of e first. This m ed in the follo lving a linear lving systems tions owth and dec aically o variables, w and the solut essions which inear or quad 0265 ndards in the hich the Dete use and effect f equation so method can be owing standar equation in o s of linear equ ay where only fa tion(s) should h result in line dratic equation ective t, and the lving e used to rds. An one uations in ctoring d be ear ns

Mind Lab Sample Math Connections for High Schools

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Page 1: Mind Lab Sample Math Connections for High Schools

 

 

 

The str

Integra

Metho

 

The De

 

In the 

proble

which 

solve m

examp

 

A.A.6 variablA.A.7 two vaA.A.8 A.A.9 A.A.10A.A.11

is requ

intege

A.A.22A.A.23A.A.24A.A.25equatioA.A.26

 

 

rategies of “R

ated Algebra 

od and Divide

etective Met

Detective Me

em is exposed

uses inverse 

most type of e

ple is illustrate

- Analyze anle or linear in- Analyze an

ariables - Analyze an- Analyze an

0 - Solve syst1 ‐ Solve a sys

uired  Note: T

rs. 

2 - Solve all t3 - Solve liter4 - Solve line5 - Solve equons in one va6 - Solve alge

330 East 85th

Rush Hour” ca

curriculum.  T

 and Conque

hod 

ethod, the pro

d and solved i

operations to

equations an

ed below. 

nd solve verbanequality in onnd solve verba

nd solve verband solve verbatems of two litem of one li

The quadratic 

types of linearral equations ear inequalitieations involvriable. ebraic proport

h Street, Suite C

an be used to

This docume

r relate to sta

oblem is defi

n a back to fr

o undo steps,

d inequalities

al problems wne variable al problems w

al problems thal problems thinear equationnear and one

equation sho

r equations infor a given va

es in one variaing fractional

tions in one v

 

C • New York, 

RUSH HOU

 address num

nt cites speci

andards unde

ned, question

ront way.  Thi

, starting with

s, including o

whose solution

whose solution

hat involve quhat involve exns in two varie quadratic eq

ould represen

n one variableariable able l expressions

variable which

 NY 10028 • Te

UR

merous NY Sta

fic examples 

er the Algebra

ns are asked b

is supports th

h the last one

nes addresse

n requires sol

n requires sol

uadratic equatxponential groiables algebraquation in tw

t a parabola 

e

Note: Expre

h result in l

el: (212) 717‐0

ate Math Stan

of ways in wh

a strand. 

based on cau

he strategy of

e first.  This m

ed in the follo

lving a linear

lving systems

tions owth and decaically o variables, w

and the solut

essions which

inear or quad

0265 

ndards in the 

hich the Dete

use and effect

f equation so

method can be

owing standar

equation in o

s of linear equ

ay

where only fa

tion(s) should

h result in line

dratic equation

ective 

t, and the 

lving 

e used to 

rds.  An 

one

uations in

ctoring 

d be 

ear

ns 

Page 2: Mind Lab Sample Math Connections for High Schools

 

 

 

1) Defi

 

 

2) Ask

achiev

 

 

3) Und

  

 

The De

ine the object

:  “How am I 

ving my objec

do the proble

330 East 85th

etective Meth

tive. 

being blocked

tive?” 

m in a back to

h Street, Suite C

hod 

d from 

o front way.

 

C • New York, 

App

 

Solve: 7x

 

The obje

 

 

a) x is be

b) 15 is b

 

 

a) To un

operatio

equation

 

7x + 15 =

      ‐ 15  

7x         =

 

b) To un

operatio

 

7x         =

7             

 

         x =

 

 NY 10028 • Te

plication of S

x + 15 = 36 

ective is to so

eing multiplie

being added 

ndo “15 is bein

ons to subtrac

n. 

= 36 

 ‐15 

= 21 

ndo “x is being

ons to divide b

= 21 

  7 

= 3 

el: (212) 717‐0

Strategy to Eq

lve for, or iso

ed by 7. 

to the 7x. 

ng added to 7

ct 15 from bo

g multiplied b

both sides of 

0265 

quation Solvin

olate x.  

7x”, use inver

oth sides of th

by 7”, use inve

the equation

ng: 

rse 

he 

erse 

n by 7. 

Page 3: Mind Lab Sample Math Connections for High Schools

 

 

Divide

 

The Di

can be

particu

solving

 

A.A.12exponeA.A.13A.A.14A.A.15A.A.16renamiA.A.17A.A.18A.A.19A.A.20factori

 

 

e and Conque

vide and Con

e dismantled i

ularly applica

g (above) sect

2 - Multiply aents Note: Us3 - Add, subtr4 - Divide a p5 - Find value6 - Simplify fing them to lo7 - Add or sub8 - Multiply a9 - Identify an0 - Factor alging a GCF)

330 East 85th

er 

nquer strategy

into its prima

ble to proble

tions of the In

and divide moUse integral ex

ract, and multpolynomial byes of a variabfractions withowest terms btract fractionand divide algnd factor the

gebraic expres

h Street, Suite C

y states that w

ary componen

ms that comb

ntegrated Alg

onomial exprexponents onlytiply monomiy a monomialle for which a

h polynomials

nal expressiongebraic fractiodifference of ssions comple

 

C • New York, 

when a probl

nts and each c

bine standard

gebra Standar

essions with ay. ials and polynl or binomial,an algebraic fs in the numer

ns with monoons and expre

f two perfect setely, includin

 NY 10028 • Te

lem is too diff

component c

ds from the p

rds.  An exam

a common ba

nomials , where the qufraction is undrator and den

omial or like bess the producsquares ng trinomials

el: (212) 717‐0

ficult to solve

can be solved

olynomial (be

mple is illustra

ase, using the

uotient has nodefined ominator by f

binomial denoct or quotient

with a lead c

0265 

e in one or tw

 separately.  

elow) and equ

ated below. 

properties of

o remainder

factoring both

ominators t in simplest f

coefficient of

wo steps, it 

This is 

uation 

f

h and

form

one (after

Page 4: Mind Lab Sample Math Connections for High Schools

 

 

 

1) Dec

 

 

2) Solv

 

 

3) Rec

 

 

  

Divide

onstruct the 

ve each comp

onstruct prob

330 East 85th

e and Conque

problem. 

ponent. 

blem. 

h Street, Suite C

er 

C • New York, 

Applicat

 

Solve for

4(2r + 1)

 

The abo

equivale

 

a) 4(2r +

 

 

Follow t

expressi

 

a) 4(2r +

       8r + 

     13r + 

 

b) 41 – (

    41 ‐   2

    34 –  2

 

 

To recon

equal to

problem

 

  13r + 4

+ 2r        

  15r + 4

          ‐ 4

 15r        

15          

 

     r = 2

 

 NY 10028 • Te

tion of Strate

the Simplif

r r: 

) + 5r = 41 – (

ove problem c

ent expression

+ 1) + 5r     an

the order of o

ion: 

+ 1) + 5r 

4   + 5r      (m

 4                (a

(6r + 21)/ 3 

2r + 7          (d

2r                 (s

nstruct the pr

o one another

m) and solve.

4 = 34 – 2r 

        + 2r 

4 = 34 

4    ‐ 4 

 = 30 

     15 

el: (212) 717‐0

egy to Equati

fication of Po

(6r + 21)/3 

can be broken

ns: 

d     b) 41 – (6

operations to s

multiplication)

ddition) 

division) 

subtraction) 

roblem, set th

r (as they wer

0265 

ion Solving In

olynomials: 

n down into tw

6r + 21)/ 3 

simplify each

he two expres

re in the origi

nvolving 

wo 

ssions 

nal 

Page 5: Mind Lab Sample Math Connections for High Schools

 

 

The str

Algebr

under 

into on

be use

planni

 

The St

 

The St

math p

always

somet

proble

 

The str

the Ro

Expres

 

A.A.12exponeA.A.13A.A.14A.A.15A.A.16renamiA.A.17A.A.18A.A.19A.A.20factori

 

rategies of “Q

ra curriculum

the Algebra s

ne strategy us

ed to generate

ng of resourc

toplight Meth

oplight Meth

problems.  St

s the correct 

imes approac

em is a multip

rategy illustra

oute, can be u

ssions section

2 - Multiply aents Note: Us3 - Add, subtr4 - Divide a p5 - Find value6 - Simplify fing them to lo7 - Add or sub8 - Multiply a9 - Identify an0 - Factor alging a GCF)

330 East 85th

Quoridor” can

.  This docum

strand.  The S

sed to solve v

e real life exa

ces. 

hod, Self‐Bloc

od, Self‐Block

udents often 

math for that

ch it as if it we

plication, rath

ated below, w

used to solve 

n of the Integr

and divide moUse integral ex

ract, and multpolynomial byes of a variabfractions withowest terms btract fractionand divide algnd factor the

gebraic expres

h Street, Suite C

n be used to a

ment cites spe

Stoplight Met

virtually any m

amples which 

cking, and Pa

king, and Pav

solve proble

t problem.  Fo

ere (7x + 3) +

er than an ad

which is a com

virtually any 

rated Algebra

onomial exprexponents onlytiply monomiy a monomialle for which a

h polynomials

nal expressiongebraic fractiodifference of ssions comple

 

C • New York, 

QUORIDO

address nume

cific example

hod, Self‐Blo

math problem

demonstrate

ving the Rou

ving the Route

ms using mat

or example, g

+ (x2 – 3x + 12)

ddition proble

mbination of t

math problem

a curriculum.

essions with ay. ials and polynl or binomial,an algebraic fs in the numer

ns with monoons and expre

f two perfect setely, includin

 NY 10028 • Te

OR

erous NY Stat

es of ways in w

cking, and Pa

m, while the E

e the necessit

ute 

e, can be inst

th that they a

given the prob

), without sto

em.  

the Stoplight 

m, such as th

a common ba

nomials , where the qufraction is undrator and den

omial or like bess the producsquares ng trinomials

el: (212) 717‐0

te Math Stand

which they re

aving the Rou

Effective Alloc

ty of math in 

trumental in c

are comfortab

blem (7x + 3)

opping to reco

Method, Self

ese from the 

ase, using the

uotient has nodefined ominator by f

binomial denoct or quotient

with a lead c

0265 

dards in the In

elate to stand

te can be inte

cation of Reso

the successfu

completing vi

ble with, whic

(x2 – 3x + 12)

ognize that th

f‐Blocking, an

Variables an

properties of

o remainder

factoring both

ominators t in simplest f

coefficient of

ntegrated 

dards 

egrated 

ources can 

ul 

irtually all 

ch is not 

, students 

he 

nd Paving 

f

h and

form

one (after

Page 6: Mind Lab Sample Math Connections for High Schools

 

 

The St

 

Red Lig

 

What t

How d

 

 

Yellow

solve t

type o

studen

metho

answe

 

What s

correc

 

 

Green 

 

 

 

toplight Meth

Self‐Blockin

ght:  Stop and

type of probl

o you know?

w Light:  Dete

the problem c

f problem an

nts block them

od and pave t

ering the ques

strategy is req

tly? 

Light:  Caref

330 East 85th

hod for Math

ng and Paving

d analyze the

em is it? 

 

rmine the str

correctly.  By 

d appropriate

mselves from

he route for 

stion. 

quired to solv

ully employ t

h Street, Suite C

h, which integ

g the Route:

e problem. 

rategy require

writing out t

e strategy, 

 using an inco

successfully 

ve this proble

he strategy.

C • New York, 

grates 

 

Prob

 

This

ther

pare

betw

 

ed to 

he 

orrect 

em 

 

Mul

by e

Then

 

Mul

 

7x(x

7x(‐3

7x(1

 

3(x2)

3(‐3

3(12

 

Com

 

7x3 –

       

 

7x3 –

 

 NY 10028 • Te

Application 

blem:  (7x + 3

s is a multiplic

re are two po

entheses and 

ween the pare

ltiply every te

every term in 

n combine lik

ltiply: 

x2) = 7x3 

3x) = ‐21x2 

12) = 84x 

) = 3x2 

3x) = ‐9x 

2) = 36 

mbine like term

– 21x2 + 84x

+  3x2  ‐   9x +

– 18x2 + 75x +

el: (212) 717‐0

of Strategy t

Problem

3)(x2 – 3x + 12

cation problem

lynomials sep

there is no op

entheses. 

erm in the firs

the second se

ke terms. 

ms: 

+ 36 

+ 36 

0265 

to Example M

m: 

2) 

m.  I know be

parated by 

peration sign

st set of paren

et of parenthe

Math 

ecause 

ntheses 

eses.  

Page 7: Mind Lab Sample Math Connections for High Schools

 

 

 

Effecti

 

In real

have a

necess

 

(Exam

 

Situati

forest 

planni

 

For ho

 

A.A.1 ‐

into an

A.A.4 mathem

 

ive Allocation

 life, we need

a sufficient su

sity of math t

ple 1) 

on:  Today th

fires and c ar

ng, an additio

omework, hav

S

‐ Translate a 

n algebraic ex

- Translate vmatical equat

330 East 85th

n of Resource

d to plan for t

pply until we

o plan for the

here are t full 

re cut down f

onal a trees r

ve students re

Standard

quantitative v

xpression 

erbal sentenctions or inequ

h Street, Suite C

es 

the use of lim

e are no longe

e effective all

grown trees 

or constructio

each full grow

esearch the a

verbal phrase

ces into ualities

 

C • New York, 

mited resource

er dependent

ocation of re

on earth.  Ea

on, paper, an

wth each yea

ctual values o

e  a) Use th

terms of 

grown tre

 

 

b) Use th

following

 

“The num

original n

that have

of trees t

 

c) Re‐writ

situation 

less than 

 

 NY 10028 • Te

es, such as tre

t on them.  Th

sources. 

ch year, f full

nd other uses

r. 

of t, f, c, and a

Example

e information

t, f, c, and a t

ees that will e

t

e variables de

g into an equa

mber of full gr

number of ful

e been destro

that have reac

n =

te the equatio

in which the 

500. 

500

t – (

el: (212) 717‐0

ees and crude

he example b

l grown trees

.  Based on cu

a. 

e Question R“Quoridor”

n above to se

that represen

exist after y y

t – (f+c)y + ay

efined above 

ation: 

rown trees, n,

ll grown minu

oyed over y ye

ched full grow

= t – (f+c)y + a

 

on in part b s

total number

0 > t – (f+c)y +

or 

(f+c)y + ay < 5

 

0265 

e oil, to ensur

elow illustrat

 are destroye

urrent analys

Related to

t up an expre

nts the numbe

years. 

to translate t

, after y years

us the number

ears plus the n

wth over y ye

ay 

so that it repr

r of full grown

+ ay 

500 

re that we 

tes the 

ed in 

is and 

ession in 

er of full 

the 

s, is the 

r of trees 

number 

ars.” 

resents a 

n trees is 

Page 8: Mind Lab Sample Math Connections for High Schools

 

A.A.5 inequa

A.A.3 algebr 

A.A.6 whoseequatioone vaA.A.22one vaA.A.24variablA.A.25expreslinear  

 

- Write algebalities that rep

- Distinguishaic expression

- Analyze an solution requon in one variariable 2 - Solve all tariable 4 - Solve linele 5 - Solve equssions Note: Eequations in

330 East 85th

braic equationpresent a situa

h the differencn and an alge

nd solve verbauires solving iable or linear

types of linear

ear inequalitie

ations involvExpressions wone variable.

h Street, Suite C

ns or ation

ce between anebraic equatio

al problems a linear r inequality in

r equations in

es in one

ing fractionalwhich result i.

 

C • New York, 

d) Set up 

be used t

remainin

 

w

 

e) Set up 

could be 

trees we 

enough tr

 

becaus

years ne

 

n n

f) What m

What ma

 

n

n

l n

g) Replac

values yo

actual nu

trees are 

 

h) Replac

values yo

number o

year to en

LEAST 30

 

 

 NY 10028 • Te

an equation 

to solve for th

g before all tr

n =

0 =

which can be 

y =

an inequality

used to solve

would need e

trees to last A

n =

0 ≤ t –

se the total n

eeds to be gre

be re‐arr

a ≥

makes the sta

akes the state

ce t, f, c, and a

ou researched

umber of year

depleted. 

ce t, f, and c, i

ou researched

of new full gro

nsure that we

00 years. 

el: (212) 717‐0

in terms of t,

he number of 

rees are depl

= t – (f+c)y + a

becomes 

= t – (f+c)y + a

re‐arranged 

= ‐t / (‐f – c + 

y in terms of t

e for the numb

each year to e

AT LEAST 300 

= t – (f+c)y + a

becomes 

– (f+c)300 + a

umber of full

eater than or

ranged to sol

≥ (t/‐300) + f +

atement in pa

ement in part 

a in the equa

d for homewo

rs we have re

in the inequa

d for homewo

own trees we

e have enoug

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Page 9: Mind Lab Sample Math Connections for High Schools

 

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Page 10: Mind Lab Sample Math Connections for High Schools

 

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Page 11: Mind Lab Sample Math Connections for High Schools

    

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Page 12: Mind Lab Sample Math Connections for High Schools

    

Types of terms: 

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Page 13: Mind Lab Sample Math Connections for High Schools

                      

   

 

   

   

   

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Page 14: Mind Lab Sample Math Connections for High Schools

     The Mov (Algebra E An equatifigure out= 29 is the 

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Page 15: Mind Lab Sample Math Connections for High Schools

    

Dragon Trsuccessfuchances opeople arTreasure the doll ca (Probabil 1) It's youdiamond of the 3 sp a) What's b) Previouand remeimprove?

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Page 16: Mind Lab Sample Math Connections for High Schools

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Page 17: Mind Lab Sample Math Connections for High Schools

Intr Mat12 o1 setblack AimThe term WarDivicategsuppcard

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Page 18: Mind Lab Sample Math Connections for High Schools

(SU eac Revimultbasestude (SU eac Ask new

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Page 19: Mind Lab Sample Math Connections for High Schools

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1

2

w Version of

m: objective of

rting Positiodents are divier), lays 12 cn in a pile in

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2. Some of variable. variable (

f SET: “Ope

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Page 20: Mind Lab Sample Math Connections for High Schools

Examthese

C

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222222223333xy

2233

mples of Sete are just som

C1 C2

x xx xx xx xx 2xx 2xx 2xx 2xx 3xx 3xx 3xx 3x

2x 2x2x 2x2x 2x2x 2x2x 3x2x 3x2x 3x2x 3x

x 3xx 3xx 3xx 3x

x yy x

2x 2y2y 2x

x 3yy 3x

ts (Note, as me of the set

2 C3

x 2xx x2 x 0 x 1 x 3xx 2x2

x -x x 1/2x 4xx 3x2

x -2xx 1/3x 4xx 4x2

x 0 x 1 x 5xx -x x 6x2

x 2/3x 6xx 9x2

x 0 x 1

y x/yx y/xy x/yx y/xy x/yx y/x

evidenced ints created ou

3 ReasoC1 (opC3 AdditiMultipSubtraDivisioAdditi

2 MultipSubtra

2 DivisioAdditi

2 Multipx Subtra3 Divisio

Additi2 Multip

SubtraDivisioAdditiMultip

2 Subtra3 Divisio

Additi2 Multip

SubtraDivisio

y Divisiox Divisioy Divisiox Divisioy Divisiox Divisio

n rule 1 abovut of the 81 c

n: peration) C2

on plication action on on

plication action on on

plication action on on

plication action on on

plication action on on

plication action on on on on on on on

ve, that therecards; there a

= C1

y y y y y y y y y y y y

2y 2y 2y 2y 2y 2y 2y 2y 3y 3y 3y 3y x y x x y y

e are multiplare many oth

C2

y y y y

2y 2y 2y 2y 3y 3y 3y 3y 2y 2y 2y 2y 3y 3y 3y 3y 3y 3y 3y 3y -1 -1

-2x -2x -2y -2y

le valid reasohers.):

C3

2y y2 0 1

3y 2y2 -y 1/2 4y 3y2 -2y 1/3 4y 4y2 0 1

5y -y

6y2 2/3 6y 9y2 0 1 -x -y

-2x2 -1/2 -2y2 -1/2

ons for each

Reason:C1 (operaC3 AdditionMultiplicaSubtractioDivisionAdditionMultiplicaSubtractioDivisionAdditionMultiplicaSubtractioDivisionAdditionMultiplicaSubtractioDivisionAdditionMultiplicaSubtractioDivisionAdditionMultiplicaSubtractioDivisionMultiplicaMultiplicaMultiplicaDivisionMultiplicaDivision

h set. Also,

ation) C2 =

ation on

ation on

ation on

ation on

ation on

ation on

ation ation ation

ation