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Sample Math Connections for High Schools. Lessons correlated to NYS Standards
Citation preview
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
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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.
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
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
h
ssions
nal
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
d
f
h and
form
one (after
The St
Red Lig
What t
How d
Yellow
solve t
type o
studen
metho
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What s
correc
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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
n
ntheses
eses.
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.
y
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
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
0265
f, c, and a th
years we hav
eted.
ay
ay,
to solve for y
a)
t, f, c, and a th
ber of new fu
ensure that w
years.
ay
a(300)
l grown trees
r equal to 0. T
lve for a:
+ c
art a an expre
b an equatio
tion in part d
ork. Solve for
maining befo
lity in part e w
ork. Solve for
e would need
gh trees to las
hat could
ve
y:
hat
ull grown
we have
in 300
This can
ession?
on?
d with the
the
ore all
with the
the
each
st AT
A.A.21solutiovariabl
1 - Determineon to a given lle or linear in
330 East 85th
e whether a gilinear equatio
nequality in on
h Street, Suite C
iven value is on in one ne variable
C • New York,
a i) Your fri
new full g
years. Do
Values fo
can be
j) Your fri
billion ne
this agree
Values fo
can be
NY 10028 • Te
iend determin
grown trees e
oes this agree
a ≥
from
2,000,000
for t, f, and c c
worked out t
iend determin
ew full grown
e with your d
In this s
(t/‐300) + f
a ≥
2,000,000
or t, f, and c c
worked out t
el: (212) 717‐0
nes that an ad
each year wou
e with your da
≥ (t/‐300) + f +
m part e becom
0,000 = (t/‐40
can b plugged
to determine
true.
nes that we n
trees each to
data?
situation, y=4
+ c ≥ 2,000,00
≥ (t/‐300) + f +
becomes
0,000 ≤ (t/‐40
can b plugged
to determine
true.
0265
dditional 2 bi
uld last exact
ata?
+ c
mes
00) + f + c
d in and the e
if it comes ou
need AT LEAST
o last 400 yea
400 and
00,000. So,
+ c
00) + f + c
d in and the in
if it comes ou
illion
tly 400
equation
ut to be
T 2
ars. Does
nequality
ut to be
The ab
A.A.2 ‐
A.A.7 two vaA.A.8 A.A.9 A.A.10A.A.11
is requ
intege
A.A.23A.A.26
For ex
Studen
situati
A.A.32A.A.33A.A.34A.A.35A.A.36A.A.37A.A.38A.A.39A.A.40A.A.4
bove example
‐ Write a verb
- Analyze anariables - Analyze an- Analyze an
0 - Solve syst1 ‐ Solve a sys
uired Note: T
rs.
3 - Solve liter6 - Solve alge
ample:
students carepresents
students caresources o
students caor decreas
nts can also b
on, questions
2 - Explain sl3 - Determine4 - Write the 5 - Write the 6 - Write the 7 - Determine8 - Determine9 - Determine0 - Determine1 - Determine
330 East 85th
e can be slight
bal expression
nd solve verba
nd solve verband solve verbatems of two litem of one li
The quadratic
ral equations ebraic proport
an be provid
an create a sover a specif
an be asked e at an expobe asked to an
s can be devis
lope as a rate e the slope of equation of aequation of aequation of ae the slope of e if two lines e whether a gie whether a gie the vertex an
h Street, Suite C
tly modified t
n that matche
al problems w
al problems thal problems thinear equationnear and one
equation sho
for a given vations in one v
ded an equati
ystem of equfic time peri
to examine tnential rate nalyze their fi
sed which ad
of change betf a line, given a line, given ita line, given tha line parallel f a line, given are parallel, given point is oiven point is ind axis of sym
C • New York,
to address th
es a given ma
whose solution
hat involve quhat involve exns in two varie quadratic eq
ould represen
ariable variable which
ion and key
uations by stiod
the depletionor quadraticndings to the
dress the foll
tween dependthe coordinat
ts slope and thhe coordinateto the x- or yits equation i
given their eqon a line, givin the solutionmmetry of a p
NY 10028 • Te
e following st
athematical e
n requires sol
uadratic equatxponential groiables algebraquation in tw
t a parabola
h result in lin
and asked fo
tudying the
n and/or proc rate e above situat
owing standa
dent and indetes of two pohe coordinate
es of two poiny-axis in any form
quations in anen the equation set of a systparabola, give
el: (212) 717‐0
tandards.
xpression
lving systems
tions owth and decaically o variables, w
and the solut
near or quadra
or the situati
availability
duction of re
tions on a coo
ards.
ependent variaints on the lin
es of a point onts on the line
ny form on of the linetem of linear en its equatio
0265
s of linear equ
ay
where only fa
tion(s) should
atic equations
ion that the e
of two or mo
esources tha
ordinate plan
ables ne on the line e
e inequalitiesn
uations in
ctoring
d be
equation
ore
at increase
ne. In this
Strategiessense, algRules”, “Tsense, alg Magic Ru Just as theexample ooperationthere is a
1) P2) Ex3) M4) A
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Types of terms:
Next, list okeep the sappropria
s taught in thgebra, geomeThe Movie Cagebra, and geo
les
ere are “Magof a “Magic Rns in a problemcertain order
arentheses xponents Multiplication/Addition/Subt
es of “Magic can always bal, correspon
Solving One S
Example – Co
problems mayFor example,
– 8b2 + 2b2 –
em may seempler. First, id
a
out all of the symbol that cate column, cr
e game Oopstry, statistics,mera Methodometry.
gic Rules” in thule” in numbm (any combr in which the
/ Division raction
Rules” in geobe defined as ding angles w
Step at a Tim
ombining Like
y seem so com, when asked
– ab2 + 4a2b –
m daunting atdentify all of t
terms from tcomes beforeross out the t
can be appli, and measurd”, and “One
he game Oopber sense is thination of mue operations n
metry are forAreatriangle = ½will always be
e
e Terms)
mplicated tha to simplify:
3a + 9a2 – 6a
first. Howevthe different t
ab2
the problem te each term wterm in the or
OOPS
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ps, there are “he order of opultiplication, dneed to be do
rmulas and an½ (base)(heig equal.
at it appears t
ab2 – 2a2b + 1
ver, if we breatypes of term
b2
that fall into ewith that termriginal proble
any math topexamples belme” to a varie
“Magic Rules”perations. Andivision, additone:
ngle relationsht) and when
there is no wa
6b2 – 19a2b,
ak it down intms in the prob
a2
each categorym.) As you fill em.
pic, including tlow illustrate ety of problem
” in every areny time theretion, and/or s
ships. For exan 2 parallel lin
ay of getting t
to its constitublem:
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those in numthe use of “M
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ea of math. Ae are multiplesubtraction),
ample, the arnes are cut by
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uents, it beco
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le: Be sure tonto the
mber Magic r
An
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omes
o
Types of terms:
Terms froproblem:
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Types of terms:
Terms froproblem:
Sum of eacolumn:
The final a (Geometr Many studgive up beproblems
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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
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ts (Note, as me of the set
2 C3
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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
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n rule 1 abovut of the 81 c
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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
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ons for each
Reason:C1 (operaC3 AdditionMultiplicaSubtractioDivisionAdditionMultiplicaSubtractioDivisionAdditionMultiplicaSubtractioDivisionAdditionMultiplicaSubtractioDivisionAdditionMultiplicaSubtractioDivisionAdditionMultiplicaSubtractioDivisionMultiplicaMultiplicaMultiplicaDivisionMultiplicaDivision
h set. Also,
ation) C2 =
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