ICL 7503: Reading Response 5

Readings: SSE Standards_Decoded (in Week 11 Module content)

Reading Response 5:

Respond to the following questions:

The section from the Standards Decoded book covers the same set of Algebra standards that you read about last week. Both documents (Standards Decoded and Standards Deconstructed) are designed to provide additional information to help teachers make sense of the standards and develop ideas of how to teach them.

1. What is one idea that you learned from the Standards Decoded document that you did not already know? How will this idea help you in teaching these standards?

2. After having reviewed both of these documents, do you think that it is important for teachers to refer to documents such as these to “unpack” the standards? If so, why? If not, why not?

3. If you were only allowed ONE of these two resources (either the Standards Decoded or the Standards Deconstructed), which would you choose to use in support of teaching these standards? Why?

A.SSE.A

A,SSE,A.1: Interpret expressions that represent a quantity in terms of its context.*

a. Interpret parts of an expression, such as terms, factors, and coefficients.

b. Interpret complicated expressions by viewing one or more of their parts as

interpret P(1 i r)” as the prod,rct of P and a factor not depending on P.

A,SSE ,A.Z: Use the structure of an expression to identify ways to rewrite it.

(rr), – (yr)r,thus recogn tzrngit as a difference of squares that can be factored

a single entity” For examPle –

For example, see xa – )’+ as

as (xz – y’)(*’ + y’).

Cluster A: lnterpret the structure of expressions’

The middle grades standards in Expression_and F,quations serve as connectors from arithmetic expressions

in elementafr, school to -or. “o*il.x
work with algebraic expressions in high school. As the complexih

“f
.-pt.rri””l i””116″r, students ,ecog_.,ir” expressi6ns as being built out of basic operations; that.is

initiaily they see expressions as sums o?t.r-, and products of factors. Students also see that complicated

“*pr”rJior,r’rre
buili up out of simpler ones. That is, students recognize different terms in an expression

(and understand what a term is) .r,d .r., recognize one or more parts
-ofthe

expression as discrete comPo-

.,.r,t . fo, example, in the expression (x + 4)'”- 6, students first see a difference of a constant and a square

and then see that inside the square term is tlre expressi on x + 4. The first way of looking at the expression

tells students that the ral,re ot’the expression is alrvays greater than or equal to -6, since a square is alwavs

gr”r;;thr;-o; “qrrl
to 0. The second way tells them that the square term is zero when x = -4. Putting

,fr.r.-G”-fro rlir+”tr can determine that this expression attains its minimum value, -6, when x =- -4′
Student! explore ways to rewrite expressions, ,uchis combinin_g liketermE that is, terms with exactly the

,r*. ,rrirli”!,r”riiti”, in them, ,,1.h ,, Zxy and 31ylt not /x and Zxz .
-fhis process. relates to the arith-

metic of radicai
“*prerriors,

for example, ZJry * ),txy anlthe.structureof co^mp-utation.Th.e Pro.gress.ions

for the Common Core State Standardi in Mhthematicti, High School, Algebra
.(2013)

stresses that-simpli-

fi..il;; ;;i , gorl of th” Algebra standards, as different iorms of an expression are useful in differen

contexts. For e*imple, viewin”g the expression for a- parabola in vertex form may be more useful in a situ-

,tio.r, ,,-,”h as finding ih. .*ti8*. ualie, *her”rs thl factored form is more useful for finding the zeros of

the expression.

Standards for Mathematical Practice

SFMP 1. Make sense of problemS and persevere in solving them’

SFMP 2. Use quantitative reasoning’
SFMp 3. Consiruct viable arguments and critique the reasoning of others.

SFMP 4. Modelwith mathematics’
SFMP 5. Use appropriate tools strategically’
SFMP 6. Attend to Precision’
SFMP 7. Look and make use of structure’
SFMP 8. Look for and express regularity in repeated reasoning.

Sfudents use different forms of algebraic expressions to solve problems. They also use quantitative
,

,.r*”iffi””aerstand relationslhips ,*o.,g different.*pr.riiort. Even though all ofthe Standards

for Mathematical practice are evident in the”se standards, Practices 7 and 8 are particularly prominent

fr.r.. ii”a.”tr are expected to see how the structure ofan algebraic expression reveals properties ofthe

***l.t*lm.i,Hrm’-::llmmffi T;’o'”1:ureasoning’suchasusingexponentsto

Related Content Standards

A.APR.A.1 A.APR.C’4 A.APR.D.6 N.CN.C.8

80 The Common Core Mathematics Companion:The Standards Decoded, High School

Students see that com-plicated expressions are built up out of simpler ones. Part of this understanding means students know what
a factor is, know how factors and coe{Ecients are relatld, and know how constants, factors, and/or colfficients relate to terms i,
an expression. Students work with the structure of a complicated expression, identifuing the parts to help understand what the
expression means. Complex formulas in science or other disciplines are a rich ,orrr”. o”f ,ppications foithese standards. The
Doppler Effect formul

^, t =(ftf,J/r, “r,-,
be seen as a product but also as including a part that is a quotient in rvhich both the

numerator and denominator )rave a”slmilar format in which the velocih,of the lr.ates in a medium, c, are affected bv the velocity
of the receiver or source to the medium. Students can determine the relative r.,alues of each and make estimates rb”ri tt . ,lr.
of the Doppler Effect’ Students may also recognize that the computational rules the-v haye learned mean that the expression

[c +
‘, ) s is not equivalent,o (t + v, ), .

[c+v”./’o [l +r,)to

‘ Requires students to understand and use vocabulary such as

factor, coefficient, term, and like terms.
. Provides representations of expressions so students may

compare terms such as w + w and 2w tn contexts-that is,
P * 2(l + w) and P – 2l + 2w. By using pictures, manipu-
latives, and symbols, students can *ak. sense of these
equivalent expressions.

. uses problems such as compound interest t(, *
fr)”n ,o

students may identify important components in context,

such as I +
fiand lZn.

o Explain, in their own words, what factor, coefficient, term,
and like terms mean in the context of expressions.

o Identify factors, coefficients, different terms, and like terms
in expressions.

. Identify individual parts of an expression as a single entity
to make use of the structure of the expression.

b
F

FnF
I
I

E5E

-r:Il

Students frequently confuse the parts of an expression. A common error is assuming a constant is not a term or that an expression
such as 3w + z – 4xyhas four terms because the student is counting variables and n”ot terms. Manipulatives and area *od.lr rr.
one strategy to use for students with this misconception- When students have difficulty with vocabulary, having them restate a
word or phrase in their own words with an exampli and a non-example is helpful. Foi example, 3x and,4xrr.iik” terms, but
7x and 4x] are not because like terms have

^exactly
the same variableTactors regardless of the’coefficient. Contextual examples

of expressions, such as the Doppler Effect formula, give students an opportuni”ty to consider what the underlying structure’of an
expression can tell them, helping students parse complicated expressions into their components.

The aforementioned Doppler Effect formula is one example of a formula that may be applied in the modeling sequence.

Related Content Standards
A.SSE.A.2 A.APR.A.1 A.APR.C.4 A.APR.D.6 N.RN.A.1

Part 3 Algebra 81

L;’s# tfi* sf.ri;r-.{,r, jr 4,

f.f; e.;s r* {: rJ;# ii i.r j r: q

/:.a,iii”8:51/i: .’

ill;’1’gre n cf

?.’. .’: -. r’ +
tLl ,ri-*- , I

Of Sqr-rirl5
ilj ;fi

;f *:; ;:
,*r. rgsiirssfe ff; For example, see t’ – t’ as rx-i – &rn
– , i.,,i* fl;xrf*r*# *s,fxz – yr)(x, + fl.

n(Zn + l)(n + 1)
, whose strucfure allows sfudents to see the expression has a degree of l

the term
]n3,

which is helpful when studying rules for summation notation and integra-

and how exponents help rewrite the

Students view expressions from a dynamic perspectire-that is, there is progress or change when using expressions fl€rdbh’in &E
ferent formats tohnd the form that is most useful in a contexfual situation. In the standard, studentr consider ra – ,+ ,t 1-1- – ‘r–;
and as the factored form (x2 – y’)(*’+ y2). This can help students consider zeros and graphing patterns. Recognizing different
forms of an expression and being able to apply the forms is an important step mentioned in the Progressions for the Common C-olt

State Standards in Mathematics for Algebra (2013). An example in that document (p. 5) concerns looking at the expression for

the sum of a series of perfect squares,

with a leading coefficient of +, that is
)

tion,providinganunderpinningofcalculus.Rewriting:y=xz+2x+Iasahinomialsquarepatterny=(x+l)2canhelpstudenb
consider the graph of the function as a kanslation of y = x2 instead of having to calculate individual ordered pairs to determine

the graph.

W*xmt &fue SYt$ffiffiruT”$ d*;
o Explain, in their own words, how specific strucfures are

seen in different expressions.
. Rewrite expressions using structure to identify important

components of the expression (wh ere zeros may occur) or
to end behavior.

i-re$fu#t ek* Tffi&{h{f fe d**s:
. Provides problems that allow students to discover special

patterns that occur with the structure of expressions, such

as forms of the difference of squares for Algebra or connec-
tions of the sine and cosine when using the Pythagorean

Theorem to discover sinz 0 + cosz 0 – 1.

. Uses problems such as compound interest.

,(t + #) (t + f;) (, + fr) (r + fr)ezntimes)
as

,(t . fr) as t[(t . #)”] where students can see

the structure of I

exPression.

Lr’12

Related Content Standards
A.SSE.A.1 A.APR.C.5 A.APR.D.7 N.RN.B.3 N.CN.C.8 F,IF.B.7.C

Addressin g,Student M isconceptions and Common Errors

Sfudents need to recognize special patterns, such as difference of squares and greatest common f4s[6r”s,-so they-can use them in
new situations. This means rt rd..rt need to have a conceptual basis for patterns, such as an area model for difference of squares.

Students may struggle with finding the greatest common factor or seeing a pattern in an expression such.as +xa. + t! +4. One

shategy usesiubstlti:tion. In the eip.essio.r, students may use Lxz = u and rewrite the expression as uz + 4u + 4 and look for a

patter-n they know to rewrite the higher level polynomial. Substitution is a useful strategy that occurs when solving equations or

systems of equations and in other contexts in mathematics.

A2 The Common Core Mathematics Companion: The Standards Decoded, High School

x
A.SsE.B

ifi$ A.SSE.B.3: Chtose and produce an equivalent form of an expression to reveal and explain properties of
the quantity represented by the expression.*

Factor a quadratic expression to reveal the zeros of the function it defines.

Complete the square in a quadratic expression to reveal the maximum or minimum value of the
function it defines.

c. Use the properties of exponents to transform expressions for exponential functions. For example, the
expression l.l5’can be rewritten as (l.l5r/r2)121 = l.0l?12t to reveal the approximate equivalent monthly
interest rate if the annual rate is L5%.

A.SSE.B.4: Derive the formula for the sum of a finite geomehic series (when the common ratio is not 1),

and use the formula to solve problems. For example, calculate mortgage payments.*

Cluster B: Write expressions in equivalent forms to solve

problems.

Students choose among different equivalent forms of an expression to ana\ze the quant\ tepresented by the exptession.
Students learn several ways to solve a quadratic expression (and its related equation), whether by using quadratic formula,
graphing, completing the square, or factoring. Students need to recognize that each of these methods has its appropriate place.
For example, solving an easily factorable expression is an efitcient way to find the zeros ofa quadratic function that can also help
students make connections between the factored form and the graph by making the zeros evident in the symbolic form.
Completing the square leads to the form 7 = a(x – h)’ + ft, which helps students determine whether the quadratic opens up or
down and which helps students identifi the vertex (and whether it is a maximum or minimum). Completing the square also may
be used to derive the quadratic formula so that students understand how the formula connects to the expression. Sfudents need
an understanding of these forms and when to use them. A student does not need to complete the square to find the vertex of
I = x2 +2x + l, nor is multiplying x(x + 1)(x – 2) = 0 and then factoring an efi5cient method. The derivation of the sum of a finite
geometric series can be a direct consequence of using equivalent forms of expressions, where (x – 1)(x” – t + x” – 2 + . . . + x + I ) =
rn – I may be used.

Standards for Mathematical Practice
SFMP 1. Make sense of problems and persevere in solving them.
SFMP 2. Use quantitative reasoning.
SFMP 3. Construct viable arguments and critique the reasoning of others.
SFMP 4. Modelwith mathematics.
SFMP 5. Use appropriate tools strategically.
SFMP 6. Attend to precision.
SFMP 7. Look for and make use of structure.
SFMP 8. Look for and express regularity in repeated reasoning.

Students construct viable arguments as to whether expressions are equivalent or not. Relating equivalent expressions also

connects to the use of mathematical structure. Deriving the formula for a finite geometric series may involve looking for and
expressing repeated reasoning. The use of equivalent expressions and formulas may be a part of modeling and quantitative
reasoning. Tasks that ask students to generalize patterns, use geometric series, and explain properties of expressions all relate to
precision, communication, and problem solving.

Related Content Standards
F.IF.C.8 F.LE .A.2 G.GPE.A. 1 G.GPE ,A2 G.GPE.A.3

n
t

E
IIl

II
E’4

II
I

f
I

Part 3 Algebra 83

i,l :-,,.,i:r.’;e ; ilCi $ l’. *JU{

r-rr-) ”{‘:,}i’itr.r} l;’r l.4t,

.$G

\t ijf:? *:iji;ii,# :,*l.1 j

*,h-i,I a’f:ti*{*’t{i i’l *

f,* e’tt:r .-1 fit#6 #-r* $ie *.pi#r’+ Ssiii i-r ici

, u:,”;=Fi.,i{, tlt.: sir##i# ;I: # i?#**it’i-;lli *r3-.,,r,.: .:,

ii*,f,;r:*s.

#se? fi** #r#p#f iges *i *rxp[ir;.#,-]f,:; t* fr**sf*r’i:;

annual rate is 1SYo,

Students learn several waysto analyzequadratic expressions and their related functions. This standard focuses on factoring anc
completing the square. Each of these methods has its appropriate place. Students understand that factoring may be an effilien:
way to analyze a quadratic expression. Completing the square yields another form of a quadratic”expression thai is sometimes
called the vettex o_r graphing form. This form, y = a(x – h)2 + k, makes it relatively easy to find the vertex of the quadratic func-
tion and apply its form to transformations from geometry. Students also apply the properties of exponents to creaie equivalent
expressions that can give insight into the quantity described by the exponential expression.

:. ‘5:”‘

Leffu*€ €*”x* Tffi&f,F4ffiffi d**s:
o Provides problems that require different forms of quadratic

expressions for their solution (factoring to find roots and
determining the extreme value by completing the square).

. Requires students to explain their solution processes and
why the form of an expression they used was appropriate.

o Provides contextual examples that require equivalent
forms of an expression for a solution or for analysis (such
as making an expression to represent the volume of a box
formed when cutting corners from a piece of paper, then
folding the paper to make a box, and then finding the
maximum volume).

. Uses exponential decay growth applications to highlight
cases in which exponential properties may be used.

if1ti*t;x,.- ? hs *ii $]-i, i i”i $: fC*f-5 *$*:
. Explain how they used equivalent forms of quadratic erpre-

sions to determine important components of a quadratic
function (and its graph).

o Solve contextual problems using equivalent forms of
expressions. For example, students may find extrema, end
behavior, growth factors, or decay factors.

Addressing Student Misronceptions and Comrnon frrrE:rs

Students may initially have difficulty learning methods for factoring. It is beneficial to connect factoring with functions. Students
may start by looking at the graph of y = y – 3 and determining the zero, slope, and y-intercept. Next students look at a graph with
both /(x) = x – 3-and g(x) = x + l. After determining the zeros, students determine what the product of the graph is by using the
y-val’res ofsets of ordered pairs- For example, f (0) = -3 and g(0) = l, so the product of /(0) ind g(0) is -3. After finding sev-eral
sets of products, students plot the points and sketch the graph of the function. Then, students compare the graph and the equiva-
lent version of the product, y x’ – 2x – 3. Students may explore one or two other examples to see the

“o.,.,ectio,
of the graphs

and the quadratic function. The next step is to have students consider the function h(x) = yz + 4x – 5 given that h(x) = f (i)g(i)
and f(x) = x – l. By using ordered pairs, graphing, or whatever solution process they find useful, students can determine thi
missing function, g(x), is x + 5. This has set up a connection among graphs, zeros, and factors for the students.

Another method that may be helpful for students when factoring is to first consider multiplication of binomials using an area
model. Algebra tiles or drawings are both helpful. i

The area model can be reversed so that students are asked to find the missing factors for a given polynomial. Students, then,
make connections among the quadratic symbolic form and the factored form, as well as with the graphing sihrations mentioned
earlier. In the figure below, students are to find the missing factor for )ab + 6a + 2b + l0 given oni factoiis )a + 5.

3ab 5b

6a 10

A4 The Common Core Mathematics Companion: The Standards Decoded, High School

T
I

T
t,

I
I
!

The use of multiPle representa,tions when approaching factoring, as well as,viewing factoring as undoing multiplication, helpsstudents grasp how the use.qf,the distributiv;;ropertyind like t”erms i, -,rttiptyir,! i;;;;;;”J.jtlitactoring. It is important to
il,’I:-x:iilffi:li::,j”r:’,f”::Tf::”;::’,tf-i,,. , concrete reason ror i”,*r,i-g io i.;;;;#;;than viewing racttring as a

f,,:il*.x;1il:l’ili::ffiT’jj’ffi:i.*T-*T]:l:Tli:”-:::tl”l.,n: square studen* rearn the process of compreting
ffi:::i:?::f,:’,.'”1**:n H:*:l*,:l*;*:l:*”,1i11*qli,iiffi”‘;i;:,:Jl””ilJ:ffii::)T:x[Ti::#?Uf*ii::ffiffiiffffip.X;,t_u*l*1*i::]Fffiffi,ffi:ffil:il:ff::::’#15,’,+.i:l”,,ittliljt,,,,as well as symbolically. (See F.IF.C.B for further discussion.)

Though students have worked with rules of exponents in Grade 8, the sophisticated approach of hansforming functions is anextension of their knowledge’ That is, students are now going beyond reri,riting repeated multiplication with exponents but arelooking at ways to write eqrlivalent.*pt”ttio”r tt rt exten? thlir use ortrr” trwJore*ponerts th-Jy-.*plor”d. contextual exampleshelp students comprehend the use of ,ules, t},e *ost accessible being halfJife and compound interest.

e*nnectlons to M$deling

Students may use completing the square when considering elliptical modeling with an echo chamber or to find the focus

H:’#m.’ff ijffili’l&:Xllr:iX**l ,Xli:*1iff;”n’;”'”.i”r’,r,i. “,;,;d
-,;;”,ii.”i”.i”,

equ,tion m,y ,rso

Related Content Standards
N.RN.B.3 F.IF.C.7c F.IF.C.B G.GPE.A. 1 G.GPE.A.2 G.GPE.A.3

b.—

Part3 lAlgebra 85

bef eTO rmUta,lfOrthe sum of,lftnite geametric series (when the cammon rat’o ts nar t )’ dt’u srE “‘s Iv’r’.”
-$S

liiiffit,as. ror eiarnple, catculat3 nortgase payments.*

Students consider geometric sequen.ces in their study of functions (F.LE.A.2: Conshuct . . . geomehic sequences’ gfuen a !Er\

a description of , ,.htio.liipl. i;i.ai’g th. ,’*;;;;*t g””*”tif” “‘i”‘1o”n”tts
to som”e cotitextual problems (the mor!6r

payment context) and exi:’;IJ;i”dj#?il;i.il;;;;iffi;q”.r,.”, “.,d
series are related and used’

one way to derive the formula forih. ,r* of a geometric series is to use this identity:

(x- I)(x”-r + {-2 + “‘ +x+ 1) =g- 1’

– –, -J – t at’t-t whereaisthefirstterm,nisthenumberofterms,andrtsthec:M
Ageometrrcserleslswrtttenas(t+ar+af+…+6f.I,whereaisthefirstterm,nisthenumberofterms,andristhecm

ratio between consecutive terms. This can be called sn, the sum of the n terms of the geometric series. Because of our theoreo”

weknow *n -rl=xn-t*f,-2+…+x+l.so,a +ar+aP+…+ af -r-a(l+r +l+” ‘+f -l) -‘a1o’
x-l

ol – ,, .This may be too abstract for some students to follow Another way to derive the formula is to first look at str = d + o +
-‘l-r

.r o –,^). rarasrrhtractir Then,S,(I-r)=a(r’-l\’
al+…+af-tandthentoconsiderrs,=ar+af+…+bf.Subtractingrs,-s,

leavesa’a-a’

which then gives us S, – 4l’- r”L-r

What the TEAC*{ER does:

. Asks students to write a short geometric series and find its
series.

Describe how to derive the formula for the sum of a

geometric series in their own words’

f,ppty the sum for a geometric series to contextual

problems.

Addr[$*ifiS’ $tudefit M isegnceptiens and Cemmsn::Erf$rs

s*I”I.’ ;;; t ;;;r,* *,* geometric lgsuence’, r-ot:* I””‘.3*:1**kt-HT’ff:;H:T,TJ:t”‘H::l
:I1″”xTiln'”:X1T,””T$#i’,-Y.,xirfiff;:T!i:::::i.;.+;ar:Hit:”‘,:-iT::*:l:x,:x11h: n’n:ffix;e’#’
H:Ht3:ff”H:[il:]H$ffHJL1i!”*:r,:”I'”],T:,rg:+:ri*IliiH*:ff:1ffi:ffi:,ILli,:::'”‘tr,'[:::1’::
fri:’i;Yffi'”::f,ffi;ii)ll-‘?ffii”1T’li#'””ffiffi;;^*'”XL:#,H1flT::*’::i::n’:;:illfl”iil”‘l;
33:i1iil3,3i’il:Tl;lJiii,1ih*jitI*ffie;ffiil-{p*l::y.*:::”‘y*Tffffi”‘ff,H:,T:ffi1,:’,’,*lf,r”h a, those with an r value of I ‘0 t 2’ as may occur rn an aPPlcauur” ^ “‘;;;i: ;# r.;; ;hat the formula siates, instead of
ililili;i”; J”t*n”a i” ti’;wr”i the reacher D”..’, j::if ll,T]$n*:ffffiilffififfi;;; rrrrthe rormula without any roundational work’