math g6 m4 module overview

8/18/2019 Math g6 m4 Module Overview

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6GRADE

New York State Common Core

Mathematics Curriculum

GRADE 6 • MODULE

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!Mo"ule # !/pressions and !uations

Table o Contents(

E$%ressions an" E&uationsMo"ule O'er'iew3

Topic A1 2elationships o the perations 46(EE(A()5.......................................................(3

Lesson (1 The 2elationship o Addition and Subtraction........................................()

Lesson '1 The 2elationship o "ultiplication and 6ivision.....................................')

Lesson 31 The 2elationship o "ultiplication and Addition.....................................3

Lesson 1 The 2elationship o 6ivision and Subtraction........................................(

Topic 71 Special Notations o perations 46(EE(A(!8 6(EE(A(*c5....................................)0

Lesson )1 !/ponents............................................................................................. )'

Lesson ,1 The rder o perations........................................................................,'

Topic C1 2eplacin* Letters and Numbers 46(EE(A(*c8 6(EE(A(5....................................93

Lesson 91 2eplacin* Letters with Numbers............................................................9)

Lesson :1 2eplacin* Numbers with Letters............................................................:)

Topic 61 !/pandin*8 ;actorin*8 and 6istributin* !/pressions 46(EE(A(*a8 6(EE(A(*+86(EE(A()8 6(EE(A(5........................................................................................................ 9

Lesson 1


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6•"odule verviewNYS COMMON CORE MA-.EMA-/CS CURR/CULUM

Mo"ule # !/pressions and !uations

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Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported Licens

Topic ;1 All perations...................................................30

Lesson '1 "ulti-Step ?roblems>All perations.................................................3'0

Topic @1 Applications o !uations 46(EE(0(18 6(EE(0(68 6(EE(0(28 6(EE(0(38 6(EE(C(45....................................................................................................................................... 330

Lesson 301 ne-Step ?roblems in the 2eal


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Mo"ule O'er'iew •NYS COMMON CORE MA-.EMA-/CS CURR/CULUM


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%rade , B "odule

E$%ressions an" E&uations

!2!<n "odule 8 students e/tend their arithmetic work to include usin* letters to representnumbers. Students understand that letters are simpl$ Dstand-insE or numbers and thatarithmetic is carried out e/actl$ as it is with numbers. Students e/plore operations interms o verbal e/pressions and determine that arithmetic properties hold true withe/pressions because nothin* has chan*ed>the$ are still doin* arithmetic with numbers.Students determine that letters are used to represent speci+c but unknown numbersand are used to make statements or identities that are true or all numbers or a ran*e o numbers. Students understand the importance o speci$in* units when de+nin* letters.Students sa$ DLet K represent Farol$nGs wei*ht in poundsE instead o DLet K

represent Farol$nGs wei*htE because wei*ht cannot be a speci+c number until it isassociated with a unit8 such as pounds8 ounces8 or *rams. The$ also determine that it isinaccurate to de+ne K as Farol$n because Farol$n is not a number. Studentsconclude that in word problems8 each letter 4or variable5 represents a number8 and itsmeanin* is clearl$ stated.

To be*in this module8 students are introduced to important identities that are useul insolvin* euations and developin* pro+cienc$ with solvin* problems al*ebraicall$. n

Topic A8 students understand the relationships o operations and use them to *enerate

euivalent e/pressions 46(EE(A()5. 7$ this time8 students have had ample e/periencewith the our operations since the$ have worked with them rom kinder*arten throu*h%rade ) 4!(OA(0()8 )(OA(0(15. The topic opens with the opportunit$ to clari$ thoserelationships8 providin* students with the knowled*e to build and evaluate identitiesthat are important or solvin* euations. n this topic8 students discover and work withthe ollowin* identities1 w− x+ x=w 8 w+ x− x=w ,a÷b ∙b=a , a ∙b÷b=a 4when b≠0 58 and 3 x= x+ x+ x . Students also discover

that i 12÷ x=4 8then 12− x− x− x− x=0 .

n Topic 78 students e/perience special notations o operations. The$ determine that3 x= x+ x+ x is not the same as x

3

8 which is x ∙ x ∙ x . Appl$in* their prior

knowled*e rom %rade )8 where whole number e/ponents were used to e/press powerso ten 41(N0-(A(*58 students e/amine e/ponents and carr$ out the order o operations8includin* e/ponents. Students demonstrate the meanin* o e/ponents to write andevaluate numerical e/pressions with whole number e/ponents 46(EE(A(!5.

Students represent letters with numbers and numbers with letters in Topic C. n past*rades8 students discovered properties o operations throu*h e/ample 4!(OA(0()8 )(OA(0(15. Now8 the$ use letters to represent numbers in order to write the propertiesprecisel$. Students realiHe that nothin* has chan*ed because the properties still remain


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statements about numbers. The$ are not properties o lettersI nor are the$ new rulesintroduced or the +rst time. Now8 students can e/tend arithmetic properties rommanipulatin* numbers to manipulatin* e/pressions. n particular8 the$ develop the

ollowin* identities1 a ∙b=b∙a 8 a+b=b+a 8 g ∙1=g 8 g+0=g 8 g÷1=g ,

g÷g=1 , and 1÷ g=1

g . Students understand that a letter in an e/pression

represents a number.


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such as the volume o a cube8 =s 8 *iven the side len*th8 or the volume o arectan*ular prism8 V =l ⋅w ⋅h 8 *iven those dimensions 46(EE(A(*c5.

n Topic %8 students are introduced to the act that euations have a structure similar to

some *rammatical sentences. Some sentences are true1 D%eor*e


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;ocus Standards

A%%l8 an" e$ten" %re'ious un"erstan"in9s o5 arithmetic to

al9e+raic e$%ressions(*

6(EE(A(!


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6(EE(0(1Understand solvin* an euation or ineualit$ as a process o answerin* auestion1 which values rom a speci+ed set8 i an$8 make the euation orineualit$ trueM Use substitution to determine whether a *iven number in aspeci+ed set makes an euation or ineualit$ true.

6(EE(0(6Use variables to represent numbers and write e/pressions when solvin* areal-world or mathematical problemI understand that a variable canrepresent an unknown number8 or8 dependin* on the purpose at hand8 an$number in a speci+ed set.

6(EE(0(2Solve real-world and mathematical problems b$ writin* and solvin*euations o the orm x+ p=q and px=q or cases in which p , q , and x are all nonne*ative rational numbers.

6(EE(0(3c or xc or x


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Un"erstan" %ro%erties o5 multi%lication an" the relationshi%+etween multi%lication an" "i'ision(

)(OA(0(1Appl$ properties o operations as strate*ies to multipl$ and divide.)

Eamples* +f 6×4=24 is no%n, then 4×6=24 is also no%n.(-ommutati'e property of multiplication.)3×5×2 can #e found #y 3×5=15 , then 15×2=30 , or #y 5×2=10

, then 3×10=30 . (Associati'e property of multiplication.) no%ing that 8×5=40 and 8×2=16 , one can nd 8×7 as8×(5+2)=(8×5)+(8×2)=40+16=56 . (/istri#uti'e property.)

Gain 5amiliarit8 with 5actors an" multi%les(

(OA(0(;ind all actor pairs or a whole number in the ran*e 1 = 100 .2eco*niHe that a whole number is a multiple o each o its actors.6etermine whether a *iven whole number in the ran*e 1 = 100 is a

multiple o a *iven one-di*it number. 6etermine whether a *iven wholenumber in the ran*e 1 = 100 is prime or composite.

Geometric measurement# un"erstan" conce%ts o5 an9le an"measure an9les(

(MD(C(1 2eco*niHe an*les as *eometric shapes that are ormed wherever twora$s share a common endpoint8 and understand concepts o an*lemeasurement1

a. An an*le is measured with reerence to a circle with its center at thecommon endpoint o the ra$s8 b$ considerin* the raction o the circular

arc between the points where the two ra$s intersect the circle. An an*lethat turns throu*h 1/360 o a circle is called a Done-de*ree an*le8E

and can be used to measure an*les.

b. An an*le that turns throu*h n one-de*ree an*les is said to have anan*le measure o n de*rees.

(MD(C(6 "easure an*les in whole-number de*rees usin* a protractor. Sketchan*les o speci+ed measure.

(MD(C(2 2eco*niHe an*le measure as additive.


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1(OA(A(*


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6(R7(A()Use ratio and rate reasonin* to solve real-world and mathematicalproblems8 e.*.8 b$ reasonin* about tables o euivalent ratios8 tapedia*rams8 double number line dia*rams8 or euations.

a. "ake tables o euivalent ratios relatin* uantities with whole-numbermeasurements8 +nd missin* values in the tables8 and plot the pairs ovalues on the coordinate plane. Use tables to compare ratios.

b. Solve unit rate problems includin* those involvin* unit pricin* andconstant speed. For eample, if it too 7 hours to mo% 4 la%ns,then at that rate, ho% many la%ns could #e mo%ed in 35 hours2 At%hat rate %ere la%ns #eing mo%ed2

Com%ute


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represent one number to make the euation true. That number is 8 8 so

a=8 .


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m∙m∙m∙m=m . These are especiall$ important when movin* rom the*eneral orm back to a speci+c value or the variable.

-erminolo98New or Recentl8 /ntro"uce" -erms

E&uation 4An euation is a statement o eualit$ between two e/pressions.5

E&ui'alent E$%ressions 4Two e/pressions are eui'alent i both e/pressionsevaluate to the same number or ever$ substitution o numbers into all thevariables in both e/pressions.5

E$%onential Notation 5or ;hole Num+er E$%onents 4Let m be a nonHero

whole number. ;or an$ number a 8 the e/pression am

is the product o m

actors o a 4i.e.8

m×¿

am=a ∙a ∙⋅⋅⋅ ∙ a⏟

¿

5. The number a is called the #ase, and

m is called the eponent or po%er o a .5

E$%ression 4An epression is a numerical e/pression8 or it is the result oreplacin* some 4or all5 o the numbers in a numerical e/pression with variables.5

Linear E$%ression 4A linear epression is an e/pression that is euivalent tothe sumdiJerence o one or more e/pressions where each e/pression is either anumber8 a variable8 or a product o a number and a variable.5

Num+er Sentence 4A num#er sentence is a statement o eualit$ between twonumerical e/pressions.5

Numerical E$%ression 4A numerical epression is a number8 or it is an$combination o sums8 diJerences8 products8 or divisions o numbers thatevaluates to a number.5

Solution o5 an E&uation 4A solution to an euation with one variable is anumber such that the number sentence resultin* rom substitutin* the numberor all instances o the variable in both e/pressions is a true number sentence.

an euation has more than one variable8 then a solution is an ordered tuple onumbers such that the number sentence resultin* rom substitutin* each numberrom the tuple into all instances o its correspondin* variable is a true number

sentence.5 -ruth >alues o5 a Num+er Sentence 4A number sentence is said to be true i

both numerical e/pressions evaluate to the same numberI it is said to be false otherwise. True and alse are called truth 'alues.5

>alue o5 a Numerical E$%ression 4The 'alue of a numerical epression is thenumber ound b$ evaluatin* the e/pression.5


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>aria+le 4A 'aria#le is a s$mbol 4such as a letter5 that is a placeholder or anumber.5

?amiliar -erms an" S8m+ols6

6istribute

!/pand

;actor

Number Sentence

?roduct

?roperties o perations 4distributive8 commutative8 associative5

Ouotient

Sum

Term

True or ;alse Number Sentence

ariable or Unknown Number

Su**ested Tools and 2epresentations 7ar model

%eometric +*ures

?rotractors

2apid


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as possible and hold their work up or their teacher to see their answers as soon as the$have the answer read$. The teacher *ives immediate eedback to each student8pointin* andor makin* e$e contact with the student and respondin* with an armationor correct work8 such as D%ood KobQE8 DResQE8 or DCorrectQE8 or respondin* with *uidance

or incorrect work such as DLook a*ain8E DTr$ a*ain8E DCheck $our work8E and so on. nthe case o the 2

math g6 m4 module overview

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