In a traditional? curriculum, the topic of proportion was often limited to the cross-multiply and divide? algorithm.? What are three ways in which the CCSS

6.RP.A*

E 'F E t t E t E E E

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r r I I I I T

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6.RP.A.1: Uriderstand the concept of a ratio and-use ratio language to describe a ratio relationship

betweentwoquantities.i;-r;;;ib,-"Tle..rg!ooliingtnbe&.s.iithebirdhouseatthezoorlasl:l' because for every 2 wings tn l, i"i |'beak." "Fo, nnury iotu candidate Areceived' candidate C receited

nearly three votes."

5.RPA.2: Understand the concept of a unit rate a/b associated with a ratio a:b with b + 0, and

use rate language in tfr" .o"t.*i t'f , ,atio relationship. For example, "This recipe has a ratio of 3

cups of flour to 4 cups of sugar, so there is 1o cup of flour for each cup of sugar'" "we paid $7 5 for 1>

hamburgers, which is a rate of $5 per hamburger'"

rExpectations for unit rates in this grade are limited to non-complex fractions'

6.Rp.A.3: Use ratio and rate reasoning to solve real-world and mathematical Problems, e'g',

;;;;–ril;t;;;irbt"; ;i"quivaleniratios, tape diagrams, double number line diagrams, or

equations.

a. Make tables of equivalent ratios relating quantities with whole-number measurements' find

missing values in,f,. ,rif* ,"a plot th"e pri* of urtu.r on the coordinate plane' Use tables to

comPare ratios'

b. Solve unit rate problems including those involving unit pricing and constant speed' For

example, yittooni iourc io *o*4 to*n ,iin, oithotritn, hiw *ary lawns couldbe mowed

in 35' horurs? At what rate were lawns being mowed?

c. Find a percent of a quantity as a rate per I00 (e'g'')0%of a quantity *t"' ffi times the

quantity); solve problems involving finding the whole' given a part and the percent'

d.Useratioreasoningtoconvertmeasurementunits;manipulateandtransformunits apfropriately whei multiplying or dividing quantities'

oMajor cluster

cluster A: understand ratio concepts and use ratio reasoning to solve problems'

Grade 5 Overview

The focus for this cluster is the study of ratio concepts and the use of proportional reasoning to solve oroblems' Students learn

how ratios and rates are used to compare t*o q.,rr,'tnni.l;;;;i;;ffi'h;;;';;"iilffi';;;;,.thim' Sixth graders find or'rt

how ratios are used i, ,";i-;Ji;ir-riior, ,.,d dir"ou., solutions to p-""""i pioblems using ratio tables' tape diagrams' and

doublenumberlines.Studentsalsoconvertbetweenstandardunitsofmeasure.

Standards for Mathematical Practice

SFMP 1. Make sense of problems and persevere in solving them'

Sixth graders interpret and solve ratio problems'

t 6 The common core Mathematics companion: The Standards Decoded' Grades 6-8

In this standard, students learn to compare two quantities or measures such as 6:l or 10:2. These compafisons are called ratios.

Students discover that ratios can be written and described in different ways. For instance, 6:l uses a colon to separate values.

Ratios can also be stated with words such as 6 to l, or as a fraction such as -6. . Standard I focuses on understanding the concept

of a ratio, however, students should use ratio language to describe real-world experiences and use their understanding for decision

making.

ot

Help students discover that a ratio is a relationship or comparison of two quantities or measures. Ratios compare two rneasures of the same types of things such as the

number of one color of socks to another color of socks or

two different things such as the number of squirrels to birds

in the park. Ratios compare parts to a whole (part:whole) such as l0 of our 25 students take music lessons. Ratios

can also compare a part of one whole to another part of the same whole (part:part) such as the ratio of white socks

in the drawer to black socks in the drawer is 4:6. Ratios are

expressed or written as d to b, a:b, or t Compare and model ratios with real-world things such as

pants to shirts or hot dogs to buns. Ratios can be stated as

the comparison of 10 pairs of pants to I B shirts and can

be written as l-0, I0 to IB, or 10:18 and simplified to, -B ;, 5 to 9, or 5:9. Ensure that students understand how

the simplified values relate to the original numbers.

Ask students to create or find simple real-world problems to use in their learning such as, "There dre 2 Thoroughbred

horses and 6 Appaloosa horses in the field. As a ratio of

Thoroughbreds to Appaloosas it is: Z or 2 to 6 or 2:6 or 6

simplified as L, I to 3, or 7:3. Or, there are 14 girls and 18 5

boys in our math class. As a ratio of girls to boys it is: # , ]B

14 to 18, ar 14:18 or simplified as 7-^ ,7 to 9, or 7:9." Invite 9

students to share their real-world examples of ratios and

use ratio language to describe their findings such as, "for

every yote candidate A receiyed, candidate C received nearly

three votes." The problems students select or write can

also be used as cyclical reviews with distributed practice throughout the school year.

Focus on the following vocabulary terms: ratio, compare,

and simplifu.

Understand that a ratio is a comparison between quantities.

Determine when a ratio is describing part-to-part or part-to-

whole comparison.

o Describe ratio relationships between two quantities using ratio language.

o Use the different ratio formats interchangeably

(4:5,4 to 5, Il )

Some sixth graders may confuse the order of the quantities such as when asked to write the ratio of boys to g_irls in the- sentence,

"There are 1-4 girls and 18 boys in our math c/ass." Instead of writing l8: 14, some 5fudents may write 14: 18. Other students may

not recognize-th" diff.r.rr.. between a part-to-part ratio and a part-to-whole ratio such as,"There are 14-gi-*.gynpared to 18

boys in tte class (14:18 part-to-part); howewr, 14 of the 32 students in our class are girls Qa32 pyt19-whole)." To address these

cornmon misconceptions, ask students to label the quantities they are comparing such as 14 girls/I8 boys.

8 The Common Core Mathematics Companion: The Standards Decoded, Grades 5-8

a – H

– H

Ea i ':siations for unit rates in this grade are limited to non-complex fractions.

T-:s standard focuses student learning on the concept of a unit rate as a special kind of ratio. Students compare different units :,: n.ieasure such as the amount of rhoney earned to the hours worked while babysitting and calculate unit rates by setting up :atios and simplifying them. Students understand a situation in ratio form and write the unit that describes the situation using appropriate rate language with words such as per and symbols such as / to compare different units or measures.

flfl

Begin by exploring the difference between a ratio and a rate. Rate is a special ratio that compares two quantities with different units of measure. Share multiple examples for students to make sense of the concept for rate such as,

"LaShanda babysat for $:S for 7 hours." Or,'oDad new truck got 400 miles on 20 gallons of gas." Then explore the unit rate that expresses a ratio as part-to-one. Generate examples such as "Lashanda is paid a unit rdte of $5 per 1

hour for babysitting (5:1)" and "My dad's new truck gets 20 miles per gallon of gas (20:1)."

Ask students to locate and share real-world examples of cost per item or distance per time in newspapers, ads, or other media. (Note that in sixth grade, students do not work with unit rates expressed as complex fractions. Both the numerator and denominator of the original ratio will be

whole numbers.)

Model how to convert rates from fraction form to word form using per, each, or @ such as 360 milesll? gallons of gas – 30 miles per gallon of gas. Allow students to talk with each other and their teacher to make sense of what they are

learning and then write and share several rate conversion examples of their own.

Focus on the following vocabulary terms: ratios, rdtes, unit rdtes, compare, and perl@. Math journals or exit slips at the end of a math class with writing prornpts such as, "An example of a ratio and a problem that goes with it is.

provide closure.

Provide cyclical, distributed practice over time to continually review simple unit rate problems.

[Jnderstand rate as a ratio that compares two quantities with different units of measure.

l]nderstand that unit rates are the ratio of two measurements or quantities in which the second term means "one" such as 60 miles per one hour.

Interpret rate language with the @ symbol or with the words per andlor eacLt.

Solve unit rate problems.

Students often confuse the terms ratio, rate, and unit rate. Try using a paper foldable with vocabulary definitions to help students

with these confusing terms. To make the foldable, divide an 8 j x I l-inch sheet of blank paper in half horizontally. Then fold

it into thirds as if a letter is being folded to fit an envelope. Unfold and write a term on each of the sections. On the inside of the foldable, write the definitions that match each term. Students may want to cut on the vertical fold lines to flip up each section to practice the definitions.

fold line

rate unit rate

9

ratio

fold and cut lines

Part 1 Ratios and Proportional Relationships

^a

"oblemt use {&tioand rafe reasonins to sorve y3{-wai'! 1i,1"T:T*'Tj :';r-rions

by reasonirtg about tables of 4 ffi

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ffi;ffi iiasreatioeoi@tierwith*:S';I?::r1:i;",:ff:::':;i,:'missins

vinq,unit pricing and cbnstant speed'

salire unitra te p,roblems includinu,tlo:9-1n31r.,;J,-^ri raiirnc rould be mowed in 35 For exarnple, if

',,fl:1ffiflf#fffifi:l?'J|lii.31liii];"#'H;;'i#;;';;id b","owed in:s hours? Atwhat

. ;IJ:l:*;;lT# itiiw 1s a rltlyie: 100 t"'s'.3.Y:J::::IY **n'lfitimes the quantitv); sotve

probtems in**'^rliii'';; 'h";;;i;' siven a part and r:"

,Y"n' d, Use ratio reasoning to eanvert.measurefflent units; manipulate:and transfarm units appropriately when

" -iitiii wng ot"dividing quantities'

In these standards, students use reasoning about murtiprication and division to solve a variety of ratio and rate problems about

cuantities. They make triil, of "q,rivaleit

ratios r:Tii,,ru6;ili;;;'tc;ole-number *"rrur"*"r,t,, find missingvalues in the

t^bl.r, .r,d plot the 0.,*'"i"ii".l ""ii,.'"""iarr#pi"?r. rhey use trur"r l""""irprr" rrir"r1"a solve unit rate and constant

speed problems. Problems involving finding th" *hol" given a part and th" p.,.".,t, such as Z0% of aquantity *.,,,, ffi , ,,"

arso a focus. For these standards, students can use equivarent ratio tabres, tape diagrams' double number lines' or equations'

Students connect ratios and fractions'

values in

it took rate we re

CD Costs

What the TfACt-lER does:

. Exprore ratios and rates used in ratio tables and graphs

to solve probl"ms. Pose a ratio situation problem with

students such as"3 CD"ost $45 'Whatwould 8 CDs

cost? How mdny cDs-cam be purchased for $ I50'007" To

solve the problem, students can use ratios, unit rates' and

multiplicative ,.,'o"i"g iy 9""1'g iid filling in the

missing values on a .f,r?t. tft.y shJuld note that if three

CDs cost $45, one Ci;;fi "ott $15' Every CP purchased

is an additional $1t' $1i iiltt the number of CDs = the

cost.TheywriteanequationsuchasC-$15n.

$1 20

$1 05

$go

$75

$oo

$45

$so

$15

Investigate unit rate problems, including unit pricing

such ,r, "gli.i. Si;i h-.a, 12'iz. drinks for $'99' Stop Here

has r6-oz.'-ariw rris r "rg . which drink cosrs the least per

ounce?" A;i;; students to create ratio and rate reasoning

examples to comPare and solve real-world problems'

Students could use newspapers, store ads, or online ads to

find th" .*I"-rpf.t ,r,a *Ltf the comparisons' Ask students

to use ,.rto"i;r to determine the better buys'

. Explore finding a percent of a quanlity as a rate per 100

such as 4A%of a quantity means # times the quantity'

NotingthataPercentisarateperl00,modelhowa (continued)

Ask students to plot the points,o" 1 coordinate plane

and draw conclisions ,bo,rt what is happening with the

pr"Ui.* ,bou.. Students should reason that for every one

movement to the right on the x-axis, the y-axis increases to

' 15x. Also, for every one movement to the left on the x-axis'

th" Y-r*is decreases bY I 5 '

10 The common core Mathematics companion: The Standards Decoded, Grades 6-8

3456 Number of GDs

e e -{

-{

Td IId IIi

; =t:::.: : i: l. repre sented with a hundreds grid by -,. -:.. ; .:,l,-, units. Have students write this as a fraction

-:

– . a ie cinral r0.40), and a percent $A%). Consider

u:rxrE a pe rcent w'heel (see Reproducible 1) or use double nunrber lines and tape diagrams in which the whole is 100 to find the rate per hundred.

o Solr.e problems involving finding the whole, given a part and the percent such as, "What is 40% of 60? 80% of what number is 300? Or 50 is 30% of what number?"

. Examine the process of how to use ratio reasoning to convert measurement units such as, "How mdny centimeters are in 5 feet?" Use the information that I inch = /.54 cm. Represent the conversion of 12 inches : I ft as a conversion lactor in ratio form, 12. ilches

.

I foot

Then multiply lZ-ilches x + = 60 inches. I foot I

Create and interpret a table of equivalent ratios.

Plot pairs of values from a table to a coordinate plane.

Use a table to compare ratios and find missing values using ratios.

Explain the difference betwden a ratio and a unit rate.

[Jnderstand that rate problems compare two different units, such as revolutions per minute.

Solve real-world problems using ratios and rates.

Reason to determine the better buy.

Write a percent as a rate over 100, including percents greater than 100 and less than I .

Find the percent of a number using rate methods.

Represent the relationship of part to whole to describe percents using models.

Convert units by multiplication or division.

ild T4

Then 60 inches x z'54 cm – 15 2.4 cm. I inch

(Iote that conversions can be made between units within a

medsurement system such as inches to feet or between systems

such as miles to centimeters.)

Allow students to talk with each other and their teacher to make sense of what they are learning.

Focus on the following vocabulary terms: ratios, rdtes, unit rdtes, equivalent ratios, percents, ratio tables, and tape diagrams.

Provide cyclical, distributed practice over time to continually practice unit rate problems.

Some sixth graders misunderstand and believe that a percent is always a natural number less than or equal to 100. To help with this misconception, provide examples of percent amounts that are greater than 100% and percent amounts that are less than I %. Try using a percent wheel for developing this understanding. See Reproducible 1.

T 'It EI E1 ts {

Efl– h != h,t b:il b

= E ET

EI i,E[-r Part 1 Ratios and Proportional Relationships 1'l

7,'nt

t'

7.RP.A*

7.RP.A.l: Compute unit rates associated with ratios of fractions, including ratios of lengths, areas

and other quantities measured in like or different :units. For example, if a person walks L mile in

each ! hour, compute the unit rate as the complex fraction Z mibs per hour, equi'talently 2 miles per t'-Lnour. 4

7.RP.A.2: Recognize and represent proportional relationships between quantities.

a. Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a

straight line through the origin.

b. Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.

c. Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased dt o constdnt pice p, the relationship between the total cost

and the number of items can be expressed as t = pn.

d. Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate.

7.RP.A.3: Use proportional relationships to solve multi-step ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent enor.

oMajor cluster

Cluster A: Analyze proportional relationships and use them to solve real-world and mathematical problems.

Grade 7 Overview

These standards extend what students learned in Grade 6 about ratios to analyzingproportions and proportional relationships. Students calculate unit rates with complex fractions and move to recognizing and representing proportional relationships in equations and on graphs. These skills and understandings are used to solve multi-step ratio and percent problems involving real-

world scenarios such as interest, tax, shopping sales, and so on.

' Standards for Mathematical Practice

SFMP 1. Make sense of problems and persevere in solving them.

Students solve multi-step ratio and real-world percent problems.

SFMP 3. Construct viable arguments and critique the reasoning of others.

Students recognize proportional relationships from non-proportional ones and discuss their reasoning with others.

E t tr E t E t t E E E E E t E E E tr E Eh

-F16 The Common Core Mathematics Companion: The Standards Decoded, Grades 5-8

r {crnpute r"l,tif raies

ir)s€,ts$red fn ifke *r :1

csrnpl*x fractinn ? e

assocrated with ratios of fractions, including ratios of lengths, areas and olher q,uantities

different units. For example, if a person walks J mile in each ] hour, compute the unit rate at the24 milesperhouLequivalently2milesperhour……i-

This standard focuses on computing unit rates using ratios of fractions known as complex fractions. In a complex fraction, the

-L

numerator, denominator, or both are fractions. In the standard, { is an example of a complex fraction. Complex fractions can

,L+ be interpreted as division statements. For example, ! "^nbe

thought of ar ] + { . Applications include situations where the

4 quantities are measured in different units such as miles per hour, pounds per square foot, feet per second, and so on.

What th* TfiACI'{ER d*es: o Explore unit rates with ratios of fractions and compare them

to unit rates with whole numbers from Grade 6.

o Treat complex fractions as division of fractions.

. Set up error analysis scenarios where students can identify errors in computing unit rates with complex fractions. For

example, Homer calculated that if a person walks + mile 2

every + hour, the unit rate is 2 miles. However, Homer,4 made an error. F ind his error, correct it, and explain to

Homer why 2 miles is not the correct answer.

o Provide opportunities for students to compute the unit rates

in real-world problems.

What the STUDENTS do: o Discover that the structure of computing unit rates with

whole numbers is the same concept as unit rates with ratios of fractions.

. Compute unit rates in real-world problems that involve complex fractions.

. In writing, explain the errors that can be made when computing unit rates with complex fractions and unlike units.

Sometimes the format of a complex

I + I . Discuss how the divisionZ4

Addressing Student Misconceptions and Common Errors

It is not uncommon to find seventh-grade students who are not fluent with fraction division.

fraction confuses them when they are used to seeing fraction division written horizontally as

bar in the complex fraction means the same as the symbol +.

For students having difficulty underStanding unit rate and those having trouble with different units such as miles per hour,

pictures and diagrams may help. Use the example from this standard: lf a person walks L mile in each hour, compute the unit' _L- 2 4

rate as the complex fraction t miles per hour, equivalently 2 miles per hour. Model with a diagram as shown. The bar represents

I hour broken into ] hour rlg.n.r,t . 4

mile I 1

I 1

I 1

I 1

mile mile mile

From this diagram, students can see that the word problem is showing mile everv J

hour.I 1

I z

L 18 The Common Core Mathematics Companion: The Standards Decoded, Grades 6-8

L- Fb.5

LHIHIH EL- I-J

E LI L L-)

Sections a-d of 7.RP.A.2 break down the standard to give guidance on ways to recognize and represent proportional relationships.

This standard emphasizes hryo metliods for deciding whether a proportional relationship exists. One method is to use equivalent ratios in a table. Ifthe ratios are equivalent, then you have a proportional relationship such as:

# of people in a room

# of hands in the room

The other method is to graph the relationship on a coordinate plane and observe whether the graph is a straight line that goes through the origin. Note that computation using cross-multiplication is not a part of this standird.

r2345 2+68?

Explore proportional reasoning scenarios with students to be sure they understand the meaning of proportional relationships in context before using the tables or graphs. While some number combinations may be proportional, the real-world example attached to the numbers may not be. [Jse examples and non-examples for students to identify and compare. An example is: "2 music downloads cost $1.98; therefore, 4 music downloads cost $3.96." A non-example ts: "Three boys cdn run d mile in 10 minutes; therefore, 6 boys cdn run d mile in 20 minutes."

Ask students to write their own examples and non-examples of proportional relationships. Student work can be shared and discussed.

Discuss equivalent ratios with the students. Ask them to suggest some equivalent pairs. Relate to equivalent

fractions. Display the pairs as the students suggest them

in the form t=i, where b + }and d * 0. Define two

equivalent ratios as a proportion.

Pose examples of proportions written with the quantities in different positions. Encourage students to decide if there is more than one correct way to set up a proportion. For exampl e: "Set up d proportion showing that 3 out of 15 students are girls is the sdme ratio as I out of 5 students are girls."

fuk students to explain how they know proportion for the example.

is not a correct

. Craph two ratios on a coordinate plane from a proportional scenario and look for a straight line that goes thrrough the origin to determine if the two ratios are proportional. For exampl e: "Mdria sells necklaces and makes a profit of $6 for each necklace. How much monq) does she make for selling 3 neckldces?"

.(3, 18)

Pose the task to students: Select other points on the graphed line and determine if they are also proportional.

3 1 15 5 r 5

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Provide examples of equivalent and non-equivalent ratios to students for them to test with a table to decide if they are proportions. Conversely, present students with a table for a

context and ask them to determine if all of the entries in the table are proportional.

Pa rt 1 Ratios a nd Proportiona I Relationsh ips 1 9

I

What the STUDENTS do: . Sort real-world exampl.Ss of proportional relationships from

non-examples. Students can create their own examples to demonstrate that they understand the concept of proportional relationships when there is a context attached.

o Communicate orally and/or in writing that a proportion is a statement of two equivalent ratios. Students apply what they know about equivalent fractions to equivalent ratios"

o Model proportional relationships by creating tables; determine if a proportional relationship exists from a given table.

Model relationships on graphs to determine if they are proportional.

Test their hypotheses about whether a proportional relationship exists between any two points on the lines graphed. Students may draw the conclusion that all points on the line are proportional toall other points on the line by relying on tables, verbal statements, or logic al arguments to draw the conclusion q

E E

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E

c d

T E

Addressing Student Misconceptions and Common Errors

While graphing, students may need to be reminded that the same types of quantities need to be graphed on the same axis. For example, when checking to determine if I0 cans of soda for $2 is proportional to 50 cans of soda for $10, the cans of soda must both be represented on the same axis and the dollar amounts must be on the other axis. Ensure students are using graph paper or graphing calc

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