limited Time Offer As a token of our appreciation you will receive a $5 Amazon Gift Card through

email if you join before May 15th 2016 amp

----~

applied mathematics practices for the 21st century

Wayne State University Invites You to Join Our Facebook Group to access NO COST Middle School Math Activities Aligned with Common Core Standards

When will I ever use this

Wayne State University through its Applied Math

Practices for the 21st Century

(AMP21) initiative is here to help answer that question We provide NO COST

activities and teacher guides to help make

mathematics more relevant Each

activity uses real -world decision

contexts

Signing up is easy

1 First please read the following pages to learn more about AMP21 and our middle school research project

2 Go to httpswwwfacebookcom groupsAMP21CM3

3 To join the group just click on the Join Group button on top of the page You can leave the group at any point of time no strings attached

4 Download any materials you would like AT NO COST

Access to our Activities is as

simple as joining our Facebook

Group Our middle

school activities involve

using ratios percentages and

proportional relationships to solve problems

grounded in real world contexts such as science

health consumer education and

more

To learn more about AMP21 contact Kenneth Chelst PhD at kchelstwayneedu or visit

httpwwwappliedmathpracticescom

AMP21 Math Activities for Middle School Classes

Here at Applied Math Practices a group of school mathematics teachers and faculty members from Wayne State University have developed a of activities middle school classes These activities help students conceptual understanding of ratios proportions also developing fluency in carrying out procedures to problems involving relationships

Our activities blend mathematics and real-world problem contexts to deliver an new curriculum

Improves aptitudes and attitudes towards mathematics We provide fOUfOfO examples with Common Core Standards

Each comes with a teachers that a reference to specific Common Core Standards Attached to this invitation you will also fmd

List of all the middle that will be over 2- A activity (Whats the better deal) and its guide and 3- Our journal article rates and percentages was published in

(GB)

AMP21 Facebook Group

Every other we post new activities to AMP2l who join our Facebook group can get access to these activities at NO COST Group members can also share their opinions about the activities discuss related and share own real-world

To join the group please go to ~~~~~~~~~~~~~~~~~~~c

The group administrators will make sure that the group content is relevant to middle math and that are helpful relevant and You can leave the group at any point of time no strings

This effort is to teachers use of social media in educational share information about effective programs such as ours

data about teachers We wi II use and discuss and make no to specific individuals If you are interested in participating in

check the following link to read the information sheet and decide if you want to

We look to meeting you in our Facebook group

Sincerely yours

Professor Kenneth PhD Engineering

Detroit MI

List of all the Middle School Activities applied mathematics practices (to be distributed over time) tor tho 2 1STCelruy

1

2

3

4

5

6

7

1

2

3

4

5

6

7

8

9

10

II

12

Short Tasks (Take less than 1 class period)

Accident Rates

Car left in the sun

How long is this song

Meeting Conflicts

Stopping Distance

Text Messaging

Will I Make the Cut

Medium Tasks (Take 1 class period)

A Sale at CVS

Candy Bar Sales

Cookies Anyone

If the Chair Fits

Jennys Lemonade Stand

My Gummy Bears All Wet

Teacher Student Ratios

The Sixth Man

To Hydrate or to Dehydrate

Whats the Better Deal

Which Coupon will Save You More Money

Which Data Plan Should I Pick

Long Tasks (Take more than 1 class period)

I Arcade Games

2 Dude whats up with Your Food

3 Earning Money

4 Enough is Enough (Sodium intake)

5 Exercise Away Big Mac

6 Lets Get Down to Kite Business

7 Light Bulbs

8 Lions and Tigers and Potholes

9 Mileage Club

10 Which Data Plan Should I Pick

11 Rainbow Loom Business

12 Read for Speed

13 Relay for Life

14 T-Shirt Sizes

15 Water Usage

16 Wilma Unlimited

Pagc l o f 6 Whats the Belte r DeaP applied malhemolics pracices

for fhe 2i sf cenfu y

Whats the Better Deal

Deal 1 (l2-ounce bottles)

Deal 2 (12-ounce bottles)

Deal 3 (48 12-ounce cans)

copy 2015 Wayne State University All rights reserved

2 of6Whats the Better Dea1

Deal 4 (2 liter bottles)

Deal 5 (4 packs with 8 bottles in each)

Which is the better deal

Deal 1

Deal 2

Deal 3

Deal 4

Deal 5

copy 2015 Wayne State University All rights reserved

Page 3 of 6Whats the Better Dea l) applied molhemalics plocHces

or the 2 st cntufY

Now support your decision by showing your work and writing a justification

Work Space for Deal 1 Work Space for Deal 2

Work Space for Deal 3 Work Space for Deal 4

Work Space for Deal 5

Explain which deal is the best deal and why

copy 2015 Wayne State Universi ty All rights reserved

Page 4 of 6Whats the Better Deal applied mathematics procHces

fOf lhe 21st c ontuf)i

Whats the Better Deal - Teacher Notes and Solutions

Grade Level 6

Common Core State Standards for Mathematics 6RPA3b Solve unit rate problems including those involving unit pricing and constant speed

6RPA3d Use ratio reasoning to convert measurement units manipulate and transform units appropriately

when multiplying or dividing quantities

Time Needed 1 class period

10 minutes - Hand out the deal illustration sheet of the deals or show on board Start a discussion

with the class of what they think the better deal would be based on intuition What is the criteria

for being a better deal (The intended criteria is cheapest price but be ready for other student

criteria) Take a hands up poll and tally results on the board

15 minutes - What do we see based on the illustrations Have students collaborate with a partner

about strategies that could be used to solve and decide which the better deal is Continue the

discussion having students share their thoughts

30 minutes - Hand out the blank worksheet and have students calculate and justify their results

Materials Required Deals Illustration Sheet

Blank worksheet for showing work and writing a justification

Focus Question Whats the best deal

Other Discussions What other factors may be considered in determining if something is a good deal

copy 2015 Wayne State University All rights reserved

Page 5 of6What s the Better Deall applied mathematics practices

or t 21st cenhuy

Original Task and Solution

Whats the Better Deal

Now support your dec ision by showing your work and writing a justification

Work Space for Deal 1

24 cans $849 J can $0354

12 oz can $0354can 1 oz $00295

Work Space for Deal 2 8 bottles $429 1 bottle (fy $0536

12 oZlbottle $0536Ibottle I oz $00447

With CARD 8 bottles $369 1 bottle $0461

12 oZlbottle $0461Ibottle 1 oz $00384

Work Space for Deal 3 Work Space for Deal 4

48 cans $1 100 2 liters = 676 ounces 1 can $0229

2 bottles $300 12 ozcan $0229can 1352 oz $300 1 oz $00190 1 oz $0022

Work Space for Deal 5

8 bottles in pack

4 packs $1 100

32 bottles $1100 I bottle $0344 J 69 ozbottle $0344Ibottle 1 oz $00204

copy 2015 Wayne State University All rights reserved

Page 6 of6Whats the Better DeltllJ applied mathemofiC$ practIces

fallne 21 st c enfuty

Explain which deal is the best deal and why

Students may try to lise a ratio sllch as cost per can However since each canbottle is a different

size they are not comparable The commonality in each is the ratio of cost per ounce They

may not all solve it this way which will lead to some terrific post-assignment discussion

Heres how each deal plays out

Deal I $0295ounce ($0378ounce if deposit is included)

Deal 2 $044701lnce ($0530ounce ifdeposit is included)

Deal 3 $0191 01lnce ($ 0274ounce ifdeposit is included)

Deal 4 $0222ounce ($ 023601lnce if deposit is included)

Deal 5 $0204oLlnce ($0263ounce if deposit is included)

Intended answer Deal 3 is the best deal because it is the cheapest per ounce Students may

want to include the deposit (making Deal 4 the best) but since you get that money back it doesnt

factor into the cost

copy 2015 Wayne State University All rights reserved

RETHINKING RATIOS RATES AND PERCENTAGES Kenneth Chelst Asli OzgOn-Koca and Thomas Edwards develop a shared understanding of these concepts

P roportional reasoning has been defined as the capstone of elementary concepts which leads to it being a cornerstone of higher

mathematics (Lesh et ai 1988) Understanding ratios and rates is fundamental to proportional reasoning The Common Core State Standards for Mathematics lists the calculation of ratios rates proportions and percentages as critical 6th and 7th grade skills It is relatively easy to write a short scenario and then ask students to calculate the ratio rate proportion or percentage All that the scenario requires is two numbers However these scenarios rarely accurately portray the ways in which these concepts are used to make a point when used in context

Two mathematics education professors and an engineering professor all with degrees in applied mathematics met to create contextual scenarios for the concepts of ratio rate and percentage We soon realized that we had neither a shared understanding of nor vocabulary for these cornerstone principles Through discussions we unpacked our knowledge to build a shared understanding for these concepts This article is written to serve both as a professional learning and growth piece As such it aims to help teachers reconsider their content knowledge of ratios rates and percentages Moreover we hope that teachers will be encouraged to think about how they might introduce these concepts and design activities that target better understanding for their students

First we categorized ratios as commensurate when the numbers being compared are measured using the same units and non-commensurate when

r 00()

QOO

0000 ~ 30000 1 shy 000

0000

0

measured using different units We saw that nonshycommensurate ratios always led to rates However commensurate ratios were more complex We were able to differentiate sub-categories leading to rates percentages or ratios While discussing these categories we had three common threads the roles of ratios and rates as they are used in real-life the role of the context and the role of units

Non-Commensurate Ratios

There are two distinct categories of ratio calculation In one context the two terms are non-commensurate with different units of measure miles and gallons cost and units deaths and miles driven Nonshycommensurate ratios are always expressed as rates The denominator may be a single unit and the rate is per unit as in miles per gallon or the denominator may be a standard number for example automobile deaths per 100 million vehicle miles driven The standard number used in the denominator is usually chosen so that the rate is greater than 1 to make the rate more meaningful see Figure 1 (below) The 2010 automotive fatality rate in the US was reported as 110 fatalities per 100 million vehicle miles traveled (VM T) Imagine using a different denominator one million miles The rate would then be 0011 fatalities per million miles travelled Most people find it easier to comprehend 11 fatalities than 0011 fatalities

Sometimes different rates are used for the same situation when viewed in a different way For example the speed of an automobile is usually given in miles per hour or kilometers per hour Doing so facilitates calculating or estimating the time

1 9

800 700

00

bull Fl1llilJOi F~IOInty Rale ~r 100M VMT

Figure 1 Fatalities and fatality rate per 100 million Vehicle Miles Travelled (VMT)

M~y 201 26

RETHINKING RATIOS RATES AND PERCENTAGES

Auto Vehicle Fatalities I 2009

Total 33883

Alcohol Impaired 10759

Fatality Rate per 100 million Vehicle 115

Miles Travelled

I 2010

32885 -998 -29

10228 -531 -49

110 -005 -43

Table 1 Auto vehicle fatality rates 2009 and 2010

needed to travel a known distance For example if you know your rate of travel in miles per hour you can easily estimate time of arrival at a location 175 miles away However when thinking about stopping time or reaction time when driving an automobile feet per second or meters per second are more meaningful For example it is much easier to visualize how far you would travel while reacting to a need to stop if you think of your speed as 88 feet per second rather than 60 miles per hour although these two rates are equivalent

Rates are often used to track change over time to compare different rates for similar entities or to make a forecast The automotive fatality rate is tracked over time to see if actions that have been taken are making driving safer (NHTSA 2012 see Figure 1 (page 26) and Table 1 (above)) The denominator used to calculate the rate may be based on the best unit of comparison or may be chosen simply for convenience Determining the automotive fatality per 100 million miles traveled (VMT) allows for comparing one year to another when the number of miles travelled may be significantly different Notice in Figure 1 the number of vehicle fatalities per year has gone up and down over a 60-year period as more people have owned cars and have travelled more miles in total However there is a strong general downward trend in the death rate per VMT

The cost per ounce is used to compare differentshysized items of the same product Miles per gallon is often used to compare two different vehicles to see which achieves better fuel economy Miles per gallon allows you to forecast how far you can travel before needing to buy fuel how much it will cost to drive a certain distance or your average annual vehicle fuel expenditure

Commensurate Ratios

In commensurate ratios the units of measure are the same or can be considered to be the same For example the number of people in prison and the population or the number of men and women are all measured in units that are essentially the same human beings

When the set whose measure is in the numerator is a subset of that whose measure is in the denominator the ratio can be presented as either a rate or as a percentage or both The rate of prison incarceration in the US is calculated by dividing the number of people in prison by the total population The US rate is 784 per 100000 of population These rates are often calculated for different subgroups For example if the numerator and denominator were limited to adults that rate would be 962 adults per 100000 adults This rate might be tracked over time or the rates of specific subgroups might be compared Sometimes these rates are converted to percentages to emphasize relative magnitude

Another type of ratio involves comparing the same two statistics over time In this instance the ratio represents the rate of growth or decline reported as a percentage For example the number of automobile fatalities is always compared from year-to-year to determine a trend as seen in Table 1 From 2009 to 2010 the number of auto vehicle fatalities decreased by 29 If adjusted for VMT the percent decline is 43

One of the most common uses of ratio involves comparing two different subsets Sometimes this type of ratio is used to emphasize the magnitude of difference For example in the US there is an imbalance in choices of STEM careers by females and males see Table 2 (page 28) Students notice this imbalance most when they see the ratio of females to males in a STEM-related course

In 2006 92 of the females entering college who chose a biological sciences major as compared to 73 of the male freshman This is equivalent to 1 female for every 08 males a reversal of the ratio found in 1976 In engineering the percentages were 24 females and 146 males a ratio of 1 female to 61 males This is an increase from the ratio in 1976 In computer science the percentages were 03 to 21 or 1 female for every 7 males a much smaller ratio than in 1976 (Sax et aI 2010)

1 IV ]01-l W atnl orguk 27

AND PERCENTAGES ~--- shy

ce 06 12 12

34

12

173

46 21

Table 2 Percent of each gender hl~~nn a stem major by year

One reason researchers track career choice data is to partially gender differences in incomes STEM careers are on average paying careers In 2009 the median income offu-time year round male workers in the US was 27 Nmnr~rI to for females The female-toshymale earnings ratio was 077 Thus females earned 77 cents for every dollar earned by a male (Majority Staff of the Joint Economic Committee of the United States 2010)

Another reason to compare ratios is to the allocation of resources For example every ten years in the US the National Highway Traffic

Administration commissions random sobriety checks of more than 6500 drivers in 60 locations across the US Blood alcohol levels are tested

between 930 am and 330 pm (2482) pJemy 11 pm to midnight (2123) and

Hnir 1 am to 3am (1948) During rtTUTltgt

hours approximately 1 in 500 drivers were alcohol impaired That rate increased to 1 in 83 in the early evening and 1 in 21 between 1 am and 3am (Lacey et al 2009) However a Nebraska highway

administrator claimed that in areas with high concentrations of bars after 11 pm the ratio increases to 1 in 4 (Duggan 2009) That means that for every four drivers stopped in the of a bar 1 of them will be found to be alcohol impaired Armed with such data police patrols can be allocated more effectively

Concluding Thoughts

We use rates ratios and percentages in daily life to make decisions and to make comparisons The purposeful use of ratios with appropriate units of measure can enable one to see the situation more clearly As the three of us poundIfIJfll()OEO a shared understanding of ratios we saw that the context solidified our understanding as we distinguished commensurate and nonshycommensurate ratios and their

contextual we were able both to communicate and to achieve a common understanding Therefore we believe authentic contextual problems and tasks are needed for middle school students to achieve a better

understanding of ratios which as previously stated is a cornerstone concept for mathematics

Kenneth Chelst Asli and Thomas Edwards Wayne State University Detroit

References

Duggan J (2009) Dodging drunk drivers Statistics on driving are sobering Lincoln Joumal Star December 312009 httpjournalstarcoml

001 cc4c002eOhtml

J H T Furr-Holden D Voas R B E Torres P A S Ramirez A Brainard K and Berning A (2009) 2007 National Roadside ofAlcohol and Use Drivers Alcohol Results NHTSA Report No (DOT HS 811 248)

Lesh R Post and Behr M (1 Proportional reasoning In J Hiebert and M Behr Number Concepts and in the Middle Grades Reston VA Lawrence Erlbaum and National Council of Teachers of Mathematics

National Highway Safety Administration (NHTSA) (2012) Motor vehicle crashes Overview Traffic Safety Facts Research Note NHTSA No (DOT HS 811 (Revised 2012) httpwww-nrdnhtsadot govPubs811552pdf

Sax L J Jacobs J A and T A (2010) Womens Representation in Science and

Fields of 1976-2006 presented at the annual meeting of the Association for the Study of Education November 2010 Indianapolis IN

Majority Staff of the Joint Economic Committee of the United States 0 December) Invest in Women Invest in America A Comprehensive Review of Women in the US Economy Washington DC

28 May

List of all the Middle School Activities applied mathematics practices (to be distributed over time) tor tho 2 1STCelruy

1

2

3

4

5

6

7

1

2

3

4

5

6

7

8

9

10

II

12

Short Tasks (Take less than 1 class period)

Accident Rates

Car left in the sun

How long is this song

Meeting Conflicts

Stopping Distance

Text Messaging

Will I Make the Cut

Medium Tasks (Take 1 class period)

A Sale at CVS

Candy Bar Sales

Cookies Anyone

If the Chair Fits

Jennys Lemonade Stand

My Gummy Bears All Wet

Teacher Student Ratios

The Sixth Man

To Hydrate or to Dehydrate

Whats the Better Deal

Which Coupon will Save You More Money

Which Data Plan Should I Pick

Long Tasks (Take more than 1 class period)

I Arcade Games

2 Dude whats up with Your Food

3 Earning Money

4 Enough is Enough (Sodium intake)

5 Exercise Away Big Mac

6 Lets Get Down to Kite Business

7 Light Bulbs

8 Lions and Tigers and Potholes

9 Mileage Club

10 Which Data Plan Should I Pick

11 Rainbow Loom Business

12 Read for Speed

13 Relay for Life

14 T-Shirt Sizes

15 Water Usage

16 Wilma Unlimited

Pagc l o f 6 Whats the Belte r DeaP applied malhemolics pracices

for fhe 2i sf cenfu y

Whats the Better Deal

Deal 1 (l2-ounce bottles)

Deal 2 (12-ounce bottles)

Deal 3 (48 12-ounce cans)

copy 2015 Wayne State University All rights reserved

2 of6Whats the Better Dea1

Deal 4 (2 liter bottles)

Deal 5 (4 packs with 8 bottles in each)

Which is the better deal

Deal 1

Deal 2

Deal 3

Deal 4

Deal 5

copy 2015 Wayne State University All rights reserved

Page 3 of 6Whats the Better Dea l) applied molhemalics plocHces

or the 2 st cntufY

Now support your decision by showing your work and writing a justification

Work Space for Deal 1 Work Space for Deal 2

Work Space for Deal 3 Work Space for Deal 4

Work Space for Deal 5

Explain which deal is the best deal and why

copy 2015 Wayne State Universi ty All rights reserved

Page 4 of 6Whats the Better Deal applied mathematics procHces

fOf lhe 21st c ontuf)i

Whats the Better Deal - Teacher Notes and Solutions

Grade Level 6

Common Core State Standards for Mathematics 6RPA3b Solve unit rate problems including those involving unit pricing and constant speed

6RPA3d Use ratio reasoning to convert measurement units manipulate and transform units appropriately

when multiplying or dividing quantities

Time Needed 1 class period

10 minutes - Hand out the deal illustration sheet of the deals or show on board Start a discussion

with the class of what they think the better deal would be based on intuition What is the criteria

for being a better deal (The intended criteria is cheapest price but be ready for other student

criteria) Take a hands up poll and tally results on the board

15 minutes - What do we see based on the illustrations Have students collaborate with a partner

about strategies that could be used to solve and decide which the better deal is Continue the

discussion having students share their thoughts

30 minutes - Hand out the blank worksheet and have students calculate and justify their results

Materials Required Deals Illustration Sheet

Blank worksheet for showing work and writing a justification

Focus Question Whats the best deal

Other Discussions What other factors may be considered in determining if something is a good deal

copy 2015 Wayne State University All rights reserved

Page 5 of6What s the Better Deall applied mathematics practices

or t 21st cenhuy

Original Task and Solution

Whats the Better Deal

Now support your dec ision by showing your work and writing a justification

Work Space for Deal 1

24 cans $849 J can $0354

12 oz can $0354can 1 oz $00295

Work Space for Deal 2 8 bottles $429 1 bottle (fy $0536

12 oZlbottle $0536Ibottle I oz $00447

With CARD 8 bottles $369 1 bottle $0461

12 oZlbottle $0461Ibottle 1 oz $00384

Work Space for Deal 3 Work Space for Deal 4

48 cans $1 100 2 liters = 676 ounces 1 can $0229

2 bottles $300 12 ozcan $0229can 1352 oz $300 1 oz $00190 1 oz $0022

Work Space for Deal 5

8 bottles in pack

4 packs $1 100

32 bottles $1100 I bottle $0344 J 69 ozbottle $0344Ibottle 1 oz $00204

copy 2015 Wayne State University All rights reserved

Page 6 of6Whats the Better DeltllJ applied mathemofiC$ practIces

fallne 21 st c enfuty

Explain which deal is the best deal and why

Students may try to lise a ratio sllch as cost per can However since each canbottle is a different

size they are not comparable The commonality in each is the ratio of cost per ounce They

may not all solve it this way which will lead to some terrific post-assignment discussion

Heres how each deal plays out

Deal I $0295ounce ($0378ounce if deposit is included)

Deal 2 $044701lnce ($0530ounce ifdeposit is included)

Deal 3 $0191 01lnce ($ 0274ounce ifdeposit is included)

Deal 4 $0222ounce ($ 023601lnce if deposit is included)

Deal 5 $0204oLlnce ($0263ounce if deposit is included)

Intended answer Deal 3 is the best deal because it is the cheapest per ounce Students may

want to include the deposit (making Deal 4 the best) but since you get that money back it doesnt

factor into the cost

copy 2015 Wayne State University All rights reserved

RETHINKING RATIOS RATES AND PERCENTAGES Kenneth Chelst Asli OzgOn-Koca and Thomas Edwards develop a shared understanding of these concepts

P roportional reasoning has been defined as the capstone of elementary concepts which leads to it being a cornerstone of higher

mathematics (Lesh et ai 1988) Understanding ratios and rates is fundamental to proportional reasoning The Common Core State Standards for Mathematics lists the calculation of ratios rates proportions and percentages as critical 6th and 7th grade skills It is relatively easy to write a short scenario and then ask students to calculate the ratio rate proportion or percentage All that the scenario requires is two numbers However these scenarios rarely accurately portray the ways in which these concepts are used to make a point when used in context

Two mathematics education professors and an engineering professor all with degrees in applied mathematics met to create contextual scenarios for the concepts of ratio rate and percentage We soon realized that we had neither a shared understanding of nor vocabulary for these cornerstone principles Through discussions we unpacked our knowledge to build a shared understanding for these concepts This article is written to serve both as a professional learning and growth piece As such it aims to help teachers reconsider their content knowledge of ratios rates and percentages Moreover we hope that teachers will be encouraged to think about how they might introduce these concepts and design activities that target better understanding for their students

First we categorized ratios as commensurate when the numbers being compared are measured using the same units and non-commensurate when

r 00()

QOO

0000 ~ 30000 1 shy 000

0000

0

measured using different units We saw that nonshycommensurate ratios always led to rates However commensurate ratios were more complex We were able to differentiate sub-categories leading to rates percentages or ratios While discussing these categories we had three common threads the roles of ratios and rates as they are used in real-life the role of the context and the role of units

Non-Commensurate Ratios

There are two distinct categories of ratio calculation In one context the two terms are non-commensurate with different units of measure miles and gallons cost and units deaths and miles driven Nonshycommensurate ratios are always expressed as rates The denominator may be a single unit and the rate is per unit as in miles per gallon or the denominator may be a standard number for example automobile deaths per 100 million vehicle miles driven The standard number used in the denominator is usually chosen so that the rate is greater than 1 to make the rate more meaningful see Figure 1 (below) The 2010 automotive fatality rate in the US was reported as 110 fatalities per 100 million vehicle miles traveled (VM T) Imagine using a different denominator one million miles The rate would then be 0011 fatalities per million miles travelled Most people find it easier to comprehend 11 fatalities than 0011 fatalities

Sometimes different rates are used for the same situation when viewed in a different way For example the speed of an automobile is usually given in miles per hour or kilometers per hour Doing so facilitates calculating or estimating the time

1 9

800 700

00

bull Fl1llilJOi F~IOInty Rale ~r 100M VMT

Figure 1 Fatalities and fatality rate per 100 million Vehicle Miles Travelled (VMT)

M~y 201 26

RETHINKING RATIOS RATES AND PERCENTAGES

Auto Vehicle Fatalities I 2009

Total 33883

Alcohol Impaired 10759

Fatality Rate per 100 million Vehicle 115

Miles Travelled

I 2010

32885 -998 -29

10228 -531 -49

110 -005 -43

Table 1 Auto vehicle fatality rates 2009 and 2010

needed to travel a known distance For example if you know your rate of travel in miles per hour you can easily estimate time of arrival at a location 175 miles away However when thinking about stopping time or reaction time when driving an automobile feet per second or meters per second are more meaningful For example it is much easier to visualize how far you would travel while reacting to a need to stop if you think of your speed as 88 feet per second rather than 60 miles per hour although these two rates are equivalent

Rates are often used to track change over time to compare different rates for similar entities or to make a forecast The automotive fatality rate is tracked over time to see if actions that have been taken are making driving safer (NHTSA 2012 see Figure 1 (page 26) and Table 1 (above)) The denominator used to calculate the rate may be based on the best unit of comparison or may be chosen simply for convenience Determining the automotive fatality per 100 million miles traveled (VMT) allows for comparing one year to another when the number of miles travelled may be significantly different Notice in Figure 1 the number of vehicle fatalities per year has gone up and down over a 60-year period as more people have owned cars and have travelled more miles in total However there is a strong general downward trend in the death rate per VMT

The cost per ounce is used to compare differentshysized items of the same product Miles per gallon is often used to compare two different vehicles to see which achieves better fuel economy Miles per gallon allows you to forecast how far you can travel before needing to buy fuel how much it will cost to drive a certain distance or your average annual vehicle fuel expenditure

Commensurate Ratios

In commensurate ratios the units of measure are the same or can be considered to be the same For example the number of people in prison and the population or the number of men and women are all measured in units that are essentially the same human beings

When the set whose measure is in the numerator is a subset of that whose measure is in the denominator the ratio can be presented as either a rate or as a percentage or both The rate of prison incarceration in the US is calculated by dividing the number of people in prison by the total population The US rate is 784 per 100000 of population These rates are often calculated for different subgroups For example if the numerator and denominator were limited to adults that rate would be 962 adults per 100000 adults This rate might be tracked over time or the rates of specific subgroups might be compared Sometimes these rates are converted to percentages to emphasize relative magnitude

Another type of ratio involves comparing the same two statistics over time In this instance the ratio represents the rate of growth or decline reported as a percentage For example the number of automobile fatalities is always compared from year-to-year to determine a trend as seen in Table 1 From 2009 to 2010 the number of auto vehicle fatalities decreased by 29 If adjusted for VMT the percent decline is 43

One of the most common uses of ratio involves comparing two different subsets Sometimes this type of ratio is used to emphasize the magnitude of difference For example in the US there is an imbalance in choices of STEM careers by females and males see Table 2 (page 28) Students notice this imbalance most when they see the ratio of females to males in a STEM-related course

In 2006 92 of the females entering college who chose a biological sciences major as compared to 73 of the male freshman This is equivalent to 1 female for every 08 males a reversal of the ratio found in 1976 In engineering the percentages were 24 females and 146 males a ratio of 1 female to 61 males This is an increase from the ratio in 1976 In computer science the percentages were 03 to 21 or 1 female for every 7 males a much smaller ratio than in 1976 (Sax et aI 2010)

1 IV ]01-l W atnl orguk 27

AND PERCENTAGES ~--- shy

ce 06 12 12

34

12

173

46 21

Table 2 Percent of each gender hl~~nn a stem major by year

One reason researchers track career choice data is to partially gender differences in incomes STEM careers are on average paying careers In 2009 the median income offu-time year round male workers in the US was 27 Nmnr~rI to for females The female-toshymale earnings ratio was 077 Thus females earned 77 cents for every dollar earned by a male (Majority Staff of the Joint Economic Committee of the United States 2010)

Another reason to compare ratios is to the allocation of resources For example every ten years in the US the National Highway Traffic

Administration commissions random sobriety checks of more than 6500 drivers in 60 locations across the US Blood alcohol levels are tested

between 930 am and 330 pm (2482) pJemy 11 pm to midnight (2123) and

Hnir 1 am to 3am (1948) During rtTUTltgt

hours approximately 1 in 500 drivers were alcohol impaired That rate increased to 1 in 83 in the early evening and 1 in 21 between 1 am and 3am (Lacey et al 2009) However a Nebraska highway

administrator claimed that in areas with high concentrations of bars after 11 pm the ratio increases to 1 in 4 (Duggan 2009) That means that for every four drivers stopped in the of a bar 1 of them will be found to be alcohol impaired Armed with such data police patrols can be allocated more effectively

Concluding Thoughts

We use rates ratios and percentages in daily life to make decisions and to make comparisons The purposeful use of ratios with appropriate units of measure can enable one to see the situation more clearly As the three of us poundIfIJfll()OEO a shared understanding of ratios we saw that the context solidified our understanding as we distinguished commensurate and nonshycommensurate ratios and their

contextual we were able both to communicate and to achieve a common understanding Therefore we believe authentic contextual problems and tasks are needed for middle school students to achieve a better

understanding of ratios which as previously stated is a cornerstone concept for mathematics

Kenneth Chelst Asli and Thomas Edwards Wayne State University Detroit

References

Duggan J (2009) Dodging drunk drivers Statistics on driving are sobering Lincoln Joumal Star December 312009 httpjournalstarcoml

001 cc4c002eOhtml

J H T Furr-Holden D Voas R B E Torres P A S Ramirez A Brainard K and Berning A (2009) 2007 National Roadside ofAlcohol and Use Drivers Alcohol Results NHTSA Report No (DOT HS 811 248)

Lesh R Post and Behr M (1 Proportional reasoning In J Hiebert and M Behr Number Concepts and in the Middle Grades Reston VA Lawrence Erlbaum and National Council of Teachers of Mathematics

National Highway Safety Administration (NHTSA) (2012) Motor vehicle crashes Overview Traffic Safety Facts Research Note NHTSA No (DOT HS 811 (Revised 2012) httpwww-nrdnhtsadot govPubs811552pdf

Sax L J Jacobs J A and T A (2010) Womens Representation in Science and

Fields of 1976-2006 presented at the annual meeting of the Association for the Study of Education November 2010 Indianapolis IN

Majority Staff of the Joint Economic Committee of the United States 0 December) Invest in Women Invest in America A Comprehensive Review of Women in the US Economy Washington DC

28 May

Pagc l o f 6 Whats the Belte r DeaP applied malhemolics pracices

for fhe 2i sf cenfu y

Whats the Better Deal

Deal 1 (l2-ounce bottles)

Deal 2 (12-ounce bottles)

Deal 3 (48 12-ounce cans)

copy 2015 Wayne State University All rights reserved

2 of6Whats the Better Dea1

Deal 4 (2 liter bottles)

Deal 5 (4 packs with 8 bottles in each)

Which is the better deal

Deal 1

Deal 2

Deal 3

Deal 4

Deal 5

copy 2015 Wayne State University All rights reserved

Page 3 of 6Whats the Better Dea l) applied molhemalics plocHces

or the 2 st cntufY

Now support your decision by showing your work and writing a justification

Work Space for Deal 1 Work Space for Deal 2

Work Space for Deal 3 Work Space for Deal 4

Work Space for Deal 5

Explain which deal is the best deal and why

copy 2015 Wayne State Universi ty All rights reserved

Page 4 of 6Whats the Better Deal applied mathematics procHces

fOf lhe 21st c ontuf)i

Whats the Better Deal - Teacher Notes and Solutions

Grade Level 6

Common Core State Standards for Mathematics 6RPA3b Solve unit rate problems including those involving unit pricing and constant speed

6RPA3d Use ratio reasoning to convert measurement units manipulate and transform units appropriately

when multiplying or dividing quantities

Time Needed 1 class period

10 minutes - Hand out the deal illustration sheet of the deals or show on board Start a discussion

with the class of what they think the better deal would be based on intuition What is the criteria

for being a better deal (The intended criteria is cheapest price but be ready for other student

criteria) Take a hands up poll and tally results on the board

15 minutes - What do we see based on the illustrations Have students collaborate with a partner

about strategies that could be used to solve and decide which the better deal is Continue the

discussion having students share their thoughts

30 minutes - Hand out the blank worksheet and have students calculate and justify their results

Materials Required Deals Illustration Sheet

Blank worksheet for showing work and writing a justification

Focus Question Whats the best deal

Other Discussions What other factors may be considered in determining if something is a good deal

copy 2015 Wayne State University All rights reserved

Page 5 of6What s the Better Deall applied mathematics practices

or t 21st cenhuy

Original Task and Solution

Whats the Better Deal

Now support your dec ision by showing your work and writing a justification

Work Space for Deal 1

24 cans $849 J can $0354

12 oz can $0354can 1 oz $00295

Work Space for Deal 2 8 bottles $429 1 bottle (fy $0536

12 oZlbottle $0536Ibottle I oz $00447

With CARD 8 bottles $369 1 bottle $0461

12 oZlbottle $0461Ibottle 1 oz $00384

Work Space for Deal 3 Work Space for Deal 4

48 cans $1 100 2 liters = 676 ounces 1 can $0229

2 bottles $300 12 ozcan $0229can 1352 oz $300 1 oz $00190 1 oz $0022

Work Space for Deal 5

8 bottles in pack

4 packs $1 100

32 bottles $1100 I bottle $0344 J 69 ozbottle $0344Ibottle 1 oz $00204

copy 2015 Wayne State University All rights reserved

Page 6 of6Whats the Better DeltllJ applied mathemofiC$ practIces

fallne 21 st c enfuty

Explain which deal is the best deal and why

Students may try to lise a ratio sllch as cost per can However since each canbottle is a different

size they are not comparable The commonality in each is the ratio of cost per ounce They

may not all solve it this way which will lead to some terrific post-assignment discussion

Heres how each deal plays out

Deal I $0295ounce ($0378ounce if deposit is included)

Deal 2 $044701lnce ($0530ounce ifdeposit is included)

Deal 3 $0191 01lnce ($ 0274ounce ifdeposit is included)

Deal 4 $0222ounce ($ 023601lnce if deposit is included)

Deal 5 $0204oLlnce ($0263ounce if deposit is included)

Intended answer Deal 3 is the best deal because it is the cheapest per ounce Students may

want to include the deposit (making Deal 4 the best) but since you get that money back it doesnt

factor into the cost

copy 2015 Wayne State University All rights reserved

RETHINKING RATIOS RATES AND PERCENTAGES Kenneth Chelst Asli OzgOn-Koca and Thomas Edwards develop a shared understanding of these concepts

P roportional reasoning has been defined as the capstone of elementary concepts which leads to it being a cornerstone of higher

mathematics (Lesh et ai 1988) Understanding ratios and rates is fundamental to proportional reasoning The Common Core State Standards for Mathematics lists the calculation of ratios rates proportions and percentages as critical 6th and 7th grade skills It is relatively easy to write a short scenario and then ask students to calculate the ratio rate proportion or percentage All that the scenario requires is two numbers However these scenarios rarely accurately portray the ways in which these concepts are used to make a point when used in context

Two mathematics education professors and an engineering professor all with degrees in applied mathematics met to create contextual scenarios for the concepts of ratio rate and percentage We soon realized that we had neither a shared understanding of nor vocabulary for these cornerstone principles Through discussions we unpacked our knowledge to build a shared understanding for these concepts This article is written to serve both as a professional learning and growth piece As such it aims to help teachers reconsider their content knowledge of ratios rates and percentages Moreover we hope that teachers will be encouraged to think about how they might introduce these concepts and design activities that target better understanding for their students

First we categorized ratios as commensurate when the numbers being compared are measured using the same units and non-commensurate when

r 00()

QOO

0000 ~ 30000 1 shy 000

0000

0

measured using different units We saw that nonshycommensurate ratios always led to rates However commensurate ratios were more complex We were able to differentiate sub-categories leading to rates percentages or ratios While discussing these categories we had three common threads the roles of ratios and rates as they are used in real-life the role of the context and the role of units

Non-Commensurate Ratios

There are two distinct categories of ratio calculation In one context the two terms are non-commensurate with different units of measure miles and gallons cost and units deaths and miles driven Nonshycommensurate ratios are always expressed as rates The denominator may be a single unit and the rate is per unit as in miles per gallon or the denominator may be a standard number for example automobile deaths per 100 million vehicle miles driven The standard number used in the denominator is usually chosen so that the rate is greater than 1 to make the rate more meaningful see Figure 1 (below) The 2010 automotive fatality rate in the US was reported as 110 fatalities per 100 million vehicle miles traveled (VM T) Imagine using a different denominator one million miles The rate would then be 0011 fatalities per million miles travelled Most people find it easier to comprehend 11 fatalities than 0011 fatalities

Sometimes different rates are used for the same situation when viewed in a different way For example the speed of an automobile is usually given in miles per hour or kilometers per hour Doing so facilitates calculating or estimating the time

1 9

800 700

00

bull Fl1llilJOi F~IOInty Rale ~r 100M VMT

Figure 1 Fatalities and fatality rate per 100 million Vehicle Miles Travelled (VMT)

M~y 201 26

RETHINKING RATIOS RATES AND PERCENTAGES

Auto Vehicle Fatalities I 2009

Total 33883

Alcohol Impaired 10759

Fatality Rate per 100 million Vehicle 115

Miles Travelled

I 2010

32885 -998 -29

10228 -531 -49

110 -005 -43

Table 1 Auto vehicle fatality rates 2009 and 2010

needed to travel a known distance For example if you know your rate of travel in miles per hour you can easily estimate time of arrival at a location 175 miles away However when thinking about stopping time or reaction time when driving an automobile feet per second or meters per second are more meaningful For example it is much easier to visualize how far you would travel while reacting to a need to stop if you think of your speed as 88 feet per second rather than 60 miles per hour although these two rates are equivalent

Rates are often used to track change over time to compare different rates for similar entities or to make a forecast The automotive fatality rate is tracked over time to see if actions that have been taken are making driving safer (NHTSA 2012 see Figure 1 (page 26) and Table 1 (above)) The denominator used to calculate the rate may be based on the best unit of comparison or may be chosen simply for convenience Determining the automotive fatality per 100 million miles traveled (VMT) allows for comparing one year to another when the number of miles travelled may be significantly different Notice in Figure 1 the number of vehicle fatalities per year has gone up and down over a 60-year period as more people have owned cars and have travelled more miles in total However there is a strong general downward trend in the death rate per VMT

The cost per ounce is used to compare differentshysized items of the same product Miles per gallon is often used to compare two different vehicles to see which achieves better fuel economy Miles per gallon allows you to forecast how far you can travel before needing to buy fuel how much it will cost to drive a certain distance or your average annual vehicle fuel expenditure

Commensurate Ratios

In commensurate ratios the units of measure are the same or can be considered to be the same For example the number of people in prison and the population or the number of men and women are all measured in units that are essentially the same human beings

When the set whose measure is in the numerator is a subset of that whose measure is in the denominator the ratio can be presented as either a rate or as a percentage or both The rate of prison incarceration in the US is calculated by dividing the number of people in prison by the total population The US rate is 784 per 100000 of population These rates are often calculated for different subgroups For example if the numerator and denominator were limited to adults that rate would be 962 adults per 100000 adults This rate might be tracked over time or the rates of specific subgroups might be compared Sometimes these rates are converted to percentages to emphasize relative magnitude

Another type of ratio involves comparing the same two statistics over time In this instance the ratio represents the rate of growth or decline reported as a percentage For example the number of automobile fatalities is always compared from year-to-year to determine a trend as seen in Table 1 From 2009 to 2010 the number of auto vehicle fatalities decreased by 29 If adjusted for VMT the percent decline is 43

One of the most common uses of ratio involves comparing two different subsets Sometimes this type of ratio is used to emphasize the magnitude of difference For example in the US there is an imbalance in choices of STEM careers by females and males see Table 2 (page 28) Students notice this imbalance most when they see the ratio of females to males in a STEM-related course

In 2006 92 of the females entering college who chose a biological sciences major as compared to 73 of the male freshman This is equivalent to 1 female for every 08 males a reversal of the ratio found in 1976 In engineering the percentages were 24 females and 146 males a ratio of 1 female to 61 males This is an increase from the ratio in 1976 In computer science the percentages were 03 to 21 or 1 female for every 7 males a much smaller ratio than in 1976 (Sax et aI 2010)

1 IV ]01-l W atnl orguk 27

AND PERCENTAGES ~--- shy

ce 06 12 12

34

12

173

46 21

Table 2 Percent of each gender hl~~nn a stem major by year

One reason researchers track career choice data is to partially gender differences in incomes STEM careers are on average paying careers In 2009 the median income offu-time year round male workers in the US was 27 Nmnr~rI to for females The female-toshymale earnings ratio was 077 Thus females earned 77 cents for every dollar earned by a male (Majority Staff of the Joint Economic Committee of the United States 2010)

Another reason to compare ratios is to the allocation of resources For example every ten years in the US the National Highway Traffic

Administration commissions random sobriety checks of more than 6500 drivers in 60 locations across the US Blood alcohol levels are tested

between 930 am and 330 pm (2482) pJemy 11 pm to midnight (2123) and

Hnir 1 am to 3am (1948) During rtTUTltgt

hours approximately 1 in 500 drivers were alcohol impaired That rate increased to 1 in 83 in the early evening and 1 in 21 between 1 am and 3am (Lacey et al 2009) However a Nebraska highway

administrator claimed that in areas with high concentrations of bars after 11 pm the ratio increases to 1 in 4 (Duggan 2009) That means that for every four drivers stopped in the of a bar 1 of them will be found to be alcohol impaired Armed with such data police patrols can be allocated more effectively

Concluding Thoughts

We use rates ratios and percentages in daily life to make decisions and to make comparisons The purposeful use of ratios with appropriate units of measure can enable one to see the situation more clearly As the three of us poundIfIJfll()OEO a shared understanding of ratios we saw that the context solidified our understanding as we distinguished commensurate and nonshycommensurate ratios and their

contextual we were able both to communicate and to achieve a common understanding Therefore we believe authentic contextual problems and tasks are needed for middle school students to achieve a better

understanding of ratios which as previously stated is a cornerstone concept for mathematics

Kenneth Chelst Asli and Thomas Edwards Wayne State University Detroit

References

Duggan J (2009) Dodging drunk drivers Statistics on driving are sobering Lincoln Joumal Star December 312009 httpjournalstarcoml

001 cc4c002eOhtml

J H T Furr-Holden D Voas R B E Torres P A S Ramirez A Brainard K and Berning A (2009) 2007 National Roadside ofAlcohol and Use Drivers Alcohol Results NHTSA Report No (DOT HS 811 248)

Lesh R Post and Behr M (1 Proportional reasoning In J Hiebert and M Behr Number Concepts and in the Middle Grades Reston VA Lawrence Erlbaum and National Council of Teachers of Mathematics

National Highway Safety Administration (NHTSA) (2012) Motor vehicle crashes Overview Traffic Safety Facts Research Note NHTSA No (DOT HS 811 (Revised 2012) httpwww-nrdnhtsadot govPubs811552pdf

Sax L J Jacobs J A and T A (2010) Womens Representation in Science and

Fields of 1976-2006 presented at the annual meeting of the Association for the Study of Education November 2010 Indianapolis IN

Majority Staff of the Joint Economic Committee of the United States 0 December) Invest in Women Invest in America A Comprehensive Review of Women in the US Economy Washington DC

28 May

2 of6Whats the Better Dea1

Deal 4 (2 liter bottles)

Deal 5 (4 packs with 8 bottles in each)

Which is the better deal

Deal 1

Deal 2

Deal 3

Deal 4

Deal 5

copy 2015 Wayne State University All rights reserved

Page 3 of 6Whats the Better Dea l) applied molhemalics plocHces

or the 2 st cntufY

Now support your decision by showing your work and writing a justification

Work Space for Deal 1 Work Space for Deal 2

Work Space for Deal 3 Work Space for Deal 4

Work Space for Deal 5

Explain which deal is the best deal and why

copy 2015 Wayne State Universi ty All rights reserved

Page 4 of 6Whats the Better Deal applied mathematics procHces

fOf lhe 21st c ontuf)i

Whats the Better Deal - Teacher Notes and Solutions

Grade Level 6

Common Core State Standards for Mathematics 6RPA3b Solve unit rate problems including those involving unit pricing and constant speed

6RPA3d Use ratio reasoning to convert measurement units manipulate and transform units appropriately

when multiplying or dividing quantities

Time Needed 1 class period

10 minutes - Hand out the deal illustration sheet of the deals or show on board Start a discussion

with the class of what they think the better deal would be based on intuition What is the criteria

for being a better deal (The intended criteria is cheapest price but be ready for other student

criteria) Take a hands up poll and tally results on the board

15 minutes - What do we see based on the illustrations Have students collaborate with a partner

about strategies that could be used to solve and decide which the better deal is Continue the

discussion having students share their thoughts

30 minutes - Hand out the blank worksheet and have students calculate and justify their results

Materials Required Deals Illustration Sheet

Blank worksheet for showing work and writing a justification

Focus Question Whats the best deal

Other Discussions What other factors may be considered in determining if something is a good deal

copy 2015 Wayne State University All rights reserved

Page 5 of6What s the Better Deall applied mathematics practices

or t 21st cenhuy

Original Task and Solution

Whats the Better Deal

Now support your dec ision by showing your work and writing a justification

Work Space for Deal 1

24 cans $849 J can $0354

12 oz can $0354can 1 oz $00295

Work Space for Deal 2 8 bottles $429 1 bottle (fy $0536

12 oZlbottle $0536Ibottle I oz $00447

With CARD 8 bottles $369 1 bottle $0461

12 oZlbottle $0461Ibottle 1 oz $00384

Work Space for Deal 3 Work Space for Deal 4

48 cans $1 100 2 liters = 676 ounces 1 can $0229

2 bottles $300 12 ozcan $0229can 1352 oz $300 1 oz $00190 1 oz $0022

Work Space for Deal 5

8 bottles in pack

4 packs $1 100

32 bottles $1100 I bottle $0344 J 69 ozbottle $0344Ibottle 1 oz $00204

copy 2015 Wayne State University All rights reserved

Page 6 of6Whats the Better DeltllJ applied mathemofiC$ practIces

fallne 21 st c enfuty

Explain which deal is the best deal and why

Students may try to lise a ratio sllch as cost per can However since each canbottle is a different

size they are not comparable The commonality in each is the ratio of cost per ounce They

may not all solve it this way which will lead to some terrific post-assignment discussion

Heres how each deal plays out

Deal I $0295ounce ($0378ounce if deposit is included)

Deal 2 $044701lnce ($0530ounce ifdeposit is included)

Deal 3 $0191 01lnce ($ 0274ounce ifdeposit is included)

Deal 4 $0222ounce ($ 023601lnce if deposit is included)

Deal 5 $0204oLlnce ($0263ounce if deposit is included)

Intended answer Deal 3 is the best deal because it is the cheapest per ounce Students may

want to include the deposit (making Deal 4 the best) but since you get that money back it doesnt

factor into the cost

copy 2015 Wayne State University All rights reserved

RETHINKING RATIOS RATES AND PERCENTAGES Kenneth Chelst Asli OzgOn-Koca and Thomas Edwards develop a shared understanding of these concepts

P roportional reasoning has been defined as the capstone of elementary concepts which leads to it being a cornerstone of higher

mathematics (Lesh et ai 1988) Understanding ratios and rates is fundamental to proportional reasoning The Common Core State Standards for Mathematics lists the calculation of ratios rates proportions and percentages as critical 6th and 7th grade skills It is relatively easy to write a short scenario and then ask students to calculate the ratio rate proportion or percentage All that the scenario requires is two numbers However these scenarios rarely accurately portray the ways in which these concepts are used to make a point when used in context

Two mathematics education professors and an engineering professor all with degrees in applied mathematics met to create contextual scenarios for the concepts of ratio rate and percentage We soon realized that we had neither a shared understanding of nor vocabulary for these cornerstone principles Through discussions we unpacked our knowledge to build a shared understanding for these concepts This article is written to serve both as a professional learning and growth piece As such it aims to help teachers reconsider their content knowledge of ratios rates and percentages Moreover we hope that teachers will be encouraged to think about how they might introduce these concepts and design activities that target better understanding for their students

First we categorized ratios as commensurate when the numbers being compared are measured using the same units and non-commensurate when

r 00()

QOO

0000 ~ 30000 1 shy 000

0000

0

measured using different units We saw that nonshycommensurate ratios always led to rates However commensurate ratios were more complex We were able to differentiate sub-categories leading to rates percentages or ratios While discussing these categories we had three common threads the roles of ratios and rates as they are used in real-life the role of the context and the role of units

Non-Commensurate Ratios

There are two distinct categories of ratio calculation In one context the two terms are non-commensurate with different units of measure miles and gallons cost and units deaths and miles driven Nonshycommensurate ratios are always expressed as rates The denominator may be a single unit and the rate is per unit as in miles per gallon or the denominator may be a standard number for example automobile deaths per 100 million vehicle miles driven The standard number used in the denominator is usually chosen so that the rate is greater than 1 to make the rate more meaningful see Figure 1 (below) The 2010 automotive fatality rate in the US was reported as 110 fatalities per 100 million vehicle miles traveled (VM T) Imagine using a different denominator one million miles The rate would then be 0011 fatalities per million miles travelled Most people find it easier to comprehend 11 fatalities than 0011 fatalities

Sometimes different rates are used for the same situation when viewed in a different way For example the speed of an automobile is usually given in miles per hour or kilometers per hour Doing so facilitates calculating or estimating the time

1 9

800 700

00

bull Fl1llilJOi F~IOInty Rale ~r 100M VMT

Figure 1 Fatalities and fatality rate per 100 million Vehicle Miles Travelled (VMT)

M~y 201 26

RETHINKING RATIOS RATES AND PERCENTAGES

Auto Vehicle Fatalities I 2009

Total 33883

Alcohol Impaired 10759

Fatality Rate per 100 million Vehicle 115

Miles Travelled

I 2010

32885 -998 -29

10228 -531 -49

110 -005 -43

Table 1 Auto vehicle fatality rates 2009 and 2010

needed to travel a known distance For example if you know your rate of travel in miles per hour you can easily estimate time of arrival at a location 175 miles away However when thinking about stopping time or reaction time when driving an automobile feet per second or meters per second are more meaningful For example it is much easier to visualize how far you would travel while reacting to a need to stop if you think of your speed as 88 feet per second rather than 60 miles per hour although these two rates are equivalent

Rates are often used to track change over time to compare different rates for similar entities or to make a forecast The automotive fatality rate is tracked over time to see if actions that have been taken are making driving safer (NHTSA 2012 see Figure 1 (page 26) and Table 1 (above)) The denominator used to calculate the rate may be based on the best unit of comparison or may be chosen simply for convenience Determining the automotive fatality per 100 million miles traveled (VMT) allows for comparing one year to another when the number of miles travelled may be significantly different Notice in Figure 1 the number of vehicle fatalities per year has gone up and down over a 60-year period as more people have owned cars and have travelled more miles in total However there is a strong general downward trend in the death rate per VMT

The cost per ounce is used to compare differentshysized items of the same product Miles per gallon is often used to compare two different vehicles to see which achieves better fuel economy Miles per gallon allows you to forecast how far you can travel before needing to buy fuel how much it will cost to drive a certain distance or your average annual vehicle fuel expenditure

Commensurate Ratios

In commensurate ratios the units of measure are the same or can be considered to be the same For example the number of people in prison and the population or the number of men and women are all measured in units that are essentially the same human beings

When the set whose measure is in the numerator is a subset of that whose measure is in the denominator the ratio can be presented as either a rate or as a percentage or both The rate of prison incarceration in the US is calculated by dividing the number of people in prison by the total population The US rate is 784 per 100000 of population These rates are often calculated for different subgroups For example if the numerator and denominator were limited to adults that rate would be 962 adults per 100000 adults This rate might be tracked over time or the rates of specific subgroups might be compared Sometimes these rates are converted to percentages to emphasize relative magnitude

Another type of ratio involves comparing the same two statistics over time In this instance the ratio represents the rate of growth or decline reported as a percentage For example the number of automobile fatalities is always compared from year-to-year to determine a trend as seen in Table 1 From 2009 to 2010 the number of auto vehicle fatalities decreased by 29 If adjusted for VMT the percent decline is 43

One of the most common uses of ratio involves comparing two different subsets Sometimes this type of ratio is used to emphasize the magnitude of difference For example in the US there is an imbalance in choices of STEM careers by females and males see Table 2 (page 28) Students notice this imbalance most when they see the ratio of females to males in a STEM-related course

In 2006 92 of the females entering college who chose a biological sciences major as compared to 73 of the male freshman This is equivalent to 1 female for every 08 males a reversal of the ratio found in 1976 In engineering the percentages were 24 females and 146 males a ratio of 1 female to 61 males This is an increase from the ratio in 1976 In computer science the percentages were 03 to 21 or 1 female for every 7 males a much smaller ratio than in 1976 (Sax et aI 2010)

1 IV ]01-l W atnl orguk 27

AND PERCENTAGES ~--- shy

ce 06 12 12

34

12

173

46 21

Table 2 Percent of each gender hl~~nn a stem major by year

One reason researchers track career choice data is to partially gender differences in incomes STEM careers are on average paying careers In 2009 the median income offu-time year round male workers in the US was 27 Nmnr~rI to for females The female-toshymale earnings ratio was 077 Thus females earned 77 cents for every dollar earned by a male (Majority Staff of the Joint Economic Committee of the United States 2010)

Another reason to compare ratios is to the allocation of resources For example every ten years in the US the National Highway Traffic

Administration commissions random sobriety checks of more than 6500 drivers in 60 locations across the US Blood alcohol levels are tested

between 930 am and 330 pm (2482) pJemy 11 pm to midnight (2123) and

Hnir 1 am to 3am (1948) During rtTUTltgt

hours approximately 1 in 500 drivers were alcohol impaired That rate increased to 1 in 83 in the early evening and 1 in 21 between 1 am and 3am (Lacey et al 2009) However a Nebraska highway

administrator claimed that in areas with high concentrations of bars after 11 pm the ratio increases to 1 in 4 (Duggan 2009) That means that for every four drivers stopped in the of a bar 1 of them will be found to be alcohol impaired Armed with such data police patrols can be allocated more effectively

Concluding Thoughts

We use rates ratios and percentages in daily life to make decisions and to make comparisons The purposeful use of ratios with appropriate units of measure can enable one to see the situation more clearly As the three of us poundIfIJfll()OEO a shared understanding of ratios we saw that the context solidified our understanding as we distinguished commensurate and nonshycommensurate ratios and their

contextual we were able both to communicate and to achieve a common understanding Therefore we believe authentic contextual problems and tasks are needed for middle school students to achieve a better

understanding of ratios which as previously stated is a cornerstone concept for mathematics

Kenneth Chelst Asli and Thomas Edwards Wayne State University Detroit

References

Duggan J (2009) Dodging drunk drivers Statistics on driving are sobering Lincoln Joumal Star December 312009 httpjournalstarcoml

001 cc4c002eOhtml

J H T Furr-Holden D Voas R B E Torres P A S Ramirez A Brainard K and Berning A (2009) 2007 National Roadside ofAlcohol and Use Drivers Alcohol Results NHTSA Report No (DOT HS 811 248)

Lesh R Post and Behr M (1 Proportional reasoning In J Hiebert and M Behr Number Concepts and in the Middle Grades Reston VA Lawrence Erlbaum and National Council of Teachers of Mathematics

National Highway Safety Administration (NHTSA) (2012) Motor vehicle crashes Overview Traffic Safety Facts Research Note NHTSA No (DOT HS 811 (Revised 2012) httpwww-nrdnhtsadot govPubs811552pdf

Sax L J Jacobs J A and T A (2010) Womens Representation in Science and

Fields of 1976-2006 presented at the annual meeting of the Association for the Study of Education November 2010 Indianapolis IN

Majority Staff of the Joint Economic Committee of the United States 0 December) Invest in Women Invest in America A Comprehensive Review of Women in the US Economy Washington DC

28 May

Page 3 of 6Whats the Better Dea l) applied molhemalics plocHces

or the 2 st cntufY

Now support your decision by showing your work and writing a justification

Work Space for Deal 1 Work Space for Deal 2

Work Space for Deal 3 Work Space for Deal 4

Work Space for Deal 5

Explain which deal is the best deal and why

copy 2015 Wayne State Universi ty All rights reserved

Page 4 of 6Whats the Better Deal applied mathematics procHces

fOf lhe 21st c ontuf)i

Whats the Better Deal - Teacher Notes and Solutions

Grade Level 6

Common Core State Standards for Mathematics 6RPA3b Solve unit rate problems including those involving unit pricing and constant speed

6RPA3d Use ratio reasoning to convert measurement units manipulate and transform units appropriately

when multiplying or dividing quantities

Time Needed 1 class period

10 minutes - Hand out the deal illustration sheet of the deals or show on board Start a discussion

with the class of what they think the better deal would be based on intuition What is the criteria

for being a better deal (The intended criteria is cheapest price but be ready for other student

criteria) Take a hands up poll and tally results on the board

15 minutes - What do we see based on the illustrations Have students collaborate with a partner

about strategies that could be used to solve and decide which the better deal is Continue the

discussion having students share their thoughts

30 minutes - Hand out the blank worksheet and have students calculate and justify their results

Materials Required Deals Illustration Sheet

Blank worksheet for showing work and writing a justification

Focus Question Whats the best deal

Other Discussions What other factors may be considered in determining if something is a good deal

copy 2015 Wayne State University All rights reserved

Page 5 of6What s the Better Deall applied mathematics practices

or t 21st cenhuy

Original Task and Solution

Whats the Better Deal

Now support your dec ision by showing your work and writing a justification

Work Space for Deal 1

24 cans $849 J can $0354

12 oz can $0354can 1 oz $00295

Work Space for Deal 2 8 bottles $429 1 bottle (fy $0536

12 oZlbottle $0536Ibottle I oz $00447

With CARD 8 bottles $369 1 bottle $0461

12 oZlbottle $0461Ibottle 1 oz $00384

Work Space for Deal 3 Work Space for Deal 4

48 cans $1 100 2 liters = 676 ounces 1 can $0229

2 bottles $300 12 ozcan $0229can 1352 oz $300 1 oz $00190 1 oz $0022

Work Space for Deal 5

8 bottles in pack

4 packs $1 100

32 bottles $1100 I bottle $0344 J 69 ozbottle $0344Ibottle 1 oz $00204

copy 2015 Wayne State University All rights reserved

Page 6 of6Whats the Better DeltllJ applied mathemofiC$ practIces

fallne 21 st c enfuty

Explain which deal is the best deal and why

Students may try to lise a ratio sllch as cost per can However since each canbottle is a different

size they are not comparable The commonality in each is the ratio of cost per ounce They

may not all solve it this way which will lead to some terrific post-assignment discussion

Heres how each deal plays out

Deal I $0295ounce ($0378ounce if deposit is included)

Deal 2 $044701lnce ($0530ounce ifdeposit is included)

Deal 3 $0191 01lnce ($ 0274ounce ifdeposit is included)

Deal 4 $0222ounce ($ 023601lnce if deposit is included)

Deal 5 $0204oLlnce ($0263ounce if deposit is included)

Intended answer Deal 3 is the best deal because it is the cheapest per ounce Students may

want to include the deposit (making Deal 4 the best) but since you get that money back it doesnt

factor into the cost

copy 2015 Wayne State University All rights reserved

RETHINKING RATIOS RATES AND PERCENTAGES Kenneth Chelst Asli OzgOn-Koca and Thomas Edwards develop a shared understanding of these concepts

P roportional reasoning has been defined as the capstone of elementary concepts which leads to it being a cornerstone of higher

mathematics (Lesh et ai 1988) Understanding ratios and rates is fundamental to proportional reasoning The Common Core State Standards for Mathematics lists the calculation of ratios rates proportions and percentages as critical 6th and 7th grade skills It is relatively easy to write a short scenario and then ask students to calculate the ratio rate proportion or percentage All that the scenario requires is two numbers However these scenarios rarely accurately portray the ways in which these concepts are used to make a point when used in context

Two mathematics education professors and an engineering professor all with degrees in applied mathematics met to create contextual scenarios for the concepts of ratio rate and percentage We soon realized that we had neither a shared understanding of nor vocabulary for these cornerstone principles Through discussions we unpacked our knowledge to build a shared understanding for these concepts This article is written to serve both as a professional learning and growth piece As such it aims to help teachers reconsider their content knowledge of ratios rates and percentages Moreover we hope that teachers will be encouraged to think about how they might introduce these concepts and design activities that target better understanding for their students

First we categorized ratios as commensurate when the numbers being compared are measured using the same units and non-commensurate when

r 00()

QOO

0000 ~ 30000 1 shy 000

0000

0

measured using different units We saw that nonshycommensurate ratios always led to rates However commensurate ratios were more complex We were able to differentiate sub-categories leading to rates percentages or ratios While discussing these categories we had three common threads the roles of ratios and rates as they are used in real-life the role of the context and the role of units

Non-Commensurate Ratios

There are two distinct categories of ratio calculation In one context the two terms are non-commensurate with different units of measure miles and gallons cost and units deaths and miles driven Nonshycommensurate ratios are always expressed as rates The denominator may be a single unit and the rate is per unit as in miles per gallon or the denominator may be a standard number for example automobile deaths per 100 million vehicle miles driven The standard number used in the denominator is usually chosen so that the rate is greater than 1 to make the rate more meaningful see Figure 1 (below) The 2010 automotive fatality rate in the US was reported as 110 fatalities per 100 million vehicle miles traveled (VM T) Imagine using a different denominator one million miles The rate would then be 0011 fatalities per million miles travelled Most people find it easier to comprehend 11 fatalities than 0011 fatalities

Sometimes different rates are used for the same situation when viewed in a different way For example the speed of an automobile is usually given in miles per hour or kilometers per hour Doing so facilitates calculating or estimating the time

1 9

800 700

00

bull Fl1llilJOi F~IOInty Rale ~r 100M VMT

Figure 1 Fatalities and fatality rate per 100 million Vehicle Miles Travelled (VMT)

M~y 201 26

RETHINKING RATIOS RATES AND PERCENTAGES

Auto Vehicle Fatalities I 2009

Total 33883

Alcohol Impaired 10759

Fatality Rate per 100 million Vehicle 115

Miles Travelled

I 2010

32885 -998 -29

10228 -531 -49

110 -005 -43

Table 1 Auto vehicle fatality rates 2009 and 2010

needed to travel a known distance For example if you know your rate of travel in miles per hour you can easily estimate time of arrival at a location 175 miles away However when thinking about stopping time or reaction time when driving an automobile feet per second or meters per second are more meaningful For example it is much easier to visualize how far you would travel while reacting to a need to stop if you think of your speed as 88 feet per second rather than 60 miles per hour although these two rates are equivalent

Rates are often used to track change over time to compare different rates for similar entities or to make a forecast The automotive fatality rate is tracked over time to see if actions that have been taken are making driving safer (NHTSA 2012 see Figure 1 (page 26) and Table 1 (above)) The denominator used to calculate the rate may be based on the best unit of comparison or may be chosen simply for convenience Determining the automotive fatality per 100 million miles traveled (VMT) allows for comparing one year to another when the number of miles travelled may be significantly different Notice in Figure 1 the number of vehicle fatalities per year has gone up and down over a 60-year period as more people have owned cars and have travelled more miles in total However there is a strong general downward trend in the death rate per VMT

The cost per ounce is used to compare differentshysized items of the same product Miles per gallon is often used to compare two different vehicles to see which achieves better fuel economy Miles per gallon allows you to forecast how far you can travel before needing to buy fuel how much it will cost to drive a certain distance or your average annual vehicle fuel expenditure

Commensurate Ratios

In commensurate ratios the units of measure are the same or can be considered to be the same For example the number of people in prison and the population or the number of men and women are all measured in units that are essentially the same human beings

When the set whose measure is in the numerator is a subset of that whose measure is in the denominator the ratio can be presented as either a rate or as a percentage or both The rate of prison incarceration in the US is calculated by dividing the number of people in prison by the total population The US rate is 784 per 100000 of population These rates are often calculated for different subgroups For example if the numerator and denominator were limited to adults that rate would be 962 adults per 100000 adults This rate might be tracked over time or the rates of specific subgroups might be compared Sometimes these rates are converted to percentages to emphasize relative magnitude

Another type of ratio involves comparing the same two statistics over time In this instance the ratio represents the rate of growth or decline reported as a percentage For example the number of automobile fatalities is always compared from year-to-year to determine a trend as seen in Table 1 From 2009 to 2010 the number of auto vehicle fatalities decreased by 29 If adjusted for VMT the percent decline is 43

One of the most common uses of ratio involves comparing two different subsets Sometimes this type of ratio is used to emphasize the magnitude of difference For example in the US there is an imbalance in choices of STEM careers by females and males see Table 2 (page 28) Students notice this imbalance most when they see the ratio of females to males in a STEM-related course

In 2006 92 of the females entering college who chose a biological sciences major as compared to 73 of the male freshman This is equivalent to 1 female for every 08 males a reversal of the ratio found in 1976 In engineering the percentages were 24 females and 146 males a ratio of 1 female to 61 males This is an increase from the ratio in 1976 In computer science the percentages were 03 to 21 or 1 female for every 7 males a much smaller ratio than in 1976 (Sax et aI 2010)

1 IV ]01-l W atnl orguk 27

AND PERCENTAGES ~--- shy

ce 06 12 12

34

12

173

46 21

Table 2 Percent of each gender hl~~nn a stem major by year

One reason researchers track career choice data is to partially gender differences in incomes STEM careers are on average paying careers In 2009 the median income offu-time year round male workers in the US was 27 Nmnr~rI to for females The female-toshymale earnings ratio was 077 Thus females earned 77 cents for every dollar earned by a male (Majority Staff of the Joint Economic Committee of the United States 2010)

Another reason to compare ratios is to the allocation of resources For example every ten years in the US the National Highway Traffic

Administration commissions random sobriety checks of more than 6500 drivers in 60 locations across the US Blood alcohol levels are tested

between 930 am and 330 pm (2482) pJemy 11 pm to midnight (2123) and

Hnir 1 am to 3am (1948) During rtTUTltgt

hours approximately 1 in 500 drivers were alcohol impaired That rate increased to 1 in 83 in the early evening and 1 in 21 between 1 am and 3am (Lacey et al 2009) However a Nebraska highway

administrator claimed that in areas with high concentrations of bars after 11 pm the ratio increases to 1 in 4 (Duggan 2009) That means that for every four drivers stopped in the of a bar 1 of them will be found to be alcohol impaired Armed with such data police patrols can be allocated more effectively

Concluding Thoughts

We use rates ratios and percentages in daily life to make decisions and to make comparisons The purposeful use of ratios with appropriate units of measure can enable one to see the situation more clearly As the three of us poundIfIJfll()OEO a shared understanding of ratios we saw that the context solidified our understanding as we distinguished commensurate and nonshycommensurate ratios and their

contextual we were able both to communicate and to achieve a common understanding Therefore we believe authentic contextual problems and tasks are needed for middle school students to achieve a better

understanding of ratios which as previously stated is a cornerstone concept for mathematics

Kenneth Chelst Asli and Thomas Edwards Wayne State University Detroit

References

Duggan J (2009) Dodging drunk drivers Statistics on driving are sobering Lincoln Joumal Star December 312009 httpjournalstarcoml

001 cc4c002eOhtml

J H T Furr-Holden D Voas R B E Torres P A S Ramirez A Brainard K and Berning A (2009) 2007 National Roadside ofAlcohol and Use Drivers Alcohol Results NHTSA Report No (DOT HS 811 248)

Lesh R Post and Behr M (1 Proportional reasoning In J Hiebert and M Behr Number Concepts and in the Middle Grades Reston VA Lawrence Erlbaum and National Council of Teachers of Mathematics

National Highway Safety Administration (NHTSA) (2012) Motor vehicle crashes Overview Traffic Safety Facts Research Note NHTSA No (DOT HS 811 (Revised 2012) httpwww-nrdnhtsadot govPubs811552pdf

Sax L J Jacobs J A and T A (2010) Womens Representation in Science and

Fields of 1976-2006 presented at the annual meeting of the Association for the Study of Education November 2010 Indianapolis IN

Majority Staff of the Joint Economic Committee of the United States 0 December) Invest in Women Invest in America A Comprehensive Review of Women in the US Economy Washington DC

28 May

Page 4 of 6Whats the Better Deal applied mathematics procHces

fOf lhe 21st c ontuf)i

Whats the Better Deal - Teacher Notes and Solutions

Grade Level 6

Common Core State Standards for Mathematics 6RPA3b Solve unit rate problems including those involving unit pricing and constant speed

6RPA3d Use ratio reasoning to convert measurement units manipulate and transform units appropriately

when multiplying or dividing quantities

Time Needed 1 class period

10 minutes - Hand out the deal illustration sheet of the deals or show on board Start a discussion

with the class of what they think the better deal would be based on intuition What is the criteria

for being a better deal (The intended criteria is cheapest price but be ready for other student

criteria) Take a hands up poll and tally results on the board

15 minutes - What do we see based on the illustrations Have students collaborate with a partner

about strategies that could be used to solve and decide which the better deal is Continue the

discussion having students share their thoughts

30 minutes - Hand out the blank worksheet and have students calculate and justify their results

Materials Required Deals Illustration Sheet

Blank worksheet for showing work and writing a justification

Focus Question Whats the best deal

Other Discussions What other factors may be considered in determining if something is a good deal

copy 2015 Wayne State University All rights reserved

Page 5 of6What s the Better Deall applied mathematics practices

or t 21st cenhuy

Original Task and Solution

Whats the Better Deal

Now support your dec ision by showing your work and writing a justification

Work Space for Deal 1

24 cans $849 J can $0354

12 oz can $0354can 1 oz $00295

Work Space for Deal 2 8 bottles $429 1 bottle (fy $0536

12 oZlbottle $0536Ibottle I oz $00447

With CARD 8 bottles $369 1 bottle $0461

12 oZlbottle $0461Ibottle 1 oz $00384

Work Space for Deal 3 Work Space for Deal 4

48 cans $1 100 2 liters = 676 ounces 1 can $0229

2 bottles $300 12 ozcan $0229can 1352 oz $300 1 oz $00190 1 oz $0022

Work Space for Deal 5

8 bottles in pack

4 packs $1 100

32 bottles $1100 I bottle $0344 J 69 ozbottle $0344Ibottle 1 oz $00204

copy 2015 Wayne State University All rights reserved

Page 6 of6Whats the Better DeltllJ applied mathemofiC$ practIces

fallne 21 st c enfuty

Explain which deal is the best deal and why

Students may try to lise a ratio sllch as cost per can However since each canbottle is a different

size they are not comparable The commonality in each is the ratio of cost per ounce They

may not all solve it this way which will lead to some terrific post-assignment discussion

Heres how each deal plays out

Deal I $0295ounce ($0378ounce if deposit is included)

Deal 2 $044701lnce ($0530ounce ifdeposit is included)

Deal 3 $0191 01lnce ($ 0274ounce ifdeposit is included)

Deal 4 $0222ounce ($ 023601lnce if deposit is included)

Deal 5 $0204oLlnce ($0263ounce if deposit is included)

Intended answer Deal 3 is the best deal because it is the cheapest per ounce Students may

want to include the deposit (making Deal 4 the best) but since you get that money back it doesnt

factor into the cost

copy 2015 Wayne State University All rights reserved

RETHINKING RATIOS RATES AND PERCENTAGES Kenneth Chelst Asli OzgOn-Koca and Thomas Edwards develop a shared understanding of these concepts

P roportional reasoning has been defined as the capstone of elementary concepts which leads to it being a cornerstone of higher

mathematics (Lesh et ai 1988) Understanding ratios and rates is fundamental to proportional reasoning The Common Core State Standards for Mathematics lists the calculation of ratios rates proportions and percentages as critical 6th and 7th grade skills It is relatively easy to write a short scenario and then ask students to calculate the ratio rate proportion or percentage All that the scenario requires is two numbers However these scenarios rarely accurately portray the ways in which these concepts are used to make a point when used in context

Two mathematics education professors and an engineering professor all with degrees in applied mathematics met to create contextual scenarios for the concepts of ratio rate and percentage We soon realized that we had neither a shared understanding of nor vocabulary for these cornerstone principles Through discussions we unpacked our knowledge to build a shared understanding for these concepts This article is written to serve both as a professional learning and growth piece As such it aims to help teachers reconsider their content knowledge of ratios rates and percentages Moreover we hope that teachers will be encouraged to think about how they might introduce these concepts and design activities that target better understanding for their students

First we categorized ratios as commensurate when the numbers being compared are measured using the same units and non-commensurate when

r 00()

QOO

0000 ~ 30000 1 shy 000

0000

0

measured using different units We saw that nonshycommensurate ratios always led to rates However commensurate ratios were more complex We were able to differentiate sub-categories leading to rates percentages or ratios While discussing these categories we had three common threads the roles of ratios and rates as they are used in real-life the role of the context and the role of units

Non-Commensurate Ratios

There are two distinct categories of ratio calculation In one context the two terms are non-commensurate with different units of measure miles and gallons cost and units deaths and miles driven Nonshycommensurate ratios are always expressed as rates The denominator may be a single unit and the rate is per unit as in miles per gallon or the denominator may be a standard number for example automobile deaths per 100 million vehicle miles driven The standard number used in the denominator is usually chosen so that the rate is greater than 1 to make the rate more meaningful see Figure 1 (below) The 2010 automotive fatality rate in the US was reported as 110 fatalities per 100 million vehicle miles traveled (VM T) Imagine using a different denominator one million miles The rate would then be 0011 fatalities per million miles travelled Most people find it easier to comprehend 11 fatalities than 0011 fatalities

Sometimes different rates are used for the same situation when viewed in a different way For example the speed of an automobile is usually given in miles per hour or kilometers per hour Doing so facilitates calculating or estimating the time

1 9

800 700

00

bull Fl1llilJOi F~IOInty Rale ~r 100M VMT

Figure 1 Fatalities and fatality rate per 100 million Vehicle Miles Travelled (VMT)

M~y 201 26

RETHINKING RATIOS RATES AND PERCENTAGES

Auto Vehicle Fatalities I 2009

Total 33883

Alcohol Impaired 10759

Fatality Rate per 100 million Vehicle 115

Miles Travelled

I 2010

32885 -998 -29

10228 -531 -49

110 -005 -43

Table 1 Auto vehicle fatality rates 2009 and 2010

needed to travel a known distance For example if you know your rate of travel in miles per hour you can easily estimate time of arrival at a location 175 miles away However when thinking about stopping time or reaction time when driving an automobile feet per second or meters per second are more meaningful For example it is much easier to visualize how far you would travel while reacting to a need to stop if you think of your speed as 88 feet per second rather than 60 miles per hour although these two rates are equivalent

Rates are often used to track change over time to compare different rates for similar entities or to make a forecast The automotive fatality rate is tracked over time to see if actions that have been taken are making driving safer (NHTSA 2012 see Figure 1 (page 26) and Table 1 (above)) The denominator used to calculate the rate may be based on the best unit of comparison or may be chosen simply for convenience Determining the automotive fatality per 100 million miles traveled (VMT) allows for comparing one year to another when the number of miles travelled may be significantly different Notice in Figure 1 the number of vehicle fatalities per year has gone up and down over a 60-year period as more people have owned cars and have travelled more miles in total However there is a strong general downward trend in the death rate per VMT

The cost per ounce is used to compare differentshysized items of the same product Miles per gallon is often used to compare two different vehicles to see which achieves better fuel economy Miles per gallon allows you to forecast how far you can travel before needing to buy fuel how much it will cost to drive a certain distance or your average annual vehicle fuel expenditure

Commensurate Ratios

In commensurate ratios the units of measure are the same or can be considered to be the same For example the number of people in prison and the population or the number of men and women are all measured in units that are essentially the same human beings

When the set whose measure is in the numerator is a subset of that whose measure is in the denominator the ratio can be presented as either a rate or as a percentage or both The rate of prison incarceration in the US is calculated by dividing the number of people in prison by the total population The US rate is 784 per 100000 of population These rates are often calculated for different subgroups For example if the numerator and denominator were limited to adults that rate would be 962 adults per 100000 adults This rate might be tracked over time or the rates of specific subgroups might be compared Sometimes these rates are converted to percentages to emphasize relative magnitude

Another type of ratio involves comparing the same two statistics over time In this instance the ratio represents the rate of growth or decline reported as a percentage For example the number of automobile fatalities is always compared from year-to-year to determine a trend as seen in Table 1 From 2009 to 2010 the number of auto vehicle fatalities decreased by 29 If adjusted for VMT the percent decline is 43

One of the most common uses of ratio involves comparing two different subsets Sometimes this type of ratio is used to emphasize the magnitude of difference For example in the US there is an imbalance in choices of STEM careers by females and males see Table 2 (page 28) Students notice this imbalance most when they see the ratio of females to males in a STEM-related course

In 2006 92 of the females entering college who chose a biological sciences major as compared to 73 of the male freshman This is equivalent to 1 female for every 08 males a reversal of the ratio found in 1976 In engineering the percentages were 24 females and 146 males a ratio of 1 female to 61 males This is an increase from the ratio in 1976 In computer science the percentages were 03 to 21 or 1 female for every 7 males a much smaller ratio than in 1976 (Sax et aI 2010)

1 IV ]01-l W atnl orguk 27

AND PERCENTAGES ~--- shy

ce 06 12 12

34

12

173

46 21

Table 2 Percent of each gender hl~~nn a stem major by year

One reason researchers track career choice data is to partially gender differences in incomes STEM careers are on average paying careers In 2009 the median income offu-time year round male workers in the US was 27 Nmnr~rI to for females The female-toshymale earnings ratio was 077 Thus females earned 77 cents for every dollar earned by a male (Majority Staff of the Joint Economic Committee of the United States 2010)

Another reason to compare ratios is to the allocation of resources For example every ten years in the US the National Highway Traffic

Administration commissions random sobriety checks of more than 6500 drivers in 60 locations across the US Blood alcohol levels are tested

between 930 am and 330 pm (2482) pJemy 11 pm to midnight (2123) and

Hnir 1 am to 3am (1948) During rtTUTltgt

hours approximately 1 in 500 drivers were alcohol impaired That rate increased to 1 in 83 in the early evening and 1 in 21 between 1 am and 3am (Lacey et al 2009) However a Nebraska highway

administrator claimed that in areas with high concentrations of bars after 11 pm the ratio increases to 1 in 4 (Duggan 2009) That means that for every four drivers stopped in the of a bar 1 of them will be found to be alcohol impaired Armed with such data police patrols can be allocated more effectively

Concluding Thoughts

We use rates ratios and percentages in daily life to make decisions and to make comparisons The purposeful use of ratios with appropriate units of measure can enable one to see the situation more clearly As the three of us poundIfIJfll()OEO a shared understanding of ratios we saw that the context solidified our understanding as we distinguished commensurate and nonshycommensurate ratios and their

contextual we were able both to communicate and to achieve a common understanding Therefore we believe authentic contextual problems and tasks are needed for middle school students to achieve a better

understanding of ratios which as previously stated is a cornerstone concept for mathematics

Kenneth Chelst Asli and Thomas Edwards Wayne State University Detroit

References

Duggan J (2009) Dodging drunk drivers Statistics on driving are sobering Lincoln Joumal Star December 312009 httpjournalstarcoml

001 cc4c002eOhtml

J H T Furr-Holden D Voas R B E Torres P A S Ramirez A Brainard K and Berning A (2009) 2007 National Roadside ofAlcohol and Use Drivers Alcohol Results NHTSA Report No (DOT HS 811 248)

Lesh R Post and Behr M (1 Proportional reasoning In J Hiebert and M Behr Number Concepts and in the Middle Grades Reston VA Lawrence Erlbaum and National Council of Teachers of Mathematics

National Highway Safety Administration (NHTSA) (2012) Motor vehicle crashes Overview Traffic Safety Facts Research Note NHTSA No (DOT HS 811 (Revised 2012) httpwww-nrdnhtsadot govPubs811552pdf

Sax L J Jacobs J A and T A (2010) Womens Representation in Science and

Fields of 1976-2006 presented at the annual meeting of the Association for the Study of Education November 2010 Indianapolis IN

Majority Staff of the Joint Economic Committee of the United States 0 December) Invest in Women Invest in America A Comprehensive Review of Women in the US Economy Washington DC

28 May

Page 5 of6What s the Better Deall applied mathematics practices

or t 21st cenhuy

Original Task and Solution

Whats the Better Deal

Now support your dec ision by showing your work and writing a justification

Work Space for Deal 1

24 cans $849 J can $0354

12 oz can $0354can 1 oz $00295

Work Space for Deal 2 8 bottles $429 1 bottle (fy $0536

12 oZlbottle $0536Ibottle I oz $00447

With CARD 8 bottles $369 1 bottle $0461

12 oZlbottle $0461Ibottle 1 oz $00384

Work Space for Deal 3 Work Space for Deal 4

48 cans $1 100 2 liters = 676 ounces 1 can $0229

2 bottles $300 12 ozcan $0229can 1352 oz $300 1 oz $00190 1 oz $0022

Work Space for Deal 5

8 bottles in pack

4 packs $1 100

32 bottles $1100 I bottle $0344 J 69 ozbottle $0344Ibottle 1 oz $00204

copy 2015 Wayne State University All rights reserved

Page 6 of6Whats the Better DeltllJ applied mathemofiC$ practIces

fallne 21 st c enfuty

Explain which deal is the best deal and why

Students may try to lise a ratio sllch as cost per can However since each canbottle is a different

size they are not comparable The commonality in each is the ratio of cost per ounce They

may not all solve it this way which will lead to some terrific post-assignment discussion

Heres how each deal plays out

Deal I $0295ounce ($0378ounce if deposit is included)

Deal 2 $044701lnce ($0530ounce ifdeposit is included)

Deal 3 $0191 01lnce ($ 0274ounce ifdeposit is included)

Deal 4 $0222ounce ($ 023601lnce if deposit is included)

Deal 5 $0204oLlnce ($0263ounce if deposit is included)

Intended answer Deal 3 is the best deal because it is the cheapest per ounce Students may

want to include the deposit (making Deal 4 the best) but since you get that money back it doesnt

factor into the cost

copy 2015 Wayne State University All rights reserved

RETHINKING RATIOS RATES AND PERCENTAGES Kenneth Chelst Asli OzgOn-Koca and Thomas Edwards develop a shared understanding of these concepts

P roportional reasoning has been defined as the capstone of elementary concepts which leads to it being a cornerstone of higher

mathematics (Lesh et ai 1988) Understanding ratios and rates is fundamental to proportional reasoning The Common Core State Standards for Mathematics lists the calculation of ratios rates proportions and percentages as critical 6th and 7th grade skills It is relatively easy to write a short scenario and then ask students to calculate the ratio rate proportion or percentage All that the scenario requires is two numbers However these scenarios rarely accurately portray the ways in which these concepts are used to make a point when used in context

Two mathematics education professors and an engineering professor all with degrees in applied mathematics met to create contextual scenarios for the concepts of ratio rate and percentage We soon realized that we had neither a shared understanding of nor vocabulary for these cornerstone principles Through discussions we unpacked our knowledge to build a shared understanding for these concepts This article is written to serve both as a professional learning and growth piece As such it aims to help teachers reconsider their content knowledge of ratios rates and percentages Moreover we hope that teachers will be encouraged to think about how they might introduce these concepts and design activities that target better understanding for their students

First we categorized ratios as commensurate when the numbers being compared are measured using the same units and non-commensurate when

r 00()

QOO

0000 ~ 30000 1 shy 000

0000

0

measured using different units We saw that nonshycommensurate ratios always led to rates However commensurate ratios were more complex We were able to differentiate sub-categories leading to rates percentages or ratios While discussing these categories we had three common threads the roles of ratios and rates as they are used in real-life the role of the context and the role of units

Non-Commensurate Ratios

There are two distinct categories of ratio calculation In one context the two terms are non-commensurate with different units of measure miles and gallons cost and units deaths and miles driven Nonshycommensurate ratios are always expressed as rates The denominator may be a single unit and the rate is per unit as in miles per gallon or the denominator may be a standard number for example automobile deaths per 100 million vehicle miles driven The standard number used in the denominator is usually chosen so that the rate is greater than 1 to make the rate more meaningful see Figure 1 (below) The 2010 automotive fatality rate in the US was reported as 110 fatalities per 100 million vehicle miles traveled (VM T) Imagine using a different denominator one million miles The rate would then be 0011 fatalities per million miles travelled Most people find it easier to comprehend 11 fatalities than 0011 fatalities

Sometimes different rates are used for the same situation when viewed in a different way For example the speed of an automobile is usually given in miles per hour or kilometers per hour Doing so facilitates calculating or estimating the time

1 9

800 700

00

bull Fl1llilJOi F~IOInty Rale ~r 100M VMT

Figure 1 Fatalities and fatality rate per 100 million Vehicle Miles Travelled (VMT)

M~y 201 26

RETHINKING RATIOS RATES AND PERCENTAGES

Auto Vehicle Fatalities I 2009

Total 33883

Alcohol Impaired 10759

Fatality Rate per 100 million Vehicle 115

Miles Travelled

I 2010

32885 -998 -29

10228 -531 -49

110 -005 -43

Table 1 Auto vehicle fatality rates 2009 and 2010

needed to travel a known distance For example if you know your rate of travel in miles per hour you can easily estimate time of arrival at a location 175 miles away However when thinking about stopping time or reaction time when driving an automobile feet per second or meters per second are more meaningful For example it is much easier to visualize how far you would travel while reacting to a need to stop if you think of your speed as 88 feet per second rather than 60 miles per hour although these two rates are equivalent

Rates are often used to track change over time to compare different rates for similar entities or to make a forecast The automotive fatality rate is tracked over time to see if actions that have been taken are making driving safer (NHTSA 2012 see Figure 1 (page 26) and Table 1 (above)) The denominator used to calculate the rate may be based on the best unit of comparison or may be chosen simply for convenience Determining the automotive fatality per 100 million miles traveled (VMT) allows for comparing one year to another when the number of miles travelled may be significantly different Notice in Figure 1 the number of vehicle fatalities per year has gone up and down over a 60-year period as more people have owned cars and have travelled more miles in total However there is a strong general downward trend in the death rate per VMT

The cost per ounce is used to compare differentshysized items of the same product Miles per gallon is often used to compare two different vehicles to see which achieves better fuel economy Miles per gallon allows you to forecast how far you can travel before needing to buy fuel how much it will cost to drive a certain distance or your average annual vehicle fuel expenditure

Commensurate Ratios

In commensurate ratios the units of measure are the same or can be considered to be the same For example the number of people in prison and the population or the number of men and women are all measured in units that are essentially the same human beings

When the set whose measure is in the numerator is a subset of that whose measure is in the denominator the ratio can be presented as either a rate or as a percentage or both The rate of prison incarceration in the US is calculated by dividing the number of people in prison by the total population The US rate is 784 per 100000 of population These rates are often calculated for different subgroups For example if the numerator and denominator were limited to adults that rate would be 962 adults per 100000 adults This rate might be tracked over time or the rates of specific subgroups might be compared Sometimes these rates are converted to percentages to emphasize relative magnitude

Another type of ratio involves comparing the same two statistics over time In this instance the ratio represents the rate of growth or decline reported as a percentage For example the number of automobile fatalities is always compared from year-to-year to determine a trend as seen in Table 1 From 2009 to 2010 the number of auto vehicle fatalities decreased by 29 If adjusted for VMT the percent decline is 43

One of the most common uses of ratio involves comparing two different subsets Sometimes this type of ratio is used to emphasize the magnitude of difference For example in the US there is an imbalance in choices of STEM careers by females and males see Table 2 (page 28) Students notice this imbalance most when they see the ratio of females to males in a STEM-related course

In 2006 92 of the females entering college who chose a biological sciences major as compared to 73 of the male freshman This is equivalent to 1 female for every 08 males a reversal of the ratio found in 1976 In engineering the percentages were 24 females and 146 males a ratio of 1 female to 61 males This is an increase from the ratio in 1976 In computer science the percentages were 03 to 21 or 1 female for every 7 males a much smaller ratio than in 1976 (Sax et aI 2010)

1 IV ]01-l W atnl orguk 27

AND PERCENTAGES ~--- shy

ce 06 12 12

34

12

173

46 21

Table 2 Percent of each gender hl~~nn a stem major by year

One reason researchers track career choice data is to partially gender differences in incomes STEM careers are on average paying careers In 2009 the median income offu-time year round male workers in the US was 27 Nmnr~rI to for females The female-toshymale earnings ratio was 077 Thus females earned 77 cents for every dollar earned by a male (Majority Staff of the Joint Economic Committee of the United States 2010)

Another reason to compare ratios is to the allocation of resources For example every ten years in the US the National Highway Traffic

Administration commissions random sobriety checks of more than 6500 drivers in 60 locations across the US Blood alcohol levels are tested

between 930 am and 330 pm (2482) pJemy 11 pm to midnight (2123) and

Hnir 1 am to 3am (1948) During rtTUTltgt

hours approximately 1 in 500 drivers were alcohol impaired That rate increased to 1 in 83 in the early evening and 1 in 21 between 1 am and 3am (Lacey et al 2009) However a Nebraska highway

administrator claimed that in areas with high concentrations of bars after 11 pm the ratio increases to 1 in 4 (Duggan 2009) That means that for every four drivers stopped in the of a bar 1 of them will be found to be alcohol impaired Armed with such data police patrols can be allocated more effectively

Concluding Thoughts

We use rates ratios and percentages in daily life to make decisions and to make comparisons The purposeful use of ratios with appropriate units of measure can enable one to see the situation more clearly As the three of us poundIfIJfll()OEO a shared understanding of ratios we saw that the context solidified our understanding as we distinguished commensurate and nonshycommensurate ratios and their

contextual we were able both to communicate and to achieve a common understanding Therefore we believe authentic contextual problems and tasks are needed for middle school students to achieve a better

understanding of ratios which as previously stated is a cornerstone concept for mathematics

Kenneth Chelst Asli and Thomas Edwards Wayne State University Detroit

References

Duggan J (2009) Dodging drunk drivers Statistics on driving are sobering Lincoln Joumal Star December 312009 httpjournalstarcoml

001 cc4c002eOhtml

J H T Furr-Holden D Voas R B E Torres P A S Ramirez A Brainard K and Berning A (2009) 2007 National Roadside ofAlcohol and Use Drivers Alcohol Results NHTSA Report No (DOT HS 811 248)

Lesh R Post and Behr M (1 Proportional reasoning In J Hiebert and M Behr Number Concepts and in the Middle Grades Reston VA Lawrence Erlbaum and National Council of Teachers of Mathematics

National Highway Safety Administration (NHTSA) (2012) Motor vehicle crashes Overview Traffic Safety Facts Research Note NHTSA No (DOT HS 811 (Revised 2012) httpwww-nrdnhtsadot govPubs811552pdf

Sax L J Jacobs J A and T A (2010) Womens Representation in Science and

Fields of 1976-2006 presented at the annual meeting of the Association for the Study of Education November 2010 Indianapolis IN

Majority Staff of the Joint Economic Committee of the United States 0 December) Invest in Women Invest in America A Comprehensive Review of Women in the US Economy Washington DC

28 May

Page 6 of6Whats the Better DeltllJ applied mathemofiC$ practIces

fallne 21 st c enfuty

Explain which deal is the best deal and why

Students may try to lise a ratio sllch as cost per can However since each canbottle is a different

size they are not comparable The commonality in each is the ratio of cost per ounce They

may not all solve it this way which will lead to some terrific post-assignment discussion

Heres how each deal plays out

Deal I $0295ounce ($0378ounce if deposit is included)

Deal 2 $044701lnce ($0530ounce ifdeposit is included)

Deal 3 $0191 01lnce ($ 0274ounce ifdeposit is included)

Deal 4 $0222ounce ($ 023601lnce if deposit is included)

Deal 5 $0204oLlnce ($0263ounce if deposit is included)

Intended answer Deal 3 is the best deal because it is the cheapest per ounce Students may

want to include the deposit (making Deal 4 the best) but since you get that money back it doesnt

factor into the cost

copy 2015 Wayne State University All rights reserved

RETHINKING RATIOS RATES AND PERCENTAGES Kenneth Chelst Asli OzgOn-Koca and Thomas Edwards develop a shared understanding of these concepts

P roportional reasoning has been defined as the capstone of elementary concepts which leads to it being a cornerstone of higher

mathematics (Lesh et ai 1988) Understanding ratios and rates is fundamental to proportional reasoning The Common Core State Standards for Mathematics lists the calculation of ratios rates proportions and percentages as critical 6th and 7th grade skills It is relatively easy to write a short scenario and then ask students to calculate the ratio rate proportion or percentage All that the scenario requires is two numbers However these scenarios rarely accurately portray the ways in which these concepts are used to make a point when used in context

Two mathematics education professors and an engineering professor all with degrees in applied mathematics met to create contextual scenarios for the concepts of ratio rate and percentage We soon realized that we had neither a shared understanding of nor vocabulary for these cornerstone principles Through discussions we unpacked our knowledge to build a shared understanding for these concepts This article is written to serve both as a professional learning and growth piece As such it aims to help teachers reconsider their content knowledge of ratios rates and percentages Moreover we hope that teachers will be encouraged to think about how they might introduce these concepts and design activities that target better understanding for their students

First we categorized ratios as commensurate when the numbers being compared are measured using the same units and non-commensurate when

r 00()

QOO

0000 ~ 30000 1 shy 000

0000

0

measured using different units We saw that nonshycommensurate ratios always led to rates However commensurate ratios were more complex We were able to differentiate sub-categories leading to rates percentages or ratios While discussing these categories we had three common threads the roles of ratios and rates as they are used in real-life the role of the context and the role of units

Non-Commensurate Ratios

There are two distinct categories of ratio calculation In one context the two terms are non-commensurate with different units of measure miles and gallons cost and units deaths and miles driven Nonshycommensurate ratios are always expressed as rates The denominator may be a single unit and the rate is per unit as in miles per gallon or the denominator may be a standard number for example automobile deaths per 100 million vehicle miles driven The standard number used in the denominator is usually chosen so that the rate is greater than 1 to make the rate more meaningful see Figure 1 (below) The 2010 automotive fatality rate in the US was reported as 110 fatalities per 100 million vehicle miles traveled (VM T) Imagine using a different denominator one million miles The rate would then be 0011 fatalities per million miles travelled Most people find it easier to comprehend 11 fatalities than 0011 fatalities

Sometimes different rates are used for the same situation when viewed in a different way For example the speed of an automobile is usually given in miles per hour or kilometers per hour Doing so facilitates calculating or estimating the time

1 9

800 700

00

bull Fl1llilJOi F~IOInty Rale ~r 100M VMT

Figure 1 Fatalities and fatality rate per 100 million Vehicle Miles Travelled (VMT)

M~y 201 26

RETHINKING RATIOS RATES AND PERCENTAGES

Auto Vehicle Fatalities I 2009

Total 33883

Alcohol Impaired 10759

Fatality Rate per 100 million Vehicle 115

Miles Travelled

I 2010

32885 -998 -29

10228 -531 -49

110 -005 -43

Table 1 Auto vehicle fatality rates 2009 and 2010

needed to travel a known distance For example if you know your rate of travel in miles per hour you can easily estimate time of arrival at a location 175 miles away However when thinking about stopping time or reaction time when driving an automobile feet per second or meters per second are more meaningful For example it is much easier to visualize how far you would travel while reacting to a need to stop if you think of your speed as 88 feet per second rather than 60 miles per hour although these two rates are equivalent

Rates are often used to track change over time to compare different rates for similar entities or to make a forecast The automotive fatality rate is tracked over time to see if actions that have been taken are making driving safer (NHTSA 2012 see Figure 1 (page 26) and Table 1 (above)) The denominator used to calculate the rate may be based on the best unit of comparison or may be chosen simply for convenience Determining the automotive fatality per 100 million miles traveled (VMT) allows for comparing one year to another when the number of miles travelled may be significantly different Notice in Figure 1 the number of vehicle fatalities per year has gone up and down over a 60-year period as more people have owned cars and have travelled more miles in total However there is a strong general downward trend in the death rate per VMT

The cost per ounce is used to compare differentshysized items of the same product Miles per gallon is often used to compare two different vehicles to see which achieves better fuel economy Miles per gallon allows you to forecast how far you can travel before needing to buy fuel how much it will cost to drive a certain distance or your average annual vehicle fuel expenditure

Commensurate Ratios

In commensurate ratios the units of measure are the same or can be considered to be the same For example the number of people in prison and the population or the number of men and women are all measured in units that are essentially the same human beings

When the set whose measure is in the numerator is a subset of that whose measure is in the denominator the ratio can be presented as either a rate or as a percentage or both The rate of prison incarceration in the US is calculated by dividing the number of people in prison by the total population The US rate is 784 per 100000 of population These rates are often calculated for different subgroups For example if the numerator and denominator were limited to adults that rate would be 962 adults per 100000 adults This rate might be tracked over time or the rates of specific subgroups might be compared Sometimes these rates are converted to percentages to emphasize relative magnitude

Another type of ratio involves comparing the same two statistics over time In this instance the ratio represents the rate of growth or decline reported as a percentage For example the number of automobile fatalities is always compared from year-to-year to determine a trend as seen in Table 1 From 2009 to 2010 the number of auto vehicle fatalities decreased by 29 If adjusted for VMT the percent decline is 43

One of the most common uses of ratio involves comparing two different subsets Sometimes this type of ratio is used to emphasize the magnitude of difference For example in the US there is an imbalance in choices of STEM careers by females and males see Table 2 (page 28) Students notice this imbalance most when they see the ratio of females to males in a STEM-related course

In 2006 92 of the females entering college who chose a biological sciences major as compared to 73 of the male freshman This is equivalent to 1 female for every 08 males a reversal of the ratio found in 1976 In engineering the percentages were 24 females and 146 males a ratio of 1 female to 61 males This is an increase from the ratio in 1976 In computer science the percentages were 03 to 21 or 1 female for every 7 males a much smaller ratio than in 1976 (Sax et aI 2010)

1 IV ]01-l W atnl orguk 27

AND PERCENTAGES ~--- shy

ce 06 12 12

34

12

173

46 21

Table 2 Percent of each gender hl~~nn a stem major by year

One reason researchers track career choice data is to partially gender differences in incomes STEM careers are on average paying careers In 2009 the median income offu-time year round male workers in the US was 27 Nmnr~rI to for females The female-toshymale earnings ratio was 077 Thus females earned 77 cents for every dollar earned by a male (Majority Staff of the Joint Economic Committee of the United States 2010)

Another reason to compare ratios is to the allocation of resources For example every ten years in the US the National Highway Traffic

Administration commissions random sobriety checks of more than 6500 drivers in 60 locations across the US Blood alcohol levels are tested

between 930 am and 330 pm (2482) pJemy 11 pm to midnight (2123) and

Hnir 1 am to 3am (1948) During rtTUTltgt

hours approximately 1 in 500 drivers were alcohol impaired That rate increased to 1 in 83 in the early evening and 1 in 21 between 1 am and 3am (Lacey et al 2009) However a Nebraska highway

administrator claimed that in areas with high concentrations of bars after 11 pm the ratio increases to 1 in 4 (Duggan 2009) That means that for every four drivers stopped in the of a bar 1 of them will be found to be alcohol impaired Armed with such data police patrols can be allocated more effectively

Concluding Thoughts

We use rates ratios and percentages in daily life to make decisions and to make comparisons The purposeful use of ratios with appropriate units of measure can enable one to see the situation more clearly As the three of us poundIfIJfll()OEO a shared understanding of ratios we saw that the context solidified our understanding as we distinguished commensurate and nonshycommensurate ratios and their

contextual we were able both to communicate and to achieve a common understanding Therefore we believe authentic contextual problems and tasks are needed for middle school students to achieve a better

understanding of ratios which as previously stated is a cornerstone concept for mathematics

Kenneth Chelst Asli and Thomas Edwards Wayne State University Detroit

References

Duggan J (2009) Dodging drunk drivers Statistics on driving are sobering Lincoln Joumal Star December 312009 httpjournalstarcoml

001 cc4c002eOhtml

J H T Furr-Holden D Voas R B E Torres P A S Ramirez A Brainard K and Berning A (2009) 2007 National Roadside ofAlcohol and Use Drivers Alcohol Results NHTSA Report No (DOT HS 811 248)

Lesh R Post and Behr M (1 Proportional reasoning In J Hiebert and M Behr Number Concepts and in the Middle Grades Reston VA Lawrence Erlbaum and National Council of Teachers of Mathematics

National Highway Safety Administration (NHTSA) (2012) Motor vehicle crashes Overview Traffic Safety Facts Research Note NHTSA No (DOT HS 811 (Revised 2012) httpwww-nrdnhtsadot govPubs811552pdf

Sax L J Jacobs J A and T A (2010) Womens Representation in Science and

Fields of 1976-2006 presented at the annual meeting of the Association for the Study of Education November 2010 Indianapolis IN

Majority Staff of the Joint Economic Committee of the United States 0 December) Invest in Women Invest in America A Comprehensive Review of Women in the US Economy Washington DC

28 May

RETHINKING RATIOS RATES AND PERCENTAGES Kenneth Chelst Asli OzgOn-Koca and Thomas Edwards develop a shared understanding of these concepts

P roportional reasoning has been defined as the capstone of elementary concepts which leads to it being a cornerstone of higher

mathematics (Lesh et ai 1988) Understanding ratios and rates is fundamental to proportional reasoning The Common Core State Standards for Mathematics lists the calculation of ratios rates proportions and percentages as critical 6th and 7th grade skills It is relatively easy to write a short scenario and then ask students to calculate the ratio rate proportion or percentage All that the scenario requires is two numbers However these scenarios rarely accurately portray the ways in which these concepts are used to make a point when used in context

Two mathematics education professors and an engineering professor all with degrees in applied mathematics met to create contextual scenarios for the concepts of ratio rate and percentage We soon realized that we had neither a shared understanding of nor vocabulary for these cornerstone principles Through discussions we unpacked our knowledge to build a shared understanding for these concepts This article is written to serve both as a professional learning and growth piece As such it aims to help teachers reconsider their content knowledge of ratios rates and percentages Moreover we hope that teachers will be encouraged to think about how they might introduce these concepts and design activities that target better understanding for their students

First we categorized ratios as commensurate when the numbers being compared are measured using the same units and non-commensurate when

r 00()

QOO

0000 ~ 30000 1 shy 000

0000

0

measured using different units We saw that nonshycommensurate ratios always led to rates However commensurate ratios were more complex We were able to differentiate sub-categories leading to rates percentages or ratios While discussing these categories we had three common threads the roles of ratios and rates as they are used in real-life the role of the context and the role of units

Non-Commensurate Ratios

There are two distinct categories of ratio calculation In one context the two terms are non-commensurate with different units of measure miles and gallons cost and units deaths and miles driven Nonshycommensurate ratios are always expressed as rates The denominator may be a single unit and the rate is per unit as in miles per gallon or the denominator may be a standard number for example automobile deaths per 100 million vehicle miles driven The standard number used in the denominator is usually chosen so that the rate is greater than 1 to make the rate more meaningful see Figure 1 (below) The 2010 automotive fatality rate in the US was reported as 110 fatalities per 100 million vehicle miles traveled (VM T) Imagine using a different denominator one million miles The rate would then be 0011 fatalities per million miles travelled Most people find it easier to comprehend 11 fatalities than 0011 fatalities

Sometimes different rates are used for the same situation when viewed in a different way For example the speed of an automobile is usually given in miles per hour or kilometers per hour Doing so facilitates calculating or estimating the time

1 9

800 700

00

bull Fl1llilJOi F~IOInty Rale ~r 100M VMT

Figure 1 Fatalities and fatality rate per 100 million Vehicle Miles Travelled (VMT)

M~y 201 26

RETHINKING RATIOS RATES AND PERCENTAGES

Auto Vehicle Fatalities I 2009

Total 33883

Alcohol Impaired 10759

Fatality Rate per 100 million Vehicle 115

Miles Travelled

I 2010

32885 -998 -29

10228 -531 -49

110 -005 -43

Table 1 Auto vehicle fatality rates 2009 and 2010

needed to travel a known distance For example if you know your rate of travel in miles per hour you can easily estimate time of arrival at a location 175 miles away However when thinking about stopping time or reaction time when driving an automobile feet per second or meters per second are more meaningful For example it is much easier to visualize how far you would travel while reacting to a need to stop if you think of your speed as 88 feet per second rather than 60 miles per hour although these two rates are equivalent

Rates are often used to track change over time to compare different rates for similar entities or to make a forecast The automotive fatality rate is tracked over time to see if actions that have been taken are making driving safer (NHTSA 2012 see Figure 1 (page 26) and Table 1 (above)) The denominator used to calculate the rate may be based on the best unit of comparison or may be chosen simply for convenience Determining the automotive fatality per 100 million miles traveled (VMT) allows for comparing one year to another when the number of miles travelled may be significantly different Notice in Figure 1 the number of vehicle fatalities per year has gone up and down over a 60-year period as more people have owned cars and have travelled more miles in total However there is a strong general downward trend in the death rate per VMT

The cost per ounce is used to compare differentshysized items of the same product Miles per gallon is often used to compare two different vehicles to see which achieves better fuel economy Miles per gallon allows you to forecast how far you can travel before needing to buy fuel how much it will cost to drive a certain distance or your average annual vehicle fuel expenditure

Commensurate Ratios

In commensurate ratios the units of measure are the same or can be considered to be the same For example the number of people in prison and the population or the number of men and women are all measured in units that are essentially the same human beings

When the set whose measure is in the numerator is a subset of that whose measure is in the denominator the ratio can be presented as either a rate or as a percentage or both The rate of prison incarceration in the US is calculated by dividing the number of people in prison by the total population The US rate is 784 per 100000 of population These rates are often calculated for different subgroups For example if the numerator and denominator were limited to adults that rate would be 962 adults per 100000 adults This rate might be tracked over time or the rates of specific subgroups might be compared Sometimes these rates are converted to percentages to emphasize relative magnitude

Another type of ratio involves comparing the same two statistics over time In this instance the ratio represents the rate of growth or decline reported as a percentage For example the number of automobile fatalities is always compared from year-to-year to determine a trend as seen in Table 1 From 2009 to 2010 the number of auto vehicle fatalities decreased by 29 If adjusted for VMT the percent decline is 43

One of the most common uses of ratio involves comparing two different subsets Sometimes this type of ratio is used to emphasize the magnitude of difference For example in the US there is an imbalance in choices of STEM careers by females and males see Table 2 (page 28) Students notice this imbalance most when they see the ratio of females to males in a STEM-related course

In 2006 92 of the females entering college who chose a biological sciences major as compared to 73 of the male freshman This is equivalent to 1 female for every 08 males a reversal of the ratio found in 1976 In engineering the percentages were 24 females and 146 males a ratio of 1 female to 61 males This is an increase from the ratio in 1976 In computer science the percentages were 03 to 21 or 1 female for every 7 males a much smaller ratio than in 1976 (Sax et aI 2010)

1 IV ]01-l W atnl orguk 27

AND PERCENTAGES ~--- shy

ce 06 12 12

34

12

173

46 21

Table 2 Percent of each gender hl~~nn a stem major by year

One reason researchers track career choice data is to partially gender differences in incomes STEM careers are on average paying careers In 2009 the median income offu-time year round male workers in the US was 27 Nmnr~rI to for females The female-toshymale earnings ratio was 077 Thus females earned 77 cents for every dollar earned by a male (Majority Staff of the Joint Economic Committee of the United States 2010)

Another reason to compare ratios is to the allocation of resources For example every ten years in the US the National Highway Traffic

Administration commissions random sobriety checks of more than 6500 drivers in 60 locations across the US Blood alcohol levels are tested

between 930 am and 330 pm (2482) pJemy 11 pm to midnight (2123) and

Hnir 1 am to 3am (1948) During rtTUTltgt

hours approximately 1 in 500 drivers were alcohol impaired That rate increased to 1 in 83 in the early evening and 1 in 21 between 1 am and 3am (Lacey et al 2009) However a Nebraska highway

administrator claimed that in areas with high concentrations of bars after 11 pm the ratio increases to 1 in 4 (Duggan 2009) That means that for every four drivers stopped in the of a bar 1 of them will be found to be alcohol impaired Armed with such data police patrols can be allocated more effectively

Concluding Thoughts

We use rates ratios and percentages in daily life to make decisions and to make comparisons The purposeful use of ratios with appropriate units of measure can enable one to see the situation more clearly As the three of us poundIfIJfll()OEO a shared understanding of ratios we saw that the context solidified our understanding as we distinguished commensurate and nonshycommensurate ratios and their

contextual we were able both to communicate and to achieve a common understanding Therefore we believe authentic contextual problems and tasks are needed for middle school students to achieve a better

understanding of ratios which as previously stated is a cornerstone concept for mathematics

Kenneth Chelst Asli and Thomas Edwards Wayne State University Detroit

References

Duggan J (2009) Dodging drunk drivers Statistics on driving are sobering Lincoln Joumal Star December 312009 httpjournalstarcoml

001 cc4c002eOhtml

J H T Furr-Holden D Voas R B E Torres P A S Ramirez A Brainard K and Berning A (2009) 2007 National Roadside ofAlcohol and Use Drivers Alcohol Results NHTSA Report No (DOT HS 811 248)

Lesh R Post and Behr M (1 Proportional reasoning In J Hiebert and M Behr Number Concepts and in the Middle Grades Reston VA Lawrence Erlbaum and National Council of Teachers of Mathematics

National Highway Safety Administration (NHTSA) (2012) Motor vehicle crashes Overview Traffic Safety Facts Research Note NHTSA No (DOT HS 811 (Revised 2012) httpwww-nrdnhtsadot govPubs811552pdf

Sax L J Jacobs J A and T A (2010) Womens Representation in Science and

Fields of 1976-2006 presented at the annual meeting of the Association for the Study of Education November 2010 Indianapolis IN

Majority Staff of the Joint Economic Committee of the United States 0 December) Invest in Women Invest in America A Comprehensive Review of Women in the US Economy Washington DC

28 May

RETHINKING RATIOS RATES AND PERCENTAGES

Auto Vehicle Fatalities I 2009

Total 33883

Alcohol Impaired 10759

Fatality Rate per 100 million Vehicle 115

Miles Travelled

I 2010

32885 -998 -29

10228 -531 -49

110 -005 -43

Table 1 Auto vehicle fatality rates 2009 and 2010

needed to travel a known distance For example if you know your rate of travel in miles per hour you can easily estimate time of arrival at a location 175 miles away However when thinking about stopping time or reaction time when driving an automobile feet per second or meters per second are more meaningful For example it is much easier to visualize how far you would travel while reacting to a need to stop if you think of your speed as 88 feet per second rather than 60 miles per hour although these two rates are equivalent

Rates are often used to track change over time to compare different rates for similar entities or to make a forecast The automotive fatality rate is tracked over time to see if actions that have been taken are making driving safer (NHTSA 2012 see Figure 1 (page 26) and Table 1 (above)) The denominator used to calculate the rate may be based on the best unit of comparison or may be chosen simply for convenience Determining the automotive fatality per 100 million miles traveled (VMT) allows for comparing one year to another when the number of miles travelled may be significantly different Notice in Figure 1 the number of vehicle fatalities per year has gone up and down over a 60-year period as more people have owned cars and have travelled more miles in total However there is a strong general downward trend in the death rate per VMT

The cost per ounce is used to compare differentshysized items of the same product Miles per gallon is often used to compare two different vehicles to see which achieves better fuel economy Miles per gallon allows you to forecast how far you can travel before needing to buy fuel how much it will cost to drive a certain distance or your average annual vehicle fuel expenditure

Commensurate Ratios

In commensurate ratios the units of measure are the same or can be considered to be the same For example the number of people in prison and the population or the number of men and women are all measured in units that are essentially the same human beings

When the set whose measure is in the numerator is a subset of that whose measure is in the denominator the ratio can be presented as either a rate or as a percentage or both The rate of prison incarceration in the US is calculated by dividing the number of people in prison by the total population The US rate is 784 per 100000 of population These rates are often calculated for different subgroups For example if the numerator and denominator were limited to adults that rate would be 962 adults per 100000 adults This rate might be tracked over time or the rates of specific subgroups might be compared Sometimes these rates are converted to percentages to emphasize relative magnitude

Another type of ratio involves comparing the same two statistics over time In this instance the ratio represents the rate of growth or decline reported as a percentage For example the number of automobile fatalities is always compared from year-to-year to determine a trend as seen in Table 1 From 2009 to 2010 the number of auto vehicle fatalities decreased by 29 If adjusted for VMT the percent decline is 43

One of the most common uses of ratio involves comparing two different subsets Sometimes this type of ratio is used to emphasize the magnitude of difference For example in the US there is an imbalance in choices of STEM careers by females and males see Table 2 (page 28) Students notice this imbalance most when they see the ratio of females to males in a STEM-related course

In 2006 92 of the females entering college who chose a biological sciences major as compared to 73 of the male freshman This is equivalent to 1 female for every 08 males a reversal of the ratio found in 1976 In engineering the percentages were 24 females and 146 males a ratio of 1 female to 61 males This is an increase from the ratio in 1976 In computer science the percentages were 03 to 21 or 1 female for every 7 males a much smaller ratio than in 1976 (Sax et aI 2010)

1 IV ]01-l W atnl orguk 27

AND PERCENTAGES ~--- shy

ce 06 12 12

34

12

173

46 21

Table 2 Percent of each gender hl~~nn a stem major by year

One reason researchers track career choice data is to partially gender differences in incomes STEM careers are on average paying careers In 2009 the median income offu-time year round male workers in the US was 27 Nmnr~rI to for females The female-toshymale earnings ratio was 077 Thus females earned 77 cents for every dollar earned by a male (Majority Staff of the Joint Economic Committee of the United States 2010)

Another reason to compare ratios is to the allocation of resources For example every ten years in the US the National Highway Traffic

Administration commissions random sobriety checks of more than 6500 drivers in 60 locations across the US Blood alcohol levels are tested

between 930 am and 330 pm (2482) pJemy 11 pm to midnight (2123) and

Hnir 1 am to 3am (1948) During rtTUTltgt

hours approximately 1 in 500 drivers were alcohol impaired That rate increased to 1 in 83 in the early evening and 1 in 21 between 1 am and 3am (Lacey et al 2009) However a Nebraska highway

administrator claimed that in areas with high concentrations of bars after 11 pm the ratio increases to 1 in 4 (Duggan 2009) That means that for every four drivers stopped in the of a bar 1 of them will be found to be alcohol impaired Armed with such data police patrols can be allocated more effectively

Concluding Thoughts

We use rates ratios and percentages in daily life to make decisions and to make comparisons The purposeful use of ratios with appropriate units of measure can enable one to see the situation more clearly As the three of us poundIfIJfll()OEO a shared understanding of ratios we saw that the context solidified our understanding as we distinguished commensurate and nonshycommensurate ratios and their

contextual we were able both to communicate and to achieve a common understanding Therefore we believe authentic contextual problems and tasks are needed for middle school students to achieve a better

understanding of ratios which as previously stated is a cornerstone concept for mathematics

Kenneth Chelst Asli and Thomas Edwards Wayne State University Detroit

References

Duggan J (2009) Dodging drunk drivers Statistics on driving are sobering Lincoln Joumal Star December 312009 httpjournalstarcoml

001 cc4c002eOhtml

J H T Furr-Holden D Voas R B E Torres P A S Ramirez A Brainard K and Berning A (2009) 2007 National Roadside ofAlcohol and Use Drivers Alcohol Results NHTSA Report No (DOT HS 811 248)

Lesh R Post and Behr M (1 Proportional reasoning In J Hiebert and M Behr Number Concepts and in the Middle Grades Reston VA Lawrence Erlbaum and National Council of Teachers of Mathematics

National Highway Safety Administration (NHTSA) (2012) Motor vehicle crashes Overview Traffic Safety Facts Research Note NHTSA No (DOT HS 811 (Revised 2012) httpwww-nrdnhtsadot govPubs811552pdf

Sax L J Jacobs J A and T A (2010) Womens Representation in Science and

Fields of 1976-2006 presented at the annual meeting of the Association for the Study of Education November 2010 Indianapolis IN

Majority Staff of the Joint Economic Committee of the United States 0 December) Invest in Women Invest in America A Comprehensive Review of Women in the US Economy Washington DC

28 May

AND PERCENTAGES ~--- shy

ce 06 12 12

34

12

173

46 21

Table 2 Percent of each gender hl~~nn a stem major by year

One reason researchers track career choice data is to partially gender differences in incomes STEM careers are on average paying careers In 2009 the median income offu-time year round male workers in the US was 27 Nmnr~rI to for females The female-toshymale earnings ratio was 077 Thus females earned 77 cents for every dollar earned by a male (Majority Staff of the Joint Economic Committee of the United States 2010)

Another reason to compare ratios is to the allocation of resources For example every ten years in the US the National Highway Traffic

Administration commissions random sobriety checks of more than 6500 drivers in 60 locations across the US Blood alcohol levels are tested

between 930 am and 330 pm (2482) pJemy 11 pm to midnight (2123) and

Hnir 1 am to 3am (1948) During rtTUTltgt

hours approximately 1 in 500 drivers were alcohol impaired That rate increased to 1 in 83 in the early evening and 1 in 21 between 1 am and 3am (Lacey et al 2009) However a Nebraska highway

administrator claimed that in areas with high concentrations of bars after 11 pm the ratio increases to 1 in 4 (Duggan 2009) That means that for every four drivers stopped in the of a bar 1 of them will be found to be alcohol impaired Armed with such data police patrols can be allocated more effectively

Concluding Thoughts

We use rates ratios and percentages in daily life to make decisions and to make comparisons The purposeful use of ratios with appropriate units of measure can enable one to see the situation more clearly As the three of us poundIfIJfll()OEO a shared understanding of ratios we saw that the context solidified our understanding as we distinguished commensurate and nonshycommensurate ratios and their

contextual we were able both to communicate and to achieve a common understanding Therefore we believe authentic contextual problems and tasks are needed for middle school students to achieve a better

understanding of ratios which as previously stated is a cornerstone concept for mathematics

Kenneth Chelst Asli and Thomas Edwards Wayne State University Detroit

References

Duggan J (2009) Dodging drunk drivers Statistics on driving are sobering Lincoln Joumal Star December 312009 httpjournalstarcoml

001 cc4c002eOhtml

J H T Furr-Holden D Voas R B E Torres P A S Ramirez A Brainard K and Berning A (2009) 2007 National Roadside ofAlcohol and Use Drivers Alcohol Results NHTSA Report No (DOT HS 811 248)

Lesh R Post and Behr M (1 Proportional reasoning In J Hiebert and M Behr Number Concepts and in the Middle Grades Reston VA Lawrence Erlbaum and National Council of Teachers of Mathematics

National Highway Safety Administration (NHTSA) (2012) Motor vehicle crashes Overview Traffic Safety Facts Research Note NHTSA No (DOT HS 811 (Revised 2012) httpwww-nrdnhtsadot govPubs811552pdf

Sax L J Jacobs J A and T A (2010) Womens Representation in Science and

Fields of 1976-2006 presented at the annual meeting of the Association for the Study of Education November 2010 Indianapolis IN

Majority Staff of the Joint Economic Committee of the United States 0 December) Invest in Women Invest in America A Comprehensive Review of Women in the US Economy Washington DC

28 May