The Mathematics of Weather - Canada.ca · 2017. 6. 22. · urban outdoor air pollution. 4 Frequently asked questions about the Air Quality Health Index (AQHI). Environment Canada,

The Mathematics of WeatherReal-World Questions That Apply Math Concepts to Weather

AcknowledgementsProject Management: Victoria Hudec (Outreach Officer: Ontario; Sky Watchers National Coordinator)Author and Publication Design: Nicole Lantz (Sprout Educational Consulting)Photography: George Lantz (Vision Photography) and iStock

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THIRD-PARTY MATERIALSSome of the information contained in this publication or product may be subject to copyrights held by other individuals or organizations. To obtain information concerning copyright ownership and restrictions, please contact:

Environment and Climate Change CanadaPublic Inquiries Centre7th Floor, Fontaine Building200 Sacré-Coeur BoulevardGatineau QC K1A 0H3Telephone: 819-997-2800Toll Free: 1-800-668-6767 (in Canada only)Email: [email protected]

ISBN: En56-255/2015E-PDFCat. No.: 978-0-660-01998-7The Mathematics of Weather - Real-World Questions That Apply Math Concepts to Weather

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Photos: © iStock

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Aussi disponible en français

Worksheets ...............................................................................................1The Math Behind AQHI ................................................................................................... 1The Math Behind AQHI: Answers ................................................................................. 2Tracking the AQHI ......................................................................................................... 3Tracking the AQHI: Answers ......................................................................................... 4Using Math to Understand the AQHI and Health ........................................................ 7Using Math to Understand the AQHI and Health: Answers ....................................... 8What is the Temperature? .............................................................................................11What is the Temperature? Answers ............................................................................ 12

Mathematics Curriculum Links ............................................................. 15

T A B L E O F C O n T E n T s

iThe Mathematics of Weather – Real-World Questions That Apply Math Concepts to Weather i

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1The Mathematics of Weather: Real-World Questions That Apply Math Concepts to Weather 1

1. The Air Quality Health Index (AQHI) isa measure of three air pollutants that affectour health. It includes ozone at groundlevel, particulate matter (PM) and nitrogendioxide (NO2).

1 A mathematical equation isused to calculate the AQHI.

A. Use pictures or words to define“equation”, and explain why it is useful.

2. PM levels should not be higher than60µg/m3. (µg means “microgram”)

A. There are 1 000 000µg in 1 g. How many millionmicrograms (µg) are in 10g?

B. Write 60µg as a decimal in grams.

C. Write the maximum level of PM asa ratio. Use m3 of air : µg pollutant.

1 National Ambient Air Quality Objectives. Health Canada, n.d. Web. 22 Jan 2013.

3. Scientists2 measured the concentrationof PM2.5 indoors near an open fire. It was693µg/m3.

PM2.5 outside on the patio was 94µg/m3.

A. Think about what the maximumPM should be compared to whatit actually was. Why might youestimate this ratio as around 6:70?

B. Could you estimate this ratioas 1:11? Why or why not?

C. Considering all theinformation about PM2.5 levels,what has a ratio of 7:1?

2 Zuk, Miriam, et al. The impact of improved wood-burning stoves on fine particulate matter concentrations in rural Mexican homes. Journal of Exposure Science and Environmental Epidemiology. (2006):1-9. Web. 22 Jan 2013.

The Math Behind AQHIReal-world questions that apply math concepts of large numbers, decimals, ratios, and equivalent ratios.

Complete the questions in your math journal or on looseleaf paper.

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2 The Mathematics of Weather – Real-World Questions That Apply Math Concepts to Weather

1. The AQHI is a measure of three airpollutants that affect our health. It includesozone at ground level, particulate matter(PM) and nitrogen dioxide. A mathematicalequation is used to calculate AQHI.

A. Use pictures or words to define“equation”, and explain why it is useful.Answers will vary.

2. PM levels should not be higher than60µg/m3 (µg means “microgram”).

A. There are 1 000 000µg in 1 g. How many millionmicrograms (µg) are in 10g?There are 1 million µg in 1g, so there are 10 million µg in 10g.

B. Write 60µg as a decimal in grams.(Move the decimal six places to the left.60 µg = 0.000006g PM.

C. Write the maximum level of PM asa ratio. Use m3 of air : µg pollutant.1:60

3. Scientists measured the concentrationof PM2.5 indoors near an open fire. It was693µg/m3. PM2.5 outside on the patio was94µg/m3.

A. Think about what the maximumPM should be compared to whatit actually was. Why might youestimate this ratio as around 6:70?The maximum desirable level of PM is 60µg/m3.

PM near the fire is 693 µg/m3.

60:693 is about 60:700, or 6:70.

B. Could you estimate this ratioas 1:11? Why or why not?Yes, 6:70 is the same as 1:11.

70 ÷ 6 is approximately 11, so they are equivalent ratios.

C. Considering all theinformation about PM2.5 levels,what has a ratio of 7:1?PM inside:PM outside = 693:94 or approximately 700:100.

This is the same as a ratio of 7:1.

The Math Behind AQHI: AnswersReal-world questions that apply math concepts of large numbers, decimals, ratios, and equivalent ratios.


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3The Mathematics of Weather – Real-World Questions That Apply Math Concepts to Weather 3

1. An Ontario study tracked the maximumAQHI each day.

They found out that the mean maximum AQHI was 4.07 in the summer and 3.18 in the fall.

A. Create an addition, subtraction,multiplication or division problem usingthese numbers. Then, work in pairsto solve another student’s problem.

2. A study in British Columbia3 trackedAQHI for six years from 2000 to 2006. Theytracked the percentage of days when AQHIwas 4 for at least one hour. This means airpollution posed a moderate health risk onthose days.

Scientists found that AQHI reached this level on 35% of the days in Prince George and on 37% of the days at the Vancouver airport.

A. How many days during thesix years was there a moderatehealth risk in Prince George?

B. Is six years a reliablesample? Why or why not?

C. If you worked at the Vancouverairport every day next year for the entireyear, would it be surprising for there tobe a moderate health risk AQHI on 30of those days? For 3 months of the year?For 6 months of the year? Explain.

3 Hasselback, Paul, and Eric Taylor. Air Quality Health Index Variation across British Columbia. B.C. Air Quality. Interior Health, Sept 2010. Web. 22 Jan 2013.

Tracking the AQHI Real-world questions that apply math concepts of statistics, decimals, operations, percent, percent as a decimal, probability and sampling.


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1. An Ontario study tracked the maximumAQHI each day.

They found out that the mean maximum AQHI was 4.07 in the summer and 3.18 in the fall.

A. Create an addition, subtraction,multiplication or division problem usingthese numbers. Then, work in pairsto solve another student’s problem.Answers will vary.

2. A study in British Columbia trackedAQHI for six years from 2000 to 2006. Theytracked the percentage of days when AQHIwas 4 for at least one hour. This means airpollution posed a moderate health risk onthose days.

Scientists found that AQHI reached this level on 35% of the days in Prince George and on 37% of the days at the Vancouver Airport.

A. How many days during thesix years was there a moderatehealth risk in Prince George?Estimating:

There are 365 days in a year.365 is closer to 400 than 300. 400 days × 6 years = 2400 days.

35% can be rounded up or down, but since we rounded up the days, let’s round down the percent to 30%.

10% would be easy because 2400 divides evenly by 10 to make 240.

Three of these would make 30%, so it would be 240 x 3.

To estimate it, use 250 x 3, or 750.

So, we estimate about 750 days over 6 years would have a moderate health risk in Prince George.

Tracking the AQHI: AnswersReal-world questions that apply math concepts of statistics, decimals, operations, percent, percent as a decimal, probability and sampling.

Complete the questions in your math journal or on looseleaf.

Worksheets


Calculating:

365 days in a year x 6 years = 2190 days.

Use 0.35 for 35%.

0.35 x 2190 days = 766.5 days with a moderate health risk in Prince George over 6 years.

B. Is six years a reliable sample? Why or why not?Data from larger samples generally produces more reliable probabilities. They are collecting data every day for six years or 2190 days. This is reliable.

C. If you worked at the Vancouver airport every day next year for the entireyear, would it be surprising for there to be a moderate health risk AQHI on 30of those days? For 3 months of the year? For 6 months of the year? Explain.It is not surprising for a six-year pattern to continue, so let’s assume approximately 37% of the days will have this AQHI.

37% is less than half, so we would expect a moderate health risk AQHI for less than half the year.

The best answer is 3 months.

Worksheets


1. The World Health Organization (WHO)recently estimated that 800 000 deaths peryear worldwide could be attributed to urbanoutdoor air pollution.4

A. Estimate: What fraction of amillion is that? Use the word “over”or “under” in your answer.

B. Let’s assume that the WHOused rounding before they reportedtheir number. List five numbersthat could have been the exactnumber of deaths worldwide fromurban outdoor air pollution.

4 Frequently asked questions about the Air Quality Health Index (AQHI). Environment Canada, 13 Nov 2013. Web. 21 Nov 2013.

2. The prevalence of asthma among people15 years of age and over has been increasingover the last 20 years:5

Year Prevalence of Asthma

1979 2.3%1988 4.9%1994 6.1%2004 8.4%

A. Make a scatterplot with year onthe x-axis and per cent of adultswith asthma on the y-axis.

B. What conclusions can youdraw from your graph?

C. What is the average (mean)prevalence of asthma?

D. How much did theprevalence of asthma increasebetween 1979 and 2004?

5 Chen, H. Asthma Heath rep 16. (2004):43-6. Asthma Facts & Statistics. Asthma Society of Canada. Web. 15 Jan 2013.

Using Math to Understand the AQHI and Health Real-world questions that apply math concepts of percent, large numbers, fractions, rounding/estimating, operations, data and data displays.


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1. The World Health Organization (WHO)recently estimated that 800 000 deaths peryear worldwide could be attributed to urbanoutdoor air pollution.

A. Estimate: What fraction of amillion is that? Use the word “over”or “under” in your answer.Over 3/4 million deaths.

B. Let’s assume that the WHO usedrounding before they reported their number. List five numbers that could have been the exact number of deaths worldwide from urban outdoor air pollution. Answers will vary.

Numbers must either round up to 800 000 or round down to 800 000.

For example, 798 953 rounds up to 800 000. 834 212 rounds down to 800 000.

2. The prevalence of asthma among people15 years of age and over has been increasingover the last 20 years:

Year Prevalence of Asthma

1979 2.3%1988 4.9%1994 6.1%2004 8.4%

A. Make a scatterplot with year on thex-axis and prevalence of asthma on they-axis.

B. What conclusions can youdraw from your graph?The prevalence of asthma has increased at a steady rate.

You can tell this because the shape of the data is smooth. It makes a fairly straight line.

0

2

4

6

8

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1970 1980 1990 2000 2010

Using Math to Understand the AQHI and Health: Answers Real-world questions that apply math concepts of percent, large numbers, fractions, rounding/estimating, operations, data and data displays.


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Some students might go a step further and conclude that air pollution in increasing steadily. Does the graph really show this? What other things besides air pollution might be causing asthma?

C. What is the average (mean) prevalence of asthma?2.34.96.18.4

21.7 ÷ 4 = 5.425

The average prevalence of asthma is 5.425 per cent.

D. How much did the prevalence of asthma increase between 1979 and 2004?8.4 - 2.3 = 6.1

An increase of 6.1 per cent.

+21.7

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1. I am a two-digit even number. The sum of my numbers is 1. The difference is 1.6I am between0˚C and 15˚C. I am the mean temperature of the warmest spring in Chilliwack, B.C. Whattemperature am I?

2. I am a one-digit negative number. Starting from 0˚C, you will reach me after three days ifeach day drops three degrees. I am the mean temperature of the coldest winter in both Sydney,N.S. and in Toronto, Ont. What temperature am I?

3. I am a two-digit negative, odd number. The number in my ones place is half the number inmy tens place. I am the mean of the coldest winter in Brandon, Man., or Timmins, Ont. Whattemperature am I?

4. I am a two digit number. The number in my ones place is 6. My tens place is 1/3 of my onesplace. I am the mean temperature of the hottest summer in Kelowna, B.C. What temperatuream I?

5. I am a two-digit even number. The sum of my digits is 10. Their difference is 2. I am thecoldest wind chill recorded in Canada in the last 30 years. I was recorded on a cold day inYellowknife, N.W.T., What temperature do I feel like?

6 Temperature facts are based on Canadian cities in the last 30 years: Canadian Cities are Weather Winners! Weather Office. Environment Canada, n.d. Web. 15 Jan 2013.

What is the Temperature?Real-world math questions6 on positive and negative integers.

Use the clues to figure out the temperature. Then indicate it on the thermometer.

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1. I am a two-digit even number. The sum of my numbers is 1. The difference is 1. I am between0˚C and 15˚C. I am the mean temperature of the warmest spring in Chilliwack, B.C. Whattemperature am I?

2. I am a one-digit negative number. Starting from 0˚C, you will reach me after three days ifeach day drops three degrees. I am the mean temperature of the coldest winter in both Sydney,N.S. and in Toronto, Ont. What temperature am I?

3. I am a two-digit negative, odd number. The number in my ones place is half the number inmy tens place. I am the mean of the coldest winter in Brandon, Man., or Timmins, Ont. Whattemperature am I?

4. I am a two digit number. The number in my ones place is 6. My tens place is 1/3 of my onesplace. I am the mean temperature of the hottest summer in Kelowna, B.C. What temperatuream I?

5. I am a two-digit even number. The sum of my digits is 10. Their difference is 2. I am thecoldest wind chill recorded in Canada in the last 30 years. I was recorded on a cold day inYellowknife, N.W.T. What temperature do I feel like?

What is the Temperature? AnswersReal-world math questions on positive and negative integers.

Use the clues to figure out the temperature. Then indicate it on the thermometer.

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1. 2. 3. 4. 5.10˚C -9˚C -21̊ C 26˚C -64

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Notes:

3. The important part of this question is to remember that it is the mean temperature,not the lowest temperature ever recorded. Mean is the average for the entirewinter. The next value that fits the description would be -63˚C, which would befar too low as a mean temperature for a winter in Canada. Even the coldest winterin the entire country had a mean of -29˚C, which was in Yellowknife, N.W.T.

5. Wind chill is -64, so the answer is that it feels like -64˚C. Note that wind chill is not marked with ˚C.


Mathematics Curriculum Links


Grade 5 NumberRepresent and describe whole numbers to 1 000 000.

Use estimation strategies, including:• front-end rounding• compensation• compatible numbers

in problem-solving contexts.

Grade 6 Number1. Demonstrate an understanding of place value, including numbers that are:

• greater than one million• less than one thousandth.

2. Solve problems involving large whole numbers and decimal numbers.

5. Demonstrate an understanding of ratio, concretely, pictorially and symbolically.

6. Demonstrate an understanding of percent (limited to whole numbers) concretely, pictorially and symbolically.

7. Demonstrate an understanding of integers, concretely, pictorially and symbolically.

Patterns and Relations 2. Represent and describe patterns and relationships, using graphs and tables.

NL

NUNTYT

BCAB MB

NBPe



PR4: Demonstrate and explain the meaning of preservation of equality concretely, pictorially and symbolically.

Statistics and Probability 1. Create, label and interpret line graphs to draw conclusions.



Grade 5 Number SenseA1 represent whole numbers to the millions

A6 read and represent numbers to millions

Grade 6 Number SenseA1 represent large numbers in a variety of forms

A2 represent fractions and decimals

A3 write and interpret ratios, comparing part-to-part and part-to-whole

A4 demonstrate an understanding of equivalent ratios

A5 demonstrate an understanding of the concept of percent as a ratio

A6 demonstrate an understanding of the meaning of a negative integer

A7 read and write whole numbers in a variety of forms

A8 demonstrate an understanding of the place-value system

C5 recognize and explain how the change in one term of a ratio affects the other term

Operation SenseB8 solve and create relevant addition, subtraction, multiplication and division problems involving decimals

B12 calculate products and quotients in relevant contexts by using the most appropriate method

NS



Data AnalysisF1 choose and evaluate appropriate samples for data collection

F2 identify various types of data sources

F6 interpret data represented in scatterplots

F7 make inferences from data displays

F8 demonstrate an understanding of the differences among mean, median, and mode

F9 explore relevant issues for which data collection assists in reaching conclusions

Chance & UncertaintyG2 evaluate the reliability of sampling results

G3 analyze simple probabilistic claims

G4 determine theoretical probabilities

QC



Cycle 3 Arithmetic: Understanding and Writing Numbersnatural numbers:

– approximation

Fractions– fractions: reading, writing, numerator, denominator, various representations, order, comparison, equivalent expressions, equivalent fractions– percentages

Decimals– up to three decimal places (tenths, hundredths, thousandths): reading, writing, various representations, order, equivalent expressions, writing numbers in expanded form

Using numbers– converting from one type of notation to another: writing fractions, decimal numbers or percentages

Integers– reading, writing, comparison, order, representation

Arithmetic: Meaning of Operations Involving Numbersnatural numbers

– meaning of an equality relation (equation), meaning of an equivalence relation Fractions

– establishing equivalent fractions

Chance & Statistics– Arithmetic mean (meaning, calculation)

Probability– Predicting the likelihood of an event (certainty, possibility or impossibility)



Grade 6Number Sense and Numeration•Read, represent, compare and order whole numbers to 1 000 000, decimal numbers to

thousandths, proper and improper fractions, and mixed numbers;•Demonstrate an understanding of relationships involving percent, ratio and unit rate.

Proportional Relationships•Represent ratios found in real-life contexts, using concrete materials, drawings and standard

fractional notation; •Represent relationships using unit rates.

Data Management and Probability•Collect and organize discrete or continuous primary data and secondary data and display the

data using charts and graphs, including continuous line graphs;•Read, describe and interpret data, and explain relationships between sets of data.

ON



Grade 5 NumberN5.1 Represent, compare, and describe whole numbers to 1 000 000 within the contexts of place value and the base ten system, and quantity. [C, CN, R, T, V]

N5.4 Develop and apply personal strategies for estimation and computation including:• front-end rounding•compensation•compatible numbers.

Grade 6 NumberN6.1 Demonstrate understanding of place value including:•greater than one million• less than one thousandth

with and without technology.N6.5 Demonstrate understanding of percent (limited to whole numbers to 100) concretely, pictorially, and symbolically.

N6.6 Demonstrate understanding of integers concretely, pictorially and symbolically.

N6.8 Demonstrate an understanding of ratio concretely, pictorially and symbolically.

Patterns and Relations P6.1 Extend understanding of patterns and relationships in tables of values and graphs.

P6.2 Extend understanding of preservation of equality concretely, pictorially, physically and symbolically.

SK



Statistics and ProbabilitySP6.1 Extend understanding of data analysis to include:• line graphs•graphs of discrete data•data collection through questionnaires, experiments, databases, and electronic media• interpolation and extrapolation.

Additional information can be obtained at:

Environment and Climate Change CanadaPublic Inquiries Centre7th Floor, Fontaine Building200 Sacré-Coeur BoulevardGatineau QC K1A 0H3Telephone: 1-800-668-6767 (in Canada only) or 819-997-2800Email: [email protected]

Documents

The Mathematics of Weather - Canada.ca · 2017. 6. 22. · urban outdoor air pollution. 4 Frequently asked questions about the Air Quality Health Index (AQHI). Environment Canada,