MATH on the Job
Trevor is a small business owner who recently opened his gallery and café in Saskatoon, SK. Before opening his business, Trevor took business administration courses at Great Plains College in his hometown of Warman. He wrote a business plan and conducted research to find a business location that was accessible, visible, and in a popular part of the city. Trevor’s gallery sells paintings, jewellery, prints, carvings, and souvenirs made by Dene and Cree artists.
A business owner uses different methods such as graphing and invoicing to track expenses and determine how profitable the business is.
As a business owner, Trevor is always busy keeping track of stock, employees’ hours and salaries, and profits. During his first year of business, after he had paid all the bills for the month, Trevor tracked his net income on a broken line graph to see how profitable his business was, and determine which months had the highest sales.
net income: income earned after all expenses have been paid
Below is Trevor’s graph of net income for the last year.
0
Net Income by Month for One Year p
m
1. In what two months did Trevor make the most net income? Why might this be so?
2. From March to August, Trevor’s net income was relatively low. Suggest two reasons why this might be so.
explore the Math
A broken line graph is a visual representation of data that usually shows change over time. Points representing values are connected by line segments on a broken line graph. In the last chapter, you used line graphs to represent slope and rate of change. The line graphs you drew in chapter 1 display numerical relationships, but broken line graphs are usually used to display data or to show a trend. They are often used by businesses to help inform them and their clients of patterns.
broken line graph:
graph that uses points joined by line segments to display data
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In the scenario on the previous page, Trevor used a simple broken line graph to visualize his profits. If he compares the profits to specific items sold during that period of time, he might be better able to manage his stock and make better choices about what to provide during different seasons to increase his profits. Trevor is able to use statistics to compare his profits during different periods of time.
statistics: the collection, presentation, and analysis of numerical data
Broken line graphs can also be used to compare sets of data using a multiple broken line graph. A multiple broken line graph needs to indicate what each line represents. This can be done by using a legend in the corner that indicates what each colour or pattern represents.
example 1
every day, fragrant, healthy bread comes out of the wood-burning ovens at Whitehorse’s Alpine bakery.
Nutritious, great-tasting food is the focus of the Alpine Bakery in Whitehorse, Yukon. The bakery makes specialty breads such as German rye, Volkenbrot, and Expedition Bread. Expedition Bread is extremely healthy and keeps well unrefrigerated because of its ingredients. Hikers and canoeists planning lengthy trips often take this bread along.
The bakery owner thinks that he generally sells more German rye bread, but he wishes to compare the number of loaves of each type of bread that were sold each month over a one-year period.
This will help him determine which products are more popular and which breads sell better at different times of the year.
sales of loaves by Month
Month J F M A M J J A S O N D
Expedition Bread 78 110 98 105 74 124 140 132 65 50 96 84
German rye 68 86 88 86 104 78 96 80 55 72 83 62
Volkenbrot 45 82 78 58 65 68 65 65 42 140 75 42
a) Using the data provided in the table, draw a broken line graph to represent sales of each type of bread.
b) Does it appear that any one type of bread sells consistently better than the others?
c) During what part of the year do bread sales generally decrease?
d) What might explain the increase in Expedition Bread sales in June, July, and August?
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Number of Loaves Sold by Type
Number of Loaves
Legend n
m Month
hInt
When graphing data, it is important to select scales for the vertical and horizontal axes that appropriately convey the information.
Note that in this graph, a legend is needed to identify what each line represents. The vertical scale begins at 0 and rises in increments of 30. The bakery owner is not looking for the exact number of loaves sold each month. Rather, he wishes to know the general pattern of sales from month to month, so this scale is appropriate.
b) The Expedition Bread and German rye generally appear to sell better than the Volkenbrot.
c) Sales appear to decrease in January and December, or during the winter.
d) The increase in sales of Expedition Bread could be caused by more people planning outdoor trips during warmer weather, and buying food to take with them.
example 2
A weather observer records, observes, and transmits weather information such as temperature and precipitation levels. Weather observers often work at local airports and provide weather information to government, media, and the public. A weather observer for Banff, AB, obtained the following information on average monthly precipitation there.
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Monthly preCIpItatIon
Month J F M A M J J A S O N D
Precipitation
(mm) 27.5 21.9 23.4 32.4 59.6 81.7 54.2 60.1 42.1 29.4 26.8
a) Graph this information on a broken line graph.
b) What month would be best for camping? Skiing? Hiking?
Justify your choices.
c) Use the graph to predict what the average amount of
precipitation might have been in December. What assumptions have you made?
About four million visitors come to banff every year.
solutIon
a)
0 10 20 30 40 50 6070 80 90 100
Dec. Nov.Oct.Sep.Aug.Jul.Jun.MayApr.Mar.Feb.Jan.
Month
Precipitation (mm)
p
Average Monthly Precipitation in Banff, Alberta
m
b) Based on the graph above, the best month for camping would be July because it is the driest month during camping season, and likely the warmest. The best month for skiing would be January because it receives the most snow during ski season. The best month for hiking would be September because it is drier and likely cooler than previous months.
c) The previous January, there had been 27.5 mm of precipitation and in November there was 26.8 mm. There is likely about the same amount of precipitation in December, especially considering the similar amounts in November, February, and March.
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dIsCuss the Ideas
extrapolatIng and InterpolatIng values
Sleet or sunshine? In the example on the previous page, a value was extrapolated from the data on precipitation levels. You may be able to
extrapolate a new value from the data shown on a broken line graph if there is a general trend of increase or decrease. It may also be possible to extrapolate from one year to the next if the time of year affects the data in a seasonal pattern.
extrapolate: to estimate a value beyond a known range of values
January February March
Month
m When you estimate
a value between two known values, you are interpolating. In a small group, look at the graph to the right. This graph was made by a health inspector whose weekly duties include checking and recording the temperature of a large walk-in cooler in a hospital kitchen.
The temperature must be at a certain level to ensure that food is stored safely. In order to deter bacteria growth, food must be stored at a temperature of 4° C or lower.
interpolate: to estimate a value between two known values
1. The cooler’s temperature measured 4° C in the first week of January, 3.9°
in the second week, and 3.7° C in the fourth week of January. What do you think the cooler’s temperature was in the third week of January?
2. The cooler’s temperature was 3.5 ° C in the second week of February and 3.3° C in the fourth week of February. What do you think the temperature was in the third week of February?
3. Predict what the cooler’s temperature will be in the second week of March.
What do you estimate it will be in the fourth week of March?
4. Discuss with a partner the accuracy you might expect when extrapolating and interpolating values.
health inspectors examine the cleanliness of commercial and industrial kitchens. they also ensure that refrigerated food is being stored at safe temperatures.
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Mental Math and estimation
Jennifer is a competitive swimmer who is part of her school’s swim team. On the wall near the pool, each member of the swim team tracks his or her performance by graphing how long it takes them to swim a certain distance, using a certain stroke. Jennifer is currently training for the 200 m freestyle race. This graph shows her performance times for the past 30 days. Based on the graph, how long do you think it will take Jennifer to complete the 200 m freestyle on day 33?
On day 36?
30 33 27
24 21 18 15 12 9 6 3 t
Jennifer’s Times, 200 m Freestyle, January
d Day of the Month
220
0 225 230 235 240 245
Time (sec)
If you are a competitive athlete, how do you keep track of your performance to determine if it is improving?
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ACtIVIty 2.1
jobs for young people
T
How do you want to spend 89 784 hours of your life? That is the amount of time a person will spend at work if they work an eight-hour day, five days a week, from the age of 22 to their retirement at age 65. As a young person, what are your chances of finding employment? This activity will help you find out.1. Working with a partner, research the youth employment rate over a 30-year period, from around 1975 to 2005, using data you can find on the Statistics Canada website. You can find relevant data by performing keyword searches, such as “youth employment rate” and “employment rate.” Because data are collected periodically, it may be difficult to find more recent statistics.
2. Next, research the general employment rate over the same period.
3. Once you have found this information, construct a multiple broken line graph that compares the youth employment rate to that of the general employment rate.
4. Once you have graphed the data, answer the following questions.
a) What population was included in the youth employment rate data?
b) Was the general employment rate higher or lower than the youth employment rate? Why might this be?
c) Has the youth employment rate recently gone up or down? What does this imply for young people looking for work?
d) Has the general employment rate recently gone up or down?
e) Using your research, who do you think has a better chance of finding a job: a young person, or someone from the general population?
During the summer, many students earn extra income by being self-employed.
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example 3
Paul was studying labour force participation rates across Canada and wanted to examine trends by gender. He found the following information graphed by Statistics Canada.
40 55 70 85 100
Women Men
200520042003200220012000199919981997199619951994199319921991199019891988198719861985198419831982198119801979197819771976
Year
y Labour Force Participation Rate, by Gender, 1976–2005
p
Percentage of Labour Force Participation
a) In 1976, what was the percentage of men who were in the labour force?
What was the percentage of women?
b) Between 1976 and 2005, what was the general trend in the percentage of men who were in the labour force? Percentage of women?
c) What might account for this change?
d) What do you notice about this graph that may be a bit misleading?
e) Think about your answer to the last question. Redraw the graph based on your conclusions. Does your new graph allow the viewer to interpret the data more clearly? Why?
f) In the graph above, the difference between the data points, or values, is difficult to see. The two lines almost appear to be flat, and most of the values look the same or similar. What could you do to make the difference between the data points more obvious?
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employees with time management and communication skills are valued in the workplace.
solutIon
a) Approximately 78% of men and 46% of women were in the labour force in 1976.
b) The percentage of men in the labour force dropped slightly to about 72% while the percentage of women in the labour force rose considerably to about 61%.
c) More women joined the labour force because of the need for a second income due to the increased cost of living. This would also mean there were fewer positions for men so that their percentage decreased.
d) The percentages start at 40%, not 0%.
e) The graph could be misleading because the values on the vertical axis do not start at zero. Redrawing the graph so that the values on the vertical axis begin at zero provides a more accurate perspective of the relationship between the two lines.
0 10 20 30 40 50 60 70 80 90 100
Men Women
200520042003200220012000199919981997199619951994199319921991199019891988198719861985198419831982198119801979197819771976
Year
y Labour Force Participation Rate, by Gender, 1976–2005
p
Percentage of Labour Force Participation
f) If the scale on the vertical axis was in units of 1 instead of units of 10, the difference between the data points would be easier to see.
Differences between data points would also be easier to see if each unit on the horizontal axis represented four years, instead of one year.
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dIsCuss the Ideas
InfluenCIng InterpretatIons
In the preceding example, you considered the scaling of the graph and how this might affect your perception of the data presented. There are many ways to influence how a viewer interprets a graph. Consider the examples below, which show two different ways to graph the same data.
0
Number of Lost/Broken Dishes
500
Number of Lost/Broken Dishes
James works for a catering company. One of his tasks is to track expenses by recording the number of dishes that are taken to an event. At the end of the event, James must record whether any dishes were lost, forgotten, or broken.
Every six months, he must present this data to his supervisor in a report summarizing kitchen expenses.
With a partner, discuss the following questions.
1. How do these graphs differ in their presentation of data? In what ways does this affect a viewer’s perception of the data?
2. Which graph do you think most accurately represents the data?
3. Why might a person want to present their data as in graph 2?
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ACtIVIty 2.2
Invest In the best
T
Enter a world full of superhuman athletes, attacking blobs of slime, or skilled warriors. You or someone you know does this each time he or she plays a video game. Which game is your favourite? While video games can be entertaining, they also earn a great deal of money for the corporations that create the games and the consoles they are played on.Would it be a wise decision to invest in a company that manufactures game-playing equipment? In this activity, you will find out. First, research the name of the company that makes your favourite game console, plus the name of one of its competitors.
Video game technology has improved greatly, but some people still prefer classic games from 25 years ago.
Next, look in the newspapers or online and locate your companies on the NASDAQ Stock Exchange. Check the closing price of your companies’ stocks each day for a week. Keep a printout or a clipping from the newspaper as a record of the prices.
1. Graph the information on a double broken line graph.
2. Which game company had a better week? Explain your reasoning.
3. If you were to buy shares in one of the companies, which one would you choose and why?
4. Compare your two companies with the companies one of your classmates researched. What general trend does there seem to be in the game console business this week?
buIld your skIlls
1. Jean-François is the caretaker for an apartment building in Estevan, Saskatchewan. He is responsible for cleaning the building’s common areas and repairing its equipment, such as washing machines, hot water tanks, and furnaces. Jean-François suspects that one of the building’s hot water tanks is not working properly, so he records the temperature of the machine’s hot water on a daily basis. The water should be 130° F. Use the data in the table to construct a broken line graph showing the hot water temperatures.
hot Water teMperatures
Day 1 Day 2 Day 3 Day 4 Day 5
90° F 82° F 78° F 81° F 69° F
building caretakers maintain and repair the interior and exterior of apartment buildings or condominiums. their jobs can include vacuuming hallways, repairing equipment, and gardening.
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2. Julie started working as a community government assistant housing manager for the city of Brandon, Manitoba, in April of last year. Her boss has been asked to write a report on the average number of people placed in community housing monthly. Julie has been given the task of recording the number of people who have been placed in community housing each month.
0 3 6 9 12 15
Number of People Placed in Community Housing
M F J D N O S A J J M A
Number of People
Month n
m
a) In which month were the most people placed? How many people were placed?
b) In which month were the fewest people placed? How many people were placed?
c) Can you make any prediction about a trend in the number of people placed in community housing, and if so, what would it be?
d) Previous statistics show that, on average, ten people were placed in community housing in Brandon each month. Compare the average of the data that Julie collected with the previous average. Predict the reasons for any differences in the averages.
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3. Consider the graph below.
600 640 680 720 760 800
200820072006200520042003200220012000199919981997
Year e
y
Earnings ($)
Average Weekly Earnings of Canadians
a) What does the graph tell you?
b) What were the approximate average weekly earnings of a Canadian during 2003?
c) Describe the general trend in weekly earnings from 1997 to 2008. Why might this trend exist?
d) How might the way this graph is constructed be misleading?
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4. Chi was trying to decide what type of work he would pursue after high school. His career counsellor had the following graph on the wall and suggested that it might help him make his decision.
0 10 15 20 25
Food services employee
Profession Data entry clerk
Retail salesperson Bookkeeper
Construction worker
Plumber CarpenterElectrician
Average Hourly Wage ($)
Average Hourly Wages in Canada, by Job, 2007–2008
p w
a) If Chi is most concerned about income, what choice do you suggest he make?
b) Which job has the highest hourly wage? The lowest? By how much?
Why do you think this is?
c) Do you know anyone who has a job that is listed on the graph? Is his or her hourly wage similar to the one displayed on the graph? Why might wages be higher or lower than those shown on the graph?
d) Before Chi tries to find a job in one of these professions, what other factors will he have to consider?
A career counsellor can help you decide where to go to college. they can also provide information on how to apply for scholarships or write a resumé.
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5. Arseno is a researcher for the Summer Olympic Organizing Committee.
In reviewing the participation of women during the twentieth century, he recorded the number of countries from which female athletes originated over the years.
nuMber of CountrIes WIth feMale olyMpIans
year 1900 1904 1908 1912 1920 1924 1928 1932 1936 1948 1952 1956 1960 1964 1968 1972 1976 1980 1984 1988 1992 1996 2000 No. of
countries 5 1 4 11 13 20 26 18 26 33 51 39 45 53 54 65 66 54 94 117 136 169 199
a) Graph the information, taking careful note of the years.
b) What is the trend in the number of countries from which women
b) What is the trend in the number of countries from which women