Data And Graphs Book

Copyright 2015 Maria Miller EDITION 8/2015 All rights reserved. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, or by any information storage and retrieval system, without permission in writing from the author. Copying permission: Permission IS granted to reproduce this material to be used with one (1) teacher’s students by virtue of the purchase of this book. In other words, one (1) teacher MAY make copies of these worksheets to be used with his/her students. Permission is not given to reproduce the material for resale. Making the file(s) available on any website for the purpose of sharing is strictly prohibited. If you have other needs, such as licensing for a school or tutoring center, contact the author at http://www.MathMammoth.com/contact.php

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Math Mammoth Data and Graphs

Contents Introduction ...................................................................

4

Reading Graphs and Charts ........................................

7

Bar Graphs and Pictographs .......................................

10

More Practice with Bar Graphs and Pictographs .....

14

Making Bar Graphs 1 ...................................................

21

Making Bar Graphs 2 ...................................................

23

Making Histograms ......................................................

25

Double Bar Graphs ........................................................

27

Line Graphs 1 .................................................................

29

Line Graphs 2 .................................................................

32

Temperature Line Graphs ............................................

36

Reading Line Graphs ....................................................

38

Double and Triple Line Graphs ...................................

40

Graphs: More Practice .................................................

42

Average ...........................................................................

47

Average (Mean) .............................................................

50

Mean, Mode, and Bar Graphs ......................................

53

Circle Graphs .................................................................

55

Review .............................................................................

57

Answers ...........................................................................

59

More from Math Mammoth ..........................................

79

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Introduction Math Mammoth Data and Graphs is a worktext that covers common statistical graphs and some related topics for grades 2-5. It is a worktext, containing both the “text” (instruction) and the “work” (exercises and problems). The book starts with the easiest topics: reading and making bar graphs, pictograms, histograms, and various kinds of line graphs. Most of these lessons are best suited for 4th and 5th grades. The goals for the study of bar graphs and line graphs are listed here. The student should learn to: z z

read bar graphs, including double bar graphs, and answer questions about already plotted data draw bar graphs and histograms from a given set of data.

In order to make histograms, it is necessary to understand how to group the data into categories (“bins”). The lesson Making Histograms explains the method we use to make categories if the numerical data is not already categorized. Toward the end of the book, we study average (also called the mean) and mode, and how these two concepts relate to line and bar graphs. This is also meant for grades 4-5. Other math curricula also introduce the median, but I decided to omit it from this book. There is plenty of time to learn that concept later, for example in Math Mammoth Statistics & Probability. Introducing all three concepts at the same time tends to jumble them together and become confusing — so that many students just grasp the calculation procedures. I feel it is better initially to just introduce and contrast the two concepts of the mean and mode, in order to give the student a solid foundation. Lastly, we study Circle Graphs which may be left to study in 6th or 7th grade because it requires that the student understand percentages. Wishing you success in teaching math, Maria Miller, the author

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Helpful Resources on the Internet Mean, Median, Mode, Range, etc. Using and Handling Data Simple explanations for finding mean, median, or mode. http://www.mathsisfun.com/data/index.html#stats Mean, Median, and Mode Lesson on how to calculate mean, median, and mode for set of data given in different ways. Also has interactive exercises. http://www.cimt.plymouth.ac.uk/projects/mepres/book8/bk8i5/bk8_5i2.htm GCSE Bitesize Mean, mode and median lessons Explanations with simple examples. http://www.bbc.co.uk/schools/gcsebitesize/maths/data/measuresofaveragerev1.shtml Measures Activity Enter your own data and the program will calculate mean, median, mode, range and some other statistical measures. http://www.shodor.org/interactivate/activities/Measures/ Graphing and Graphs Bar Chart Virtual Manipulative Build your bar chart online using this interactive tool. http://nlvm.usu.edu/en/nav/frames_asid_190_g_1_t_1.html?from=category_g_1_t_1.html An Interactive Bar Grapher Graph data sets in bar graphs. The color, thickness and scale of the graph are adjustable. You can input your own data, or you can use or alter pre-made data sets. http://illuminations.nctm.org/Activity.aspx?id=4091 Create a Graph Kids can create bar graphs, line graphs, pie graphs, area graphs, and xyz graphs to view, print, and save. http://nces.ed.gov/nceskids/createagraph/default.aspx Circle Grapher A tool to graph data sets in a circle graph. You can input your own data or alter a pre-made data set. http://illuminations.nctm.org/activitydetail.aspx?id=60 Graphs Quiz from ThatQuiz.org This quiz asks questions about different kinds of graphs (bar, line, circle graph, multi-bar, stem-and-leaf, boxplot, scattergraph). You can modify the quiz parameters to your liking, such as to plot the graph, answer different kinds of questions about the graph, or find mean, median, or mode based on the graph. http://www.thatquiz.org/tq-5/math/graphs Thatquiz.org Quiz for Graphs A 10-question quiz involving bar graphs and pictographs. http://www.thatquiz.org/tq-5/?-j40v0h-l1-p0

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Survey game A game where you do a survey in which you ask children their favorite hobby or color. Make a frequency table, a bar graph, and a pictogram from the results. http://www.kidsmathgamesonline.com/numbers/mathdata.html Stem-and-Leaf Enter the values and this web page creates your stem-and-leaf plot for you. http://www.mrnussbaum.com/graph/sl.htm Stem-and-Leaf Plots Quiz This is from Glencoe mathematics, an online multiple-choice quiz that is created randomly. Refresh the page (or press F5) to get another quiz.

http://www.glencoe.com/sec/math/studytools/cgi-bin/msgQuiz.php4?isbn=0-07-825200-8&chapter=12&lesson=1&&headerFile=4&state=na

Data Analysis and Probability gizmos from Explorelearning.com Interactive exploration activities online, with lesson plans. Topics include box-and-whisker plots, histograms, stem-and-leaf plots, lines of best fits using least squares, scatter plots; probability topics; and more. This is an excellent resource. The gizmos work for 5 minutes for free. You can also sign up for a free trial account. http://www.explorelearning.com/index.cfm?method=cResource.dspChildrenForCourse&CourseID=129

Statistics Interactive Activities from Shodor A set of interactive tools for exploring and creating different kinds of graphs and plots. You can enter your own data or explore the examples. http://www.shodor.org/interactivate/activities/BarGraph/ http://www.shodor.org/interactivate/activities/Histogram/ http://www.shodor.org/interactivate/activities/CircleGraph/ http://www.shodor.org/interactivate/activities/MultiBarGraph/ http://www.shodor.org/interactivate/activities/PlopIt/

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Reading Graphs and Charts 1. The graph shows the number of books some children read during a vacation reading assignment. Read the graph and fill in the blanks.

a. Who read the most books? ________________________

How many books? ________ b. Who read the fewest books? _______________________

How many books? ________ c. How many more books did Jack read than Jake?

________ books

d. Altogether, the four girls read _______ books. e. Altogether, the four boys read _______ books. f. Did the girls or the boys read more books? _______________________

How many more? ________

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Math Mammoth Data and Graphs (Blue Series)

2. Below you see a pictogram that shows how many vegetables were used in certain places. Each represents 5 kilograms. Vegetable use in one week a. Draw carrots in the pictograph for the Jacksons, the Joneses, and the Millers. Jacksons The Jacksons used 15 kg. The Joneses used 5 kg. The Millers used 10 kg.

Joneses Millers

b. Fill in.

Restaurant A

Restaurant A used _______ kg of vegetables.

Restaurant B

Restaurant B used _______ kg of vegetables.

= 5 kilograms of vegetables c. How many more kilograms of vegetables

did Restaurant B use than the Joneses?

d. How many kilograms of vegetables

did the two restaurants use in all?

__________ kg

__________ kg

3. The graph shows how many newspapers Jack sold at his newspaper stand from Monday through Sunday. a. For each day, find about how

many newspapers Jack sold. Look how tall each column is, and round to the nearest ten.

Day

Mon

Tues

Wed

Thurs

Fri

Sat

Sun

about

about

about

about

about

about

about

______

______

______

______

______

______

______

Newspapers

b. About how many newspapers did Jack

sell on Saturday and Sunday together? ______ newspapers c. About how many more newspapers did

Jack sell on Sunday than on Monday? ______ newspapers

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4. The teacher made a chart that shows how many books the children read each month. Books read

Jan

Feb

Mar

Apr

Annie

13

21

18

14

Freddie

8

5

11

9

Lisa

8

13

16

18

Jonathan

10

8

14

15

Total

a. How many books did Jonathan read in March? b. Who read the most books in February?

In April?

c. How many more books did Annie read

in February than in January?

d. How many more books did Lisa read

in April than Freddie? e. Find how many books each child read in all.

Put the number in the “total” column in the table above. Annie

+

Freddie

Lisa

+

+

Jonathan

+

f. How many more books in all did

Annie read than Freddie?

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Bar Graphs and Pictographs Bar graphs use “bars” or rectangles in them to show some information.

1. This bar graph shows how many hours some second grade students slept last night.

a. How many students slept 8 hours last night? b. How many students slept 10 hours last night? c. How many more students slept 9 hours than the ones who slept 10 hours? d. A school nurse said that children need to sleep well for at least 8 hours.

How many students slept less than 8 hours last night? e. How many students slept at least 8 hours last night? f. Make a pictograph. Draw ONE sleepy face

to mean 2 students. Students

Students who slept less than 8 hours Students who slept at least 8 hours

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Math Mammoth Data and Graphs (Blue Series)

2. Below, you see page counts for 14 different second grade math books. 217 388 365 290 304 315 243 352 289 392 346 308 329 323 Count how many books have between 200 and 249 pages.

Page count Number of books 200-249

Count how many books have between 250 and 299 pages.

250-299

Continue. Write your counts in the chart. 300-349 350-399 After that, draw a bar graph using the numbers in the above chart.

a. How many books had their page count between 350 and 399 pages? b. How many books had 300 pages or more? c. How many books had less than 250 pages? d. What was the lowest page count?

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3. The pictograph shows how many people visited the fairgrounds on different days. Each

symbol means 100 people. Half a symbol means 50 people. Draw a bar graph.

Day Thursday Friday Saturday Sunday

a. What was the most popular day of the fair?

How many people visited on that day? b. How many more people visited on Sunday than on Friday? c. What was the total number of visitors on Thursday and Friday? d. Which day would you have gone, if you didn’t like to be in a crowd?

Which day would you have gone, if you liked to be in a crowd? 4. Joe practiced basketball. Make a pictograph showing how many baskets he made each day. First choose a picture. Then choose how many baskets that picture represents. Day Baskets

Day

Mon

80

Mon

Tue

60

Wed

100

Thu

30

Baskets

Tue Wed Thu

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5. The bars in a bar graph can be this way too, (sideways) like they are lying down.

These numbers are in scrambled order, and they tell us how many households are in different parts of town: 275, 658, 256, 308, 286. Write the correct number after each bar on the graph. 6. (Optional) If you would like, make a survey among your class or friends. A survey means you ask many people the same question and write down what they answer. Then you make a graph. Some ideas: z

z

z

z

Ask many people what their favorite color is. Then make a bar graph. Ask many people what their eye color is. Then make a bar graph. Ask many people if they have a pet, and what pet it is. Then make a bar graph. Ask many people what their favorite game or sport is. Then make a bar graph.

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More Practice with Bar Graphs and Pictographs 1. The pictograph shows how many fish the family members caught when they went fishing. Dad Mom Amy Chris a. Who caught the most fish?

b. How many more fish did Chris catch than Dad? c. How many fewer fish did Amy catch than Mom? d. How many did Amy and Chris catch together? 2. The table lists the eye colors of some children. Draw the bars for the bar graph.

9

4

10

16

11

blue

green

gray

brown

hazel

a. How many children have either brown or hazel eyes? b. How many more children have brown eyes than have green eyes? c. How many children do not have blue eyes? 14

Math Mammoth Data and Graphs (Blue Series)

3. The bar graph shows how many toy cars some kids have. Chloe has 14 cars. Draw a bar for her in the graph.

a. How many cars does Ethan have? b. How many more cars does Tony have than Ethan? c. How many cars do Ryan and Chloe have together? d. If Ethan gives Ryan 5 cars, will Ryan then have more than Tony? 4. Jack is a fisherman. The pictograph shows how many kilograms of fish he caught last week. Each

Fish Caught Monday Wednesday

represents 200 kg of fish.

Friday Sunday a. How many kilograms of fish did he catch on Wednesday? b. How many kilograms of fish did he catch on Friday? c. How many more kilograms of fish did he catch on Friday than on Monday? d. How many kilograms of fish did he catch in total during this week?

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5. The bar graph below tells us the counts for different kinds of animals in a zoo. Answer the questions. Use the grids for additions and subtractions if you need to.

a. How many rainforest birds does the zoo have? b. How many Australian animals are there? c. How many Australian and African animals does the zoo have together? d. How many more African animals does the zoo have than Australian animals? e. What is the total number of Australian, European, and African animals?

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6. The bar graph shows how much money the Riley family spent for groceries in four different weeks. a. Mark above each bar how much they spent for groceries in dollars.

b. How much more did they pay for week 3 than for week 4?

c. How much more did they pay for week 2 than for week 1?

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7. Alex checked the price of a certain TV in four different stores.

Price Bob’s TV Store

a. Draw a bar graph from his results.

$525

The Nerdy Store $564

b. How much is the difference between

the most and the least expensive TV?

Home Express

$632

Lion Appliances $599

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8. The children picked fruit from Grandpa Jerry’s fruit trees. They picked 25 oranges, 10 mangos, 40 bananas and 25 apples. Make a pictograph to show this! Draw one fruit picture to mean 10 fruits. Draw a picture of a half fruit to mean 5 fruits. Fruits we Picked = 10 oranges oranges = 10 mangos mangos = 10 bananas bananas = 10 apples apples

9. a. Which is the most popular club? b. How many more students are in the

sports club than in the math club?

c. How many students are in the art, music,

and sports clubs in total?

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Math Mammoth Data and Graphs (Blue Series)

10. a. How much does the volleyball set cost? b. How much more does the snorkeling set

cost than the swim rings set?

c. How much do the sand toys and

beach ball cost together? d. What is the total cost if you buy

the two cheapest items?

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Math Mammoth Data and Graphs (Blue Series)

Making Bar Graphs 1 1. Beverly asked her classmates how many hours they watch the TV each day. The results are below; she already organized them in order. 001111111111122223333444556 Each number above is someone's answer to Beverly’s question. So two people answered that they watched TV for 0 hours. Quite a few answered that they watch TV about 1 hour per day. With such lots of numbers, first we need to make a frequency table. In a frequency table, we count how frequently or how often a certain number was in our list of data. After counting all of that, we can make a bar graph. In Beverly’s data above, the number zero (0 hours of TV) appeared two times. The number two (2 hours of TV) appeared four times. Finish the frequency table and the bar graph. Hours of TV Frequency

0h

2

1h 2h

4

b. How many classmates did Beverly question? c. What was the most common response to Beverly's question? d. How many of these kids watch TV one hour or less? e. How many kids watch TV three hours or more? f. Are there more kids who watch TV three hours a day than kids who watch TV two hours a day? g. Are there more kids watching TV two hours or more, than kids watching TV less than two hours?

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Math Mammoth Data and Graphs (Blue Series)

2. a. Beverly also asked some people about their favorite color. Make a bar graph. Color Frequency red

2

orange

1

yellow

4

green

5

blue

7

purple

4

black

2

white

2

b. How many people did Beverly question? c. Were the “warm” colors or the “cold” colors more popular?

(Warm colors are red, orange, and yellow. Cold colors are green, blue, and purple.)

3. These numbers are students’ quiz scores. 1 3 5 3 6 4 9 8 6 4 8 7 5 3 9 8 6 2 1 8 9 10 2 9 7 6 a. Make a frequency table and a bar graph. Test score Frequency

b. What was the most common quiz score?

How many students got that score?

c. What was the least common quiz score?

How many students got that score?

d. How many students got a score from five to eight? e. How many students did excellent (got a score of nine or 10)? f. The teacher said after the test, “Anyone with a score of four or less will need to retake the test.” How many students need to do the test again?

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Math Mammoth Data and Graphs (Blue Series)

Making Bar Graphs 2 Bar graphs are used if the data can be separated into distinct groupings or categories. For example, if you study children’s eye color, the categories are “blue,” “green,” “brown,” “hazel,” etc. The graph on the right shows the number of U.S. households that own a dog, cat, bird, or a horse. A household owning, say, both a dog and a cat would be included in both numbers. Note that the vertical axis scale is in million households. Note how the data values are recorded above each bar. To get the true number, multiply that by 1,000,000. 1. According to the graph above, how many U.S. households own a cat? A horse?

2. a. Draw a bar graph from the data on the right. Notice you need to figure out the scale on the horizontal axis (miles). Hint: make sure the largest number in the river lengths fits on the grid, and that there isn’t lots of “empty space” left over beyond that.

b. About how many times longer is the Mississippi-Missouri than the Ohio-Allegheny? c. About how many times longer is the Mississippi-Missouri than the Yukon?

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River

Length (miles)

Mississippi-Missouri 3,902 Yukon

1,980

Rio Grande

1,900

Columbia

1,450

Colorado

1,450

Ohio - Allegheny

1,306

Snake

1,038

Math Mammoth Data and Graphs (Blue Series)

3. The table lists the number of U.S. households that own certain exotic pets. a. Make a bar graph from the data. Notice that the vertical axis is scaled for thousands of households. b. Estimate the number of households that own either a turtle, a lizard, or a snake. c. Which is more popular: to own a hamster, a guinea pig, or a gerbil or to own a turtle, a lizard, or a snake? Justify your answer.

Number of Households (in 1,000) Rabbits 1,870 Turtles 1,106 Hamsters 826 Lizards 719 Guinea Pigs 628 Ferrets 505 Snakes 390 Gerbils 187 Pet

Source: 2007 U.S. Pet Ownership & Demographics Sourcebook

4. Serena asked 20 people in a class how many brothers and sisters they had. Here is the data: 0 3 2 1 1 0 1 1 2 2 1 2 5 3 2 6 1 1 0 2 (Each number is the response from one person.) Draw a bar graph. First count how many people had zero siblings, how many had one sibling, and so on (the frequencies). Number of frequency siblings 0 1 2 3 4 5 6

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Math Mammoth Data and Graphs (Blue Series)

Making Histograms Histograms are like bar graphs, but the bars are drawn so they touch each other. Histograms are used with numerical data. For example, let’s look at the math test scores of a fifth grade class: 13 40 32 38 32 28 21 30 45 17 22 26 33 25 27 36 42 19 21 First we need to make categories or “bins” for this data, and after that make a histogram. 1. Let’s make five categories or bins for the test score data. These bins are shown on the right. The first bin is from 12 to 18 points, the second bin is from 19 to 25 points, and so on.

point count frequency 12-18

How did we come up with those limits for the bins?

19-25

First, we find the smallest and the greatest value among the data. Those are 13 and 45. The first bin has to include 13 and the last bin has to include 45.

26-32

If we want five bins, we find the difference between those numbers and divide that by 5. The result will give us the width of each bin.

33-39 40-46

45 − 13 = 32 and 32 ÷ 5 = 6.4. The width could be 6.4 points. However, often it is more reasonable to use a whole number for the bin width, so instead of 6.4, let’s make the bins 7 points “wide.” We can start the first category at 13 (the lowest score) or even a little bit before that, at 12. The important thing is that the last bin has to be able to include 45, our highest number. Each bin starts at 7 points higher than the previous one: the second at 19, the third at 26, and so on. Next, count how many individual test scores “fall into,” or belong in, each bin—that is the frequency. Now you are ready to draw the histogram! Draw it so that the bars touch each other without leaving gaps in between.

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Math Mammoth Data and Graphs (Blue Series)

2. The following data describe the weights of 15 healthy female German Shepherd dogs (in pounds). 65 72 62 60 66 67 65 73 70 64 66 63 68 58 63 Draw a histogram. Make four categories. First determine the bin width: find the largest and the smallest numbers in the data, calculate the difference between them, divide that difference by four, and round the result up to the next whole number.

weight

frequency

3. Researchers determined the age of 26 African elephants, living in three herds, by their molars. Here is the data (each number is the age of one elephant): 3 0 6 23 12 0 1 15 9 8 43 2 4 10 22 38 5 17 3 8 18 27 19 7 4 Draw a histogram. Make five categories. Determine the bin width for the categories as above.

age

frequency

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Math Mammoth Data and Graphs (Blue Series)

Double Bar Graphs Double-bar graphs are used to compare two sets of data. Both data sets have to have the same categories. For example, the chart here shows how many students are able to swim in a certain elementary school. The categories are the five different grades. There is a separate bar for boys and girls for each grade. We can see the following: z

In grades 1, 2, and 3, more boys are able to swim than girls.

z

In grades 4 and 5 about the same number of girls can swim as boys.

z

As students get older, more students are able to swim. This is probably because the school has swimming instruction in grades 2, 3, and 4.

1. a. Write in the table how many students can swim, and how many cannot swim, for grades 1 to 5. Grade

can cannot swim swim

b. How does the number of students who can swim change during the grades 1 - 5? c. How does the number of students who cannot swim change during the grades 1 - 5?

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Math Mammoth Data and Graphs (Blue Series)

2. a. List the genres which had an increase in the number of loans from 2006 to 2007. b. Estimate the total number of loans for the three most popular genres in 2006. c. Estimate the total number of loans for the three least popular genres in 2007.

3. A researcher asked some people over 50 years of age and some people between 30 and 50 years of age about their favorite type of books. Favorite book genres by age over 50 30-50 Biographies & Memoirs

245

36

Comics

45

126

Mysteries

357

186

Poetry

56

22

Romance

128

215

Science Fiction & Fantasy

37

267

a. Draw a double-bar graph of the data. You also need to fill in the legend. b. Find the genre that holds the last place in the one age group while holding the first place in the other age group.

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Math Mammoth Data and Graphs (Blue Series)

Line Graphs 1 A line graph shows how something changes over time, such as over several hours, days, weeks, months, or years. The data values are often drawn as dots. Then the dots are connected with lines. The x-axis and the y-axis are the two lines that frame the picture. The time units are written under the x-axis. To read a line graph, look “up” from the time unit until you find the dot. Then draw an imaginary line from that dot to the y-axis. For example, in July Amy had saved $90. 1. Look at the line graph about Amy’s savings. a. How many dollars had Amy saved in May? b. How many dollars had Amy saved in August? c. How many dollars had Amy saved in September? d. In which month had she saved up $75? e. In September Amy used up her savings to buy a used bike. How much did the bike cost? 2. The graph shows a puppy’s weight for 10 days after birth. Notice how the two axes are named as “day” and “grams”. a. About how many grams did the puppy weigh on day 1? ________ Day 2? ________ Day 3? ________ Day 4? ________

b. What is the first day that the puppy weighed 600 g or more? c. What is the first day that the puppy weighed 700 g or more?

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Math Mammoth Data and Graphs (Blue Series)

3. Look at the graph about the monthly retail prices of strawberries in 2004, given in dollars per pound. The retail price is the price you see in a grocery store or the price the customers pay.

a. Describe the price changes as the year progresses. Do you know why the price is lower in the summer?

Month

Price ($ per lb)

Jan

2.48

Feb

2.33

Mar

2.12

Apr

1.66

May

1.67

Jun

1.85

Jul

1.63

Aug

1.82

Sep

1.84

Oct

2.60

Nov

3.19

Dec

3.60

b. Find the highest price per pound and the lowest price per pound. What is the difference of these two? c. How much did it cost to buy 2 lb of strawberries in August? In November?

4. Becca’s mom wrote down an “x” mark for every bad behavior she did during the day. The table shows the list of her x-marks. a. Make a line graph. Remember to name one axis as “days” and the other as “x-marks”. b. Did Becca’s behavior improve?

Day

x-marks

Mon

10

Tue

8

Wed

9

Thu

6

Fri

3

Sat

4

Sun

2

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Math Mammoth Data and Graphs (Blue Series)

5. The table gives the average maximum temperatures for each month in New York.

Month

Max. Temp.

Month

Max. Temp.

Month

Max. Temp.

Jan

3°C

May

20°C

Sep

26°C

Feb

3°C

Jun

25°C

Oct

21°C

Mar

7°C

Jul

28°C

Nov

11°C

Apr

14°C

Aug

27°C

Dec

5°C

a. Make a line graph. Three values are already done for you. b. What are the coldest months? c. What are the warmest months? d. What is the difference in maximum temperature between the coldest and the warmest month? 6. Do a line graph from some data that you gather yourself! Just remember, it has to be something that changes over time. You can also “make up” data from your own head. Here are some ideas: z

outside temperature from the morning till the evening

z

your savings in the past 6 months, or an imaginary child’s savings in 6 or 8 or 12 months

z

how many hours of schoolwork (or housework or playing, etc.) you do each day of the week

z

how many pages of a book you read each day of the week

z

your height from year 0 to year 9 of your life

You can also use this neat online tool for creating your graph: http://nces.ed.gov/nceskids/createagraph/ To use it, you need to have your data ready. It will not give you any data. It just draws the graph.

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Math Mammoth Data and Graphs (Blue Series)

Line Graphs 2 Mary sold muffins every day at 2 pm in the school cafeteria. She recorded her sales in the table. Muffin Sales, Week 11 Day Muffins sold Mon Tue Wed Thu

24 36 41 33

Fri

17

We can draw a line graph out of this data because the data is organized by time (days of the week). To do that, we first plot the individual data points in a coordinate grid. Then we draw lines to connect neighboring points. Besides that, the line graph also needs: z z z z

a title on top for the whole graph labels for the tick marks on the two axes a label for the vertical axis (the y-axis) a label for the horizontal axis (the x-axis) unless it is very clear what it is about. In the graph above, the labels “Mon,” “Tue,” and so on show very clearly that they are days of the week, so we don’t necessarily need a title, “Days of the week,” for the horizontal axis.

Use a line graph for data that is organized by some unit of time (hours, days, weeks, years, etc.) 1. a. Add a label for the vertical axis that says “Rainfall (mm)”. The “mm” stands for millimeters. b. Add five more data points to the graph according to this data: Day 11 12 13 14 15 rainfall (mm) 9 0 0 13 2 c. Draw a line between each two consecutive points. d. How many “dry” days were there in the first half of April?

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Math Mammoth Data and Graphs (Blue Series)

2. Jessie recorded the temperature of his fridge every 30 minutes during the day. a. Finish drawing the line graph. b. How did the temperature change around noon? Give a possible reason for that change. c. How did the temperature change around 5PM? Give a possible reason for that change. 7:00 7:30 8:00 8:30 9:00 9:30 10:00 10:30 11:00 11:30 12:00 12:30 Temperature (°F) 46 47 52 50 49 47 50 50 48 56 62 55 Time

13:00 13:30 14:00 14:30 15:00 15:30 16:00 16:30 17:00 17:30 18:00 18:30 19:00 Temperature (°F) 50 48 47 51 50 48 53 55 64 61 55 50 46 Time

3. Robert recorded his total savings at the end of each month. Draw a line graph of that data. Note: YOU need to choose the scaling for the vertical axis so that the largest number, $107, will fit on the grid. Will the gridlines go by five? By ten? By twenty? By fifteen or some other number? Month Total savings Apr

$8

May

$22

Jun

$46

Jul

$61

Aug

$78

Sep

$95

Oct

$107

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Math Mammoth Data and Graphs (Blue Series)

4. The table below shows the monthly visitor count to Juanita’s blog. Three of the data points are already plotted. Your task is to plot the rest and finish the line graph. Note: Since the vertical gridlines go by 200s, you cannot make an exact dot at, say, 1442. You need to round the numbers first. Round them to the nearest 50. Then plot the points. rounded to Month Visitors the nearest 50

Jan

1039

1050

Feb

1230

1250

Mar

1442

Apr

1427

May

1183

Jun

823

Jul

674

Aug

924

Sep

1459

Oct

1540

Nov

1638

Dec

1149

1200 650

In the ____________________ Juanita’s blog had many fewer visitors than in the spring or fall. The three months with the fewest visitors were _____________, _____________, and ____________. The three months with most visitors were ______________, ______________, and ______________. 5. A car travels with constant speed so that each second it travels 30 meters. Time 0s 1s 2s 3s 4s 5s

Distance 0m 30 m 60 m 90 m 120 m 150 m

a. Fill in the table, and continue the graph till 12 seconds. b. When will the car have traveled 3 km?

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Math Mammoth Data and Graphs (Blue Series)

6. a. Draw a line graph of this data: z

z

z

z

z

z

After-School Sports Club

First draw the two axes, one at the bottom and the other at the left side. Use a ruler, so the graph looks neat and tidy.

Year Members

Label the axes. Label the horizontal axis as “year” (not as “x”). Label the vertical axis as “members” (not as “y”).

1998

56

Label the whole graph by writing at the top: “After-School Sports Club Members from 1998 to 2005.”

1999

63

2000

60

Since the horizontal axis is for the years, draw tick marks on that axis for the years, but use three squares between each tick mark because the numbers for the years are so long (four digits).

2001

35

2002

27

Then choose a scaling for the vertical axis: Because it has numbers between 27 and 63, it makes sense to mark it in fives, starting from 0. In other words, let each grid square be 5 members.

2003

32

2004

57

2005

63

Now you are ready to plot the points and draw the line graph.

b. What do you think might have caused the drop in membership in 2001 - 2003?

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Math Mammoth Data and Graphs (Blue Series)

Temperature Line Graphs 1. Read the chart and fill in the table. Month

Jan

Feb

Mar

Apr

May

Jun

Jul

Aug

Sep

Oct

Nov

Dec

Max Temperature

a. What is the hottest month? b. What is the coldest month? c. Find two months that have the same maximum average temperature. d. What is the difference between the maximum temperature of May and that of June? e. What is the difference between the maximum temperatures of June and July? f. What is the difference between the maximum temperature of the coldest month and that of the hottest month?

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Math Mammoth Data and Graphs (Blue Series)

2. Draw a line graph with the data.

Month

Minimum Temperature

Month

Minimum Temperature

Jan

-10

Jul

7

Feb

-9

Aug

6

Mar

-8

Sep

3

Apr

-2

Oct

-4

May

-1

Nov

-5

Jun

5

Dec

-7

a. Which month is colder, January or March? b. What is the difference between the minimum temperature of May and that of June? c. What is the difference between the minimum temperatures of October and November? d. How many degrees does the minimum temperature change from January to June?

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Math Mammoth Data and Graphs (Blue Series)

Reading Line Graphs The graph shows how many people were living on farms in the United States during 1900-2000. You can see how dramatically the number has dropped! The question (a) in exercise 1 asks you to estimate the farm population in year 2010. Do it by tracing over the graph and continuing the graph in a natural way till the year 2010. The plain numbers listed in the table do not really help with estimation (without further mathematical tools). Notice that the table lists the farm population in thousands of people. For example, in year 1970 there were 9712 thousand people—or 9,712,000 people—living on farms. In other words, you need to tag three zeros onto each of those numbers. Note also that these numbers are actually rounded to the nearest thousand—no population remains an exact number of so many thousand people, year after year. Year

Farm Population (thousands of people)

1900

29875

1910 1920

32077 31974

1930

30529

1940

30547

1950 1960

23048 13445

1970

9712

1980

6051

1990 2000

4591 2993

Source: Census of Agriculture

1. a. Consider the graph above. Estimate the U.S. farm population in the year 2010. b. In which two decades were the greatest drops in farm population? c. How many people did the farm population decrease during those two decades (separately)? d. What was the first year when the farm population dropped below 10 million? e. When approximately did the farm population drop below 5,000,000?

38

Math Mammoth Data and Graphs (Blue Series)

2. The International Union for Conservation of Nature (IUCN) produces a report every few years called IUCN Red List of Threatened Species. This report lists the number of animal and plant species that are considered endangered and extinct. The term “threatened” actually means the species can either be considered “Critically Endangered,” “Endangered,” or “Vulnerable.” Study the data and the graph below, and answer the questions. Numbers of threatened species by major groups of organisms (1996–2012) Number of in in in in in in in in in in in in threatened 1996/98 2000 2002 2003 2004 2006 2007 2008 2009 2010 2011 2012 species → Mammals

1,096

1,130 1,137 1,130 1,101 1,093 1,094 1,141 1,142 1,131 1,134 1,139

Birds

1,107

1,183 1,192 1,194 1,213 1,206 1,217 1,222 1,223 1,240 1,240 1,313

Reptiles

253

296

293

293

304

341

422

423

469

594

664

807

Fishes

734

752

742

750

800 1,171 1,201 1,275 1,414 1,851 2,011 2,058

(Data from 2012 IUCN Red List)

a. How many reptile species were considered “threatened” in 2003? In 2012? b. How many fish species were considered “threatened” in 2003? In 2012? c. In which major animal group has the number of threatened species stayed approximately the same between 2000 and 2012? d. In which major animal groups has the number of threatened species nearly tripled from 2000 to 2012? e. Find a period of time when the number of threatened species decreased in a certain species. for several years. Which species was it, and when was it?

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Math Mammoth Data and Graphs (Blue Series)

Double and Triple Line Graphs A double-line graph shows data for two different things that occur in the same time period. It simply has one line for one set of data, and another line for the other. We can distinguish these two lines by marking the data points in different manners and by using different colors. For example, we can use circles for one data set and triangles for the other. Usually double-line graphs also have a legend that explains which line belongs to which data set. The same principles apply to triple-line graphs or quadruple line graphs. In the example here, we can see that Mom sent many more messages than Dad.

1. Refer to the double line graph above about the text messages Mom and Dad sent with their cell phones. a. Find how many messages total Mom sent, and how many Dad sent. b. How many more messages did Mom send on Thursday than Dad? c. Find the day with the greatest difference between the number of messages Mom sent and the number of messages Dad sent. d. Find the day with the least difference between the number of messages Mom sent and the number of messages Dad sent.

2. This graph shows the total number of tropical storms and hurricanes in three Atlantic hurricane seasons. a. Find the total number of storms for 2005, 2006, and 2007.

b. Which year was unusually active? c. Based on this graph, which month is the most active month?

40

Math Mammoth Data and Graphs (Blue Series)

3. Project. (optional) Go to http://en.wikipedia.org/wiki/Timeline_of_the_2008_Atlantic_hurricane_season and find out how many hurricanes & tropical storms there were for each month from May till December 2008. Count a storm that extends into two months by its starting month. For example, Gustav is counted for August. Month

May

Jun

Jul

Aug

Sep

Oct

Nov

Dec

Number of hurricanes & tropical storms Add that data to the line graph on the previous page. Also, answer the questions: - Was the 2008 Atlantic hurricane season more or less active than the season in 2005? In 2007? - Which month of the 2008 season was the most active month? - Is the same month also the most active month for 2005, 2006, and 2007 seasons? (You can also extend this project to include 2009, 2010, and so on, up to the year prior to doing these exercises.)

4. The table records Anna’s and Alex’s test scores in five science tests. a. Draw a double-line graph of the data. Note the legend that is already given. b. Describe Anna’s performance over time. (Did she improve? Get worse? Stay about the same?) c. In which test was the difference between Anna’s and Alex’s point count the greatest? In which test was it the smallest?

Anna Alex Test 1

65

72

Test 2

62

66

Test 3

77

71

Test 4

85

59

Test 5

82

68

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Math Mammoth Data and Graphs (Blue Series)

Graphs: More Practice 1. Maria studied the weight of 1 year old roosters. She went to a farm and weighed 20 roosters. The numbers below are their weights, in ounces. 96, 94, 90, 101, 84, 102, 101, 95, 108, 113, 87, 95, 97, 84, 90, 99, 89, 93, 92, 100. a. Make a bar graph. Draw the bars touching each other (no gaps between the bars).

Weight (ounces) Frequency

83..88 89..94 95..100 101..106 107..112 113..118

b. Maria calculated the average several times and got different results from her calculator. She must have made errors in pushing the buttons! Use the graph and the data to figure out which one is the correct average weight: 89 ounces, 95 1/2 ounces, or 100 1/2 ounces? 2. The bar graph lists the number of loans in the Charleston library in the weeks of May and June. From this graph, you cannot read the exact numbers of loans, but you can find the approximate numbers. Estimate, to the nearest ten, the total number of loans for: a. weeks 18-21 b. weeks 22-25

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Math Mammoth Data and Graphs (Blue Series)

3. The graph shows the weekly strawberry sales of a small strawberry farm.

a. What week did the farm sell the most strawberries? About how much were the sales that week? b. What week did the farm sell the least strawberries? About how much were the sales that week? c. Estimate the total sales for the weeks 25-27. 4. Make a line graph of the baby’s weight. Week

Weight

0

6 lb 14 oz

1

6 lb 12 oz

2

6 lb 14 oz

3

7 lb

4

7 lb 2 oz

5

7 lb 4 oz

6

7 lb 6 oz

7

7 lb 7 oz

Weight in ounces

43

Math Mammoth Data and Graphs (Blue Series)

5. The table lists the marital status of people in the United States aged 15 years and older in the 2000 census. MARITAL STATUS Never married 59,913,370 Now married 120,231,273 (not separated) Separated

4,769,220

Widowed

14,674,500

Divorced

21,560,308

Source: From Census 2000 data, www.census.gov.

a. Round the numbers to the nearest million. Then use the rounded numbers to complete the graph.

b. Estimate the total number of people who are either separated, widowed, or divorced.

c. Estimate the number of people who are not married (the sum of all the categories except “Now married”).

44

Math Mammoth Data and Graphs (Blue Series)

6. The table shows how many adult and child visitors a small art museum had during one week.

Museum visitors Day

a. Calculate the total visitor counts. b. What was the difference in the total visitor count between the busiest day and the least busy day?

c. Find the average number of adult visitors in a day. Give your answer to one decimal digit. Use your notebook for the long division.

d. Find the average number of child visitors in a day. Give your answer to one decimal digit. Use your notebook for the long division.

Adults Children

Monday

29

14

Tuesday

23

10

Wednesday

34

18

Thursday

38

19

Friday

35

19

Saturday

57

25

Sunday

63

31

Total Visitors

Totals

e. Make a double-bar graph of this data.

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Math Mammoth Data and Graphs (Blue Series)

7. a. Make a histogram from the data in the frequency table on the right.

Height in cm Number of people

Height in cm Number of people

120...129

4

160...169

95

130...139

10

170...179

61

140...149

41

180...189

39

150...159

82

190...199

6

b. How many people were short (less than 140 cm tall)? c. How many were tall (180 cm or taller)? d. Most adults are 160 cm tall or taller. Use this fact to guess (estimate) how many children and how many adults were in this group. e. Could this data come from... z

a group of elementary school children?

z

a group of people who were at a swimming pool?

z

a group of elderly women in an old people’s home? Explain your reasoning.

46

Math Mammoth Data and Graphs (Blue Series)

Average The Millers went on a trip. The first day, they drove 110 miles, the second day, 142 miles, the third day, 126 miles, and the last day, 82 miles. The Millers drove a total of 460 miles. In the diagram, we have put those distances as sticks one after another, though of course in reality they did not drive just straight stretches of roads. IF they had driven 115 miles each day, it would have totaled the same 460 miles. On average, the Millers drove 115 miles a day, or their average daily mileage was 115 miles. What is the average of 20, 32, 27, 37, and 24? First find the total by adding. Then, divide that into equal parts.

20 + 32 + 27 + 37 + 24 = 140.

140 ÷ 5 = 28. So, the average of 20, 32, 27, 37, and 24 is 28.

If these numbers were the ages of club members, we would say the average age of the members is 28 years. However, they could also be distances, weights, volumes, or just plain numbers. 1. Judith’s test scores were 78, 87, 69, and 86. Find her average score.

2. John measured the temperature five times during a day. These are his measuring results: 18°C, 22°C, 26°C, 23°C, and 16°C. Find the average temperature for the day.

3. Dad drove a 414 km stretch in six hours. How many kilometers did he drive, on the average, in one hour?

47

Math Mammoth Data and Graphs (Blue Series)

You can also use the average “backwards.” During a 20-hour drive from Denver to Dallas, Dad’s average speed was 40 miles per hour. How far is Denver from Dallas? You can multiply to find the answer: 20 hours × 40 miles/hour = 800 miles. Note that in reality, he did not drive with a totally even speed all of the time because he had to stop at crossings, slow down on curves, stop for a snack and so on. We do not know how his speed varied on the trip. All we are given is that his average speed was 40 miles per hour. (And, of course the average speed was calculated by dividing the length of the trip by the total number of hours the trip took.)

4. The package of eggs says that the average weight per egg is 55 grams. How much would a dozen eggs weigh?

5. For her hospital stay, Mom was charged an average of $76 daily. What was the total cost of her one-week stay?

6. Mom’s weekly grocery bills in June were $234, $178, $250, and $198. What was her average weekly grocery bill?

7. Some children ran a race. These are the resulting times: Ann Judy Rose Elizabeth Grace Nancy

12 min 15 min 14 min 19 min 12 min 18 min

Michael Greg James Caleb Hans

12 min 10 min 11 min 15 min 17 min

Find the girls’ average running time and the boys’ average running time separately. Girls’ average: ______________ Boys’ average: ______________ Are boys or girls quicker on average? What is the difference of the two averages?

48

Math Mammoth Data and Graphs (Blue Series)

8. Here are the science quiz scores for ten fourth-graders: 24 20 24 16 28 30 14 22 23 19 a. Make a frequency table and a bar graph. Quiz score Frequency

13..15 16..18 19..21 22..24 25..27 28..30 b. Calculate the average score.

c. Both the bar graph and the average tell us what the “middle” or “typical” result in the test was. Explain how you can guess what the average is approximately, just using the graph.

9. These are the ages of the members of a bird watching club: 18 28 25 33 29 17 44 37 30 a. Calculate the average age.

b. The club gained a new member, 79-year old Jim. What is the average age now?

If 213 ÷ 17 = 12 R9, what is 213 ÷ 12?

49

Math Mammoth Data and Graphs (Blue Series)

Average (Mean) Example. Five children earned these amounts of money for a job: $12, $27, $18, $9, and $22. The graph below shows visually how much each child earned. Together, they earned $88. If this $88 had been divided equally among the children, each child would have gotten $18. (Of course it was not, because the children got paid according to how much they worked.) This $18 is the average pay. Average =

$12 + $27 + $18 + $9 + $22 5

=

$88 = $18. 5

The graph on the bottom shows the situation IF each child had received the average earning ($18). Notice that $18 is sort of in the “middle” or in between the lowest and highest earnings.

z

You calculate the average of a data set by summing all the numbers in the data set, and dividing the sum by the number of entries in the data set. In other words average =

z z

sum of all data entries number of data entries

The average is always somewhere in the “middle” of the data set’s numbers. The average is also called the mean. We will use both terms in this lesson so you get used to both of them.

1. Calculate the average of the data sets. Do not use a calculator. a. 2, 4, 5, 9, 0, 4, 1, 7

b. 13, 16, 20, 22, 16, 13, 17, 12, 15

2. Calculate the mean of the data sets to the nearest tenth. This time use a calculator. a. 2, 4.3, 5, 9, 4.7, 9.4, 3.7, 5.1

b. 312, 288, 284, 329, 293, 302

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Math Mammoth Data and Graphs (Blue Series)

Average and Line Graphs Remember that line graphs show how data changes over time. Let’s look at the muffin sales again. Muffin Sales, Week 11 Day Muffins sold Mon 24 Tue Wed Thu Fri

The average is

36 41 33 17

24 + 36 + 41 + 33 + 17 5

= 30.2 muffins.

On average, she sold 30.2 muffins or about 30 muffins. See how it is plotted on the line graph, using a dashed line. Notice that the average is somewhere in the middle of the data: some data points are above, some are below it. If Mary had sold 30.2 (or 30 1/5) muffins every day, she would have sold the same total amount as she actually did: 151 muffins. If Mary had sold 30.2 muffins every day, what would the line graph have looked like? It would have been a totally horizontal line, with every day’s data point at the same level of 30.2. 3. Find the average visitor count to Juanita’s blog in the year 2008. Then plot the average in the line graph with a dashed line, like in the example above.

Month Visitors Jan 1039 Feb 1230 Mar 1442 Apr 1427 May Jun Jul Aug

1183 823 674 924

Sep Oct Nov Dec

1459 1540 1638 1149

b. Find the average visitor count for the summer months June through August only. c. Find the average visitor count for September through December.

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Math Mammoth Data and Graphs (Blue Series)

4. The average rainfall in the first 15 days of April was 5.067 mm. What was the total rainfall in the first 15 days of April (in millimeters)?

5. The birth weights of a certain piglet litter were: 1,400 g 1,480 g 1,250 g 1,710 g 1,630 g 1,250 g 1,700 g 1,820 g 1,500 g a. Find the average to the nearest gram. b. How many grams below the average were the lightest (two) piglets? c. How many grams above the average was the heaviest piglet? d. Remove the two lightest piglets’ weights from the data. Now calculate the average again. Did the average change? If it did, by how much?

6. The data below gives the monthly salaries of StarMop Inc. employees: $1,146 $1,178 $1,189 $1,209 $1,209 $1,210 $1,213 $1,215

$3,400

a. Calculate the mean. b. Remove the person with the highest salary from the data set. Calculate the mean again. How did it change?

Joan checked the price of a certain plasma TV in four different stores. In three of the stores the price was $549, $589, and $599. She calculated that the average price was $567. What was the price in the fourth store? Choose the right answer: a. $609 b. $531 c. $460

52

d. $567

Math Mammoth Data and Graphs (Blue Series)

Mean, Mode, and Bar Graphs Do you think you could calculate the average from the data shown in the bar graph? After all, there are numbers involved. Actually, we cannot. To see why, you need to think what kind of original data produced this graph. What was asked of the people in the study? What did they respond? The people were asked something like, “What pets do you have?” The people would have answered, “cat,” or “dog,” and so on. The original data set consists simply of the words “cat,” “dog,” “bird,” and “horse”—each one listed many times, because each mention of a “cat” would mean the answer of one particular household. cat, cat, dog, dog, dog, bird, dog, dog, bird, cat, dog, horse, dog, cat, dog,… We cannot calculate anything from this kind of data set because it is not numerical data! However, we CAN find the most commonly occurring item, and that is called the mode. In this case, the mode is dog. It made the highest bar on the graph. The mode is the most commonly occurring item in a data set. z

z

Sometimes a set of data has two or more modes. For example, the data set green, green, blue, blue, black, brown, hazel has two modes: both green and blue are equally common. If none of the items occurs twice or more, there is no mode. For example, this data: green, blue, pink, red, black, brown, purple has no mode.

1. Find the mode of the data set shown in the bar graph on the right.

2. a. Find the mode of this data: water, pop, juice, pop, juice, water, milk, water, pop, pop, juice, pop b. If the above are the answers of 12 people to some question, what could have been the question?

53

Math Mammoth Data and Graphs (Blue Series)

3. Nineteen children were asked about their favorite ice cream flavor. Here are their responses: strawberry, vanilla, chocolate, vanilla, chocolate chip, chocolate, pecan, pecan, vanilla, vanilla, strawberry, chocolate chip, vanilla, chocolate, chocolate, vanilla, strawberry, chocolate chip, vanilla.

a. Find the mode. b. Draw a bar graph. c. If possible, calculate the mean. 4. These are the spelling test scores of a fifth grade class: 4 5 7 9 9 10 10 11 11 12 12 12 13 14 17 18 18 18 19 19 19 20 24 25 a. Find the mode. b. Draw a bar graph. c. If possible, calculate the mean. Test Score Frequency <8 8..10 11..13 14..16 17..19 20..22 23..25

5. a. Find the mode. b. Draw a bar graph.

Grades of a math class Grade Frequency

c. If possible, calculate the average. d. There were ____ students in all. What fraction of the students got grade B?

F D C B A

3 8 12 17 10

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Math Mammoth Data and Graphs (Blue Series)

Circle Graphs A circle graph is made up of sectors, and each sector has a central angle. A circle graph (or “pie chart”) that represents various percentages will have a sector for each percentage. Therefore, we need to calculate how many degrees the central angle is for each sector. To do that, we simply calculate that percentage of 360 degrees (the full circle). For example, 25% of some total corresponds to 0.25 × 360º = 90º. Similarly, 67% of some total corresponds to 0.67 × 360º = 241.2º. 1. Sketch a circle graph that shows ... a. 50%, 25%, and 25%

b. 33.3%, 33.3%, 1/6, and 1/6

c. 20%, 20%, 10%, and 50%

2. The table shows different kinds of specialty breads that a grocery store ordered. Fill in the table. Make a circle graph. (Note: You’ll need a protractor to draw the angles.) Type

Quantity Fraction Percentage

white bread

50

bran bread

25

rye bread

30

corn bread

40

4-grain bread

55

TOTAL

200

Central Angle

1/4

1

100%

360º

3. a. Make a bar graph of the quantities of each type of bread from the table above. → b. Does the bar graph show percentages?

55

Math Mammoth Data and Graphs (Blue Series)

4. Think of fractions. Estimate how many percent the sectors of the circle graphs represent.

a.

b.

c.

5. The table shows how many of the different flavors of protein powders a company sold. Draw a circle graph showing the percentages. You will need a protractor and a calculator. Flavor

Amount sold

Chocolate

67

Vanilla

34

Strawberry

16

Blueberry

26

Percentage of total

Central Angle

100%

360º

TOTAL

6. Mark polled 6th graders about their favorite hobbies. Below are his results. Draw a circle graph to show the percentages. Round the angles to whole degrees. You will need a protractor and a calculator. Favorite hobby

Percentage Central Angle

Reading

12.3%

TV

24.5%

Computer games

21%

Sports

22.3%

Pets

7.1%

Collecting

8.1%

No hobby

4.7%

TOTAL

100%

360º

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Math Mammoth Data and Graphs (Blue Series)

Review 1. Plot the points from the “number rule” on the coordinate grid. Fill in the rest of the table first, using the rule given. The rule is: y = 9 − x. x

0

1

2

3

4

5

6

7

8

9

y x y

2. Find the mean and mode of this data set to the nearest hundredth: 5, 9, 13, 12, 16, 10, 19, 11, 10.

3. a. Estimate what the amount of tractors might have been in the year 2010. b. During which decade did the amount of tractors rise the quickest? What was the approximate amount of increase in tractors during that decade? c. Describe the trend in the amount of tractors between 1970 and 1995. d. About how many-fold was the increase in tractors between 1930 and 1960?

Source: Census of Agriculture

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Math Mammoth Data and Graphs (Blue Series)

4. A department store was tracking the sales of many items, including umbrellas. a. In 2007, which months were the sales less than 40 umbrellas? How about in 2008? b. Find the month with the greatest difference between 2007 and 2008 sales.

5. Four hundred eight students were asked about how many hours they had slept the previous night. The results are summarized in the table below: Hours of Sleep

Frequency

6 7 8 9 10 11 12 total

2 15 56 148 137 40 10 408

a. Draw a bar graph. Note you need to choose the scaling on the vertical axis. b. Find the mode. c. Which of the following could be a part of the original data set? (Hint: Imagine the original data set. Think: What was asked? What did the students answer?) 2, 15, 56, 148, 137, 40, 10, 2, 15, … 6, 10, 8, 8, 9, 7, 11, 10, 9, 10, 11, … d. Which of the following is the average for this data set? 9.4 hours

8.5 hours

11.1 hours 10.7 hours

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Math Mammoth Data and Graphs (Blue Series)

Math Mammoth Data and Graphs Answer Key Reading Graphs and Charts, p. 7 1. a. Jane read the most books. She read 18 books. b. Jim read the fewest books. He read 8 books. c. Three more books. 16 − 13 = 3. d. The girls read a total of 18 + 15 + 9 + 12 = 54 books. e. The boys read a total of 14 + 8 + 16 + 13 = 51 books. f. The girls read more books; three more books.

2. a. Vegetable use in one week Jacksons Joneses Millers

2. a. See the chart on the right. b. Restaurant A used 35 kg of vegetables. Restaurant B used 40 kg of vegetables. c. 35 kg more d. 75 kg total 3. a.

Day

Mon

Tues

Restaurant A Restaurant B

Wed

Thurs

Fri

Sat

Sun

Newspapers about 30 about 40 about 40 about 40 about 50 about 30 about 70 b. about 100 papers c. about 40 more newspapers 4. a. Jonathan read 14 books. b. Annie in February; Lisa in April c. 8 more books d. 9 more books e. Books read Jan Feb Mar Apr Total Annie

13

21

18

14

66

Freddie

8

5

11

9

33

Lisa

8

13

16

18

55

Jonathan

10

8

14

15

47

f. Annie read 33 more books than Freddie. Annie 1 3

+

2

1

1

8

1

4

6

6

Freddie 8 1 +

8

Jonathan 1 0 8

5

1

3

1

1

6

1

8

5

5

9 3

Lisa

3

+

+

59

1

4

1

5

4

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Math Mammoth Data and Graphs (Blue Series)

Bar Graphs and Pictographs, p. 10 1. a. 5 students b. 4 students c. 7 students d. 4 students e. 20 students f.

Students

Students who slept less than 8 hours Students who slept at least 8 hours 2. Page count Number of books 200-249

2

250-299

2

300-349

6

350-399

4

a. 4 books

b. 10 books c. 2 books d. 217 pages

3.

a. Saturday; 950 people b. 650 – 500 = 150 more people c. 350 + 500 = 850 people d. If you don’t like crowds: Thursday. If you like crowds: Saturday.

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Math Mammoth Data and Graphs (Blue Series)

Bars and Pictographs, cont. 4. Check the student’s work. Example: 1 Day

= 10 baskets

Baskets

Mon Tue Wed Thu 5.

6. Answers will vary.

More Practice with Bar Graphs and Pictographs, p. 14 1. a. Chris b. 5 more c. 4 fewer fish

d. 15 together

2.

a. 27 children

b. 12 children c. 41 children

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Math Mammoth Data and Graphs (Blue Series)

More Practice with Bar Graphs and Pictographs, cont. 3.

a. 17 cars b. 8 cars

c. 35 cars d. Yes, he will have 26 and Tony 25.

4. a. Wednesday he caught 1,300 kg of fish. b. Friday, he caught 1,100 kg of fish. c. He caught 500 kg more. d. He caught 3,700 kg of fish during the week. 5. a. 90 birds b. 25 Australian animals c. 85 together d. 35 more

e. 135 animals

6. a.

b. They paid $40 more for week 3 than week 4. 200 – 160 = 40 c. They paid $20 more for week 2 than week 1. 170 – 150 = 20

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Math Mammoth Data and Graphs (Blue Series)

More Practice with Bar Graphs and Pictographs, cont. 7. a.

b. The difference is $632 – $525 = $107. 8.

9. a. The Sports Club is the most popular. b. There are 15 more students. c. There are 68 students in the three clubs. 10. a. The volleyball set costs $24. b. The snorkeling set costs $4 more than the swim ring set. c. They cost $13. d. The cheapest would be $13.

Making Bar Graphs 1, p. 21 1. a.

Hours of TV Frequency

b. 27 classmates c. 1 hour of TV. d. 13 e. 10 f. no g. yes.

63

0h

2

1h

11

2h 3h

4 4

4h

3

5h

2

6h

1

Math Mammoth Data and Graphs (Blue Series)

Making Bar Graphs 1, cont. 2. a.

b. 27 people c. “Cold” colors. 3. a.

b. 6, 8, and 9; four students each c. 10; 1 student d. 12 e. 5 f. 9

Test score Frequency 1 2 2 2 3 3 4 2 5 2 6 4 7 2 8 4 9 4 10 1

Making Bar Graphs 2, p. 23 1. There are 37,460,000 households that own a cat, and 2,087,000 that own a horse. 2. a. Assuming that the student chooses the scaling for the horizontal axis to go from 0 to 4,000 with tick marks at every 100, the graph will look like the one below. If the student chooses some other scale, such as each tick mark 150 or 200 units apart, then the bars in the graph will appear smaller and there will be lots of empty space in the right part of the graph.

b. About three times as long c. About two times as long

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Math Mammoth Data and Graphs (Blue Series)

Making Bar Graphs 2, cont. 3. a.

b. About 1,100,000 + 720,000 + 390,000 = 2,210,000 households. c. The number of households owning a hamster, a guinea pig, or a gerbil is approximately: 830,000 + 630,000 + 190,000 = 1,650,000. Since about 2,210,000 households own a turtle, a lizard, or a snake, the latter is more popular. 4. Number of frequency siblings 0

3

1

7

2

6

3

2

4

0

5

1

6

1

Making Histograms, p. 25 1. point count frequency 12-18

2

19-25

5

26-32

6

33-39

3

40-46

3

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Math Mammoth Data and Graphs (Blue Series)

Making Histograms, cont. 2. The bin width of 4 works well: (73 − 58)/4 = 15/4 = 3.75, rounded up to 4. weight frequency 58-61

2

62-65

6

66-69

4

70-73

3

3. The bin width is 9: (43 − 0) / 5 = 8.6, rounded up to 9. Age

Frequency

0-8

13

9-17

5

18-26

4

27-35

1

36-44

2

Double Bar graphs, p. 27 1. a.

Grade 1 2 3 4 5

can swim 20 26 38 46 48

cannot swim 30 24 12 4 2

b. It increases. c. It decreases. 2. a. biographies, mysteries, and poetry b. About 25,000 + 32,000 + 26,000 = 83,000 loans. (The three most popular genres in 2006 were mysteries, children’s, and romance.) c. About 13,000 + 7,000 + 6,000 = 26,000 loans (The three least popular genres in 2007 were comics, poetry, and biographies.)

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Math Mammoth Data and Graphs (Blue Series)

Double Bar graphs, cont. 3. a.

b. science fiction and fantasy

Line Graphs 1, p. 29 1. a. $60

b. $140

c. $70

d. June

e. $70

2. a. Day 1: 500 grams; Day 2: 525 grams; Day 3: 550 grams; Day 4: 575 grams b. Day 5. c. Day 8.

b. Highest price was in December, $3.60 per pound, and the lowest price was in July, $1.63 per pound. The difference is $1.97. c. In August, $3.64. In November, $6.38.

3. a. The price lowers in the summer and is higher in the winter, because in the summer there is an abundance of strawberries; all stores and markets are selling strawberries. Nobody can keep the price high because if they did, people would go elsewhere to buy. 4. a.

b. yes.

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Math Mammoth Data and Graphs (Blue Series)

Line Graphs 1, cont. 5. a.

b. December, January, and February c. June, July, August, and September d. 25 degrees 6. Answers will vary. Check the student’s answers.

Line Graphs 2, p. 32 1. a, b, c:

d. 6 days. 2. a.

b. The temperature went up. The family probably ate lunch then and had to open the fridge several times before and after eating, which made the fridge temperature go up. c. The temperature went up again. The family probably ate supper then and had to open the fridge several times before and after eating, which made the fridge temperature go up.

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Math Mammoth Data and Graphs (Blue Series)

Line Graphs 2, cont. 3. It is easiest to make the gridlines go by 20s.

4.

Month Visitors

rounded to the nearest 50

Jan Feb

1039 1230

1050 1250

Mar

1442

1450

Apr

1427

1450

May Jun

1183 823

1200 800

Jul

674

650

Aug

924

900

Sep Oct

1459 1540

1450 1550

Nov

1638

1650

Dec

1149

1150

In the summer Juanita’s blog had many fewer visitors than in the spring or fall. The three months with the fewest visitors were June , July , and August. The three months with the most visitors were September , October , and November. Note: To find the three months with most visitors, look at the actual numbers given in the table. 5. a. Time 0s 1s 2s 3s 4s 5s 6s 7s 8s 9s 10 s 11 s 12 s

Distance 0m 30 m 60 m 90 m 120 m 150 m 180 m 210 m 240 m 270 m 300 m 330 m 360 m

b. The car will have traveled 3 km in 100 seconds, or 1 min 40 s.

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Math Mammoth Data and Graphs (Blue Series)

Line Graphs 2, cont. 6. a.

b. Answers may vary. Maybe during 2001-2003 the sports club had a leader that the students didn’t like. Or maybe during those years some other activity was offered that was much more popular.

Temperature Line Graphs, p. 36 Month

Jan Feb Mar Apr May Jun

Jul

Aug Sep

Oct

Nov Dec

Max Temperature 6°C 7°C 10°C 13°C 17°C 20°C 22°C 21°C 19°C 14°C 10°C 7°C 1. a. July b. January c. March and November; or February and December. d. 3 degrees Celsius e. 2 degrees Celsius f. 16 degrees Celsius 2.

a. January b. 6° c. 1° d. 15°

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Math Mammoth Data and Graphs (Blue Series)

Reading Line Graphs, p. 38 1. a. If you continue the line on the line graph in a similar trend as from 1990 to 2000, the farm population could be about 2,000,000. Of course, this does not guarantee that actually happened. Looking at the numbers given, the decrease from 1990 to 2000 was about 1,500,000 persons. We cannot expect it to drop at the same rate, but perhaps if the farm population dropped at a slower rate, maybe it dropped by a half million or by a million, and was about 1.5 to 2 million people. b. In the 1940s and 1950s. c. In the 1940s, the farm population decreased by about 7,499,000 people. From 1950 to 1960, it decreased by about 9,603,000 people. d. The farm population decreased to under 10 million people in about 1969. e. The farm population decreased to under 5 million people in about 1986. 2. a. 293; 807 b. 750; 2,058 c. mammals

d. fishes and reptiles e. mammals; from 2002 to 2006

Double and Triple Line Graphs, p. 40 1. a. Mom sent 12 + 6 + 8 + 11 + 5 + 6 + 10 = 58, and Dad sent 4 + 2 + 2 + 6 + 2 + 3 + 1 = 20. b. Mom sent 11 − 6 = 5 more messages. c. On Sunday the difference was 10 − 1 = 9 messages. d. On Friday (5 − 2) and on Saturday (6 − 3) the difference was only 3 messages. 2. a. In 2005 there were 2 + 4 + 9 + 5 + 7 + 3 + 1 = 31 storms; in 2006 1 + 3 + 2 + 4 + 0 = 10 storms; and in 2007 1 + 1 + 0 + 3 + 8 + 1 + 0 + 1 = 15 storms. b. The 2005 season was unusually active. c. September, since the total number of storms for September is the highest (5 + 4 + 8 = 17 storms). 3. Answers will vary. 4. a.

b. Anna improved quite a bit. c. The difference was the greatest (85 − 59 = 26 points) in test 4. It was the smallest (66 − 62 = 4 points) in test 2.

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Math Mammoth Data and Graphs (Blue Series)

Graphs: More Practice, p. 42 1. a.

Weight (ounces) Frequency 83..88

3

89..94

6

95..100

6

101..106

3

107..112

1

113..118

1

b. The average is 95 1/2 ounces. You can see it in the bar graph because the number 95 1/2 is near the middle and near the peak of the graph. You could also see it from the data itself, noting that lots of the weights are 90-something. 2. a. 340 + 360 + 320 + 320 = 1,340 b. 300 + 290 + 290 + 260 = 1,140 3. a. They sold the most strawberries during week 26. About $4,500. b. They sold the least strawberries during week 23. The sales were $1,500. c. About $12,000. 4.

0

Weight in ounces 6 lb 14 oz 110

1

6 lb 12 oz

108

2

6 lb 14 oz

110

3 4

7 lb 7 lb 2 oz

112 114

5

7 lb 4 oz

116

6

7 lb 6 oz

118

7

7 lb 7 oz

119

Week Weight

5. a. MARITAL STATUS Never married

60,000,000

Now married 120,000,000 (not separated) Separated

5,000,000

Widowed

15,000,000

Divorced

22,000,000

Source: From Census 2000 data, www.census.gov.

b. The estimated number of people who are either separated, widowed, or divorced is ≈ 42,000,000. c. The estimated number of people who are not married is ≈ 102,000,000.

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Math Mammoth Data and Graphs (Blue Series)

Graphs: More Practice, cont. 6.a.

e.

Museum's visitors Day

Adults Children

Total Visitors

Monday Tuesday

29 23

14 10

43 33

Wednesday

34

18

52

Thursday

38

19

57

Friday Saturday

35 57

19 25

54 82

Sunday

63

31

94

279

136

415

Totals

b. Sunday is the busiest day with 94 visitors, and Tuesday the least busy with 33 visitors. The difference in the total visitor count between those two days is 94 − 33 = 61. c. 279 ÷ 7 = 39.9 d. 136 ÷ 7 = 19.4 7. a.

b. 14 people c. 45 people d. About 82 + 41 + 10 + 4 = 137 children, about 95 + 61 + 39 + 6 = 201 adults e. It could be a group of people that were at the swimming pool at 5 pm on a certain Tuesday because there were both children and adults.

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Math Mammoth Data and Graphs (Blue Series)

Average, p. 47 1. (78 + 87 + 69 + 86) ÷ 4 = 80. Judith’s average score is 80. 2. (18 + 22 + 26 + 23 + 16) ÷ 5 = 21. The average temperature for the day was 21°C. 3. 414 ÷ 6 = 69. Dad averaged 69 km in one hour. 4. 12 × 55 = 660. A dozen eggs would weigh 660 grams. 5. 7 × 76 = 532. It cost $532. 6. (234 + 178 + 250 + 198) ÷ 4 = 215. Her weekly average grocery bill was $215. 7. The girls’ average time was 15 minutes. The boys’ average time was 13 minutes. The boys are faster. The difference is two minutes. 8. a. Quiz score Frequency 13..15

1

16..18

1

19..21

2

22..24

4

25..27

0

28..30

2

b. The average score is 22. c. Look at the “peak” of the graph. The average is usually near that point. 9. a. The average age is 29.

b. Now the average age is 34.

Puzzle corner: 213 ÷ 12 is 17 R9.

Average (Mean), p. 50 1. a. 4 2. a. 5.4

b. 16 b. 301.3

3. a. The average is 1,210.7

b. 807

c. 1,446.5

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Math Mammoth Data and Graphs (Blue Series)

Average (Mean), cont. 4. The total rainfall calculates to 5.067 mm/day × 15 days = 76.005 mm. Actually, it probably was exactly 76 mm, and the calculated average given in the problem is rounded to three decimals. 5. a. The average weight is 1,527 g. b. 1,527 − 1,250 = 277 g c. 1,820 − 1,527 = 293 g d. 1,606 g. The average increased by 1,606 − 1,527 = 79 grams 6. a. Mean = $12,969 ÷ 9 = $1,441. b. Mean = ($12,969 − $3,400) ÷ 8 = $1,196. The mean decreased by $245 when the highest salary was not included. This shows how “sensitive” the mean can be for small changes in the actual data. Puzzle corner. $531. Guess and check works here. You can also think logically that since the average is $567, and two of the given prices are higher than the average, then the last unknown price is not more than $567, so it has to be $531.

Mean, Mode, and Bar Graphs, p. 53 1. “Now married” 2. a. “Pop” b. “What is your favorite drink?” or “What did you drink yesterday at suppertime?” or “What is your least favorite drink?” etc. 3. a. “Vanilla.” b.

c. It isn’t possible. 4. a. There are three modes: 12, 18, and 19. b. Test Score Frequency <8

3

8..10 11..13

4 6

14..16

1

17..19

7

20..22 23..25

1 2

c. The average is: 336 / 24 = 14

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Math Mammoth Data and Graphs (Blue Series)

Mean, Mode, and Bar Graphs, cont. 5. a. The mode is “B.” b.

c. It isn’t possible. d. There were 50 students in all. 17/50 of the students got a “B.”

Circle Graphs, p. 55 1.

a.

2.

b.

c.

Bread

Amount

Fraction

Percentage

Central Angle

white bread

50

1/4

25%

90º

bran bread

25

1/8

12.5%

45º

rye bread

30

3/20

15%

54º

corn bread

40

1/5

20%

72º

4-grain bread

55

11/40

27.5%

99º

TOTAL

200

1

100%

360º

3. a.

b. No, it doesn’t.

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Math Mammoth Data and Graphs (Blue Series)

Circle Graphs, cont. 4. Answers will vary. The answers below are actually the exact percentages. However, the student’s answers may differ from these a little and be totally acceptable. (Each circle should sum to 100%.) Starting from 12:00 and going clockwise: a. About 45%, about 5%; about 40%, about 10%. (The first two and the last two should each sum to 50%.) b. About 20%, 20%, 10%; 25%, 12.5%, 12.5%. (The first three and the last three should each sum to 50%.) c. About 65%, about 20%, about 15%. 5. Flavor

Amount sold

Percentage of total

Central Angle

chocolate

67

46.9%

169º

vanilla

34

23.8%

86º

strawberry

16

11.2%

40º

blueberry

26

18.2%

65º

TOTAL

143

100%

360º

Note: The “Percentage of total” column actually totals to 100.1% because the numbers that were rounded up moved a little bit farther in the upward direction than the numbers that were rounded down moved in the downward direction. That’s typical for calculations that round several numbers.

6. Favorite hobby

Percentage

Central Angle

Reading

12.3%

44º

TV

24.5%

88º

Computer games

21%

76º

Sports

22.3%

80º

Pets

7.1%

26º

Collecting

8.1%

29º

no hobby

4.7%

17º

TOTAL

100%

360º

Review, p. 57 1. The rule is: y = 9 − x. x

0

1

2

3

4

y

9

8

7

6

5

x y

5 4

6 3

7 2

8 1

9 0

2. The mean is 11.67; the mode is 10. 3. a. Estimate: 3,750,000 tractors in 2010. b. From 1940 to 1950, the increase was about 1,750,000 tractors c. Slowly declining (but at a slightly increasing rate of decline). d. In 1930 there were about 1 million tractors; in 1960 about 4 1/2 million. So the increase was 4 1/2-fold.

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Math Mammoth Data and Graphs (Blue Series)

Review, cont. 4. a. In 2007: June, July, August, and November. In 2008: March, May, July, August, and November. b. June.

5. a. b. The mode is 9 hours. c. The latter (6, 10, 8, 8, 9, 7, 11, 10, 9, 10, 11,...) d. 3827 hours ÷ 408 students = 9.38 hours/student ≈ 9.4 hours

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Math Mammoth Data and Graphs (Blue Series)

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