Grade 6, Unit 8 - Practice Problems (2024)

Tyler asked 10 students at his school how much time in minutes it takes them to get from home to school. Determine if each of these dot plots could represent the data Tyler collected.Explain your reasoning for each dot plot.

Here is a list of questions.For each question, decide if the responses will producenumerical data or categorical data and give two possible responses.

1. What is your favorite breakfast food?
2. How did you get to school this morning?
3. How many different teachers do you have?
4. What is the last thing you ate or drank?
5. How many minutes did it take you to get ready this morning—from waking up to leaving for school?

1. Writetwo questions that you could ask the students in your class that would result in categorical data.For each question, explain how you know that responses to it would producecategorical data.
2. Writetwo questions that you could ask the students in your class that would result in numerical data.For each question, explain how you know that responses to it would produce numerical data.

Triangle $DEF$ has vertices $D=(\text-4, \text-4), E=(\text-2, \text-4)$, and $F=(\text-3, \text-1)$.

Lin and her friends went out for ice cream after school. The following questions came up during their trip. Select all the questions that are statistical questions.

1. How far are we from the ice cream shop?
2. What is the most popular ice cream flavor this week?
3. What does a group of 4 people typically spend on ice cream at this shop?
4. Do kids usually prefer to get a cup or a cone?
5. How many toppings are there to choose from?

Here is a list of questions about the students and teachers at a school. Select all the questions that are statistical questions.

1. What is the most popular lunch choice?
2. What school do these students attend?
3. How many math teachers are in the school?
4. What is a common age for the teachers at the school?
5. About how many hours of sleep do students generally get on a school night?
6. How do students usually travel from home to school?

Here is a list ofstatistical questions. What data would you collect and analyze to answer each question?For numerical data, includethe unit of measurement that you would use.

1. What is a typical height offemale athletes on a team in the most recent international sporting event?
2. Are most adults in the school football fans?
3. How long do drivers generally need to wait at a red lightin Washington, DC?

Describe the scale you would use on the coordinate planeto plot each set of points. What value would you assign to each unit ofthe grid?

Noah’s water bottle contains more than 1 quart of water but less than $1 \frac{1}{2}$ quarts. Let $w$ be the amount of water in Noah’s bottle, in quarts. Select all the true statements.

1. $w$ could be $\frac{3}{4}$.
2. $w$ could be 1.
3. $w > 1$
4. $w$ could be $\frac 4 3$.
5. $w$ could be $\frac 5 4$.
6. $w$ could be $\frac 5 3$.
7. $w > 1.5$

Order these numbers from least to greatest:

A teacher drew a line segment that was 20 inches long on the blackboard. She askedeach of her students to estimate the length of the segment and usedtheir estimates to draw thisdot plot.

1. How many students were in the class?
2. Were students generally accurate in their estimates of the length of the line? Explain your reasoning.

Here are descriptions of data sets. Select alldescriptions of data sets thatcouldbe graphed asdot plots.

1. Class size for the classes at an elementary school
2. Colors of cars in a parking lot
3. Favorite sport of eachstudentin a sixth-grade class
4. Birth weights for the babies born during October at a hospital
5. Number of goals scored in each of 20 games played by a school soccer team

Priya recorded the number of attempts it took each of 12 of her classmates to successfully throw a ball into a basket. Make a dot plot of Priya’s data.

Solve each equation.

Find the quotients.

Find the area of each triangle.

Clare recordedthe amounts of time spent doing homework, in hours per week, bystudents in sixth, eighth, and tenth grades. She madea dot plotof the data foreach grade and provided the following summary.

• Students in sixth grade tendto spend less time on homework than students in eighth and tenth grades.

• The homework times for the tenth-grade students are more alike than the homework times for the eighth-grade students.

Use Clare's summary to match each dot plot to the correct grade (sixth, eighth, or tenth).

Mai played10 basketball games. She recordedthe number of points she scoredand made a dot plot. Mai said that she scored between 8 and 14 points in most of the 10 games, butone gamewas exceptional.During that gameshescored more than double her typical score of 9 points. Use the number lineto make a dot plotthat fits the description Mai gave.

A movie theater is showing three different movies. The dot plots represent the ages of the people who were at the Saturday afternoon showing of each of these movies.

1. One of these movies was an animated movie rated G for general audiences. Do you think it was Movie A, B, or C? Explain your reasoning.
2. Which movie has a dot plot with ages that that center at about 30 years?
3. What is a typical age for the people who were at Movie A?

Find the value of each expression.

1. $3.727+ 1.384$
2. $3.727 -1.384$
3. $5.01\boldcdot4.8$
4. $5.01\div4.8$

Three sets of data about ten sixth-grade students were used to make three dot plots.The person who made these dot plots forgot to label them. Match each dot plot withthe appropriate label.

The dot plots show the time it takes to get to school for ten sixth-grade students from the United States, Canada, Australia, New Zealand, and South Africa.

1. List the countries in order of typical travel times, from shortest to longest.
2. List the countries in order of variability in travel times, from the least variability to the greatest.

Twenty-five students were asked to rate—on a scale of 0 to 10—how important it is to reduce pollution.A rating of 0 means“not at all important” and a rating of 10 means “very important.” Here is a dot plot of their responses.

Explain why a rating of 6 is not a good description of the center of this data set.

Tyler wants to buy some cherries at the farmer’s market. He has \$10 and cherries cost \$4 per pound.

1. If $c$ is the number of pounds of cherries that Tyler can buy, write one or more inequalities or equationsdescribing $c$.
2. Can 2 be a value of $c$? Can 3 be a value of $c$?What about -1? Explain your reasoning.
3. If $m$ is the amountof money, in dollars,Tyler can spend, write one or moreinequalities or equationsdescribing $m$.
4. Can 8 be a value of $m$? Can 2 be a value of $m$? What about 10.5? Explain your reasoning.

Match histogramsA through E to dot plots 1 through 5 so that each match represents the same data set.

Here is a histogram that summarizes the lengths, in feet, of a group of adult female sharks.Select all the statements that are true, according to thehistogram.

1. A total of 9 sharks were measured.
2. A total of 50 sharks were measured.
3. The longest shark that was measured was 10 feet long.
4. Most of the sharks that were measured were over 16 feet long.
5. Two of the sharks that were measured were less than 14 feet long.

This table shows the times, in minutes, it took 40 sixth-grade students to run 1 mile.

These two histograms show the number of text messages sent in one week by two groups of 100 students. The first histogram summarizes data from sixth-grade students. The second histogram summarizes data from seventh-grade students.

3. Overall, which group of students—sixth- or seventh-grade—sent more text messages?

Forty sixth-grade students ran 1 mile. Here is a histogram that summarizes their times,in minutes. The center of the distributionisapproximately10 minutes.

On the blank axes,draw a secondhistogram that has:

• a distribution of times for a different group of 40 sixth-grade students.
• a center at10 minutes.
• less variability than the distribution shown in the first histogram.

Jada has $d$ dimes. She has more than 30 cents but less than a dollar.

1. Write twoinequalities that represent how many dimes Jada has.
2. Can $d$ be 10?
3. How many possible solutions make both inequalities true? If possible, describeor list the solutions.

The histogram summarizes the data on the body lengths of 143 wild bears. Write a few sentences describing the distribution of body lengths.

Which data set is more likely to producea histogram with a symmetric distribution? Explain your reasoning.

• Data on thenumber of seconds on a track of music in a pop album.

• Data on the number of seconds spent talking on the phone yesterday by everyone in the school.

Decide if each data set might produce one or more gaps when represented by a histogram. For each data set that you think might produce gaps, briefly describe or give an example of how the values in the data set might do so.

1. The ages of students in a sixth-grade class.
2. The ages of people in an elementary school.
3. The ages of people eating in a family restaurant.The ages of people who watch football.
4. The ages of runners in a marathon.

Evaluate the expression $4x^3$ for each value of $x$.

Jada drank 12 ounces of water from her bottle. This is 60% of the water the bottle holds.

1. Write an equation to representthis situation. Explain the meaning of any variables you use.
2. How much water does the bottle hold?

A preschool teacher is rearrangingfour boxes of playing blocks so that each box contains an equal number of blocks. CurrentlyBox 1 has 32 blocks, Box 2 has 18, Box 3 has 41, and Box 4 has 9.

Select all the ways he couldmake each box have the same number of blocks.

1. Remove all the blocks and make four equal piles of 25, then put each pile in one of the boxes.
2. Remove 7 blocks from Box 1 and place them in Box 2.
3. Remove 21 blocks from Box 3 and place them in Box 4.
4. Remove 7 blocks from Box 1 and place them in Box 2, and remove 21 blocks from Box 3 and place them in Box 4.
5. Remove 7 blocks from Box 1 and place them in Box 2, and remove 16 blocks from Box 3 and place them in Box 4.

In a round of mini-golf, Clare records the number of strokes it takes to hit the ball into the hole of each green. She saidthat, if she redistributed the strokes on different greens, she could tell thather average number of strokes per hole is 3.

Explain how Clare is correct.

Three sixth-grade classes raised \$25.50, \$49.75, and \$37.25 for their classroom libraries. Theyagreed to share the money raised equally. What is each class’s equal share? Explain or show your reasoning.

An earthworm farmer examinedtwo containers of a certain species of earthworms so that he could learn about their lengths. He measured 25 earthworms in each container and recorded their lengthsin millimeters.

Here are histograms of the lengths for each container.

1. Which container tends to have longer worms than the other container?
2. For which container would 15 millimetersbe a reasonable description of a typical length of the worms in the container?
3. If length is related to age, which container had the most young worms?

On school days, Kiran walks to school. Here arethe lengths of time, in minutes, forKiran’s walks on5 school days.

2. Without calculating, decide if 15 minutes would be a good estimate of the mean. If you think it is a good estimate, explain your reasoning. If not, give a better estimate and explain your reasoning.
3. Calculate the mean for Kiran’s data.
4. In the table, record the distance of each data point from the mean and its location relative to the mean.

   	timein minutes	distancefrom themean	leftorright ofthemean?
   row 1	16
   row 2	11
   row 3	18
   row 4	12
   row 5	13

5. Calculate thesum ofall distances to the left of the mean, then calculatethesum ofdistances to the right of the mean. Explain how these sums show that the mean is a balance point for the values in the data set.

Compare the numbers using >, <, or =.

1. Plot $\frac{2}{3}$ and $\frac{3}{4}$ on a number line.

2. Is $\frac{2}{3} < \frac{3}{4}$, or is $\frac{3}{4} < \frac{2}{3}$? Explain how you know.

Select allexpressions that represent the total area of the large rectangle.

Han recordedthe number of pages that he read each day forfive days. Thedot plot shows his data.

2. Find the mean number of pages that Han read duringthefive days. Drawa triangle to mark the mean on the dot plot.
3. Use the dot plot and the mean tocomplete the table.

   	numberofpages	distancefrommean	leftorrightofmean
   row 1	25		left
   row 2	28
   row 3	32
   row 4	36
   row 5	42

4. Calculate the mean absolute deviation (MAD)of the data. Explain or showyour reasoning.

Ten sixth-grade students recorded the amounts of time each tookto travelto school. The dot plot shows their travel times.

1. Which number of minutes—9 or 4.2—is a typical amount of time for theten sixth-grade students to travel to school? Explain your reasoning.
2. Based on the mean and MAD, Jada believes thattravel times between 5 and 13 minutes are common for this group. Do you agree? Explain your reasoning.
3. A different group of ten sixth-grade students also recorded their travel timesto school. Their mean travel time was also 9 minutes, but the MAD was about 7 minutes. What could the dot plot of this second data set be?Describe or draw how it might look.

In an archery competition, scores for each round are calculated by averaging the distance of 3 arrows from the center of the target.

An archer has a mean distance of 1.6 inches and a MAD distance of 1.3 inches in the first round. In the second round, the archer's arrows are farther from the centerbut are more consistent. What values for the mean and MAD would fit this description for the second round? Explain your reasoning.

The dot plots show the amounts of timethatten U.S.students and ten Australian studentstook to get to school. Which statement is true about the MAD of the Australian data set?

1. It is significantly less than the MAD of the U.S.data set.
2. It is exactly equal to the MAD of the U.S.data set.
3. It is approximately equal to the MAD of the U.S.data set.
4. It is significantly greater than the MAD of the U.S.data set.

The dot plots show the amounts of timethatten South African students and ten Australian students took to get to school. Without calculating, answer thequestions.

Two high school basketball teams have identical records of 15 wins and 2 losses. Sunnyside High School's mean scoreis 50 pointsand itsMAD is 4 points. Shadyside High School's mean scoreis 60 pointsand itsMAD is 15 points.

Lin read the records of each team’s score. She likes the team that had nearly the same score for every game it played. Which team do you think Lin likes? Explain your reasoning.

Jada thinks the perimeter of this rectangle can be represented with the expression $a+a+b+b$. Andre thinks it can be represented with $2a+2b$.

Draw a number line.

1. Plot and label three numbers between -2and -8(not including -2and -8).
2. Use the numbers you plotted and the symbols $<$ and $>$ to write three inequality statements.

Here is a table that showsstudent'sscores for 10 rounds of a video game.

What is the median score?

1. 125
2. 145
3. 147
4. 150

The medians of the following dot plots are 6, 12, 13, and 15, but not in that order. Match each dot plot with its median.

Ten sixth-grade students reported the hours of sleep they get on nights before a school day. Their responses are recorded in the dot plot.

Which estimatedo you think isbest? Explain how you know.

In one study of wild bears, researchers measured the weights, in pounds, of 143 wild bears that ranged in age from newborn to 15 years old. Thedata were used to make this histogram.

3. Do more than half of the bears in this group weigh less than 250 pounds?
   
4. If weight is related to age, with older bears tending to have greater body weights, would you say that there were more old bears or more young bears in the group? Explain your reasoning.

Here is a dot plot that shows the ages of teachers at a school.

1. The mean is less than the median.
2. The mean is approximately equal to the median.
3. The mean is greater than the median.
4. The mean cannot be determined.

Priya asked each of fivefriends toattempt to throwa ball in a trash can until they succeeded. She recorded the number of unsuccessful attemptsmade by each friend as: 1, 8, 6, 2, 4. Priya made a mistake: The 8 in the data set shouldhave been 18.

How would changing the 8 to 18 affect the mean and median of the data set?

1. The mean would decrease and the median would not change.
2. The mean would increase and the median would not change.
3. The mean would decrease and the median would increase.
4. The mean would increase and the median would increase.

In his history class, Han's homework scores are:

The history teacher uses the mean to calculate the grade for homework. Write an argument for Han to explain why median would be a better measure to use for his homework grades.

The dot plots show how much time, in minutes, students in a class took to complete each of five different tasks. Select all the dot plots of tasks for which themean time is approximately equal to themedian time.

Suppose that there are 20 numbers in a data set and that they are all different.

1. How many of the values in this data set are between the first quartile and the third quartile?
2. How many of the values in this data set are between the first quartile and the median?

In a word game, 1letter is worth 1point. This dot plot shows the scoresfor20 common words.

3. What is the third quartile (Q3)?
4. What is the interquartile range (IQR)?

Here are fivedot plots that show the amounts of time thatten sixth-grade students in five countries took to get to school. Match each dot plot with the appropriate median and IQR.

Each studentin a class recorded how many books they read duringthe summer. Here is a box plot that summarizes their data.

2. What is the median number of books read by thestudents?
3. What is the interquartile range (IQR)?

Use thisfive-number summary to draw a box plot. All values are in seconds.

The table shows the number of hours per week that each of 13seventh-grade students spent doinghomework. Create a box plot to summarize the data.

The box plot displays the data onthe response times of 100 mice to seeing a flash of light. How many mice are represented by the rectangle between 0.5 and 1 second?

Here is a dot plot that represents a data set. Explain why the mean of the data set is greater than itsmedian.

Jada earns money from babysitting, walking her neighbor’s dogs, and running errands for her aunt. Every four weeks, she combines her earnings and divides them into three equal parts—one for spending, onefor saving, and onefor donating to a charity. Jada donated \$26.00 of her earnings from the past four weeks to charity.

How much could she have earned from each job? Make two lists of how muchshe could have earned from the three jobsduring the pastfour weeks.

Here are box plots that summarize the heights of 20 professional male athletes inbasketball, football, hockey, and baseball.

1. In which two sports are the players’ height distributions most alike? Explain your reasoning.
2. Which sport shows the greatest variability in players’ heights? Which sport shows the least variability?

Here is a box plot that summarizesdata forthe time, in minutes, thata fire department took to respond to 100 emergency calls.

Pineapples were packed in three large crates. For each crate, the weight of everypineapple in the cratewas recorded. Here are three box plots that summarizethe weights in each crate.

Two TV shows each asked 100 viewers for their ages. For oneshow, the mean age of the viewerswas 35 years and the MAD was20 years. For the other show, the mean age of the viewerswas 30 years and the MAD was 5 years.

A sixth-grade student says he watches one of the shows. Which show do you think he watches? Explain your reasoning.

Twenty students timed how long each of them took to solve a puzzle. Here is adot plot oftheir solution times.

2. Calculate the mean and the median. Was your prediction correct?
3. Which pair of measures of center and variability—mean and MAD, or median and IQR—would you choose to describe the distribution of the solutiontimes? Explain your reasoning.

A local library showeda movie during a children’s festival. Jada recordedthe ages of 100 children who watched the movie. Here is a histogram for her data.

2. Can the histogram be used to estimate the median age of the movie watchers?
   
3. Canthe histogram be used to find the exact median age of the movie watchers?
   
4. Is it possible totell from the histogram if there was at least one 3-year-old who watched the movie?
   

In a game, players are shown a picture that has 14 butterfly shapes hidden in it. They aregiven 3 minutes to find as manyhidden butterfly shapes as they can.

Twenty fourth-grade students and twentysixth-grade students played this game. The dot plots represent the numbers of butterfly shapes the two groups found.

Write a few sentences comparing the numbers of hidden butterflies found for these two groups of students.

In a round of mini-golf, Elena records the number of strokes it takes to hit the ball in the hole for each green.Find the range, median, and interquartile range for the number of strokes needed on each green.