Students should be fairly comfortable with a number line. This problem is intended to familiarize students with benchmark fractions and decimals between 0 and 1, which are useful in probability.
If most students struggle with this item, plan to take extra time for the Launch of Lesson 2, Activity 4. Display a blank number line and call on students to place benchmark fractions, decimals, and percents on the number line. Ask students to share their strategies for making the placement. Connect this to the questions “Which is closer to 1?” and “Which has the largest value?”
2.
Students did push-ups for a fitness test. Their goal was 20 push-ups. For each student, determine what percentage of the goal they achieved.
Elena did 16 push-ups.
Jada did 42 push-ups.
Lin did 13 push-ups.
Andre did 21 push-ups.
Answer:
80%. 16 is 80% of 20, because 2016=10080.
210%. 42 is 210% of 20, because 2042=100210.
65%. 13 is 65% of 20, because 2013=10065.
105%. 21 is 105% of 20, because 1 is 5% of 20, and 20 is 100% of 20.
Teaching Notes
Calculating and making sense of percentages will come up throughout the unit as students study probability and sampling.
Watch for students struggling with the percentages that are greater than 100%. Encourage students to use fractions, benchmarks, or proportional reasoning to help. Since 1 push-up is 5% of the goal, any percentage can be determined by multiplying the number of push-ups by 5.
If most students struggle with this item, plan to emphasize the Activity Synthesis of Lesson 2, Activity 3, where percentages are used and interpreted with probability. During the unit, students have choices of reporting probabilities as fractions or percentages, but they need to interpret percentages. For example, a 30% chance means we expect success in 3 out of 10 trials. Offer tape diagrams and double number lines as tools for helping students recall and calculate percentages. Additional practice can be found in Grade 6, Unit 3, Lessons 11–16.
3.
Andre surveyed a random sample of 20 students and found that 13 of them were in favor of having school start one hour later.
The school has 250 students. Make an estimate for the number of students in the school who are in favor of having school start one hour later.
Do you think it would be surprising if 150 students out of 250 were in favor? Explain your reasoning.
Do you think it would be surprising if 100 students out of 250 were in favor? Explain your reasoning.
Answer:
Answers vary. Sample response: 163 students. 65% of the students surveyed were in favor, and 65% of 250 is 162.5.
No, this would not be surprising. Sample reasoning: 150 is 60% of 250, so having 60% in favor instead of 65% would not be surprising.
Yes, this would be surprising. Sample reasoning: 100 is 40% of 250, so having 40% in favor instead of something close to the 65% from the sample would be surprising.
Teaching Notes
This problem mainly tests students’ understanding of proportional relationships, while preparing for some of the concepts about using samples to draw inferences that will be presented in the upcoming unit.
If most students struggle with this item, it is not necessary to take any specific action. Students start off Lesson 3 with some additional examples of samples from a population so that they can start to practice this new concept. Attend to students who struggle with this problem and make sure they are able to follow the proportional reasoning they are using. More practice with this concept is found in Lesson 16.
4.
Select all the measures of variability or spread.
A.
mean
B.
IQR (interquartile range)
C.
MAD (mean absolute deviation)
D.
median
E.
range
Answer:B, C, E
Teaching Notes
This item tests students’ ability to distinguish measures of center from measures of spread. This is also an opportunity to determine whether students are familiar with these terms from their work in grade 6. All of these terms are needed in the second half of the unit, both when using samples to make inferences about populations and when testing whether there is a significant difference between two populations. Review of mean, MAD, and dot plots will come in this unit.
Students who do not select choice B or choice C have a significant gap in understanding about the meaning and use of IQR and MAD, or they simply may not remember the terms. Students who select choice A or choice D do not fully understand the differences between measures of center and measures of spread. Students who select choice E may not know what range is or may not recognize it as a measure of spread.
If most students struggle with this item, plan to spend additional time on the Lesson Synthesis of Lesson 11, Activity 2. Review both MAD and IQR as well as mean and median. Focus on what MAD and IQR tell us about data sets, not how to calculate them. If extra practice interpreting MAD and IQR is needed, Grade 6, Unit 8, has additional work on calculating mean, median, MAD, and IQR.
5.
Ten students each attempted 10 free throws. This list shows how many free throws each student made.
8, 5, 6, 6, 4, 9, 7, 6, 5, 9
What is the median number of free throws made?
What is the IQR (interquartile range)?
Answer:
6 free throws. The ordered list is 4,5,5,6,6,6,7,8,9,9. The two middle terms in the ordered list are both 6.
3 free throws. The first half of the data is 4,5,5,6,6, and its median is 5. The second half of the data is 6,7,8,9,9, and its median is 8. The IQR is 3, since 8−5=3.
Teaching Notes
This question more specifically tests whether students remember how to calculate median and IQR. Watch for students attempting to answer the question without first sorting the data, and for students who have significant trouble understanding the question because they do not recognize or understand the vocabulary.
If most students struggle with this item, plan to review it, or a similar item using small data sets, during the Launch of Lesson 18, Activity 3.
6.
Two groups went bowling. Here are the scores from each group.
Group A: 70, 80, 90, 100, 110, 130, 190
Group B: 50, 100, 107, 110, 120, 140, 150
Construct two box plots, one for the data in each group.
Which group shows greater variability? Explain how you know.
Answer:
See image. Group A is on the top, and Group B is on the bottom.
Group A shows greater variability. Sample reasoning: It has a wider range (from 70 to 190) and a wider IQR (from 80 to 130).
Teaching Notes
This problem tests students' prior knowledge of box plots and provides an opportunity for students to demonstrate how they are thinking about variability.
If students have difficulty, they will need a quick refresher on constructing box plots when this concept first appears in the unit. Students may argue variability by appealing to range (whether they use the term or not), but the IQR is the more important measure of spread for the work of this unit.
If most students struggle with this item, plan to review it during the Launch of Lesson 18, Activity 4. If students are struggling to understand how to construct box plots, Grade 6, Unit 8, Lesson 16, has activities to help students understand and construct them.
7.
This dot plot shows the number of coins in 10 students’ pockets.
What is the mean and MAD (mean absolute deviation) of this data? Explain how you know.
Answer:
The mean is 3 coins. The MAD is 2.2 coins.
Sample reasoning: First, calculate the mean. The total number of coins is 30. The mean is 3 coins, because 1030=3. The absolute deviations are 3, 3, 3, 2, 0, 0, 2, 2, 3, 4. The total of these is 22. The MAD is 1022.
Teaching Notes
This problem tests students’ fluency with mean, MAD, and dot plots.
Look at students’ explanations to see which incorrect answers come from a misunderstanding of mean and MAD and which are simply arithmetic errors.
If most students struggle with this item, plan to review it during the Activity Synthesis of Lesson 11, Activity 2, to help students recall how to use the dot plot to find the mean and MAD. If students struggle with both this item and Item 4, plan to use activities from grade 6 to do a more thorough review of interpreting and estimating measures of center and spread to compare data sets represented in box plots.