Comparing Numerical Data Using Box Plots

Warm Up OBJECTIVE: SWBAT compare numerical data using box plots. Language Objective: SWBAT verbally analyze and compare data using content specific vocabulary. If your job is to recommend a ski resort by comparing the annual snowfall of two mountains for the past 50 years, how would you compare all the data? (Hint: What are some measures we have been using to compare sets of data?) 1962 1963 1964 1965 1966 1967 … Powder Valley Mad Mountain 217 in. 132 in. 310 in. 104 in. 186 in. 287 in. … (5 min) 0 – 5 In-Class Notes Students may take a variety of approaches to solve this problem. The question is open-ended purposely. Students may compare the two sets of data using the maximum and minimum values. They may choose to use a measure of center such as the median or mean. They also might choose to look at the values of Q1 and Q3 (this is unlikely). The goal is to have students analyze the two sets of data using measures that have been introduced to over the course of the previous 4 lessons. Push students to prove their answer in more than one way. If none of the students answer the question by comparing the lower quartile (Q1) and the upper quartile (Q3), push them to do so. It will be helpful for later on in the lesson when they compare Q1 and Q3 using box plots. Preparation Notes 107 in. 233 in. 207 in. 106 in. 229 in. 37 in. … Mean, Median, Minimum, Maximum, Q1, Q3 Agenda

Launch 1962 1963 1964 1965 1966 1967 … Powder Valley Mad Mountain 217 in. 132 in. 310 in. 104 in. 186 in. 287 in. … 107 in. 233 in. 207 in. 106 in. 229 in. 37 in. … What are some of the ways we can represent this data visually so we can compare it more easily? (2 mins) 6 – 8 In-Class Notes Poll students to see if they prefer to compare data sets by just looking at the data (like it is presented on this slide) or if they prefer to use tables/graphs. This will help highlight the purpose of box plots. Then ask students to turn and talk about the question posed on the slide. As students discuss possible representations, they should see that line plots, bar graphs, and circle graphs (the graphs they are familiar with) would be neither effective nor efficient. Discuss student ideas as a whole class. Preparation Notes Agenda

Launch Think-Pair-Share 1962 1963 1964 1965 1966 1967 … Powder Valley Mad Mountain 217 in. 132 in. 310 in. 104 in. 186 in. 287 in. … 107 in. 233 in. 207 in. 106 in. 229 in. 37 in. … We have been using box plots to summarize sets of data. How could we show 2 sets of data on a box plot? Sketch what that might look like. (box plots on next slide) (2 min) 8 – 10 In-Class Notes Students should first think about this question independently, then discuss with a peer, and lastly, share out to the class. To push student thinking, ask what a box plot might look like with two sets of data. Would it be possible to show two sets of data on one number line? Preparation Notes

Launch Whole Class Powder Valley Mad Mountain Annual Snowfall (inches) (1 min) 10 – 11 In-Class Notes Inform students that this is what a box plot displaying two sets of data looks like. This box plot uses some of the data from the warm-up. Students may be confused because they saw only six pieces of data for the two ski resorts. Inform them that this box plot was constructed using data for fifty years (students just didn’t see all fifty years). Tell students that the goals of today’s lesson are to be able to construct a box plot for two sets of data like this and also to compare two sets of data using a box plot. Preparation Notes 0 50 100 150 200 250 300 350 400 Annual Snowfall (inches) Agenda

Launch Powder Valley Mad Mountain Annual Snowfall (inches) How can we use this box plot to compare the two ski resorts? (Hint: Is there a way to use the box plot to compare the values from the five number summary for each resort?) 0 50 100 150 200 250 300 350 400 Annual Snowfall (inches) Powder Valley Mad Mountain (2 min) 11 – 13 In-Class Notes Students should discuss this question in small groups (or with a partner depending on the classroom set-up). Essentially students should be looking at the box plot and hypothesizing how it might be read. After discussing this question in small groups, groups should share out their ideas with the class. Preparation Notes Agenda

Explore Whole Class From the box plot, you can easily see the median snowfall for each resort. Powder Mad Valley Mountain Median 175 inches 225 inches Annual Snowfall (inches) Powder Valley Mad Mountain 50 100 150 200 250 300 350 400 (1 min) 13 – 14 In-Class Notes Ask a student to come up to the board to point out the median for both sets of data. Follow-up questions: What does a median snowfall of 175 inches mean for Powder Valley? What does a median snowfall of 225 inches mean for Mad Mountain? Click to show the median for each resort. Preparation Notes This slide is connected to Math PS 2 - Reason Abstractly and Quantitatively: Attend to the meaning of quantities, not just how to compute them. Agenda

Explore Whole Class Using the medians to compare the resorts, which resort appears to be better? Powder Mad Valley Mountain Median 175 inches 225 inches Annual Snowfall (inches) Powder Valley Mad Mountain 50 100 150 200 250 300 350 400 (1 min) 14 – 15 In-Class Notes Have a student read the question aloud. Before having students answer the question, ask them, For a ski resort, is it desirable to have a lot of snow? Incorporate student movement by asking students to physically show their answer. For example, those who think Powder Valley is better based on the median could put two hands on their heads. Students who think Mad Mountain is better based on the median could put two hands on their shoulders. If there is disagreement on which is better, have one person supporting Powder Valley and one person supporting Mad Mountain explain their reasoning. Once an agreement has been made, click to show answer. Preparation Notes This slide is connected to Math PS 3 – Construct Viable Arguments and Critique the Reasoning of Others: Understand and use stated assumptions, definitions, and previously established results in constructing arguments. Agenda

Explore Whole Class Using the box plot, you can identify the record high (maximum) and record low (minimum) annual snowfalls for each resort. Powder Mad Valley Mountain Median 175 inches 225 inches Record Low 75 inches 0 inches Record High 325 inches 400 inches Annual Snowfall (inches) Powder Valley Mad Mountain 50 100 150 200 250 300 350 400 (1 min) 15 – 16 In-Class Notes Ask a student to come up to the board to point out the record low/minimum for both sets of data. Ask another student to come up to the board to point out the record high/maximum for both sets of data. Follow-up questions: What does a minimum snowfall of 75 inches mean for Powder Valley? What does a maximum snowfall of 400 inches mean for Mad Mountain? Click to show record low/record high for each resort. Before moving on, ensure that students understand that the more snow a ski resort has, the more attractive it is to potential skiers. Preparation Notes: This slide is connected to Math PS 2 - Reason Abstractly and Quantitatively: Attend to the meaning of quantities, not just how to compute them. Agenda

Explore Whole Class Using the minimum values to compare the resorts, which resort appears to be better? Powder Mad Valley Mountain Median 175 inches 225 inches Record Low 75 inches 0 inches Record High 325 inches 400 inches Annual Snowfall (inches) Powder Valley Mad Mountain 50 100 150 200 250 300 350 400 (1 min) 16 – 17 In-Class Notes Have a student read the question aloud. Incorporate student movement by asking students to physically show their answer. For example, those who think Powder Valley is better based on the record low/minimum could put two hands on their heads. Students who think Mad Mountain is better based on the record low/minimum could put two hands on their shoulders. If there is disagreement on which is better, have one person supporting Powder Valley and one person supporting Mad Mountain explain their reasoning. Once an agreement has been made, click to show answer. Preparation Notes This slide is connected to Math PS 3 – Construct Viable Arguments and Critique the Reasoning of Others: Understand and use stated assumptions, definitions, and previously established results in constructing arguments. Agenda

Explore Whole Class Using the maximum values to compare the resorts, which resort appears to be better? Powder Mad Valley Mountain Median 175 inches 225 inches Record Low 75 inches 0 inches Record High 325 inches 400 inches Annual Snowfall (inches) Powder Valley Mad Mountain 50 100 150 200 250 300 350 400 (1 min) 17 – 18 In-Class Notes Have a student read the question aloud. Incorporate student movement by asking students to physically show their answer. For example, those who think Powder Valley is better based on the record high/maximum could put two hands on their heads. Students who think Mad Mountain is better based on the record high/maximum could put two hands on their shoulders. If there is disagreement on which is better, have one person supporting Powder Valley and one person supporting Mad Mountain explain their reasoning. Once an agreement has been made, click to show answer. Point out that Mad Mountain seems attractive, because it has the largest record high snowfall, but it also has the smallest record low. Preparation Notes This slide is connected to Math PS 3 – Construct Viable Arguments and Critique the Reasoning of Others: Understand and use stated assumptions, definitions, and previously established results in constructing arguments. Agenda

Explore Think-Pair-Share What does the distance between points in the box plot tell you about how spread out the data is? The greater the distance between points in the box plot, the more spread out the annual snowfall data is. Powder Valley Mad Mountain (2 min) 18 – 20 In-Class Notes Have a student read the question aloud. Students should think about this question independently, then discuss it with a peer, and lastly, share out in a whole group discussion. Encourage struggling students to compare the size of the colored arrows to help them formulate an answer. Once students have presented the idea that the greater the distance between points in the box plot, the more spread out the annual snowfall data is, click to show answer. Follow-up question: How does the distance between the points relate to the range of the data? Preparation Notes This slide is connected to Math PS 2 - Reason Abstractly and Quantitatively: Attend to the meaning of quantities, not just how to compute them. 0 50 100 150 200 250 300 350 400 Agenda Hint

Explore Think-Pair-Share What does the distance between points in the box plot tell you about how spread out the data is? The greater the distance between points in the box plot, the more spread out the annual snowfall data is. Powder Valley Mad Mountain (2 min) 18 – 20 In-Class Notes Have a student read the question aloud. Students should think about this question independently, then discuss it with a peer, and lastly, share out in a whole group discussion. Encourage struggling students to compare the size of the colored arrows to help them formulate an answer. Once students have presented the idea that the greater the distance between points in the box plot, the more spread out the annual snowfall data is, click to show answer. Follow-up question: How does the distance between the points relate to the range of the data? Preparation Notes This slide is connected to Math PS 2 - Reason Abstractly and Quantitatively: Attend to the meaning of quantities, not just how to compute them. 0 50 100 150 200 250 300 350 400 Agenda Hint

Explore Whole Class Which mountain varies less in terms of the amount of snowfall from year to year? Powder Mad Valley Mountain Median 175 inches 225 inches Record Low 75 inches 0 inches Record High 325 inches 400 inches Variation small large Annual Snowfall (inches) Powder Valley Mad Mountain 50 100 150 200 250 300 350 400 (1 min) 20 – 21 In-Class Notes Have a student read the question aloud. Before having the students answer the question, do a quick check for understanding regarding the wording of the question. Ask a student to rephrase the question in sixth grade language. Ensure that a student has appropriately rephrased the question before moving on. Incorporate student movement by asking students to physically show their answer. For example, those who think Powder Valley varies less could put two hands on their heads. Students who think Mad Mountain varies less could put two hands on their shoulders. If there is disagreement on which varies less, have one person supporting Powder Valley and one person supporting Mad Mountain explain their reasoning. Once an agreement has been made, click to show answer. Follow-up questions: In the case of ski resorts, is it better to have a little or a lot of variation in the amount of snowfall? In general, is variability within data sets good or bad? (Have students come up with examples to support their answers) Preparation Notes: This slide is connected to Math PS 2 - Reason Abstractly and Quantitatively: Attend to the meaning of quantities, not just how to compute them. This slide is connected to Math PS 3 – Construct Viable Arguments and Critique the Reasoning of Others: Understand and use stated assumptions, definitions, and previously established results in constructing arguments. Agenda

Explore Whole Class Which resort has a greater chance of receiving more than 300 inches of snow? Powder Mad Valley Mountain Median 175 inches 225 inches Record Low 75 inches 0 inches Record High 325 inches 400 inches Variation small large Chance of >300 in. lesser greater Annual Snowfall (inches) Powder Valley Mad Mountain 50 100 150 200 250 300 350 400 (2 min) 21 – 23 In-Class Notes Have a student read the question aloud. Ask a student to come up to the board to point out where on the graph a chance of more than 300 inches of snow is shown for both mountains. Then incorporate student movement by asking students to physically show their answer. For example, those who think Powder Valley has a greater chance could put two hands on their heads. Students who think Mad Mountain has a greater chance could put two hands on their shoulders. If there is disagreement on which has a greater chance, have one person supporting Powder Valley and one person supporting Mad Mountain explain their reasoning. Once an agreement has been made, click to show answer. Students should see that Mad Mountain has a greater chance of receiving more than 300 inches of snow since every value in its fourth quartile is greater than 300 inches. Preparation Notes This slide is connected to Math PS 3 – Construct Viable Arguments and Critique the Reasoning of Others: Understand and use stated assumptions, definitions, and previously established results in constructing arguments. Agenda

Explore Small Group Which resort would you recommend? Valley Mountain Powder Mad Valley Mountain Median 175 inches 225 inches Record Low 75 inches 0 inches Record High 325 inches 400 inches Variation small large Chance of >300 in. lesser greater (1 min) 23 – 24 In-Class Notes Students should discuss this question in small groups (or with a partner depending on the classroom set-up). Students should make a decision based on all five variables. After discussing this question in small groups, groups should share out their ideas with the class. Powder Valley may seem like a better choice, because the amount of snowfall does not vary as much from year to year. Therefore, it is more consistent in the amount of snow it has. However, students could also argue for Mad Mountain since its median snowfall is higher and it has a greater chance of having more than 300 inches of snow. Preparation Notes This slide is connected to Math PS 4 - Model With Mathematics: Apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. Agenda

Explore Think-Pair-Share How does the data gathered below relate to the pieces of a five number summary? Powder Mad Valley Mountain Median 175 inches 225 inches Record Low 75 inches 0 inches Record High 325 inches 400 inches Variation small large Chance of >300 in. lesser greater Annual Snowfall (inches) Powder Valley Mad Mountain 50 100 150 200 250 300 350 400 (1 min) 24 – 25 In-Class Notes Have a student read the question aloud. Students should think about this question independently, then discuss it with a peer, and lastly, share out in a whole group discussion. Students should see that the median, record low, and record high are all parts of the five number summary. Follow-up questions: What pieces of the five number summary are we missing for each resort? (Q1 and Q3) How could we use the box plot to find Q1 and Q3? Preparation Notes This slide is connected to Math PS 3 – Construct Viable Arguments and Critique the Reasoning of Others: Understand and use stated assumptions, definitions, and previously established results in constructing arguments. Agenda

Explore Whole Class Original Question: If your job is to recommend a ski resort by comparing the annual snowfall between two mountains for the past 50 years, how would you compare all the data? How did we use the box plot to answer this question? Annual Snowfall (inches) Powder Valley Mad Mountain 50 100 150 200 250 300 350 400 (2 min) 25 – 27 In-Class Notes Students should recognize that by using a box plot, it was easy to compare the two sets of data. By looking at the box plot, one could compare the median, minimum, maximum, Q1, Q3, variability, and even the occurrence of certain events. These are all measures that cannot be gathered by just looking at data that is written in a list format. Preparation Notes Agenda

Practice – Sharing Question #1 A farmer starts 9 tomato plants in a greenhouse several weeks before spring. The seedlings look a little small this year so the farmer decides to compare this year’s growth with last year’s growth. This year’s growth is measured in inches as: 12 8.4 10 9.8 14 7.9 11 12.7 13.7 Last year’s growth was measured in inches as: 11.7 9 17 10.5 13.4 15.2 16.8 11.5 15 Should the farmer be concerned about the tomato plants this year? Why or why not? Support your answer by creating a box plot. (1 min) 42 – 43 In-Class Notes Ask a student to read the question aloud. Teacher’s choice: Either ask students to share out and explain how they arrived at their answer or show the solution by going through the next few slides. Accommodation: Modify numbers so that they are whole numbers if students have limited experience working with decimals. Preparation Notes Agenda

Practice – Sharing Question #1 This year’s growth is measured in inches as: 12 8.4 10 9.8 14 7.9 11 12.7 13.7 7.9 8.4 9.8 10 11 12 12.7 13.7 14 9.1 13.2 Median = 11 inches Minimum = 7.9 inches Lower Quartile (Q1) = 9.1 inches Maximum = 14 inches Upper Quartile (Q3) = 13.2 inches (1 min) 43 – 44 In-Class Notes Remind students to follow along on their handout, which they use to take notes if they do not already have similar answers recorded from their Practice time. This slide reviews #1 from the Class Work handout. Preparation Notes Agenda

Practice – Sharing Question #1 Last year’s growth was measured in inches as: 11.7 9 17 10.5 13.4 15.2 16.8 11.5 15 9 10.5 11.5 11.7 13.4 15 15.2 16.8 17 11 16 Median = 13.4 inches Minimum = 9 inches Lower Quartile (Q1) = 11 inches Maximum = 17 inches Upper Quartile (Q3) = 16 inches (1 min) 44 – 45 In-Class Notes Remind students to follow along on their handout, which they use to take notes if they do not already have similar answers recorded from their Practice time. This slide reviews #1 from the Class Work handout. Preparation Notes Agenda

Practice – Sharing Question #1 Last Year’s Growth This Year’s Growth (1 min) 45 – 46 In-Class Notes Remind students to follow along on their handout, which they use to take notes if they do not already have similar answers recorded from their Practice time. This slide reviews #1 from the Class Work handout. Give students about 30 seconds to compare their box plot to what it should look like. Poll students to see what they chose to use as the range and scale of their number lines. Ensure that students understand that this is just one range and one scale that could be used. There are many options. Ask students what this box plot is missing (title and a label on the x-axis). Preparation Notes The purpose of this slide is to give students an opportunity to check their work. Agenda

Practice – Sharing Question #1 This Year Last Year Minimum 7.9 9 Q1 9.1 11 Median 11 13.4 Q3 13.2 16 Maximum 14 17 Last Year’s Growth This Year’s Growth (2 min) 46 – 48 In-Class Notes Remind students to follow along on their handout, which they use to take notes if they do not already have similar answers recorded from their Practice time. This slide reviews #1 from the Class Work handout. Preparation Notes The farmer should be concerned. The box plots show that this year’s seedlings are smaller than last year’s seedlings. All of the five-number summary values are less. Agenda

Practice – Sharing Question #2a Your job: Make a peanut butter recommendation for grocery shoppers. Suppose price is the only factor a buyer considers. Is natural peanut butter or regular peanut butter a better choice? Explain. Grocery shoppers should purchase regular peanut butter if price is the only factor, as all of the five-number summary values are less. (1 min) 48 – 49 In-Class Notes Remind students to follow along on their handout, which they use to take notes if they do not already have similar answers recorded from their Practice time. This slide reviews #2a from the Class Work handout. Call on a student to share his/her ideas using the popsicle stick jar. This holds students accountable, particularly since every student has an opportunity to preview the question before being called on. Follow-up questions: Which peanut butter brand is deemed to be cheaper? How do you know? What was the median price for the natural brand? What was the maximum price for the regular brand? What was the difference in the lowest/minimum price between the natural and regular brand? Based on the box plot, which brand would you buy? Have students complete the following prompts: ______% of the natural brands were priced between 22 and 26 cents per serving. ______% of the regular brands were priced below 22 cents per serving. The range of the _________brand prices is 12 cents per serving. The middle 50% of the _________brands are clustered more closely than the middle 50% of the _________ brands. Preparation Notes Have popsicle sticks with students’ names on them in a jar prepared. Agenda

Practice – Sharing Question #2b Your job: Make a peanut butter recommendation for grocery shoppers. Suppose quality is the only factor a buyer considers. Is natural peanut butter or regular peanut butter a better choice? Explain. Grocery shoppers should purchase natural peanut butter if quality is the only factor, as all of the five-number summary values are greater. (1 min) 49 – 50 In-Class Notes Remind students to follow along on their handout, which they use to take notes if they do not already have similar answers recorded from their Practice time. This slide reviews #2b from the Class Work handout. Call on a student to share his/her ideas using the popsicle stick jar. This holds students accountable, particularly since every student has an opportunity to preview the idea before being called on. Follow-up questions: Which peanut butter brand is deemed to be of better quality? How do you know? What was the median rating for the natural brand? What was the maximum rating for the regular brand? What was the difference in minimum ratings between the natural and regular brand? Based on the box plot, which brand would you buy? Have students complete the following prompts: ______% of the natural brands were rated between 58 and 70. ______% of the regular brands were rated below 54. The range of the _________brand ratings is 72. The middle 50% of the _________brands are clustered more closely than the middle 50% of the _________ brands. Preparation Notes Have popsicle sticks with students’ names on them in a jar prepared. Agenda

Summary Think-Pair-Share 1962 1963 1964 1965 1966 1967 … Powder Valley Mad Mountain 217 in. 132 in. 310 in. 104 in. 186 in. 287 in. … 107 in. 233 in. 207 in. 106 in. 229 in. 37 in. … Methods we could have used to compare these two sets of data during our warm-up today: Mean, Median, Minimum, Maximum, Lower Quartile (Q1) and Upper Quartile (Q3) (2 min) 50 – 52 In-Class Notes Before posing the question, quickly review the warm-up question and the statistical measures that were used to answer it. Click to show summary question. Students should spend about 15 seconds thinking about their answer independently. Then they should spend about 30 seconds talking with a peer. Have students share out ideas. The two main ideas here are: 1) Box plots allow the reader to easily see the spread of a set of data and 2) Box plots can be used to more easily compare two sets of data, as the similarities and differences are much more evident in comparison to sets of data written as lists. Preparation Notes If we already have all of these strategies for comparing two sets of data, why did we learn about using box plots to compare sets of data today? Agenda

The two main ideas here are: 1) Box plots allow the reader to easily see the spread of a set of data and 2) Box plots can be used to more easily compare two sets of data, as the similarities and differences are much more evident in comparison to sets of data written as lists.

Assessment – Exit Ticket! Complete and hand in the Exit Ticket before you leave! (8 min) 52 – 60 In-Class Notes Distribute Exit Tickets. Remind students to clearly show all of their thinking and work. All work should be independent! Preparation Notes The rationale of using this Exit Ticket is to (1) collect individual student data assessing their understanding of the concept of box plots and ability to create box plots that compare two sets of data and (2) to assess student understanding of the median and its relationship to box plots. Agenda