Measures of Variation - Range and IQR
Lesson 6 of 11
Objective: SWBAT calculate range and interquartile range and use them to describe the distribution of data.
Students enter silently according to the Daily Entrance Routine. Over half of the 7th grade scored “far below basic” on the most recent unit test, “Percent applications”. This means that over half of the grade earned a score below 60%. Results like these are deeply devastating, but it is important to analyze data to identify the main problem. My students seem to be struggling with two main issues: wording of the problems in the new “common core” aligned assessments created internally by our organization/partners AND ratio and proportion skills rooted in percent applications. I chose to focus more on remediating basic skills and understandings around proportion and rate, rather than breaking down words given in ONE test. The way I see it, the language of the standards in the new curriculum leaves room for lots of interpretation of how to assess understanding. The skills themselves are delineated more specifically. As organizations and publishers begin to work out how to best assess, the wording and complexity of these word problems will surely continue to change and be debated. Again, the mathematical skills themselves are clear AND important.
The most commonly misunderstood standard measured in this last unit test is 7.RP.3, 55% are scoring below basic. Questions like #1 in the Do Now aim to address the basic understanding of proportional relationships in a word problem as well as including a multi-step (multi-information) component. The wording however, is not as confusing as I have noted in past Unit tests. In past years, before common core, my students were confronted with direct questions, like “Jake spent $20 at the movies. The tickets cost $12. What percent of $20 is $12?” This type of question may have been the “higher order thinking” question in the old curriculum due to the way in which the question itself was framed, but as publishers continue to explore “multi-step” problems as prescribed by the new curriculum, they add more information and more indirectly suggested steps, as shown in #1. When I search through my resources for example questions to give my students, I am looking for this multi-step facet, with more of an emphasis on skills. I am NOT looking for wordy problems. Lastly, formatting can have a deep impact in understanding. As students move away from the old standards and cope with new word problems which emphasize problem solving and multiple steps, it may be helpful to consider formatting when creating assessments. For example, this is a copy of a commonly missed question on the most recent Unit 6 test:
Notice how the formatting could easily lend itself to mistakes in reading and corresponding facts about each coat. As we guide our students through these types of word problems, it would be useful to present the information in a way that avoids confusion and leads students to a more direct way of organizing this information. Kids mimic what they see. If we present the information in a more direct format (as seen below), students will begin to form their own organizational schema and apply it when faced with a complex situation. Notice the difference between the problem above and the one below is formatting – the words are mostly the same.
Question #2 serves to review creating and understanding stem and leaf plots. Some students showed through checked homework (Day 123) that they are still struggling to understand how to identify the stem, the leaves, and/or how to read a diagram already created. These two concepts are important for review, so I make sure to leave extra wiggle room in this lesson to check for understanding in this section and answer any questions that may linger.
Problems are reviewed by checking students’ work on the blackboard. I ask students to raise their hand if they agree (and then disagree) with work they see on the board. Students are selected before class begins and their names are displayed on the powerpoint so that they know they must put their work up on the board as soon as they enter the classroom. For the second question, student pairs have been predetermined to complete the work quickly and efficiently.
After reviewing the Do Now, students are asked to take out their Homework so that we can review the answers. The first two slides display the list of data in order from least to greatest to visually show students the locations of the quartiles. More discussion is needed for finding these quartiles and we cycle back to the general notion of odd number and even number of values.
When reviewing the problem about the heights of players on two different teams, we discuss test taking strategies. In this problem, we discuss the strategy of reading each answer choice and finding the measure of central tendency we see in most choices. Since they all reference the median, this is the clear measure we need to calculate to identify the answer.
The last question reviews percent change, a skill that also needs review according to the data from the last unit test. Again, the standard that seems to bind all of these skills together is 7.RP.3. Ask the following essential questions to review this standard and engage students in MP4, the ability to model the problem or question in the form of a fraction or percent.
- What is a percent?
A part of a whole
- What are some ways we can represent percents?
Using the symbol %, as a fraction, as a decimal
- What is the “whole” in this problem? What does it represent?
The total; the initial value; give students different examples (i.e I have 8 bills in my hand. 2 of them are one dollar bills. The whole is the total number of bills in my hand)
- What is the “part” they are asking to find? What does it represent?
A given amount; a change or difference (i.e I have 8 bills in my hand. 2 of them are one dollar bills. The part is the number of bills in my hand that are one dollar bills)
- What is a ratio?
A comparison of two quantities; give examples and ask about “part” and” whole”
The ratio of boys to girls in a class is 2 to 3. The whole is 5, or all of the students in the class; the part is either the number of boys or girls, depending on the question
- What is a proportion?
A comparison of two equivalent ratios
- When should we use a proportion?
To find unit rate, to find equivalent ratios, to convert to percent
Class Notes + Practice
We spend more time practicing how to identify quartiles in lists of numbers during the Class Notes . After copying the definition of interquartile range I ask students to discuss how interquartile range can be useful when analyzing and comparing data. I ask them to review examples we covered in past days. They are given 3 minutes to discuss and then each group of 4 will share out one unified thought. I ask one group to look at the data displayed in the 8th slide of the power point and discuss how the IQR gives us more accurate data (omitting the outliers). They will be asked to share out their thoughts as well.
I then give students three different lists of data (PPT slide 9) and ask them to find the quartiles and the IQR in the worksheet attached (a copy of the ppt slide with room for work). The fourth example is given as a stem and leaf plot. I give no direction for showing the work for this final problem. Instead, as I walk around while students are working, I note the different methods being used so that these students may be asked to display their work on the board. I see students re-listing the numbers and some figure out how to use the structure of the stem and leaf plot to find the quartiles.
Closing: Got it?
After reviewing the answers and answering questions, students are given a “got it?” sheet to complete independently. After checking the answers, I will be able to give students feedback on some of the details they must continue to study. It will also give me a good picture of the details I need to keep spiraling in do nows and homework. I plan to use the data to work with a small group during the following lesson. Creating stem and leaf plots as well as box and whisker plots can be very confusing for some students while others may be able to complete the task independently. This is information I need to gather before the next lesson as well as the next assessment. Working with small groups to show them how to create these intricate diagrams will pay off for that small group of students in the end.