### Clarifications

*Clarification 1*: Within the Probability and Statistics course, instruction includes the use of spreadsheets and technology.

**Subject Area:**Mathematics (B.E.S.T.)

**Grade:**912

**Strand:**Data Analysis and Probability

**Standard:**Summarize, represent and interpret categorical and numerical data with one and two variables.

**Date Adopted or Revised:**08/20

**Status:**State Board Approved

## Benchmark Instructional Guide

### Connecting Benchmarks/Horizontal Alignment

### Terms from the K-12 Glossary

- Categorical Data
- Numerical Data

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

In grade 7, students created and interpreted different displays of univariate numerical and categorical data. In grade 8, they created scatter plots and began to interpret them by consider lines of fit. In Algebra I, students interpret the components of data displays for numerical and categorical data, both univariate and bivariate. In later courses, they will use data displays to compare distributions of data sets to one another and to theoretical distributions.- It is the intention of this benchmark to include cases where students must calculate measures of center/variation to interpret
*(MTR.3.1)*. - For students to have full understanding of numerical/categorical, univariate/bivariate data sets and their displays, instruction should include MA.912.DP.1.1. These benchmarks are not intended to be separated. One is reinforced by the other.
- Numerical univariate includes histograms, stem-and-leaf plots, box plots and line plots.
- Numerical bivariate includes scatter plots and line graphs.
- Categorical univariate includes bar charts, line plots, circle graphs, frequency tables and relative frequency tables.
- Categorical bivariate includes segmented bar charts, joint frequency tables and joint relative frequency tables.

- Instruction includes identifying the measures of center and spread from different scenarios.
- Instruction includes explaining that an outlier is extremely smaller or larger than the rest of the data set.
- Teacher provides opportunities to analyze the effects on the measures of center and spread when the outlier is the minimum and maximum.
- This benchmark reinforces the importance of the use of questioning within instruction.
- Does this display univariate or bivariate data?
- Is the data numerical or categorical?
- What do the different quantities within the data display mean in terms of the context of the situational data?

### Common Misconceptions or Errors

- Students may not be able to properly distinguish between numerical and categorical data or between univariate and bivariate data.
- Students may misidentify or misinterpret the quantities in data displays.
- Students may not be able to distinguish between the measures of center (mean, median) and measures of spread (range, IQR).
- Students may not completely grasp the effect of outliers on the data set; or incorrectly conclude a point is an outlier.
- Students may not be able to distinguish the differences between frequencies and relative frequencies.
- Students misidentify the condition that determines a conditional or relative frequency in a joint table.

### Strategies to Support Tiered Instruction

- Instruction includes a graphic organizer to complete collaboratively.
- For example, teacher can provide the graphic below and have students match the vocabulary terminology with the correct definition. Then, have students create an example that can help with remembering the vocabulary terminology.

- For example, teacher can provide the graphic below and have students match the vocabulary terminology with the correct definition. Then, have students create an example that can help with remembering the vocabulary terminology.

- Teacher provides students with definitions and co-creates examples for frequency and relative frequency.
- For example, have students draw a definition chart in their interactive notebook. Give them the opportunity to create an example that will help them remember the definition.

### Instructional Tasks

*Instructional Task 1 ( MTR.3.1, MTR.4.1)*

- The histogram below shows the efficiency level (in miles per gallons) of 110 cars.
- Part A. Does this display univariate or bivariate data?
- Part B. Is the data numerical or categorical?
- Part C. What do the different quantities within the data display mean in terms of the context of the situational data?
- Part D.
- How many cars have an efficiency between 15 and 20 miles per gallon?
- How many cars have an efficiency more than 20 miles per gallon?
- What percentage of cars have an efficiency less than 20 miles per gallon?

*Instructional Task 2 ( MTR.3.1, MTR.4.1) *

- A police department tracked the number of ticket writers and number of tickets issued for each of the past 8 weeks. The scatter plot shows the results.

- Part A. Does this display univariate or bivariate data? Is the data numerical or categorical?
- Part B. What do the different quantities within the data display mean in terms of the context of the situational data?
- Part C. Which statement is an appropriate interpretation of the data?
- a. More ticket writers result in fewer tickets being issued.
- b. There were 50 tickets issued every week.
- c. When there are 10 ticket writers, there will be 600 tickets issued.
- d. When there are more ticket writers, more tickets are being issued.

### Instructional Items

*Instructional Item 1 *

- The scatter plot shows the amount of sleep needed per day by age.

- Part A. Does this display univariate or bivariate data? Is the data numerical or categorical?
- Part B. What is a possible trend that is shown by the data?

**The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.*

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## Student Resources

## Problem-Solving Tasks

The purpose of this task is to allow students to demonstrate an ability to construct boxplots and to use boxplots as the basis for comparing distributions.

Type: Problem-Solving Task

This problem could be used as an introductory lesson to introduce group comparisons and to engage students in a question they may find amusing and interesting.

Type: Problem-Solving Task

This problem solving task asks students to examine the relationship between shops and crimes by using a correlation coefficient. The implications of linking correlation with causation are discussed.

Type: Problem-Solving Task

The task provides a context to calculate discrete probabilities and represent them on a bar graph.

Type: Problem-Solving Task

This task can be used as a quick assessment to see if students can make sense of a graph in the context of a real world situation. Students also have to pay attention to the scale on the vertical axis to find the correct match. The first and third graphs look very similar at first glance, but the function values are very different since the scales on the vertical axes are very different. The task could also be used to generate a group discussion on interpreting functions given by graphs.

Type: Problem-Solving Task

This task asks the student to gather data on whether classmates play an instrument and/or participate in a sport, summarize the data in a table and decide whether there is an association between playing a sport and playing an instrument. Finally, the student is asked to create a graph to display any association between the variables.

Type: Problem-Solving Task

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Type: Problem-Solving Task

## Parent Resources

## Problem-Solving Tasks

The purpose of this task is to allow students to demonstrate an ability to construct boxplots and to use boxplots as the basis for comparing distributions.

Type: Problem-Solving Task

This problem could be used as an introductory lesson to introduce group comparisons and to engage students in a question they may find amusing and interesting.

Type: Problem-Solving Task

This problem solving task asks students to examine the relationship between shops and crimes by using a correlation coefficient. The implications of linking correlation with causation are discussed.

Type: Problem-Solving Task

The task provides a context to calculate discrete probabilities and represent them on a bar graph.

Type: Problem-Solving Task

This task can be used as a quick assessment to see if students can make sense of a graph in the context of a real world situation. Students also have to pay attention to the scale on the vertical axis to find the correct match. The first and third graphs look very similar at first glance, but the function values are very different since the scales on the vertical axes are very different. The task could also be used to generate a group discussion on interpreting functions given by graphs.

Type: Problem-Solving Task

This task asks the student to gather data on whether classmates play an instrument and/or participate in a sport, summarize the data in a table and decide whether there is an association between playing a sport and playing an instrument. Finally, the student is asked to create a graph to display any association between the variables.

Type: Problem-Solving Task

Students are asked to examine data given in table format and then calculate either row percentages or column percentages and state a conclusion about the meaning of the data. Either calculation is appropriate for the solution since there is no clear relationship between the variables. Whether the student sees a strong association or not is less important than whether his or her answer uses the data appropriately and demonstrates understanding that an association means the distribution of favorite subject is different for 7th graders and 8th graders.

Type: Problem-Solving Task