Students with visual impairments may face challenges when working on the Mathematics standards in the Common Core State Standards (CCSS). As a response to this, Perkins School for the Blind convened a panel of experts to identify specific standards that would be a potential challenge to students who are blind or visually impaired, and then proposed ideas for materials, foundational skills, tips and strategies, and lesson ideas to help to address these challenges.

This post is part of a series about different parts of the Mathematical standards.

### Topics:

- What is a student likely to be working on in this area
- What are the particular challenges for a student who has a visual impairment?
- Foundational skills
- Materials
- Tips and strategies
- Lesson ideas

### What is a student likely to be working on in this area:

- Students generally begin studying Statistics & Probability in the 6th grade and continue through High School.
- Probability is a study of chance and its application in daily life.
- Statistics involves the collection of data and the various methods of analysis.

### What are the particular challenges for a student who has a visual impairment?

- Statistical concepts rely heavily on the presentation of data utilizing visual and spatial skills. The student’s study of Statistics and Probability will rely greatly on the tactile representation of graphs and materials.

### Foundational Skills:

These skills are listed in a general order of skills required by the student as they advance through the grades.

- Understand the rules of addition and multiplication and apply them to statistical models
- Understand the purpose of statistics, information generalization, and random samples
- The ability to use data from random samples to draw inferences
- The ability to informally assess the degree of visual overlap
- Build on prior experience with dot plots to compare/contrast data displayed on two dot plots
- Make inferences from data
- Use measures of center and variability for numerical data from random samples or draw inferences about two populations
- Understanding of how to graph points in a coordinate system
- Understand that a probability close to 1 means something is very likely
- Read a segmented bar graph
- Determine the probability of a chance event
- Understand how to collect data and put it into a table format
- Understand how to find slope and intercept
- Understand surveys, experiments, and observational studies
- Understand what kind of data is derived from each type of study
- Understand how samples are chosen for each kind of study
- Be able to compute mean, median, and set up proportions
- Be able to perform needed calculations on an accessible calculator
- Describe, identify, and apply deviation, standard deviation, absolute deviation, measures of central tendency, measures of variability
- Be able to use probability to predict relative frequency
- Develop a probability model and use it to find probabilities
- Using Nemeth Code understand a table of values as a way to represent multiple points
- Ability to use models and simulate a variety of events
- Understand the probabilities of compound events
- Have the ability to compare simple events with compound events
- Know how to recognize gaps, clusters, and trends in the data set
- Be able to recognize extreme data points (outliers) and their impact on center
- Be able to effectively communicate what data reveals
- Understand that when comparing distributions there must be common scales and units
- Understand the characteristics of categorical data
- Be familiar with reading and the use of a two-way frequency tables.
- Use and compute joint, marginal, and conditional relative frequencies
- Recognize linear relationships displayed in a scatter plot
- Be able to determine an equation for the line of best fit for a set of data points
- Be able to create a graphic display of residuals
- Be able to recognize patterns in residual plots
- Calculate error margins (residuals) with a calculator
- Represent data on two quantitative variables on a scatter plot and describe how the variables are related
- Recognize the types of relationships that lend themselves to linear and exponential models
- Be able to create and use regression models to represent a contextual situation
- Understand what inference means and how statistics use samples to make inferences
- Understand what margin of error means and how to test for it
- Be able to compute findings for statistical experiments and compare the results
- Understand why one result shows significance and another does not
- Be able to evaluate reports based on data and use data to talk about study results
- Understand independence and probability as it relates to statistics
- Understand and interpret conditional probability
- Be able to describe a sample space and use set notation, key vocabulary, and graphic organizers
- Understand what two-way frequency tables are and be able to construct one using tactile media or a computer
- Be able to discuss what conditional probability and independence and provide examples

### Materials

- Braille notetakers
- Large print paper and pencils
- Computer with JAWS or VoiceOver
- Braille graph paper
- Braille dots
- Braille and Large Print rulers
- Tactile graphics, textured objects, and bags of tactile objects for experiments
- Braille spinner
- Talking calculator
- Recorder
- Orion TI-84 Plus Talking Graphing calculator or iPad graphing calculator
- Tracing paper
- Spreadsheet software
- Braille charts
- Pegboard
- APH Draftsman
- APH Graphing Aid for Mathematics

### Tips and Strategies

- When using textured objects, students can conduct experiments with the objects chosen out of a paper bag.
- When using braille spinners, students can determine probabilities.
- When using tactile graphics, students can create tree diagrams and conduct simulations.
- When using tracing paper, students can trace shapes of distribution with glue.
- When using a notetaker, students can record and compare the mean.
- When using a notetaker, students can record and compare data points and spreads.
- When using braille charts, students can input data for identifying frequencies.
- When using braille graphics with points plotted, if there is too much complexity in the amount of points it is advisable for students to create tactile scatter plots. Tactile grid paper along with multiple thumbtacks can be used to establish the concept.
- When using the APH Graphic Aid for Mathematics, students can manually plot points with a ruler stood on its edge. Place approximately half the points on one side and half on the other to form a line of best fit.
- When using pegboards, start with concrete rate of change problems, such as walking speeds and then graph with a pegboard.
- When using braille graph paper, student can graph rate of change problems on braille graph paper and estimate correlation coefficient based on slope.
- Students can define slope using slides and blocks.
- When working on correlations and causation problems, students will benefit from group activities and discussions using student relevant topics.
- When selecting tactile objects, students could use coins or bags of tactile objects.
- When using notetakers, students can record results of statistical problems.
- When using computers, students can define and record information, as well as use the Internet to look up definitions and provide examples.
- When using computers, students can define problems.
- Students can work in cooperative group activities, where each group chooses a location and estimates the mean age of the population in that location. In this group work, each group finds their mean, margin of error, and creates simulation problem to perform calculations.
- When using bags of tactile objects, students can conduct one trial versus multiple trials with multiple bags.
- Students can use spinners to conduct multiple trials
- When using tracing paper, students can use glue for drawing unions and intersections. Students can draw all possibilities with glue.

### Lesson Ideas

- Activities to gather data from students about preferences to discuss samples and statistics in general.
- Internet activity: Go to five websites and search for a pre-selected word on a page. Make inferences about the probability that every website has that word.
- Start with classroom measurements. Determine the mean of student heights. Estimate heights of same age students. Research accuracy on Internet.
- Internet activity: Each group chooses two different music genres. Use the Internet to determine which artists sell the most albums in each genre.
- Group activities with two different textured items in paper bags.
- Write probability as a fraction.
- Group activities with different number of wedges on spinners and impact on probability of specific frequency of outcomes.
- Use a spinner activity to predict probabilities of other activities.
- Given a linear set of points where the x-coordinates increase by one, identify what the y-coordinates are increasing by as a rate of prices rising or falling, height of a person growing, etc.
- Construct a graph or table from given categorical data, and compare data categorized in the graph or table.
- Crate a tactile graphic of a bell curve shown as a normal distribution.
- Have a student create a simple bar graph using a real life example.
- Find the mean of a set of numbers using a calculator.
- Show the student tactile graphics of various real-life situations and illustrate how they are similar to parabola or a straight line. For example, show a rocket taking off, curving, and falling back to earth, or a student diving into a swimming pool, or Niagara Falls.
- Show the student a tactile graphic of the diver ending up doing a belly flop instead of a smooth parabolic dive.
- Give the student a tactile graphic scatter plot drawn on APH Draftsman film and placed in the APH Draftsman. Give them a ruler and have them draw a straight line through the points so that approximately half is on one side of the ruler and half is on the other.
- Start with concrete rate of change problems like walking speeds. Graph on a pegboard.
- Define slope using slides and blocks.
- Transfer slopes from slides to paper and discuss coefficients.
- Group activities and discussions using student relevant topics.
- Define random in relevant ways like weather, flavors of a box of chocolates, ages of people on a particular bus.
- Use computers for defining and recording information.
- Use the Internet to look up definitions and examples.
- Use computers to define problem.
- As a group activity, have groups choose a location and estimate the mean age of the population in the location. Each group finds their mean, margin of error, and creates simulation problems to perform calculations.
- Separate students into two groups. One uses cottonand the other uses ear plugs. Experiment with the effectiveness of treatments on noise blocking solutions. Afterward evaluate the experiment for validity and reliability then make recommendations.
- Record results of choosing objects from multiple bags of tactile objects versus multiple trials with a single bag of objects.
- Using bags of 2 or more tactually different objects record what the chances are of picking Object A our of 12 objects or if there are 6 of each in the bag. Next, if you remove an Object A what are the changes the next object selected will also be an Object A? What are the changes it will be an Object B?