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:

- The study of Geometry deals with points, lines, shapes, and space
- Plane Geometry is about flat shapes, lines, and circles
- Solid Geometry is about three dimensional objects such as spheres and cubes
- Students study Geometry beginning in Kindergarten (possibly pre-school) through High School where a student will most likely spend one school year studying the subject
- In the earliest grades students are learning to identify shapes, lines and their relationship in space

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

Geometry relies heavily on the visual representation of shapes and their relationship in space. Student will need tactile drawings and physical representations to support their studies.

### Foundational Skills:

- Understanding rotations, reflections and translations in order to experiment with them and explain the results
- Calculate expressions using order of operations
- Understand required Nemeth code including parenthesis, baseline indicator, congruent, shapes, shape indicators, segments, rays, lines, angles, degrees, parallel, prime for the transformed image, shape indicator, termination indicator, radical, superscript, multi-purpose indicator, Greek later indicator, Pi, fractions indicator
- Understand graphing, measuring angles, length, and direction
- Understand graphed points, how to read tactile graphics, and how dilations, translations, rotations, and reflections affect the coordinates
- Be able to conduct experiments which show that rotations, reflections, and translations of lines and line segments, angles, and parallel lines are rigid
- Understand graphed points, how to read tactile graphics, translations, rotations, and reflections
- Be able to use a sequence of transformations and map on figure to a second figure to show congruency
- Be able to describe a sequence of transformations needed to generate the image given its pre image
- Verbally describe the location on a coordinate grid of an image with respect to the pre-image
- Discuss the difference between rigid and non-rigid transformations
- Demonstrate that congruency is a special case of similarity
- Demonstrate that similar figures maintain shape but alter size through dilation (scale factor)
- Use and apply facts that result from parallel lines cut by a transversal
- Understanding of triangles and areas of a square
- Understand legs vs. hypotenuse on a right triangle and how to solve related problems
- Understand what cones, cylinders, and spheres are including where length, width, height and radii are on those figures
- Recall surface area formula
- Understand how to graph the points in a coordinate system
- Integrate manual and technological methods and relate the scenarios to authentic student centered situations
- Understand a table of values as a way to represent multiple points and then understand how to graph the points in a coordinate system
- Graph points of a scatter lot that seem to have a linear association and observe a trend line
- Know the difference between shapes and transformations
- Understand congruence
- Understand the concept of corresponding pairs, sides, and angles and their measurements
- Be able to complete proofs in both narrative forms and by drawing figures
- Know how to use a compass, ruler, and protractor
- Understand the physical characteristics of shapes
- Understand ratios and proportions
- Understand how to solve algebraic equations
- Understand the parts of a circle

### Materials

- Braille/Large Print Number Line
- APH Geometry Tactile Graphics Kit (Braille/Large Print)
- APH Graphic Aid for Mathematics
- Graph paper (tactile and bold lined)
- Talking Calculator
- Orion TI-84 Plus Talking graphic Calculator or iPad graphic calculator
- Braille Notetaker
- Perkins Brailler
- Electronic Notepad with Scientific Calculator
- Nemeth Code
- Tactile Graphics
- Tactile dots
- APH Draftsman
- Wheatley board
- Wikki Stix or graphic art tape
- Manipulative circles
- Braille and or Large Print Protractor

### Tips and Strategies

- When using a Braille protractor student can practice drawing shapes and angles
- When using tactile graphics make sure they contain the appropriate components
- Use 2D manipulatives with the appropriate components
- Using heavy card stock shapes can be made to represent triangles, cubes and other shapes
- When using 3D models of cones, cylinders, and spheres, include those that can be filled with a liquid such as water
- When using tactile graphics of scatter plots, use of the APH Graphic Aid for Mathematics to create scatter plots manually.
- Use of the Orion TI-84+ Talking Graphing Calculator to create scatter plots and latent to the individual points to gather information
- Place ruler on edge and try to fit it in between points such that about half of the points are about the ruler and half are below.
- When using APH Geometry Tactile Graphics Kit have the student rotate and reflect polygons
- When using the APH Mathematics Graphic Aid (with Black Rubber Board) students can draw shapes based on transformations. The same things can be completed using APH grid paper or the APH Draftsman
- When using tactile diagrams, with any new shape, students need access to tactile graphics of triangles in order to compare corresponding pairs of sides and angles. Additionally, there are some mainstream manipulatives that are triangles where the students can move the angles and how they impact other angles
- When using tactile diagrams, students can use tactile symbols to mark diagrams
- When using the Wheatley Board, students can use string, or paper folding with thicker paper rather than reflective devices or dynamic geometric software for Braille users
- When using magnification or contrasting color with the dynamic geometric software it will be easier to see for low vision students
- When solving problems students need to draw geometric figures. Students need to be able to use ruler (with caliper), compass, and protractor
- When using tactile graphics, students need access to ones that show similar shapes for introduction materials. Students could be given one shape and be asked to create a similar shape using something like the Wheatley. This would require them being able to construct shapes with appropriate angles and side lengths
- When solving proofs, students need access to materials that allow them to draw or construct the figures. They also need quality tactile graphics to explore concepts
- When using tactile grids, plot points on a tactile grid to compute area by counting the squares in the grid
- When using 3-dimensional manipulatives, have multiple shapes as well as nets of the polyhedrons to better understand the parts of the figures. Tools such as the Geometro would be beneficial

### Lesson Ideas

- Recognize translations, rotations, and reflections of shapes.
- Find objects in the environment that are translations, rotations, and reflections such as tiles, clock hands at different times, opposite pages in a spiral notebook
- Use shapes that can be picked up and placed on top of each other
- Using real world examples, turn shapes around on a graph and have students predict what will happen once it lands on a different part of the graph.
- Practice with 3D shapes that are the same and different
- Identify similar shapes with and without rotation.
- Use real shapes and objects in their environment that students can relate to
- Compare any angle to a right angle, and describe the angle as greater than, less than, or congruent to a right angle.'
- Use a protractor, corner of a piece of cardboard or index card (something that is a right angle) to do the comparisons. You can also relate it to how open a door is, etc.
- Form right triangles by connecting the origin, a point on the x-axis, and a point on the y-axis. (or a point with another point vertical and another point horizontal)
- Understanding of tracking vertically and horizontally
- Graph Paper (tactile or bold-line) and tactile dots and lines
- Graphic Aid for Mathematics
- Use the formulas for perimeter, area, and volume to solve real-world and mathematical problems (limited to perimeter and area of rectangles and volume of rectangular prisms).
- Understanding of what rectangles and rectangular prisms are, including where length, width, and height are on those figures.
- Interpret whether data seems to be rising or falling
- Use a rubber band, Wikki Stix, or other straight object to estimate a line through a linear system of points
- Describe similarities between opposite parts of items (reflection) or how an object can be turned and look the same (rotation)
- Students can learn the basic terms and understand that when things turn that they are rotating and when they "flip", then they are reflections. This would be a basic terminology-focused only
- Given a geometric figure and a rotation, reflection, or translation of that figure, identify the components of the two figures that are congruent
- Identify corresponding congruent and similar parts of shapes
- Functionally, the concept of similarity may be focused more on how things "look" instead of exact measurement. For many students, being able to say that two shapes are similar (even if they are not exactly similar) may be an appropriate skill. If we want them to be able to find a smaller size of detergent that has the same shape (but is smaller), this concept could be applied
- Students need to know that all circles are similar and possible the main labels. Coins are an excellent example of where students will need to able to differentiate sizes of circles.
- Find perimeters and areas of squares and rectangles to solve real-world problems.
- Make a prediction about the volume of a container, the area of a figure, and the perimeter of a figure, and then test the prediction using formulas or models.
- Identify the shapes of two-dimensional cross-sections of three dimensional objects.
- Use properties of geometric shapes to describe real-life objects.