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:

- Expressions and Equations are generally studied from 5th or 6th grade through High School
- An equation is a mathematical sentence that says two things are equal such as 2x + 3 = 15
- An expression is a phrase that stands for a single number such as 2x + 3

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

- Development of number sense is essential to helping solve problems
- It is important to help the student develop an organized system to solving the problem. Often teachers look for the method student used to solve the problem.
- Learning how to detail, step by step, how the problem is solved helps the student document the solution and can be of aid if there is a mistake.

### Foundational Skills:

- Apply properties of operations to add, subtract, factor, and expand linear expressions with rational coefficients
- Understand that rewriting an expression in different forms clarifies problem
- Solve real-life multi-step problems using all tools learned
- Graph a solution set of an inequality
- Read a word problem and form an equation from the information presented
- Nemeth: superscript indicator, baseline indicator, scientific notation, parentheses, fraction indicators, triangle, segment (directly over, horizontal line, termination indicator)
- Understand basic operations on integers, an exponent as repeated multiplication and simplifying like factors in a fraction
- Understand powers of 10 and how multiplication by 10s or division by 10s affects a number
- Understand how to graph points and reading coordinate graphs
- Relate and compare graphic, symbolic, numerical representations of proportional relationships
- Calculate constant rate of change/slope of a line
- Understand that similar right triangles can be used to establish that slope is constant for a non-vertical line â€¢ graphically derive equations y = mx and y = mx + b
- Differentiate between zero slope and undefined slope
- Understand how the y-intercept translates a line along the y-axis (families of graphs)
- Understand how to graph points and reading coordinate graphs
- Solve linear equations
- Understand the properties of operations to generate equivalent expressions, identifying when 2 expressions are equivalent
- Proficient braille reading and writing skills
- Interpret expressions that represent a quantity in terms of context
- Choose and produce an equivalent form of an expression to reveal
- Explain properties of quantity represented by expression se the structure of an expression to identify ways to rewrite it
- Derive the formula for the sum of a finite geometric series
- Understand that polynomials for a system analogous to integers closed under operations of addition, subtraction, multiplication and division
- Apply Remainder Theorem
- Identify zeros of polynomials through factoring and graph
- Prove polynomial identities and use them to describe numerical relationships
- Apply Binomial Theorem
- Rewrite simple rational expressions
- Understand that rational expressions form a system analogous to the rational numbers
- Create equations and inequalities in one variable and use them to solve problems
- Create equations in two or more variables to represent relationships
- Graph equations
- Represent constraints by equations or inequalities and by systems of equations or inequalities
- Rearrange formulas to highlight a quantity of interest
- Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step
- Solve simple rational and radical equations in one variable
- Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters
- Solve quadratic equations in one variable by completing the square and quadratic formula
- Demonstrate strong foundational knowledge in algebra
- Solve quadratic equations in one variable by completing the square and quadratic formula
- Demonstrate strong foundational knowledge in algebra
- Prove that replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions
- Understand how to construct spreadsheets with formulas to complete some column entries in terms of given column entries
- Solve quadratic equations and linear equations and use substitution
- Use matrices
- Solve linear equations using matrices
- Use assistive technology for solving matrices
- Graph equations with two variables

### Materials

- Talking calculator
- Perkins brailler
- Notetaker
- Computer with JAWS or VoiceOver
- Braille ruler
- Positive and negative tokens
- Standardized braille number line graphs
- APH Mathbuilder Unit 1- sorting (braille/large print)
- Unit 8- data collection (braille/large print)
- Tactile graphics for any number lines
- Tactile sketchpad
- Number line device
- Graphing calculator
- Orion TI-36X Talking Scientific Calculator, notetaker, or iPad scientific calculator
- Graph paper (tactile or bold-line) and tactile dots
- APH Graphic Aid for Mathematics
- Pegboards and Braille cubes
- Objects to create equations
- Raised-line drawing kit

### Tips and Strategies

- When using a computer with a screen reader students can practice rewriting expressions
- When using braille number line graphs, teachers can prep the graphs to show inequalities and students can practice showing inequalities
- When using tactile graphics for any graphs or similar triangles
- When using graph paper (tactile or bold-line) and tactile dots students can create models of proportional relationships
- APH's graphic Aid for Mathematics has additional suggested materials/methods for functional academics (APH)
- When using a notetaker, students can practice writing and rewriting expressions
- When working on problems, students can practice listening and speaking mathematically to describe factors and coefficients.
- When working on problems, students must have access to examples/exercises written in hard copy Braille and access to Perkins Brailler for completing assignments
- When using Nemeth Code, students look for patterns in Nemeth operators to help them remember what is a single term in an expression, and what is not. Operators that have horizontal dots in a single braille cell are addition and subtraction, both with dots 3 and 6. Terms or numbers separated by addition or subtraction symbols cannot be combined.
- When using tactile symbols, students can practice with them to learn about the meaning of combining terms and analyzing expressions. For example, a square next to a triangle can make 1 single shape. That would be a term. A square next to a circle cannot make one single shape. That would be an expression.
- When using computer with JAWS or VoiceOver, students can listen to expressions, while using notetaker to write and rewrite or complete the square.
- When using pegboards and braille cubes, students can build triangles. Additionally, braille cubes can be used to practice beginning with an exponent of 1 and building triangles of increasing height
- When working on problems, have students guess coefficients and exponents for increasing exponents in original expression.
- When using a Perkins Brailler, it can be difficult to do true Pascal Triangle, the most effective method is to start each row at left margin. Although numbers don’t line up properly in this representation, student can keep track of the different coefficients to write the nth row in terms of coefficients in the previous row.
- When using objects to create equations, students may need lots of oral interpretation of simple word problems to increase understanding.
- When using pegboard or graph board, students can talk through the equation. Conversely, instead of graphing, show equations and their values on spreadsheets. Construct spreadsheets with formulas to complete some column entries in terms of given column entries.
- When sighted students are using graphing calculators, students with visual impairments should use spreadsheets.
- When using talking graphic calculator, students can create equations and check their solutions
- When using notetakers, students can manipulate expressions
- When using a Perkins Brailler, students can maintain steps in the process of solving an equation. Additionally, a calculator can be used to substitute the same random number into side of equality at each step. Students must have access to examples/exercises written in hard copy Braille
- When using a notetaker, students need access to a printer to display solutions in print
- When students are learning problem solving, use real world examples
- When sighted students are using graphing calculators, students with visual impairments should be using spreadsheets
- When using a Perkins Brailler, systems of linear equations may be difficult to express when four or more variables are involved. For systems of equations in four of more variables, it may be best to use a computer algebra system
- When using computer with JAWS or Voiceover, start with concrete examples using spreadsheets
- When using a computer with JAWS or VoiceOver, students can use spreadsheets to plot pairs (x,y), which may be examined
- When solving problems, students can discuss using large panels
- When using a raised-line drawing kit, have students graph the lines of the two linear equations and shade the appropriate intersecting half planes

### Lesson Ideas

- Use different colored bags of the same types of items to explain coefficients. A blue bag of paint brushes, for example, means everybody who gets that brush will paint blue. The bag is like parentheses.
- Work with interest rates. A rate on items in the parentheses and a different rate on other items.
- Put everything together for writing equations for geometric shape dimensions, buying and selling items in a store, and earning interest.
- Identify the meaning of an exponent (limited to exponents of 2 and 3).
- Express area of a square or volume of a cube using exponents and then calculate the area or volume
- Identify a geometric sequence of whole numbers with a whole number common ratio.
- Use repeated multiplication to form a geometric sequence.
- Given an area of square or volume of a cube, find the length of a side
- Compose and decompose whole numbers up to 999
- Graph a simple ratio by connecting the origin to a point representing the ratio in the form of y/x.
- Solve simple algebraic equations with one variable using addition and subtraction.
- Factoring and expression is like taking it apart. Discuss what that means.
- Tactile symbol shapes as with academic students.
- Using food - separating a single loaf of bread into several pieces of bread can be done. The loaf can be outside a pot of soup. Everyone will get some of what is outside the pot and some of what is inside the pot, much like factoring terms in parentheses.
- Taking apart a sandwich into all it's factors.
- A sandwich bag can be like parentheses.
- All the carrots in the bag will get dipped into the ranch outside the bag. The ranch becomes like a coefficient.
- Practice seeing all components of an expression. Discuss how things can be rearranged but retain the same outcomes.Why do you have to disassemble and reassemble items? Like you may have to disassemble two outfits to create 4 different outfits.
- Discuss time value of money â€“ how to determine present value and future value using properties of geometric series some of this can be introduced with shapes, where a square is divided into quarter and each quarter is divided into quarters. This can continue to infinity. Students can use cut paper to feel this concept.
- Use pegboards to find patterns created by increasing coefficients in expressions.
- A lot of oral interpretation of simple word problems that have to do with shopping.
- Use actual objects when possible to act out word problems. He had twice as many pennies as she did... etc.
- Pennies to begin with. Add nickels and start with simple equations.
- Pegboard or graph board for graphing.
- Move on to writing equations with constraints from standard word problems.
- Use a talking graphics calculator to help with creating equation and checking.
- Use a notetaker to manipulate expressions.
- Use the Perkins braille writer to maintain steps in process of solving an equation.
- Use a calculator to substitute the same random number into side of equality at each step.
- Act out equations. Use listening. How high does the volume dial have to be turned to hear a word easily at 5 feet, 10 feet, etc. How is it different for different students.
- Graph results on pegboard or tactile graph paper.
- A lot of concrete examples of problems that involve many solutions. Like students can plan a bake sale and try to get $500 profits. They can sell more items at a lower margin or fewer items at a higher margin.