Curriculum Resource Guide: Equations

BACK TO Curriculum Resource Guides

Curriculum Resource to Prepare Students for AA-AAS Mathematics Content: Equations The purposes of the Curriculum Resource Guides Are:

• To provide guidance for teaching the Common Core State Standards (CCSS) to students with Significant Cognitive Disabilities (SWSCD) that both aligns with these standards and provides differentiation for individual student needs
• To provide examples for differentiating instruction for a wide range of SWSCD. These examples can be used in planning specific lessons, alternate assessment items, and professional development.
• To serve as a companion document to the Progress Indicators for the CCSS found in the NCSC Learning Progressions
• To help educators build knowledge of the essential content reflected in these Progress Indicators of the CCSS
• To delineate the necessary skills and knowledge students need to acquire to master these indicators

1. What are "equations" and how are they taught in general education settings?

1a.1 The essential knowledge in this content area

Equations are a statement that the values of two mathematical expressions are equal. Expressions can be thought of as a phrase while an equation is a complete sentence.

Comparison of Terms
Expression	Equation
mathematical phrase: $x + 3$	mathematical sentence: $x + 3 = 9$
word phrase: a number plus three	word sentence: some number plus three equals nine
number, operation, variable	number, operation, variable, equal sign
evaluate: substitute simplify	solve for/isolate the variable
	one solution

Students can use models (objects or drawings) to represent expressions or equations.

$4*7\rightarrow$	$\heartsuit\heartsuit\heartsuit\heartsuit\heartsuit\heartsuit\heartsuit$
	$\heartsuit\heartsuit\heartsuit\heartsuit\heartsuit\heartsuit\heartsuit$
	$\heartsuit\heartsuit\heartsuit\heartsuit\heartsuit\heartsuit\heartsuit$
	$\heartsuit\heartsuit\heartsuit\heartsuit\heartsuit\heartsuit\heartsuit$

Multiplication and Division Equations

The standards require students to understand that "x" or "times" means "groups of" and 4 x 7 means 4 groups of 7. Also, "÷" or "divided by" means "how many in each group" or "how many groups can you make?" Example: The children rode in 4 cars to the museum. There were 3 children in each car. How many children went to the museum? 4 groups of 3, 4 x 3 = 12. When given word problems, students will need to be able to identify the key word in order to determine what operation is required to represent and solve the problem. Key words for multiplication include: product, of, multiplied, times, as much, by, and twice. Key words for division include: divide evenly, cut, split, each, every, average, equal pieces, out of, ratio, share, and quotient. Students need practice reading word problems to identify the key words and match the correct operation to the problem.

Examples: Circle the key word and write the correct operation. (Key words are highlighted yellow)

Jon gets a $12 allowance per month. How much allowance does he get each week? (4 weeks in a month) Operation: ÷

Nasir wants to play cars with his friends. He has 9 cars that he wants to share with his 3 friends. How many cars will each friend receive? Operation: ÷

Esteban finished 4 math problems. Cecily finished 2 times as much. How many math problems did Cecily finish? Operation: *

Identify when two expressions are equivalent

The expressions on either side of the equal sign must represent the same quantity. Students can first be taught this as a rule (e.g., "You must have the same amount on both sides of the equal symbol.") Then provide students with practice in determining whether sets are equal. Another way to describe equality is that there must be "fair shares" on either side of the equals sign. One way to teach equality is to use a balance. For example, provide the following chart then use a balance and weights to represent the amounts. Then discuss whether the numbers are equal (fair shares) or not equal and fill in the third column.

Left side	Right side	Equation
3	2	3 ≠ 2
6	6	6 = 6
2	2	2 = 2
1	4	1 ≠ 4

Once students are firm on their understanding of equality using single whole numbers, then they can better determine whether expressions are equivalent. Students can do this by simplifying or solving the expression.

For example:

Are the expressions equivalent? 8 – 4 and 3 + 2

Expression	Simplifying Expression
8 – 4	4
3 + 2	5

4 ≠ 5 No, the expressions are not equivalent.

Solving for Variables in Equations

In equations, variables are often used as placeholders for unknown quantities. When given a word problem or real life situation, students can be taught to assign a variable to the unknown quantity. In an equation, this variable represents a specific value. For example: Shelby wrote some thank you letters in the afternoon. She wrote 3 more that night. She wrote 10 thank you letters in all. How many letters did Shelby write that afternoon? Students can use "n" to represent the unknown quantity of letters written in the afternoon. "In all" is a key word that indicates this is an addition problem.

$n + 3 = 10$

Using inverse operations to solve equations

In order to solve an equation by determining the value of the variable, students must learn to isolate the variable, or work the problem so that the variable is "alone" on one side of the equal symbol. To isolate a variable, students need to use inverse operations. Inverse operations can be thought of as "opposite" operations. The inverse operation of addition is subtraction and the inverse operation of multiplication is division. For example:

	$:\ n+3 = 10$
$inverse\ operation \rightarrow$	$:\ \ \ \ -3\ \ \ -3$
	$:\ \ \ \ \ \ n=7$

In the example above, the inverse operation of minus 3 was used on both sides of the equation. Students must be taught the rule that "what you do on one side of the equals symbol, you must do on the other side." To help students comprehend this, you can use the scale example again. Teachers can set up a scale with equal amounts of items so that the scale is balanced. Have students take away a few items from one side and observe that the scale becomes unbalanced (i.e., the sides are unequal). Then have the students take away the same amount from the other side and observe that the scale is balanced again (i.e., the side are equal).

1a.2 Common misunderstandings in this content area

Students often think that the equals sign means "get an answer." It is important to provide repeated opportunities to demonstrate equality (as described above) to ensure they comprehend the meaning of equality.

1a.3 Prior knowledge/skills needed (can be taught concurrently)

• Performing basic operations (addition, subtraction, multiplication, and division)
• Number and symbol identification

2. What are some of the types of activities general educators will use to teach this skill?

2.1 Activities from General Education Resources

(CCR & Standards for Mathematical Practice Reference Table)

• 2 2"Story translations" – students are given a math story and asked to write an equation that means the same thing.
• 4 2 "Names for numbers" – students are given a number and asked to write several expressions to represent the number (e.g., given number 10, student writes the following expressions: 2+8, 20÷2, 17-7)
• 3 1 "Tilt or balance" – draw a balance on the board and write an expression above each pan and ask students whether it will tilt or balance. (e.g., left pan says 2 x 7 and right pan says 6 + 6; students call out "tilt")
• 2 1 Given a story problem and a list of items/concepts related to the story problem, the student is asked to identify which items are variables (unknown quantities). For example, if the story problem says "Nina worked for thirty minutes, and then she took a break. Then she worked for another 45 minutes. She was at work a total of one and a half hours." Circle the variable: first work session, break, or second work session.
• 5 3 Given a formula and batting statistics for a player, the student is asked to substitute the variables for the correct values and then solve the problem to determine a player's batting average.

Links Across Content Areas

• Science – use formulas, substitute values for variables, and solve equation to determine: acceleration, mass, volume, friction, orbit, etc.

3. What Connectors to the Common Core Standards Are Addressed in Teaching "Equations"?

Grade Differentiation	Core Content Connectors	Common Core State Standards
3rd grade	3.NO.2b Use the relationships between addition and subtraction to solve problems	3.NBT.A.2
	3.NO.2c1 Solve multi-step addition and subtraction problems up to 100	3.NBT.A.2
	3.NO.2d1 Find the total number of objects when given the number of identical groups and the number of objects in each group neither number larger than 5	2.OA.C.4 3.OA.A.1
	3.NO.2d2 Find total number inside an array with neither number in the columns or rows larger than 5	2.OA.C.4 3.OA.A.1
	3.NO.2d3 Solve multiplication problems with neither number greater than 5	3.OA.A.1
	3.NO.2d4 Determine how many objects go into each group when given the total number of objects and the number of groups where the number in each group or number of groups is not greater than 5	3.OA.A.2
	3.NO.2d5 Determine the number of groups given the number of total number of objects and the number of objects in each group where the number in each group and the number of groups is not greater than 5	3.OA.A.2
	3.NO.2e1 Solve and check one or two-step word problems requiring addition, subtraction, or multiplication with answers up to 100.	3.OA.D.8
4th grade	4.NO.2c2 Solve multi digit addition and subtraction problems up to 1000	3.NBT.A.2
	4.NO.2d6 Find total number inside an array with neither number in the columns or rows larger than 10	3.OA.A.1
	4.NO.2d7 Determine how many objects go into each group when given the total number of objects and the number of groups where the number in each group or number of groups is not greater than 10	3.OA.A.2
	4.NO.2d8: Match an accurate addition and multiplication equation to a representation	3.OA.A.1
	4.NO.2e2 Solve or solve and check one or two step word problems requiring addition, subtraction or multiplication with answers up to 100	4.OA.A.3
	4.PRF.1d2 Use objects to model multiplication and division situations involving up to 10 groups with up to 5 objects in each group and interpret the results	3.OA.A.1
	4.PRF.1e3 SOLVE multiplicative comparisons with an unknown using up to 2-digit numbers with information presented in a graph or word problem (e.g., an orange hat cost $3. A purple hat cost 2 times as much. How much does the purple hat cost? [3 * 2 = p])	4.OA.A.2
5th grade	5.SE.1a1 Given a real world problem, write an equation using 1 set of parentheses	5.OA.A.1
	5.SE.1b Evaluate whether or not both sides of an equation are equal	6.EE.A.4
	5.NO.2a1 Solve problems or word problems using up to three digit numbers and addition or subtraction	4.OA.A.3
	5.NO.2a5 Solve word problems that require multiplication or division	5.NBT.B.6
6th grade	6.SE.1a2 Repeated Given a real world problem, write an equation using 1 set of parentheses	6.EE.A.2c 6.EE.B.6
	6.SE.1a3 Write expressions for real-world problems involving one unknown number
	6.PRF.2a2 Use variable to represent numbers and write expressions when solving real world problems	6.EE.B.6
	6.PRF.2a3 Use variables to represent two quantities in a real-world problem that change in relationship to one another	6.EE.C.9
	6.PRF.1d1 Solve real world single step linear equations	6.EE.B.7
7th grade	7.SE.1f1 Set up equations with 1 variable based on real world problems	7.EE.B.4
	7.SE.1f2 Solve equations with 1 variable based on real world problems	7.EE.B.4
	7.PRF.1g2 Use variables to represent quantities in a real‐world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities	7.EE.B.4
	7.PRF.2a5 Repeated Use variables to represent two quantities in a real-world problem that change in relationship to one another	6.EE.C.9
	7.PRF.2d Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers	7.EE.B.4b
8th grade	8.PRF.1f2 Describe OR SELECT THE relationship between the two quantities Given a line graph of a situation	8.EE.B.5
	8.PRF.1g3 Solve linear equations with 1 variable	8.EE.C.7
Grades 9-12	HS.NO.1a1 Simplify expressions that include exponents	HSN-RN.A.2, HSA-SSE.B.3
	HNO.2a Solve simple equations using rational numbers with one or more variables	HSA-REI.A.2
	H.NO.3a2 Rewrite mathematical statements (e.g., an expression) in multiple forms
	H.PRF.2b1 Translate a real-world problem into a one variable linear equation	HSA-CED.A.1
	H.PRF.2b2 Solve equations with one or two variables using equations or graphs	HSA-REI.A.1 HSA-REI.B.3 HSA-CED.A.2

Performance Examples for Priority CCCs
Grade 3
CCC	Performance Example	Essential Understandings: Concrete Understandings and Representations
Numbers: 3.NO.2d3 Solve multiplication problems with neither number greater than 5. (3.OA.1)	Student solves simple multiplication. "This says four times five. What is four times five?" $4 * 5$	Concrete Understandings: Create an array of sets (e.g., 3 rows of 2 objects)	Representation: Identify or draw pictorial representation of an array that matches the multiplication problem State what the numbers represent (ex. first number is number of sets, second number is number within each set)
Numbers: 3.NO.2e1 Solve and check one or two-step word problems requiring addition, subtraction, or multiplication with answers up to 100. (3.OA.8)	Student selects expression that matches word problem, solves problem, then selects equivalent expression that can be used to check work. Kunius had 3 weeks to sell cookies for school. He sold 6 boxes each week. Which of these will show how many boxes Kunius sold? $3+3\ \ \ \ \ \ \ \ 3*6\ \ \ \ \ \ \ \ 3-6$ How many boxes did Kunius sell? Which one of these can be used to check your work? $3 + 3 + 3\ \ \ \ \ \ 18-3\ \ \ \ \ \ 6 + 6 + 6$	Concrete Understandings: Combine (+), decompose (-), and multiply (*) with concrete objects; use counting to get the answers Match the action of combining with vocabulary (i.e., in all; altogether) or the action of decomposing with vocabulary (i.e., have left; take away; the difference) in a word problem	Representation: Draw or use a representation of the word problem Understand symbols: +, =, -, * Add on or count back depending upon the words in the problem
Grade 4
CCC	Performance Example	Essential Understandings: Concrete Understandings and Representations
Patterns: 4.PRF.1d2: Use objects to model multiplication and division situations involving up to 10 groups with up to 5 objects in each group and interpret the results (3.OA.1)	Present a paper with the following printed on it and read it aloud: "Ms. Smith is an art teacher. She is preparing to teach an art lesson to five students. Each student will need four markers to complete the art activity. You need to find out how many markers Ms. Smith will need all together." Give the student 24 markers. "Use these markers to show me how five students would each get four markers. You may not use all the markers." If the student makes an error, model the correct answer and say "There should be five groups of four markers, like this." "How many markers does the teacher need all together?"	Concrete Understandings: Create an array (e.g., show me 2 groups/rows of 3; or 2*3)	Representation: Use an array to represent a multiplication or division problem Select a numeral to place under each representation in the modeled equation Select a pictorial representation of an array that matches the multiplication or division problem Understand concepts, vocabulary and symbols: =, *, ÷ , groups, objects, set, equal groups, combination, comparison, multiplication, division, array, row, column, equation
Numbers: 4.NO.2d7: Determine how many objects go into each group when given the total number of objects and the number of groups where in each group or number of groups is not greater than 10 (3.OA.2)	"Bethany and her friends decided to start a dog walking business after school to earn some spending money. "Bethany has three friends, for a total of four people who want to walk dogs." Point to the picture of Bethany and her friends. "There are eight dogs that need to be walked." Point to the picture of the dogs. "Each person will walk an equal number of dogs. How many dogs will each person walk? You can use these blocks to help solve the problem."	Concrete Understandings: Create an array of objects given a specific number of rows and the total number, place one object in each group/row at a time	Representation: Draw an array using the given information Understand symbols: ÷, =
Numbers: 4.NO.2d8: Match an accurate addition and multiplication equation to a representation (3.OA.1)	Present the student with the football pictures and say "Here is a picture of a bunch of footballs. When things are lined up like this it helps us think about addition and multiplication." ♪ ♪ ♪ ♪ ♪ ♪ ♪ ♪ ♪ ♪ ♪ ♪ ♪ ♪ ♪ "Now here are three addition equations. Which equation shows what you see in this picture?" $3+3+3+3=12\ \ \ \ \ \ 5+5+5=15\ \ \ \ \ \ 5+5+3=13$ "Now here are three multiplication equations. Which equation shows what you see in this picture?" $1*3=3\ \ \ \ \ 3*4=12 \ \ \ \ \ \ 3*5=15$	Concrete Understandings: Select the representation of manipulatives on a graphic organizer to show addition/multiplication equation. Match to same for representations of equations (may be different objects but same configuration)	Representation: Select a representation to place under each numeral in addition equation State what the numbers represent in multiplication equation (ex. first number is number of sets, second number is number within each set) Select a pictorial representation of an array that matches the multiplication problem Understand symbols +, *, =
Grade 5
CCC	Performance Example	Essential Understandings: Concrete Understandings and Representations
Symbols 5.SE.1b Evaluate whether both sides of an equation are equal. (6.EE.4)	Student indicates whether an equation is true. I am going to show you some equations. Tell me if each equation is true. An equation is true if the expressions on both sides of the equation are equal. Here is the first equation. Is this equation true? Is $3 + 8$ equal to $10 + 1?$ $3 + 8 = 10 + 1$ Is $12-3$ equal to $5 + 4?$ $12-3 = 5 + 4$ Is $18 \div 6$ equal to $2 * 3?$ $18 \div 6 = 2 * 3$	Concrete Understandings: Determine if sets are equal/ not equal Model an equation with objects	Representation: Understand symbols for =, ≠, +, -, ÷, *
Numbers: 5.NO.2a1: Solve problems or word problems using up to three digit numbers and addition or subtraction. (4.OA.3)	Present the note card with the following word problem on it and read aloud to the student "The baseball team has decided to sell candles to raise money to go on a trip to play baseball at a tournament. They have 450 candles to sell. After one day, they had sold 324 candles. How many candles are left to sell? Show me your work and how you would solve this equation." If the student does not generate a correct equation show them an already written equation: $450-324 = \_\_\_\_\_\_$ "Solve the equation, how many candles do they have left to sell?"	Concrete Understandings: Combine (+) or decompose (-) with concrete objects; use counting to get the answers Match the action of combining with vocabulary (i.e., in all; altogether) or the action of decomposing with vocabulary (i.e., have left; take away, difference) in a word problem	Representation: Draw or use a representation of the word problem Understand symbols +, =, -, Add on or count back depending upon the words in the problem Translate wording into numeric equation
Numbers: 5.NO.2a5: Solve word problems that require multiplication or division. (5.NBT.6)	Problem 1: Present the note card with the following word problem on it and read aloud to the student "Ms. Wood's class is going on a field trip to the zoo. There are 9 people going to the zoo. It costs $11 per person for admission into the zoo. How much money does the class need in all? Show me your work as you solve the problem." If the student does not generate a correct equation show them an already written equation: $9 * 11 = \_\_\_\_\_\_$ "Solve the equation, how many much money do they need?" Problem 2: Present the note card with the following word problem on it and read aloud to the student "While at the zoo, they decide to feed the donkey. The zoo sells bags of 36 carrots to feed to the donkeys. Remember, there are nine people on the trip. If they split it up evenly, how many carrots does each person get to feed to the donkey? Show me your work as you solve the problem." If the student does not generate a correct equation show them an already written equation: $36=9x$ "Solve the equation, how many carrots does each person get to feed to the donkey?" $\frac{36}{9}=x$ $x=4$	Concrete Understandings: Combine (*) or decompose (÷) with concrete objects; use counting to get the answers Match the action of combining with vocabulary (i.e., in all; altogether) or the action of decomposing with vocabulary (i.e., have left; take away) in a word problem Understand concept of division: Sharing or grouping numbers into equal parts Understand concept of multiplication: the result of making some number of copies of the original	Representation: Draw or use a representation of the word problem Symbols ÷, =, * Identifying purpose to either find a total (multiplication) or one component (number of sets or number within each set for division) depending upon the problem Translate wording into numeric equation
Grade 6
CCC	Performance Example	Essential Understandings: Concrete Understandings and Representations
Patterns: 6.PRF.1d1: Solve real world single step linear equations (6.EE.7)	Show students the following word problem and read it aloud: "Hans has $10 to spend while playing mini-golf. He spent $7 on the ticket and spent the rest on candy. Write an equation to show how much Hans spent on snacks. Use the letter S to represent the amount he spent on snacks." Student writes an appropriate equations (e.g., 10-s=7, 10-7=s, 10=7+s etc.). After student write the equation (if student gets this portion wrong, write an appropriate equation for the student) ask: "Solve this equation to see how much money Hans spent on snacks. Show your work."	Concrete Understandings: Recognize the intended outcome of a word problem based on a linear equation	Representation: Match a representation of an equation with a variable to a real world problem Set up an equation in which both sides are equal (adding or subtracting the same number/value from both sides of the equation) Understands vocabulary and symbols: +, -, *, ÷, = Understands concepts and vocabulary: variable, solution, equation
Grade 7
CCC	Performance Example	Essential Understandings: Concrete Understandings and Representations
Patterns: 7.PRF.1g2: Use variables to represent quantities in a real‐world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities (7.EE.4)	Show the students the following word problem and read it aloud: Barney wants to buy a new video game. He has 24 dollars. He needs 50 dollars to buy the new video game. Present the student with the equation template and cut out response cards (below) and say: "Use these cards to make the equation. Use the letter D to represent how much more money he needs because that's an unknown variable." 24 50 D + = Say: "Use the equation to solve for how much more money Barney needs to buy the video game."	Concrete Understandings: Record/replace a variable in an equation with a fact from a story on a graphic organizer.	Representation: Create a pictorial array of a simple equation to translate wording Understand concepts, vocabulary and symbols: +, -, *, ÷, =, ≠, <, >, equation, equal, inequality
Grade 8
CCC	Performance Example	Essential Understandings: Concrete Understandings and Representations
Patterns: 8.PRF.1g3: Solve linear equations with 1 variable (8.EE.7)	Present the equation to the student and say "You are going to solve a problem using a variable. Remember, a variable is a letter that represents an unknown number." Read the equation to the student and have them solve it. $54 = 9x$	Concrete Understandings: Use manipulatives or graphic organizer to solve a problem.	Representation: Create a pictorial array of a simple equation to translate wording to solve for x or y Understand concepts, vocabulary and symbols: +, -, *, ÷, =, variable, equation
High School
CCC	Performance Example	Essential Understandings: Concrete Understandings and Representations
Numbers: H.NO.3a2 Rewrite mathematical statements (e.g., an expression in multiple forms) (No CCC listed)	Show students the following word problem and read it aloud "Rita went to buy some snack foods for her friends. She is buying packages of Skittles and bags of pretzels. She doesn't know how many Skittles or pretzels are in each bag, but it is always the same amount. She asked Francis and Dan how many they wanted. Francis wants 3 bags of Skittles and 2 bags of pretzels. Dan wants 2 bags of Skittles and 5 bags of pretzels." Which expression shows how many things Rita bought all together? $3s + 2p + 2s + 5p$ $6s + 6p + 1s + 9p$ $3s + 2p$ Once student makes a selection, remove the incorrect responses, give students paper to write on, and say "Simplify the expression here."
Patterns: H.PRF.2b1: Translate a real-world problem into a one-variable equation (A.CED.1)	Show the student the following word problem and read it aloud: Omar picked 7 baskets of apples. He gave 20 apples to his teacher, 40 apples to his the debate team, and 80 apples to the football team. Write an equation to show how many apples were in each basket. Use the letter "a" to represent the unknown variable. You do not need to solve the equations, just write it.	Concrete Understandings: Match an equation with one variable to the real world context.	Representation: Create a pictorial array of a simple equation to translate wording Symbols: +, -, *, ÷, =

4. What are Some Additional Activities That Can Promote Use of this Academic Concept in Real World Contexts?

• 5 4 Use a formula and solve the equation to determine the cost of additional text message charges.
• 5 4 Solve an equation to determine amount of paint needed to paint a room.
• 5 4 Solve an equation to determine how much a sale item will cost.
• 5 4 Solve an equation to determine how much mulch is needed to cover a section of lawn.
• 5 4 Given a recipe, determine how long a roast must be cooked based on the weight of the meat and the time per pound that is required to cook the meat to a safe temperature.

5. How Can I Further Promote College and Career Readiness when Teaching "Equations"?

Ideas for Promoting Career/ College Ready Outcomes

Communicative competence

Students will increase their vocabulary to include concepts related to "equations." In addition, they will be learning concepts such as: "equal", "multiply", "divide", "add", "subtract", "balance", "same", "each", "times", "more", and "take away."

Fluency in reading, writing, and math

Students will have an opportunity to increase their numeracy and sight word fluency while participating in problem solving related to "equations" such as number recognition, symbol identification, reading comprehension, composing equations, and identifying key words.

Age appropriate social skills

Students will engage in peer groups to solve problems related to "equations" that will provide practice on increasing reciprocal communication and age appropriate social interactions. For example, students might work together with their peers to develop equations based on story problems and substitute values for variables when given a science formula and values.

Independent work behaviors

By solving real life problems related to "equations" students will improve work behaviors that could lead to employment such as landscaping, working as a cashier, stocking shelves, or a chef. When providing opportunities for real life problems, leave some materials out and prompt/teach the students to determine who they should ask and what they should ask for to be able to solve the problem.

Skills in accessing support systems

At times, students will need to ask for assistance to complete activities related to "equations" which will give them practice in accessing supports. Students will gain practice asking for tools such as talking calculators, number lines, graphic organizers, and formulas. They can ask a peer to complete the physical movements of the tasks they are not able to do themselves. Be sure to teach students to ask versus having items or supports automatically given to them.

6. How Do I Make Instruction on "Equations" Accessible to ALL the Students I Teach?

6.1 Teach Prerequisites and BasicEquation Skills Concurrently: Remember that students can continue to learn basic numeracy skills in the context of this grade level content.

Basic numeracy skills that can be worked on as a part of a lesson relating to equations:

• Number identification
• Equal and/or same
• Symbol identification (+, -, =, *, ÷)
• Addition and subtraction
• Creating sets

6.2 Incorporate Universal Design for Learning (UDL in planning, and provide for additional differentiated instruction when teaching equations.)

Some examples of options for teaching equations to students who may present instructional challenges due to:
Sensory Differences such as Blindness, Visual Impairment, Deafness, or Deaf/Blindness	Physical Disability or Motor Differences (such as weakness or motor planning difficulty)	Extremely limited evidence of experience/ skill or motivation/attention.	Lack of or extremely limited use of speech.
Options for Representation
Provide auditory options Talking calculator when solving equations Text-to-speech software or voice recordings to read aloud story problems Single message sequence voice–output devices to count aloud Captioning software that presents auditory information visually Provide tactile options: Object cues, using miniature objects or other tangible symbols to assist with problem comprehension and operations Tactile equation mat Create numbers and symbols out of tactile materials such as sandpaper or wiki stix Provide visual and manipulative options to scaffold representation of concepts: Color code equations and corresponding parts of calculator to support students correctly entering equations Provide manipulatives for quantities, such as Cuisenaire rods.	Reduce Physical Effort When reading word problems, student can scan array of key math operation words and select correct key word and operation for equation Place equations and graphic organizers on slant board or eye gaze board Display flip chart, interactive white board or other teaching materials at student eye level Utilize a switch instead of a computer mouse or software that allows the mouse to be controlled with the students' head rather than their hands	Illustrate through multiple media Utilize interactive whiteboard Incorporate interactive websites that provide nonlinguistic tools for exploring math concepts: Illuminations http://illuminations.nctm.org/ActivitySearch.aspx Math Open Reference http://www.mathopenref.com/ There are many resources listed here: http://www.udlcenter.org/implementation/examples Use virtual manipulatives and technology to show equations Incorporate computer representations, videos, and animations	Provide customized display of information Consistent model by utilizing modes of communication used by students (point to symbols representing concepts, operations) Teacher model competent use of AAC during instruction
Options for Expression
Vary the methods for response by: Student states answer or scans raised numbers to select correct answer; use voice output devices for student to select the correct answer Provide manipulatives for student to respond or contribute to interaction Student states answer by selecting picture or symbol Allow students who are deaf to videotape their answers/ process descriptions.	Provide options for responses/expression: Student selects numbers versus writing them; matches numerals and operation symbols to equation Choose response by pointing to, eye gazing, or selecting object or item Place operations and symbols and/or equations on electronic whiteboard and have student use switch to select correct answer or create equation Optimize access to tools/ alternatives for responding: Provide symbols, objects, manipulatives, and pictures for matching/ student responses	Provide multimedia options for responses/expression: Allow the student to make selections by pointing to, gazing at, or selecting answers on the interactive white board Utilize a switch or adapted computer mouse	Provide options for modes of communication: Incorporate responses into student's AAC device or eye gaze array Phrase questions so that they require a "yes/no" response, these can easily be answered using an eye gaze, head turn, two switches, etc Choose response by pointing to or selecting object or item Use a blink response to count tiles or select answer; count tiles/cubes out loud having student move in some voluntary way (e.g., nod head, tap hand, tap foot) to count along
Options for Engagement
Recruit interest by providing choices: Digital/talking representations, videos, interactive websites Increase personal relevance: Use items that are familiar and reinforcing to students. Incorporate high preference items into story problems, as well as student names	Recruit interest by increasing personal relevance: Ensure that engaging and high preference content is visible and accessible to student Highlight key words in story problems When creating response options make them large enough and separate them far enough so that student can make clear eye gaze or head nod to make intentional selection Provide opportunities to work with typically developing peer on items (teach peer how to interpret student's responses)	Recruit interest by providing choices: Digital/talking representations, videos, talking calculators Use of computer representations, videos Provide manipulatives that may be of high interest to the student and use high interest scenarios in word problems Provide options for sustaining effort and persistence: Break tasks down to maximize student attention Token economy system that embeds equations (You have 2 Justin Bieber tokens. You need 5 total. How many more do you need to earn before you can listen to a song?) Vary demands and materials to maintain interest Follow equation unit with a community-based instruction field trip which require the skills learned to be used	Recruit interest with modes of communication: Allow students to choose items or subjects that are relevant to them via AAC devices, symbols, or eye gaze array

CCR & Standards for Mathematical Practice Table

Promoting Career and College Readiness	Standards for Mathematical Practice
1 Communicative Competence	1 Make sense of problems and persevere in solving them.
2 Fluency in reading, writing, and math	2 Reason abstractly and quantitatively.
3 Age appropriate social skills	3 Construct viable arguments and critique the reasoning of others.
4 Independent work behaviors	4 Model with mathematics
5 Skills in accessing support systems	5 Use appropriate tools strategically.
	6 Attend to precision.
	7 Look for and make use of structure.
	8 Look for and express regularity in repeated reasoning