Equations
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==Curriculum Resource Guide <font color ="#000066"> - '''Mathematics Content: Equations''' == | ==Curriculum Resource Guide <font color ="#000066"> - '''Mathematics Content: Equations''' == | ||
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|rowspan="5"| 6th grade | |rowspan="5"| 6th grade | ||
|6.SE.1a2 ''Repeated'' Given a real world problem, write an equation using 1 set of parentheses | |6.SE.1a2 ''Repeated'' Given a real world problem, write an equation using 1 set of parentheses | ||
− | |6.EE. | + | |6.EE.A.2c |
6.EE.B.6 | 6.EE.B.6 | ||
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|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.A.2) | |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.A.2) | ||
− | |“Bethany and her friends decided to start a dog walking business after school to earn some spending money. | + | |“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.” |
|• 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 | |• 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 | ||
|• Draw an array using the given information | |• Draw an array using the given information | ||
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|• Understand symbols for =, ≠, +, -, ÷, × | |• Understand symbols for =, ≠, +, -, ÷, × | ||
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− | |Numbers: 5.NO.2a1: Solve problems or word problems using up to three digit numbers and addition or subtraction. (4.OA.3) | + | |Numbers: 5.NO.2a1: Solve problems or word problems using up to three digit numbers and addition or subtraction. (4.OA.A.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.” | |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.” | ||
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===<u>'''4. What are Some Additional Activities That Can Promote Use of this Academic Concept in Real World Contexts?'''</u>=== | ===<u>'''4. What are Some Additional Activities That Can Promote Use of this Academic Concept in Real World Contexts?'''</u>=== | ||
− | * | + | * [[File:Lightbulb.png|baseline]] <span style="font-size:x-large; font-family:arial;"><b>5</b></span>[[File:Gradcap.png|baseline]] <span style="font-size:x-large; font-family:arial;"><b>4</b></span> Use a formula and solve the equation to determine the cost of additional text message charges. |
− | * | + | * [[File:Lightbulb.png|baseline]] <span style="font-size:x-large; font-family:arial;"><b>5</b></span>[[File:Gradcap.png|baseline]] <span style="font-size:x-large; font-family:arial;"><b>4</b></span> Solve an equation to determine amount of paint needed to paint a room. |
− | * | + | * [[File:Lightbulb.png|baseline]] <span style="font-size:x-large; font-family:arial;"><b>5</b></span>[[File:Gradcap.png|baseline]] <span style="font-size:x-large; font-family:arial;"><b>4</b></span> Solve an equation to determine how much a sale item will cost. |
− | * | + | * [[File:Lightbulb.png|baseline]] <span style="font-size:x-large; font-family:arial;"><b>5</b></span>[[File:Gradcap.png|baseline]] <span style="font-size:x-large; font-family:arial;"><b>4</b></span> Solve an equation to determine how much mulch is needed to cover a section of lawn. |
− | * | + | * [[File:Lightbulb.png|baseline]] <span style="font-size:x-large; font-family:arial;"><b>5</b></span>[[File:Gradcap.png|baseline]] <span style="font-size:x-large; font-family:arial;"><b>4</b></span> 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. |
===<u>'''5. How Can I Further Promote College and Career Readiness when Teaching “Equations”?'''</u>=== | ===<u>'''5. How Can I Further Promote College and Career Readiness when Teaching “Equations”?'''</u>=== | ||
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!width="1250"| Standards for Mathematical Practice | !width="1250"| Standards for Mathematical Practice | ||
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− | | | + | |[[File:Gradcap.png|baseline]] <span style="font-size:x-large; font-family:arial;"><b>1</b></span>Communicative Competence |
− | | | + | |[[File:Lightbulb.png|baseline]] <span style="font-size:x-large; font-family:arial;"><b>1</b></span>Make sense of problems and persevere in solving them. |
|- | |- | ||
− | | | + | |[[File:Gradcap.png|baseline]] <span style="font-size:x-large; font-family:arial;"><b>2</b></span>Fluency in reading, writing, and math |
− | | | + | |[[File:Lightbulb.png|baseline]] <span style="font-size:x-large; font-family:arial;"><b>2</b></span>Reason abstractly and quantitatively. |
|- | |- | ||
− | | | + | |[[File:Gradcap.png|baseline]] <span style="font-size:x-large; font-family:arial;"><b>3</b></span>Age appropriate social skills |
− | | | + | |[[File:Lightbulb.png|baseline]] <span style="font-size:x-large; font-family:arial;"><b>3</b></span>Construct viable arguments and critique the reasoning of others. |
|- | |- | ||
− | | | + | |[[File:Gradcap.png|baseline]] <span style="font-size:x-large; font-family:arial;"><b>4</b></span>Independent work behaviors |
− | | | + | |[[File:Lightbulb.png|baseline]] <span style="font-size:x-large; font-family:arial;"><b>4</b></span>Model with mathematics |
|- | |- | ||
− | | | + | |[[File:Gradcap.png|baseline]] <span style="font-size:x-large; font-family:arial;"><b>5</b></span>Skills in accessing support systems |
− | | | + | |[[File:Lightbulb.png|baseline]] <span style="font-size:x-large; font-family:arial;"><b>5</b></span>Use appropriate tools strategically. |
|- | |- | ||
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− | | | + | |[[File:Lightbulb.png|baseline]] <span style="font-size:x-large; font-family:arial;"><b>6</b></span>Attend to precision. |
|- | |- | ||
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− | | | + | |[[File:Lightbulb.png|baseline]] <span style="font-size:x-large; font-family:arial;"><b>7</b></span>Look for and make use of structure. |
|- | |- | ||
| | | | ||
− | | | + | |[[File:Lightbulb.png|baseline]] <span style="font-size:x-large; font-family:arial;"><b>8</b></span>Look for and express regularity in repeated reasoning. |
|} | |} | ||
[[Category:Curriculum Resource Guides]] | [[Category:Curriculum Resource Guides]] |
Latest revision as of 14:19, 29 January 2014
BACK TO Curriculum Resource Guides
[edit] Curriculum Resource Guide - Mathematics Content: Equations
The Curriculum Resource Guides describe how to teach the content to students with the most significant cognitive disabilities. |
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Used in conjunction with the Content Modules that describe the content (what is being taught) in general education they provide teachers with the necessary background knowledge to prepare students for the NCSC alternate assessment. |
The Curriculum Resource Guides offer examples of how the content is taught in general education, ideas for real life use, examples of universal design for learning, and ways to promote college and career readiness. |
[edit] Questions about:
- Equations
- Equations Teaching Activities
- Equations and Common Core Connectors
- Real World Equation Activities
- Promoting College and Career Readiness through Equations
- Making Equations Accessible to ALL Students
[edit] 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 vales of two mathematical expressions are equal. Expressions can be thought of as a phrase while an equation is a complete sentence.
(insert graphic here)
Students can use models (objects or drawings) to represent expressions or equations.
(insert graphic here)
- 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. 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: x
- 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.
(insert graphic here) (insert graphic here)
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
(insert graphic here)
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 -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
[edit] 2. What are some of the types of activities general educators will use to teach this skill?
2.1 Activities from General Education Resources
- (insert graphic) “Story translations” – students are given a math story and asked to write an equation that means the same thing.
- (insert graphic) “Names for numbers”2 – 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)
- (insert graphic) “Tilt or balance”2 – 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”)
- (insert graphic) 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.
- (insert graphic) 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.
[edit] 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 x 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 | Essential Understandings: Representations |
---|---|---|---|
Numbers: 3.NO.2d3 Solve multiplication problems with neither number greater than 5.
(3.OA.A.1) |
Student solves simple multiplication.
“This says four times five. What is four times five?” (4x5) |
• Create an array of sets (e.g., 3 rows of 2 objects) | • 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.D.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 + 6 3 x 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 |
• Combine (+), decompose (-), and multiply (x) 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 |
• Draw or use a representation of the word problem
• Understand symbols: +, =, -, x • Add on or count back depending upon the words in the problem |
Grade 4 |
CCC | Performance Example | Essential Understandings: Concrete Understandings | Essential Understandings: 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.A.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?” |
• Create an array (e.g., show me 2 groups/rows of 3; or 2X3) | • 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: =, X, ÷ , 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.A.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.” | • 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 | • 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.A.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?” 1x3=3 3x4=12 3x5=15 |
• 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) |
• 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 +, x, = |
Grade 5 |
CCC | Performance Example | Essential Understandings: Concrete Understandings | Essential Understandings: Representations |
---|---|---|---|
Symbols 5.SE.1b Evaluate whether both sides of an equation are equal.
(6.EE.A.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 ÷ 6 equal to 2 x 3? 18 ÷ 6 = 2 x 3 |
• Determine if sets are equal/ not equal
• Model an equation with objects |
• Understand symbols for =, ≠, +, -, ÷, × |
Numbers: 5.NO.2a1: Solve problems or word problems using up to three digit numbers and addition or subtraction. (4.OA.A.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?” |
• 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 |
• 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.B.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 x 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= ______ (9) “Solve the equation, how many carrots does each person get to feed to the donkey?” |
• Combine (x) 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 |
• Draw or use a representation of the word problem
• Symbols ÷, =, x • 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 | Essential Understandings: Representations |
---|---|---|---|
Patterns: 6.PRF.1d1: Solve real world single step linear equations (6.EE.B.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 writes 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.” |
• Recognize the intended outcome of a word problem based on a linear equation | • 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) • Understand vocabulary and symbols: +, -, X, ÷, = • Understand concepts and vocabulary: variable, solution, equation |
Grade 7 |
CCC | Performance Example | Essential Understandings: Concrete Understandings | Essential Understandings: 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.B.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.” Say: “Solve this equation to see how much more money Barney needs to buy the video game.” |
• Record/replace a variable in an equation with a fact from a story on a graphic organizer. | • Create a pictorial array of a simple equation to translate wording
• Understand concepts, vocabulary and symbols: +, -, X, ÷, =, ≠, <, >, equation, equal, inequality |
Grade 8 |
CCC | Performance Example | Essential Understandings: Concrete Understandings | Essential Understandings: Representations |
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Patterns: 8.PRF.1g3: Solve linear equations with 1 variable (8.EE.C.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 |
• Use manipulatives or graphic organizer to solve a problem | • Create a pictorial array of a simple equation to translate wording to solve for x or y
• Understand concepts, vocabulary and symbols: +, -, X, ÷, =, variable, equation |
High School |
CCC | Performance Example | Essential Understandings: Concrete Understandings | Essential Understandings: Representations |
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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?
6s + 6p + 1s + 9p 3s + 2p
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Patterns: H.PRF.2b1: Translate a real-world problem into a one-variable equation (HSA-CED.A.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. |
• Match an equation with one variable to the real world context. | • Create a pictorial array of a simple equation to translate wording
• Symbols: +, -, X, ÷, = |
[edit] 4. What are Some Additional Activities That Can Promote Use of this Academic Concept in Real World Contexts?
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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.
[edit] 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.
[edit] 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 (+, -, =, x, ÷)
• 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: |
Options for Representation |
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 |
---|---|---|---|
Provide auditory options
Provide tactile options:
Provide visual and manipulative options to scaffold representation of concepts:
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Reduce Physical Effort
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Illustrate through multiple media
Illuminations http://illuminations.nctm.org/Search.aspx?view=search&type=ac Math Open Reference There are many resources listed here: http://www.udlcenter.org/implementation/examples
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Provide customized display of information
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Options for Expression |
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 |
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Vary the methods for response by:
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Provide options for responses/expression:
Optimize access to tools/ alternatives for responding:
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Provide multimedia options for responses/expression:
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Provide options for modes of communication:
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Options for Engagement |
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 |
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Recruit interest by providing choices:
Increase personal relevance:
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Recruit interest by increasing personal relevance:
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Recruit interest by providing choices:
Provide options for sustaining effort and persistence:
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Recruit interest with modes of communication:
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