Equations

>> "Extensive experience making up and solving number stories at the primary grades may also make later conceptual uses of mathematics, and the school analogue - solving story problems - more effective and meaningful at higher grades" (Carroll, Fuson, & Diamond, 2000, p. 50)

=Overview= On this wikipage, you be introduced to the concept of an equation in a variety of formats.

Equation Examples
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Domain: Operations & Algebraic Thinking (13)

 * Kindergarten (1)**
 * **Standard:** Understand addition, and understand subtraction.
 * **//Performance Expectation://** Represent addition and subtraction with objects, fingers, mental images, drawings1, sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations.

>> >> >>
 * Grade 1 (4)**
 * **Standard:** Represent and solve problems involving addition and subtraction.
 * **//Performance Expectation://** Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem
 * **//Performance Expectation://** Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.
 * **Standard:** Work with addition and subtraction equations.
 * **//Performance Expectation://** Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. //For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 – 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2.//
 * **//Performance Expectation://** Determine the unknown whole number in an addition or subtraction equations relating three whole numbers. //For example, determine the unknown number that makes the equations true in each of the equations 8 + ? = 11, 5 = _ – 3, 6 + 6 = _//


 * Grade 2 (1)**
 * **Standard:** Represent and solve problems involving addition and subtraction.
 * **//Performance Expectation://** Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem

>> >>
 * Grade 3 (3)**
 * **Standard:** Represent and solve problems involving multiplication and division.
 * **//Performance Expectation://** Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.
 * **//Performance Expectation://** Determine the unknown whole number in a multiplication or division equation relating three whole numbers. //For example, determine the unknown number that makes the equation true in each of the equation 8 × ? = 48, 5 = _ ÷ 3, 6 × 6 = ?//
 * **Standard:** Solve problems involving the four operations, and identify and explain patterns in arithmetic.
 * **//Performance Expectation://** Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.

>> >>
 * Grade 4 (4)**
 * **Standard:** Use the four operations with whole numbers to solve problems.
 * **//Performance Expectation://** Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 × 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations.
 * **//Performance Expectation://** Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison.
 * **//Performance Expectation://** Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.

Domain: Number & Operations in Base Ten (3)
>>
 * Grade 4 (2)**
 * **Standard:** Use place value understanding and properties of operations to perform multi-digit arithmetic.
 * **//Performance Expectation://** Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
 * **//Performance Expectation://** Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.


 * Grade 5 (1)**
 * **Standard:** Perform operations with multi-digit whole numbers and with decimals to hundredths.
 * **//Performance Expectation://** Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

Domain Number & Operations—Fractions (6)
>>
 * Grade 4 (2)**
 * **Standard:** Build fractions from unit fractions.
 * **//Performance Expectation://** Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem.
 * **//Performance Expectation://** Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem.

>> >> >>
 * Grade 5 (4)**
 * **Standard:** Use equivalent fractions as a strategy to add and subtract fractions.
 * **//Performance Expectation://** Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem.
 * **Standard:** Apply and extend previous understandings of multiplication and division.
 * **//Performance Expectation://** Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem.
 * **//Performance Expectation://** Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.
 * **//Performance Expectation://** Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem.

Domain: Measurement & Data (1)

 * Grade 2 (1)**
 * **Standard:** Relate addition and subtraction to length.
 * **//Performance Expectation://** Use addition and subtraction within 100 to solve word problems involving lengths that are given in the same units, e.g., by using drawings (such as drawings of rulers) and equations with a symbol for the unknown number to represent the problem.

Domain: Ratios & Proportional Relationships (1)

 * Grade 6 (1)**
 * **Standard:** Understand ratio concepts and use ratio reasoning to solve problems.
 * **//Performance Expectation://** Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.

Domain: The Number System (1)

 * Grade 6 (1)**
 * **Standard:** Apply and extend previous understandings of multiplication and division to divide fractions by fractions.
 * **//Performance Expectation://** Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc.) How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi?.

Domain: Expressions & Equations (1)

 * Grade 6 (1)**
 * **Standard:** Reason about and solve one-variable equations and inequalities.
 * **//Performance Expectation://** Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.

Equation References by Grade Level
>> >> >> >> >> >> >> >> >> >> >> >> >> >> >> >> >> >> >>
 * Kindergarten (1)**
 * **Standard:** Understand addition, and understand subtraction.
 * **//Performance Expectation://** Represent addition and subtraction with objects, fingers, mental images, drawings1, sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations.
 * Grade 1 (4)**
 * **Standard:** Represent and solve problems involving addition and subtraction.
 * **//Performance Expectation://** Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.
 * **//Performance Expectation://** Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.
 * **Standard:** Work with addition and subtraction equations.
 * **//Performance Expectation://** Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 – 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2.
 * **//Performance Expectation://** Determine the unknown whole number in an addition or subtraction equations relating three whole numbers. For example, determine the unknown number that makes the equations true in each of the equations 8 + ? = 11, 5 = _ – 3, 6 + 6 = _
 * Grade 2 (2)**
 * **Standard:** Represent and solve problems involving addition and subtraction.
 * **//Performance Expectation://** Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.
 * **Standard:** Relate addition and subtraction to length.
 * **//Performance Expectation://** Use addition and subtraction within 100 to solve word problems involving lengths that are given in the same units, e.g., by using drawings (such as drawings of rulers) and equations with a symbol for the unknown number to represent the problem.
 * Grade 3 (3)**
 * **Standard:** Represent and solve problems involving multiplication and division.
 * **//Performance Expectation://** Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.
 * **//Performance Expectation://** Determine the unknown whole number in a multiplication or division equation relating three whole numbers. //For example, determine the unknown number that makes the equation true in each of the equations 8 × ? = 48, 5 = _ ÷ 3, 6 × 6 = ?//
 * **Standard:** Solve problems involving the four operations, and identify and explain patterns in arithmetic.
 * **//Performance Expectation://** Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.
 * Grade 4 (8)**
 * **Standard:** Use the four operations with whole numbers to solve problems.
 * **//Performance Expectation://** Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 × 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations.
 * **//Performance Expectation://** Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison.
 * **//Performance Expectation://** Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.
 * **Standard:** Use place value understanding and properties of operations to perform multi-digit arithmetic.
 * **//Performance Expectation://** Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
 * **//Performance Expectation://** Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
 * **Standard:** Build fractions from unit fractions.
 * **//Performance Expectation://** Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem.
 * **//Performance Expectation://** Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem.
 * Grade 5 (5)**
 * **Standard:** Perform operations with multi-digit whole numbers and with decimals to hundredths.
 * **//Performance Expectation://** Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
 * **Standard:** Use equivalent fractions as a strategy to add and subtract fractions.
 * **//Performance Expectation://** Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem.
 * **Standard:** Apply and extend previous understandings of multiplication and division.
 * **//Performance Expectation://** Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem.
 * **//Performance Expectation://** Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.
 * **//Performance Expectation://** Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem.
 * Grade 6 (3)**
 * **Standard:** Understand ratio concepts and use ratio reasoning to solve problems.
 * **//Performance Expectation://** Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
 * **Standard:** Apply and extend previous understandings of multiplication and division to divide fractions by fractions.
 * **//Performance Expectation://** Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc.) How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi?.
 * **Standard:** Reason about and solve one-variable equations and inequalities.
 * **//Performance Expectation://** Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.

Equation Lessons
1. Casey had 4 cookies. Molly gave her 7 more cookies and Nicole gave her 1 cookie from her lunch box. How many cookies does Casey have altogether?

2. Natalie loves ants. She has 9 ants in her ant "house." Emily loves ants too and she loves to share. She gave Natalie 6 ants. When Kelly heard that Emily gave Natalie ants, she wanted to give her ants too, so Kelly gave Natalie 4 ants. How many ants does Natalie have altogether?

3. Sarah brought 4 tissues out to the playground because she had a runny nose. Sarah used all the tissues and placed them on the picnic table next to her. Kaela saw that Sarah was out of tissues and her nose was still running, so she gave her 3 tissues from her pocket. Sarah used all the tissues and her nose was still running. When Jenna came to the picnic table where Sarah was sitting, she saw all the used tissues and Sarah's nose. She took 6 tissues out of her pocket and gave them to Sarah. Sarah used all of them too. How many tissues did Sarah use altogether?