ALEX Classroom Resource

  

Grade 6 Mathematics Module 4, Topic G: Solving Equations

  Classroom Resource Information  

Title:

Grade 6 Mathematics Module 4, Topic G: Solving Equations

URL:

https://www.engageny.org/resource/grade-6-mathematics-module-4-topic-g-overview

Content Source:

EngageNY
Type: Audio/Video

Overview:

In Module 4, Topic G, students are introduced to the fact that equations have a structure similar to some grammatical sentences. Some sentences are true:  “George Washington was the first president of the United States.” or “2 + 3 = 5.” Some are clearly false:  “Benjamin Franklin was a president of the United States.” or “7 + 3 = 5.” Sentences that are always true or always false are called closed sentences. Some sentences need additional information to determine whether they are true or false. The sentence “She is 42 years old” can be true or false depending on who “she” is. Similarly, the sentence “x + 3 = 5” can be true or false depending on the value of x. Such sentences are called open sentences. An equation with one or more variables is an open sentence. The beauty of an open sentence with one variable is that if the variable is replaced with a number, then the new sentence is no longer open:  it is either clearly true or clearly false. For example, for the open sentence x + 3 = 5:

If is replaced by 7, the new closed sentence, 7 +3 = 5 is false because 10 ≠ 5. 

If x is replaced by 2, the new closed sentence, 2 + 3 = 5 is true because 5 = 5.

From here, students conclude that solving an equation is the process of determining the number(s) that, when substituted for the variable, result in a true sentence (6.EE.B.5). In the previous example, the solution for x + 3 = 5 is obviously 2. The extensive use of bar diagrams in Grades K–5 makes solving equations in Topic G a fun and exciting adventure for students. Students solve many equations twice, once with a bar diagram and once using algebra. They use identities and properties of equality that were introduced earlier in the module to solve one-step, two-step, and multistep equations. Students solve problems finding the measurements of missing angles represented by letters (4.MD.C.7) using what they learned in Grade 4 about the four operations and what they now know about equations. 

Content Standard(s):
Mathematics
MA2015 (2016)
Grade: 6
17 ) Use variables to represent numbers, and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number or, depending on the purpose at hand, any number in a specified set. [6-EE6]


NAEP Framework
NAEP Statement::
8A4a: Solve linear equations or inequalities (e.g., ax + b = c or ax + b = cx + d or ax + b > c).

NAEP Statement::
8A4b: Interpret "=" as an equivalence between two expressions and use this interpretation to solve problems.



Alabama Alternate Achievement Standards
AAS Standard:
M.AAS.6.17- Match a phrase to the corresponding one- step one-variable expression (e.g., "a number plus 3" matches "x + 3").


Mathematics
MA2019 (2019)
Grade: 6
18. Determine whether a value is a solution to an equation or inequality by using substitution to conclude whether a given value makes the equation or inequality true.
Unpacked Content
Evidence Of Student Attainment:
Students:
Given situations that have been modeled with equations or inequalities:
  • Substitute given specified values for the variables and the evaluate expressions.
  • Determine if the resulting numerical sentence is true when the specified values are substituted for the variables.
  • Explain with mathematical reasoning why a specified value is or is not a solution to a given equation or inequality.
Teacher Vocabulary:
  • Substitution
  • Equation
  • Inequality
Knowledge:
Students know:
  • Conventions of order of operations.
  • The solution is the value of the variable that will make the equation or inequality true.
  • That using various processes to identify the value(s) that when substituted for the variable will make the equation true.
Skills:
Students are able to:
  • Substitute specific values into algebraic equation or inequality and accurately perform operations of addition, subtraction, multiplication, division and exponentiation using order of operation.
Understanding:
Students understand that:
  • Solving an equation or inequality means finding the value or values (if any) that make the mathematical sentence true.
  • The solution to an inequality is often a range of values rather than a specific value.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
M.6.18.1: Define exponent, numerical expression, algebraic expression, variable, base, power, square of a number, and cube of a number.
M.6.18.2: Compute a numerical expression with exponents, with or without a calculator.
M.6.18.3: Restate exponential numbers as repeated multiplication.
M.6.18.4: Choose the correct value to replace each variable in the expression (Substitution).
M.6.18.5: Calculate the multiplication of single or multi-digit whole numbers, with or without a calculator.

Prior Knowledge Skills:
  • Use comparison symbols.
    Examples: >, =, or < .
  • Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.
  • Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits using >, =, and < symbols to record the results of comparisons.
  • Comparison symbols.
    Examples: >, =, and < .
  • Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits using >, =, and < symbols to record the results of comparisons.
  • Compare two fractions with the same numerator or the same denominator by reasoning about their size.
  • Recognize that comparisons are valid only when the two fractions refer to the same whole.
  • Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.
  • Convert fractions to decimals.
  • Compare two decimals to tenths.
  • Compare whole numbers.
  • Identify comparison symbols.
    Examples: >, <, and = .

Alabama Alternate Achievement Standards
AAS Standard:
M.AAS.6.19 Match equations and inequalities to real-world situations.


Mathematics
MA2019 (2019)
Grade: 6
19. Write and solve an equation in the form of x+p=q or px=q for cases in which p, q, and x are all non-negative rational numbers to solve real-world and mathematical problems.

a. Interpret the solution of an equation in the context of the problem.
Unpacked Content
Evidence Of Student Attainment:
Students:
Given contextual or mathematical situations which may be modeled by x + p = q or px = q (p,q, and x are rational and non-negative),
  • Explain the role of the variable as a place holder where the variable stands for a particular number (y + 7 = 12) or a value in a formula (A = L × W) and where values are substituted for one or more variables another variable assumes different values.
  • Write and solve equations modeling the situation, solve the resulting equations, and justify the solutions.
Teacher Vocabulary:
  • Variable
  • Equation
  • Non-negative rational numbers
Knowledge:
Students know:
  • Correct translation between verbally stated situations and mathematical symbols and notation.
  • How to write and solve a simple equation using non-negative rational numbers to solve mathematical and real-world problems.
Skills:
Students are able to:
  • Translate fluently between verbally stated situations and algebraic models of the situation.
  • Use inverse operations and properties of equality to produce solutions to equations of the forms x + p = q or px = q.
  • Use logical reasoning and properties of equality to justify solutions, reasonableness of solutions, and solution paths.
Understanding:
Students understand that:
  • Variables may be unknown values that we wish to find.
  • The solution to the equation is a value for the variable which, when substituted into the original equation, results in a true mathematical statement.
  • A symbolic representation of relevant features of a real-world problem can provide for resolution of the problem and interpretation of the situation.
  • The structure of mathematics present in the properties of the operations and equality can be used to maintain equality while rearranging equations, as well as justify steps in the solutions of equations.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
M.6.19.1: Define equation and variable.
M.6.19.2: Set up an equation to represent the given situation, using correct mathematical operations and variables.
M.6.19.3: Solve the equation represented by the real-world situation.
M.6.19.4: Identify the unknown variable in a given situation.
M.6.19.5: List given information from the problem.
M.6.19.6: Explain the solution in the context of the problem.

Prior Knowledge Skills:
  • 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.
  • Recognize key terms to solve word problems.
    Examples: times, every, at this rate, each, per, equal/equally, in all, total.
  • Define simple expression.
  • Recall simple equations.
  • Recognize properties of addition and multiplication.
  • Recall addition, subtraction, multiplication, division symbols.
  • Define parentheses, braces, and brackets.
  • Define numerical expression.
  • Recognize expressions.
  • Apply properties of operations as strategies to add and subtract.
  • Recall properties of operations as strategies to add and subtract.
  • Represent addition and subtraction with objects, mental images, drawings, expressions, or equations.

Alabama Alternate Achievement Standards
AAS Standard:
M.AAS.6.19 Match equations and inequalities to real-world situations.


Tags: equation, expressions, inequality, rational, realworld, unknown, variables
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Comments

There are seven lessons on this topic.

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  This resource provided by:  
Author: Hannah Bradley