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Grade 6 Mathematics Module 4, Topic H: Applications of Equations

  Classroom Resource Information  

Title:

 Grade 6 Mathematics Module 4, Topic H: Applications of Equations

URL:

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

Content Source:

 ReadWriteThink
Type: Lesson/Unit Plan

Overview:

In Module 4, Topic H, students use their prior knowledge from Module 1 to construct tables of independent and dependent values in order to analyze equations with two variables from real-life contexts. They represent equations by plotting the values from the table on a coordinate grid (5.G.A.1, 5.G.A.2, 6.RP.A.3a, 6.RP.A.3b, 6.EE.C.9). The module concludes with students referring to true and false number sentences in order to move from solving equations to writing inequalities that represent a constraint or condition in real-life or mathematical problems (6.EE.B.5, 6.EE.B.8). Students understand that inequalities have infinitely many solutions and represent those solutions on number line diagrams. 

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.

Mathematics
MA2019 (2019)
Grade: 6
21. Identify, represent, and analyze two quantities that change in relationship to one another in real-world or mathematical situations.

a. Use tables, graphs, and equations to represent the relationship between independent and dependent variables.
Unpacked Content
Evidence Of Student Attainment:
Students:
Given a real-world problem involving two quantities that change in relationship to one another,
  • Represent the context using graphs, tables, and equations.
  • Explain the connections among the representations using mathematical vocabulary including dependent and independent variables.
Teacher Vocabulary:
  • Dependent variables
  • Independent variables
  • Equations
Knowledge:
Students know:
  • Roles of dependent and independent variables.
  • Correct translation between verbally stated situations and mathematical symbols and notation.
Skills:
Students are able to:
  • Represent real-world problems involving two quantities that change in relationship to one another using equations, graphs, and tables,
  • Use mathematical vocabulary to explain connections among representations of function contexts.
  • Analyze and interpret the relationship between the independent and the dependent variable in a given situation.
Understanding:
Students understand that:
  • Equations with two variables represent mathematical relationships in which the value of the dependent variable varies with changes in the independent variable.
  • A symbolic or visual representation of relevant features of a real-world problem can aid in interpretation of the situation.
  • Translating between language, a table, an equation, or a graph represents the same relationship and provides a different perspective on the function.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
M.6.21.1: Define dependent variable, independent variable, ordered pairs, input, output, and coordinate plane.
M.6.21.2: Examine the graph and table to determine any relationship between the variables.
M.6.21.3: Recall how to draw a number line.
M.6.21.4: Draw and label a coordinate plane.
M.6.21.5: Analyze the pattern represented by the values in the table and develop an equation to express the relationship.
M.6.21.6: Relate the table and graph to the equation.
M.6.21.7: Plot independent (input) and dependent (output) values on a coordinate plane.
M.6.21.8: Create a table of independent and dependent values from the equation.

Prior Knowledge Skills:
  • Model writing ordered pairs.
  • Identify the x- and y- values in ordered pairs.
  • Label the vertical axis (y).
  • Label the horizontal axis (x).
  • Define ordered pair of numbers, quadrant one, coordinate plane, and plot points.
  • Locate positive numbers on a vertical number line.
    Examples: thermometer, map.
  • Locate positive numbers on a horizontal number line.
  • Locate negative numbers on a horizontal number line.
  • Label x- and y-axis and zero on a coordinate.
  • Illustrate vertical and horizontal number lines.
  • Specify locations on the coordinate system.
  • Define x-axis, y-axis, and zero on a coordinate.
  • Define ordered pair of numbers.
  • 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.
  • Recognize that comparisons are valid only when the two fractions refer to the same whole.
  • Recognize arithmetic patterns (including geometric patterns or patterns in the addition table or multiplication table).
  • Recall basic multiplication facts.
  • Interpret data using graphs including bar, line, and circle graphs, and Venn diagrams.
  • Identify the parts of a line plot.
  • Recognize a line plot.
  • Draw a scaled picture graph and a scaled bar graph to represent a data set.
  • Recognize a fraction as a number on the number line.
Alabama Alternate Achievement Standards
AAS Standard:
M.AAS.6.21 Identify the independent and dependent variables among two quantities that change in relationship to one another in real-world situations (e.g. purchase total depends on number of items purchased).
Tags: coordinate grid, dependent, equation, expressions, graph, independent, inequality, number line, rational, realworld, unknownequation, variables
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There are five lessons on this topic.

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Author: Hannah Bradley