In Module 3, Topic B, students use linear equations and inequalities to solve problems. They continue to use bar diagrams from earlier grades where they see fit but will quickly discover that some problems would more reasonably be solved algebraically (as in the case of large numbers). Guiding students to arrive at this realization on their own develops the need for algebra. This algebraic approach builds upon work in Grade 6 with equations (6.EE.B.6, 6.EE.B.7) to now include multi-step equations and inequalities containing rational numbers (7.EE.B.3, 7.EE.B.4). Students solve problems involving consecutive numbers, total cost, age comparisons, distance/rate/time, area and perimeter, and missing angle measures. Solving equations with a variable is all about numbers, and students are challenged with the goal of finding the number that makes the equation true. When given in context, students recognize that a value exists, and it is simply their job to discover what that value is. Even the angles in each diagram have a precise value, which can be checked with a protractor to ensure students that the value they find does indeed create a true number sentence.

Content Standard(s):

Mathematics MA2019 (2019) Grade: 7

8. Solve multi-step real-world and mathematical problems involving rational numbers (integers, signed fractions and decimals), converting between forms as needed. Assess the reasonableness of answers using mental computation and estimation strategies.

Unpacked Content

Evidence Of Student Attainment:

Students:

Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form.

Interpret solutions of problems with rational numbers in the context of the problem.

Assess the reasonableness of answers using mental computation and estimation strategies.

Use variables to represent quantities in a real-world or mathematical problem.

Teacher Vocabulary:

Rational numbers

Integers

Estimation

Knowledge:

Students know:

techniques for converting between fractions, decimals, and percents.

Techniques for estimation, mental computations, and how to assess the reasonableness of their answers.

Skills:

Students are able to:

convert between different forms of a rational number.

Add, subtract, multiply and divide rational numbers.
-translate verbal forms of problems into algebraic symbols, expressions, and equations.

Use estimation and mental computation techniques to assess the reasonableness of their answers.

Understanding:

Students understand that:

One form of a number may be more advantageous than another form, based on the problem context.

Using estimation strategies helps to determine the reasonableness of answers.

Diverse Learning Needs:

Essential Skills:

Learning Objectives: M.7.8.1: Define estimation, rational numbers, and reasonable.
M.7.8.2: Recall mental calculation strategies.
M.7.8.3: Recall estimation strategies.
M.7.8.4: Analyze the given word problem to set up a mathematical problem.
M.7.8.5: Recognize the mathematical operations of rational numbers in any form, including converting between forms. (Ex. 0.25=1/4 =25%).
M.7.8.6: Recognize the rules of operations of positive and negative numbers.
M.7.8.7: Recognize properties of numbers (Distributive, Associative, Commutative).
M.7.8.8: Recall problem solving methods.

Prior Knowledge Skills:

Represent addition and subtraction with objects, mental images, drawings, expressions, or equations.

Define integers, positive and negative numbers.

Define rational number.

Define equivalent, simplify, term, distributive property, associative property of addition and multiplication, and the commutative property of addition and multiplication.

Alabama Alternate Achievement Standards

AAS Standard: M.AAS.7.8 Add and subtract integers in a real-world situation.

Mathematics MA2019 (2019) Grade: 7

9. Use variables to represent quantities in real-world or mathematical problems and construct algebraic expressions, equations, and inequalities to solve problems by reasoning about the quantities.

a. Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach.

b. 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. Graph the solution set of the inequality, and interpret it in the context of the problem.

Unpacked Content

Evidence Of Student Attainment:

Students:

Write and solve mathematical equations (or inequalities) to model real-world problems.

Interpret the solution to an equation in the context of a problem

Interpret the solution set of an inequality in the context of a problem.

Graph the solution to an inequality on a number line.

Teacher Vocabulary:

Algebraic expressions

Equations

Inequalities

Greater than

Greater than or equal to

less than

less than or equal to

Knowledge:

Students know:

p(x + q) = px + pq, where p and q are specific rational numbers.

When multiplying or dividing both sides of an inequality by a negative number, every term must change signs and the inequality symbol reversed.

In the graph of an inequality, the endpoint will be a closed circle indicating the number is included in the solution set (≤ or ≥) or an open circle indicating the number is not included in the solution set ( < or >).

Skills:

Students are able to:

use variables to represent quantities in a real-world or mathematical problem.

Construct equations (px + q = r and p(x + q) = r) to solve problems by reasoning about the quantities.

Construct simple inequalities (px + q > r or px + q < r) to solve problems by reasoning about the quantities.

Graph the solution set of an inequality.

Understanding:

Students understand that:

Real-world problems can be represented through algebraic expressions, equations, and inequalities.

Why the inequality symbol reverses when multiplying or dividing both sides of an inequality by a negative number.

Diverse Learning Needs:

Essential Skills:

Learning Objectives: M.7.9.1: Define equation, inequality, and variable.
M.7.9.2: Set up equations and inequalities to represent the given situation, using correct mathematical operations and variables.
M.7.9.3: Calculate a solution or solution set by combining like terms, isolating the variable, and/or using inverse operations.
M.7.9.4: Test the found number or number set for accuracy by substitution.
M.7.9.5: Recall solving one step equations and inequalities.
M.7.9.6: Recognize properties of numbers (Distributive, Associative, Commutative).
M.7.9.7: Define equation and variable.
M.7.9.8: Set up an equation to represent the given situation, using correct mathematical operations and variables.
M.7.9.9: Calculate a solution to an equation by combining like terms, isolating the variable, and/or using inverse operations.
M.7.9.10: Test the found number for accuracy by substitution.
Example: Is 5 an accurate solution of 2(x + 5)=12?.
M.7.9.11: Identify the unknown, in a given situation, as the variable.
M.7.9.12: List given information from the problem.
M.7.9.13: Recalling one-step equations.
M.7.9.14: Explain the distributive property.
M.7.9.15: Define inequality and variable.
M.7.9.16: Set up an inequality to represent the given situation, using correct mathematical operations and variables.
M.7.9.17: Calculate a solution set to an inequality by combining like terms, isolating the variable, and/or using inverse operations.
M.7.9.18: Test the solution set for accuracy.
M.7.9.19: Identify the unknown, of a given situation, as the variable.
M.7.9.20: List information from the problem.
M.7.9.21: Recall how to graph inequalities on a number line.
M.7.9.22: Recall how to solve one step inequalities.

Prior Knowledge Skills:

Define inequality.

Define equivalent, simplify, term, distributive property, associative property of addition and multiplication, and the commutative property of addition and multiplication.

Define equation, solution of an equation, solution of an inequality, and inequality.

Compare and contrast equations and inequalities.

Determine if an inequality is by replacing the variable with a given number.

Determine if an equation is true by replacing the variable with a given number.

Simplify a numerical sentence to determine equivalence.

Recognize the symbols for =, >, <, ?, and ?.

Define equation and variable.

Set up an equation to represent the given situation, using correct mathematical operations and variables.

Identify the unknown, in a given situation, as the variable.

Alabama Alternate Achievement Standards

AAS Standard: M.AAS.7.9 Use the properties of operations to solve one-step equations and inequalities from real-world and mathematical problems.

Mathematics MA2019 (2019) Grade: 7

21. Use facts about supplementary, complementary, vertical, and adjacent angles in multi-step problems to write and solve simple equations for an unknown angle in a figure.

Unpacked Content

Evidence Of Student Attainment:

Students:

Find the values of angles using complementary and supplementary angle relationships and equations.

Identify angle relationships in angle diagrams involving vertical, supplementary, and complementary angles.

Write equations to represent relationships between known and unknown angle measurements.

Determine the measures of unknown angles and judge the reasonableness of the measures.

Teacher Vocabulary:

Supplementary angles

Complementary angles

vertical angles

Adjacent angles

Knowledge:

Students know:

supplementary angles are angles whose measures add to 180 degrees.

Complementary angles are angles whose measures add to 90 degrees.

vertical angles are opposite angles formed when two lines intersect.

Adjacent angles are non-overlapping angles which share a common vertex and side.

Skills:

Students are able to:

write a simple equation to find an unknown angle.

Identify and determine values of angles in complementary and supplementary relationships.

Identify pairs of vertical angles in angle diagrams.

Identify pairs of complementary and supplementary angles in angle diagrams.

Use vertical, complementary, and supplementary angle relationships to find missing angles.

Understanding:

Students understand that:

vertical angles are the pair of angles formed across from one another when two lines intersect, and that the measurements of vertical angles are congruent.

Complementary angles are angles whose measures add up to 90^{o}, and supplementary angles are angles whose measures add up to 180^{o}.

Relationships between angles depends on where the angles are located.

Diverse Learning Needs:

Essential Skills:

Learning Objectives: M.7.21.1: Define supplementary angles, complementary angles, vertical angles, adjacent angles, parallel lines, perpendicular lines, and intersecting lines.
M.7.21.2: Discuss strategies for solving multi-step problems and equations.
M.7.21.3: Identify all types of angles.
M.7.21.4: Identify right angles and straight angles.

Prior Knowledge Skills:

Model using a protractor to draw angles.

Draw points, lines, line segments, and parallel and perpendicular lines, angles, and rays.

Define vertex/vertices and angle.

Alabama Alternate Achievement Standards

AAS Standard: M.AAS.7.21 Classify angles as acute, obtuse, right, or straight.