Seventh Grade

 

Algebra

 

Standard 7-3:  The student will demonstrate through the mathematical processes an understanding of proportional relationships.

 

The indicators for this standard are grouped by the following major concepts:

 

The indicators that support each of those major concepts and an explanation of the essential learning for each major concept follows.

 

 

Patterns, Relationships, and Functions

 

Indicators

7-3.1      Analyze geometric patterns and pattern relationships.

7-3.7  Classify relationships as either directly proportional, inversely proportional,    or nonproportional.

 

          Students have used patterns all their lives and began to learn about and study patterns as early as kindergarten.  In kindergarten through 2nd grade, students progress from identifying patterns to translating patterns into rules.  The emphasis on the creation of numeric patterns begins in the 3rd grade, and in 4th and 5th grade, transitions to analyzing patterns, then representing these patterns in words, expressions, and equations.  Middle school continues this study of patterns by placing emphasis on numeric and algebraic patterns in the 6th grade, with the 7th grade focus on geometric patterns. Geometric patterns could include triangular numbers and square numbers.

          In 7th grade, emphasis should be placed on the concept of change itself, describing the rate of change and determining if the rate of change is constant or not.  Students are also expected to represent change in a variety of ways tables, graphs, and equations. In addition, 7th graders should begin to examine the relationship between variables.  Ultimately, they should be able to identify positive correlations (where both variables are increasing or decreasing) and negative correlations (where one variable is increasing and the other is decreasing). 

          From the study of patterns discussed above, 7th grade students should advance towards learning to classify relationships as directly proportional (when one quantity always changes by the same factor as another, = k or y=kx, where k is a constant), inversely proportional (when one quantity decreases by the same factor as the other increases, xy=k or y= where k is a constant), or nonproportional.  Please note that directly proportional is also known as direct variation and inversely proportional as inverse variation.  Because 7th grade also includes the introduction of slope, students should be led to discover the connection between slope and relationships that are directly proportional, the constant k being the same as the slope. This is the first time that students have been introduced to the terms directly and inversely proportional, and instruction should enable them to differentiate between the two, both numerically and graphically.

 

Teacher Note:  When using two order pairs (x1, y1) and (x2, y2), teachers may refer to these alternative forms:

Directly proportional is =. Inversely proportional is .

 

 

Representations, Properties, and Proportional Reasoning

 

Indicator

7-3.6 Represent proportional relationships with graphs, tables, and equations.

 

          In 6th grade, students determine whether two ratios are equivalent and use proportions to determine unit rates.  In 7th grade, the curriculum is extended to include: using ratios and proportions to solve problems involving scale factors and rates, explaining the proportional relationship among similar shapes, and to apply proportional reasoning to find a missing value within similar figures.  This indicator takes it a step further by having students investigate proportional relationships with graphs, tables, and equations.

          Seventh grade students will classify relationships as either directly proportional, inversely proportional, or nonproportional.  In addition, students will represent these relationships through graphs, tables, and equations.  An overall curriculum theme seen throughout the 7th grade Algebra strand is the concept of a constant rate of change (slope) and the tables, equations, and graphs that result from a relationship that has a constant rate of change.  It is important that students be given ample opportunities to discover the connection between direct proportionality and the table, equation, and graph this relationship produces as it leads to the gentle introduction of slope and linear functions.  As students graph directly proportional relationships, they should be able to identify the unit rate as the slope of the related line. 

          Representing inversely proportional relationships in tables, equations, and graphs allows students to understand that not all tables and equations produce similar graphs and that slope only exists when there is a "constant" rate of change.  This understanding is important as the focus in 8th grade will be on the table, equations, and graphs derived from linear functions. 

 

 

Solve Mathematical Situations

 

Indicators

7-3.4 Use inverse operations to solve two-step equations and two-step inequalities.

7-3.5 Represent on a number line the solution of a two-step inequality.

 

          In 6th grade, students use inverse operations to solve one-step equations that involve only whole numbers.  Although 6th graders do represent algebraic relationships with variables in simple inequalities, they have not yet had any instruction in solving inequalities, so 7th grade will be the first time that solving any type of inequality (both one-step and two-step) is introduced and the first time to be exposed to solving two-step equations.  As 8th graders prepare for Algebra I, a strong foundation in solving equations is a necessity.  The foundation begins to be built in the 6th grade with simple one step equations (with whole numbers only), transitions to one and two-step equations (with rational numbers) and inequalities in 7th grade, and the process continues into 8th grade with the focus on solving inequalities and multi-step equations. 

           Please note that students in 7th grade are to use inverse operations to solve

equations and inequalities.  A connection can be made here to order of operations

in that when solving equations or inequalities (particularly two-step), we

proceed in isolating the variable by doing the order of operations in reverse order.

See 6th grade Algebra Indicator 6-3.2 (Apply order of operations to simplify whole-

number expressions) for information on prior knowledge for order of operations.

          Caution should be exercised when introducing solving inequalities that include negative numbers.  The tendency is to simply tell students to reverse the inequality symbol when multiplying or dividing by a negative number.  However, without understanding why, a student will soon forget that “rule” of inequalities. It is important that students understand "why" the sign is reversed when multiplying or dividing by a negative number.

                   A more in-depth look at the concept of equality/inequality began in the 6th grade and continues throughout 7th and 8th grade.  If students have a solid foundation with the concepts of equality/inequality and that understanding is applied to solving equations and inequalities, the notion of “balancing” both sides of an equation or inequality should not present a problem for students.  

          In 7th grade, students should also understand that solutions to inequalities can be written as an inequality, in set notation, or graphed on a number line.  It is important to distinguish between the meaning of < verses ≤ and > verses ≥, particularly in regards to their graphs. 

 

 

Change in Various Contexts

 

Indicators

7-3.2   Analyze tables and graphs to describe the rate of change between and among quantities.

7-3.3 Understand slope as a constant rate of change.

 

          As stated earlier, patterns continue to be explored in the 7th grade, but the focus becomes more symbolic.  Seventh grade students should examine patterns in tables and graphs, describe the change among quantities, and connect their observations to a rate of change and determining if the rate of change is constant or not.  Students should be provided with opportunities to discover that the rate of change and slope are one in the same. Once this observation is made students can use this understanding to solve problems as they analyze tables and graphs.