Variables & Patterns
Standards & Objectives | |
File Size: | 216 kb |
File Type: |
Goals for Variables & Patterns
Students should...
- Develop understanding of variables and how they are related
- Develop understanding of expressions and equations
Investigation 1
Vocabulary | |
File Size: | 156 kb |
File Type: |
The goal of this Investigation is to develop students’ ability to look for quantities that change over time, especially distance and speed of moving objects, and their ability to construct and interpret data tables and coordinate graphs representing patterns in that change. These mathematical goals are pursued through work on four Problems in the context of planning and pilot‑testing a three‑day cross‑country bicycle trip from Atlantic City, NJ, to Williamsburg, VA.
|
Investigation 2
Vocabulary | |
File Size: | 157 kb |
File Type: |
The goal of this Investigation is to extend students’ understanding and skill in working with relationships between quantitative variables as they are expressed in written stories, tables, and graphs. In particular, the Problems of this Investigation focus on cause-and-effect relationships where changes in one variable can reasonably be thought to cause changes in another. In these cases it is natural to call the first quantity the independent variable and the responding quantity the dependent variable (though this designation is sometimes arbitrary).
The first two Problems of the Investigation focus on questions about the Ocean Bike Tours business plans, with the key variables being demand (number of customers), price, income, operating cost, and profit. Although students will likely to be able to understand those concepts, the terms will be new to most students. In Problem 2.3 students examine the profit potential of the Ocean Bike Tours business. They extend their graphing skills to deal with negative coordinates. Question A requires graphing in quadrants I and IV. Questions B and C of Problem 2.3, as well as Questions E and G in Problem 2.4, extend graphing to all four quadrants to meet the requirement of the Common Core State Standards. |
Investigation 3
Vocabulary | |
File Size: | 158 kb |
File Type: |
This Investigation continues development of student understanding and skill in analyzing relationships between quantitative variables by use of algebraic expressions and equations to represent those relationships. The first Problem focuses on algebraic equations involving only one arithmetic operation in the form y=mx or y=x+a. The second Problem focuses on special properties of proportional relationships with rules in the form y=mx. The third Problem extends the ideas in the first two Problems to relationships expressed with two-operation rules. The relationships represented by the equations are also represented with tables and graphs. Students use numeric reasoning, tables, graphs, and equations to find the value of a variable in a given relationship. The fourth Problem returns to the Order of Operations rules in an algebraic context.
For the most part, equations are limited to finding the value of the dependent variable given the value of the independent variable. Students do use tables and graphs to find the value of the independent variable given a value of the dependent variable. The next Investigation further develops students’ skill in solving equations. All four Problems address explicit algebra objectives in the Common Core State Standards for Mathematics (CCSSM). |
Investigation 4
Vocabulary | |
File Size: | 159 kb |
File Type: |
The purpose of this Investigation is to address a number of CCSSM objectives in algebra. These include developing understanding of the term algebraic expression and some beginning skill in checking for equivalence of expressions. The second major focus continues work from Investigation 3 on solving equations using tables, graphs, or equations to find the value of a variable for equations of the form y=px and y=p+x. The third objective is to address inequalities and their solutions in simple cases.In Problems 4.1, 4.2, and 4.3 students work on the objectives related to expressions. The Problems ask students to use context clues to decide when two expressions should be equivalent and then check that equivalence by inspection of values for dependent variables that arise from substitution of equal values for the independent variables. The Distributive Property is revisited and used to show equivalence of simple expressions, such as 4n+5n or 3(n+1).
In Problem 4.4 students work on objectives related to equations. The first Question focuses on equations that involve a single operation and can be solved by use of fact family or inverse operation ideas. The Problems tie together the earlier work with multiple representations for relationships and lays useful groundwork for the careful treatment of linear functions and equations in Grade 7. Problem 4.5 addresses solutions of inequalities and their graphs. This is called for explicitly by the CCSSM. The last part of the Problem poses a situation that ties together all the ideas of the unit. |