Rules for solving one-step, two-step, and multi-step equations, and equations with variables on both sides are presented. By the end of the fourth lesson, students solve applied problems involving general linear equations. Students reason with ratios and percents, construct proportions and apply them to scaled drawings, and use formulas as a problem-solving strategy.
One-Step Equations -
Two-Step Equations -
Multi-Step Equations -
Equations with Variables on Both Sides -
Ratios and Proportions -
Scale and Indirect Measurement -
Percent Problems -
Problem-Solving Strategies: Use a Formula -
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Problem-Solving Strand for Mathematics
Use a Formula is the problem-solving strategy highlighted in this unit. Many teachers and students believe that knowing what formula to use or apply to a given problem turns what may have been a difficult “problem” into more of a straightforward exercise. Substituting in known quantities and using a formula, they reason, is simple.
For some students, however, following the steps of a formula is not a simple process, especially if there’s an unknown quantity in the formula. Still for others, using a formula with unfamiliar terms such as ohms or finding the principal using a formula such as , is simply baffling. The first case, of course, is an ideal time to encourage students to apply a previously practiced skill like Working Backwards. By using opposite operations to “undo” an equation, students acquire a new formula, one they have generated themselves.
When presenting formulas to students, an excellent practice is to help students understand where the formula comes from in the first place. A straightforward example of this would be the formula . Students can easily understand that this formula for the perimeter of a rectangle matches the physical process of “marching around” the figure in fact or in their mind. Similarly, a mental image of a grid on which a rectangle is outlined verifies why the area formula for a rectangle is .
The challenge for us as teachers is to relate as many new formulas as possible to formulas that have already become self-evident to the students:
- One classic strategy is to encourage students to create a visual image of the area of a parallelogram in relation to a rectangle, which can enclose the parallelogram when a triangular portion is shifted left or right, and which, therefore, exhibits an equivalent area.
- Another strategy is to illustrate the formula for the area of a circle by cutting a circle into pie-slice wedges and rearranging them into a near-parallelogram, which gives students real reason to believe that the area of a circle really is “pi times radius squared.” (This particular model makes an excellent poster for the classroom and can be an even more effective poster when it is the well-executed work of a classmate.)
- Yet another strategy is to demonstrate how percentages work in retail stores by using mark-ups and discounts as examples. Having different groups of students compute two different scenarios -- such as those presented in Example 14 of the Percent Lesson – can be an easy and convincing way for students to understand the algebraic mathematics involved in such situations.
Helping students make sense of mathematical formulas can teach them to “trust” the formulas they are asked to use and cannot yet prove (such as Ohm’s Law which relates volts, amps, and resistance).
Aligning with the NCTM Process Standards
The NCTM Process Standards that are addressed in the use and understanding of formulas include: connections, representation, and reasoning and proof. The connections between mathematics and science become readily apparent in this unit as students recognize and apply mathematics in contexts outside of mathematics (CON.3) -- e.g. questions about the speed of sound, resistance, voltage, and the time required for a jet plane to climb a given distance. Several formulas illustrate how business and finance use mathematics to relate costs, show profit and loss margins, and calculate interest rates. These formulas also encourage students to recognize and use connections among mathematical ideas in the use of opposite operations to solve problems (CON.1).
As students strive to understand the derivation of various formulas, many connections occur (CON.2). Students use representations repeatedly to model and interpret physical, social and mathematical phenomena (R.3). Conjectures are made and investigated (RP.2), arguments are developed and evaluated (RP.3), and various types of reasoning and methods of proof (RP.4) are utilized.
- CON.1 - Recognize and use connections among mathematical ideas
- CON.2 - Understand how mathematical ideas interconnect and build on one another to produce a coherent whole
- CON.3 - Recognize and apply mathematics in contexts outside of mathematics
- RP.2 - Make and investigate mathematical conjectures
- RP.3 - Develop and evaluate mathematical arguments and proofs
- RP.4 - Select and use various types of reasoning and methods of proof
- R.3 - Use representations to model and interpret physical, social, and mathematical phenomena