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## Technology Tips: Linear Programming with More Than Two Decision Variables

In a 1999 article in Mathematics Teacher, we demonstrated how graphing systems of linear inequalities could be motivated using real-world linear programming problems (Edwards and Chelst 1999). At that time, the graphs were drawn by hand, and the corner-point principle was applied to find the optimal solution. However, that approach limits the number of decision variables to two, and problems with only two decision variables are often transparent and inauthentic.

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## Cracking Codes and Launching Rockets

This historically significant real-life application of a cryptographic coding technique, which incorporates first-year algebra and geometry, makes mathematics come alive in the classroom.

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## Problems before Procedures: Systems of Equations

Students today come to first-year algebra with considerable prior experience and a wide range of skills. Teachers need to modify their instructional strategies accordingly.

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## The Back Page: My Favorite Lesson – The Purple Milk Problem

In this favorite lesson, students must engage in cooperative problem solving and think outside the algebra box as they work to make sense of the Purple Milk problem.

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## Delving Deeper: Differentiated Insurance Contracts

By examining pricing for insurance for a moped, students can explore the theory of systems of inequalities and the topic of distributions in statistics. Fair systems for determining the premium (taking into account cautious and reckless drivers) are considered.

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## Authenticity of Mathematical Modeling

The choice and context of authentic problems—such as designing a staircase or a soda can—illustrate the modeling process in several stages.

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## Punch Up Algebra with POWs

An action research study focuses on the teaching strategies used to facilitate Problems of the Week.

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## Tiling the Cube

The Tiling Tubs task is a middle school activity published in NCTM's Navigating through Algebra in Grades 6-8. Students examine a drawing that shows a square hot tub with side length s feet. A border of square tiles surrounds the tub, with each border tile a 1-foot square. Students determine the number of border tiles required to surround the tub and express that number in as many ways as they can, thereby creating various equivalent algebraic representations. A valuable component of Tiling Tubs is its requirement that students contextually justify their algebraic representations. Students also realize that more than one correct representation and justification exists.

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## Fostering Algebraic Understanding through Math Magic

Meaningful context motivates students to appreciate the usefulness of variables, expressions, equations, and symbolic transformations.

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## Multiple Representations and Connections with the Sierpinski Triangle

A task using multiple representations helps students write explicit algebraic equations.