 Secondary Mathematics III

Strand: ALGEBRA - Creating Equations (A.CED)

Create equations that describe numbers or relationships, using all available types of functions to create such equations (Standards A.CED.1�4).
• ALGEBRA - Creating Equations (A.CED) - Sec Math III Core Guide
The Utah State Board of Education (USBE) and educators around the state of Utah developed these guides for the Secondary Mathematics III - Creating Equations (A.CED).
This task provides a simple but interesting and realistic context in which students are led to set up a rational equation (and a rational inequality) in one variable, and then solve that equation/inequality for an unknown variable.
• Bernardo and Sylvia Play a Game
This task presents a simple but mathematically interesting game whose solution is a challenging exercise in creating and reasoning with algebraic inequalities. The core of the task involves converting a verbal statement into a mathematical inequality in a context in which the inequality is not obviously presented, and then repeatedly using the inequality to deduce information about the structure of the game.
The emphasis in this task is not on complex solution procedures. Rather, the progression of equations, from two that involve different values of the sales tax, to one that involves the sales tax as a parameter, is designed to foster the habit of looking for regularity in solution procedures, so that students don't approach every equation as a new problem but learn to notice familiar types.
• Cash Box
The purpose of this task is to gives students an opportunity to engage in Mathematical Practice #3 Construct viable arguments and critique the reasoning of others. This task gives a teacher the opportunity to ask students not only for a specific answer of whether the dollar came from in the cash box or not, but for students to construct an argument as to how they came to their solution.
• Clea on an Escalator
This task has students create equations to model a physical scenario, and then reason with those equations to come up with a solution.
• Dimes and Quarters
This task does not actually require that the student solve the system but that they recognize the pairs of linear equations in two variables that would be used to solve the system.
• Equations and Formulas
This task asks students to use inverse operations to solve the equations for the unknown variable, or for the designated variable if there is more than one.
This task is the last in a series of three tasks that use inequalities in the same context at increasing complexity in 6th grade, 7th grade and in HS algebra. Students write and solve inequalities, and represent the solutions graphically.
• Global Positioning System I
This question examines the algebraic equations for three different spheres. The intersections of each pair of spheres are then studied, both using the equations and thinking about the geometry of the spheres.
• Growing coffee
This task is designed to make students think about the meaning of the quantities presented in the context and choose which ones are appropriate for the two different constraints presented. In particular, note that the purpose of the task is to have students generate the constraint equations for each part (though the problem statements avoid using this particular terminology), and not to have students solve said equations.
• How Much Folate?
This task a could be used as an introduction to writing and graphing linear inequalities. Part (a) includes significant scaffolding to support the introduction of the ideas. Part (b) demonstrates that, in some situations, writing down all possible combinations is not feasible.
• Inscribing and Circumscribing Right Triangles
This lesson unit is intended to help educators assess how well students are able to use geometric properties to solve problems.
• Introduction to Polynomials - College Fund
This task could serve as an introduction to polynomials or as an application after students are familiar with this type of function.
• Introduction to the Materials (Math 3)
Introduction to the Materials in the Mathematics Three of the The MVP classroom experience begins by confronting students with an engaging task and then invites them to grapple with solving it. As students ideas emerge, take form, and are shared, the teacher orchestrates the student discussions and explorations towards a focused mathematical goal. As conjectures are made and explored, they evolve into mathematical concepts that the community of learners begins to embrace as effective strategies for analyzing and solving problems.
• Module 3: Polynomial Functions - Student Edition (Math 3)
The Mathematics Vision Project, Secondary Math Three Module 3, Polynomial Functions, begins with a task that links linear, quadratic, and cubic functions together by highlighting the rates of change of each function type and using a story context to show that a linear function is the sum of a constant, a quadratic function is the accumulation or sum of a linear function, and a cubic function is the sum of a quadratic function.
• Module 3: Polynomial Functions - Teacher Edition (Math 3)
The Mathematics Vision Project, Secondary Math Three Module 3, Polynomial Functions, begins with a task that links linear, quadratic, and cubic functions together by highlighting the rates of change of each function type and using a story context to show that a linear function is the sum of a constant, a quadratic function is the accumulation or sum of a linear function, and a cubic function is the sum of a quadratic function.
• Module 4: Rational Expressions and Functions - Student Edition (Math 3)
The Mathematics Vision Project, Secondary Math Three Module 4, Rational Expressions and Functions, students work with the fractions that are ratios of polynomials, rational expressions and functions.
• Module 4: Rational Expressions and Functions - Teacher Edition (Math 3)
The Mathematics Vision Project, Secondary Math Three Module 4, Rational Expressions and Functions, students work with the fractions that are ratios of polynomials, rational expressions and functions.
• Optimization Problems: Boomerangs
This lesson unit is intended to help educators assess how well students are able to interpret a situation and represent the constraints and variables mathematically, select appropriate mathematical methods to use, explore the effects of systematically varying the constraints, and interpret and evaluate the data generated and identify the optimum case, checking it for confirmation.
• Paper Folding
This is a very open-ended task designed for students to develop some of the basic ideas surrounding exponential growth.
• Paying the rent
This simple conceptual task focuses on what it means for a number to be a solution to an equation, rather than on the process of solving equations.
• Planes and wheat
This is a simple exercise in creating equations from a situation with many variables. By giving three different scenarios, the problem requires students to keep going back to the definitions of the variables, thus emphasizing the importance of defining variables when you write an equation.
• Regular Tessellations of the plane
This task examines the ways in which the plane can be covered by regular polygons in a very strict arrangement called a regular tessellation. The goal of the task is to use algebra in order to understand which tessellations of the plane with regular polygons are possible.
• Rewriting equations
The goal of this task is to manipulate equations in order to solve for a specified variable.
• Silver Rectangle
This task provides a geometric context for working with ratios and algebraic equations. Students will create and then solve an algebraic equation describing a remarkable shape, the silver rectangle.
• Sum of angles in a polygon
This problem provides students with an opportunity to discover algebraic structure in a geometric context. More specifically, the student will need to divide up the given polygons into triangles and then use the fact that the sum of the angles in each triangle is 180.
• Throwing a Ball
Although this task is quite straightforward, it has a couple of aspects designed to encourage students to attend to the structure of the equation and the meaning of the variables in it.
• Uranium 238
The goal of this task is to represent an exponential relationship by an equation and identify, using knowledge of the context and the structure of the equation, possible graphs for the equation.
• Writing constraints
The purpose of this task is to give students practice writing a constraint equation for a given context. http://www.uen.org - in partnership with Utah State Board of Education (USBE) and Utah System of Higher Education (USHE).  Send questions or comments to USBE Specialist - Lindsey Henderson and see the Mathematics - Secondary website. For general questions about Utah's Core Standards contact the Director - Jennifer Throndsen.

These materials have been produced by and for the teachers of the State of Utah. Copies of these materials may be freely reproduced for teacher and classroom use. When distributing these materials, credit should be given to Utah State Board of Education. These materials may not be published, in whole or part, or in any other format, without the written permission of the Utah State Board of Education, 250 East 500 South, PO Box 144200, Salt Lake City, Utah 84114-4200.