This module contains the an algebraic expressions and equations from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr.
Students connect polynomial arithmetic to computations with whole numbers and integers. Students learn that the arithmetic of rational expressions is governed by the same rules as the arithmetic of rational numbers. This unit helps students see connections between solutions to polynomial equations, zeros of polynomials, and graphs of polynomial functions. Polynomial equations are solved over the set of complex numbers, leading to a beginning understanding of the fundamental theorem of algebra. Application and modeling problems connect multiple representations and include both real world and purely mathematical situations.
Students learn how to use and graph real-world stream gage data to create event and annual hydrographs and calculate flood frequency statistics. Using an Excel spreadsheet of real-world event, annual and peak streamflow data, they manipulate the data (converting units, sorting, ranking, plotting), solve problems using equations, and calculate return periods and probabilities. Prompted by worksheet questions, they analyze the runoff data as engineers would. Students learn how hydrographs help engineers make decisions and recommendations to community stakeholders concerning water resources and flooding.
The purpose of this task is to provide an opportunity for students to reason about equivalence of equations. The instruction to give reasons that do not depend on solving the equation is intended to focus attention on the transformation of equations as a deductive step.
In addition to the associated lesson, this activity functions as a summative assessment for the Using Stress and Strain to Detect Cancer unit. In this activity, students will create a 1-D strain plot in Microsoft Excel depicting the location of a breast tumor amidst healthy tissue. The results of this activity will function as proof of the accuracy and reliability of the students' breast cancer detection design.
This lesson unit is intended to help teachers assess how well students are able to solve quadratics in one variable. In particular, the lesson will help teachers identify and help students who have the following difficulties: making sense of a real life situation and deciding on the math to apply to the problem; solving quadratic equations by taking square roots, completing the square, using the quadratic formula, and factoring; and interpreting results in the context of a real life situation.
This lesson unit is intended to help teachers assess how well students are able to use linear inequalities to create a set of solutions. In particular, the lesson will help teachers identify and assist students who have difficulties in: representing a constraint by shading the correct side of the inequality line; and understanding how combining inequalities affects a solution space.
Students create model elevator carriages and calibrate them, similar to the work of design and quality control engineers. Students use measurements from rotary encoders to recreate the task of calibrating elevators for a high-rise building. They translate the rotations from an encoder to correspond to the heights of different floors in a hypothetical multi-story building. Students also determine the accuracy of their model elevators in getting passengers to their correct destinations.
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.
In this activity, students learn about creating a design directly from a CAD (computer-aided design) program. They will design a tower in CAD and manufacture the parts with a laser cutter. A competition determines the tower design with the best strength:weight ratio. Students also investigate basic structural truss concepts and stress concentrations. Partnership with a local college or manufacturing center is necessary for the completion of this project.
Students explore the properties of composites using inexpensive materials and processing techniques. They create beams using Laffy Taffy and water, and a choice of various reinforcements (pasta, rice, candies) and fabricating temperatures. Student groups compete for the highest strength beam. They measure flexure strength with three-point bend tests and calculations. Results are compared and discussed to learn how different materials and reinforcement shapes affect material properties and performance.
The purpose of this task is to continue a crucial strand of algebraic reasoning begun at the middle school level (e.g, 6.EE.5). By asking students to reason about solutions without explicily solving them, we get at the heart of understanding what an equation is and what it means for a number to be a solution to an equation. The equations are intentionally very simple; the point of the task is not to test technique in solving equations, but to encourage students to reason about them.
The students discover the basics of heat transfer in this activity by constructing a constant pressure calorimeter to determine the heat of solution of potassium chloride in water. They first predict the amount of heat consumed by the reaction using analytical techniques. Then they calculate the specific heat of water using tabulated data, and use this information to predict the temperature change. Next, the students will design and build a calorimeter and then determine its specific heat. After determining the predicted heat lost to the device, students will test the heat of solution. The heat given off by the reaction can be calculated from the change in temperature of the water using an equation of heat transfer. They will compare this with the value they predicted with their calculations, and then finish by discussing the error and its sources, and identifying how to improve their design to minimize these errors.
Students learn about the water cycle and its key components. First, they learn about the concept of a watershed and why it is important in the context of engineering hydrology. Then they learn how we can use the theory of conservation of mass to estimate the amount of water that enters a watershed (precipitation, groundwater flowing in) and exits a watershed (evaporation, runoff, groundwater out). Finally, students learn about runoff and how we visualize runoff in the form of hydrographs.
Using a household fan, cardboard box and paper towels, student teams design and build their own evaporative cooler prototype devices. They learn about the process that cools water during the evaporation of water. They make calculations to determine a room's cooling load, and thus determine the swamp cooler size. This activity adds to students' understanding of the behind-the-scenes mechanical devices that condition and move air within homes and buildings for human health and comfort.
This lesson unit is intended to help teachers 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; interpret and evaluate the data generated and identify the optimum case, checking it for confirmation; and communicate their reasoning clearly.
The purpose of the task is to show students a situation where squaring both sides of an equation can result in an equation with more solutions than the original one. The reason for this is that it is possible to have two unequal numbers whose squares are equal.
This is a task from the Illustrative Mathematics website that is one part of a complete illustration of the standard to which it is aligned. Each task has at least one solution and some commentary that addresses important aspects of the task and its potential use.
Modeling Our World with Mathematics Unit 1: Health & Fitness Topic 1 - A Healthier You!