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 asects of the task and its potential use. Here are the first few lines of the commentary for this task: The following is a student solution to the inequality \frac{5}{18} - \frac{x-2}{9} \leq \frac{x-4}{6}. \begin{align} \frac{5}{18} - \frac{x-2}{9} & \le...

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 asects of the task and its potential use. Here are the first few lines of the commentary for this task: Enrico has learned a geometric technique for ''completing the square'' to find the solutions of quadratic equations. To solve the equation $x^2 + 6x + ...

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.

The problem presents a context where a quadratic function arises. Careful analysis, including graphing, of the function is closely related to the context. The student will gain valuable experience applying the quadratic formula and the exercise also gives a possible implementation of completing the square.

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 asects of the task and its potential use. Here are the first few lines of the commentary for this task: Michelle, Hillary, and Cory created a YouTube video, and have a plan to get as many people to watch it as possible. They will each share the video with...

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 asects of the task and its potential use. Here are the first few lines of the commentary for this task: A company uses two different-sized trucks to deliver concrete blocks. The first truck can transport $x$ blocks per trip, and the second can transport $...

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 asects of the task and its potential use. Here are the first few lines of the commentary for this task: The profit, $P$ (in thousands of dollars), that a company makes selling an item is a quadratic function of the price, $x$ (in dollars), that they charg...

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 asects of the task and its potential use. Here are the first few lines of the commentary for this task: Given the height $h$ and volume $V$ of a certain cylinder, Jill uses the formula r=\sqrt{\frac{V}{\pi h}} to compute its radius to be 20 meters. If a s...

At this point in the unit, students have learned about Pascal's law, Archimedes' principle, Bernoulli's principle, and why above-ground storage tanks are of major concern in the Houston Ship Channel and other coastal areas. In this culminating activity, student groups act as engineering design teams to derive equations to determine the stability of specific above-ground storage tank scenarios with given tank specifications and liquid contents. With their floatation analyses completed and the stability determined, students analyze the tank stability in specific storm conditions. Then, teams are challenged to come up with improved storage tank designs to make them less vulnerable to uplift, displacement and buckling in storm conditions. Teams present their analyses and design ideas in short class presentations.

The purpose of this task is to help students interpret signed numbers in a context as a magnitude and a direction and to make sense of the absolute value of a signed number as its magnitude.

Students work as physicists to understand centripetal acceleration concepts. They also learn about a good robot design and the accelerometer sensor. They also learn about the relationship between centripetal acceleration and centripetal force governed by the radius between the motor and accelerometer and the amount of mass at the end of the robot's arm. Students graph and analyze data collected from an accelerometer, and learn to design robots with proper weight distribution across the robot for their robotic arms. Upon using a data logging program, they view their own data collected during the activity. By activity end , students understand how a change in radius or mass can affect the data obtained from the accelerometer through the plots generated from the data logging program. More specifically, students learn about the accuracy and precision of the accelerometer measurements from numerous trials.

This task examines, from a mathematical and statistical point of view, how scientists measure the age of organic materials by measuring the ratio of Carbon 14 to Carbon 12. The focus here is on the statistical nature of such dating. This task addresses a very important issue about precision in reporting and understanding statements in a realistic scientific context.

This task is a refinement of ``Carbon 14 dating'' which focuses on accuracy. Because radioactive decay is an atomic process modeled by the laws of quantum mechanics, it is not possible to know with certainty when half of a given quantity of Carbon 14 atoms will decay. This type of question is very important in science and it also provides an opportunity to study the very subtle question of how errors behave when applying a function: in some cases the errors can be magnified while in others they are lessened.

This problem involves solving a system of algebraic equations from a context: depending how the problem is interpreted, there may be one equation or two.

This task is a somewhat more complicated version of "Accurately weighing pennies I'' as a third equation is needed in order to solve part (a) explicitly. Instead, students have to combine the algebraic techniques with some additional problem-solving (numerical reasoning, informed guess-and-check, etc.)

The solutions unit consists of the following: General points for discussion relating to the teaching of the mathematical content in the activities. Step-by-step mathematical solutions to the activities. Annotations to the solutions to assist teachers in their understanding the maths as well as teaching issues relating to the mathematical content represented in the activities. Suggestions of links to alternative activities for the teaching of the mathematical content represented in the activities.

In this activity, students explore the effect of chemical erosion on statues and monuments. They use chalk to see what happens when limestone is placed in liquids with different pH values. They also learn several things that engineers are doing to reduce the effects of acid rain.

Students conduct a simple experiment to model and explore the harmful effects of acid rain (vinegar) on living (green leaf and eggshell) and non-living (paper clip) objects.

Students are introduced to the differences between acids and bases and how to use indicators, such as pH paper and red cabbage juice, to distinguish between them.

Students play and record the “Mary Had a Little Lamb” song using musical instruments and analyze the intensity of the sound using free audio editing and recording software. Then they use hollow Styrofoam half-spheres as acoustic mirrors (devices that reflect and focus sound), determine the radius of curvature of the mirror and calculate its focal length. Students place a microphone at the acoustic mirror focal point, re-record their songs, and compare the sound intensity on plot spectrums generated from their recordings both with and without the acoustic mirrors. A worksheet and KWL chart are provided.