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.
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The problem statement describes a changing algae population as reported by the Maryland Department of Natural Resources. In part (a), students are expected to build an exponential function modeling algae concentration from the description given of the relationship between concentrations in cells/ml and days of rapid growth (F-LE.2).
In this module, students synthesize and generalize what they have learned about a variety of function families. They extend the domain of exponential functions to the entire real line (N-RN.A.1) and then extend their work with these functions to include solving exponential equations with logarithms (F-LE.A.4). They explore (with appropriate tools) the effects of transformations on graphs of exponential and logarithmic functions. They notice that the transformations on a graph of a logarithmic function relate to the logarithmic properties (F-BF.B.3). Students identify appropriate types of functions to model a situation. They adjust parameters to improve the model, and they compare models by analyzing appropriateness of fit and making judgments about the domain over which a model is a good fit. The description of modeling as, the process of choosing and using mathematics and statistics to analyze empirical situations, to understand them better, and to make decisions, is at the heart of this module. In particular, through repeated opportunities in working through the modeling cycle (see page 61 of the CCLS), students acquire the insight that the same mathematical or statistical structure can sometimes model seemingly different situations.
This task provides a real world context for interpreting and solving exponential equations. There are two solutions provided for part (a). The first solution demonstrates how to deduce the conclusion by thinking in terms of the functions and their rates of change. The second approach illustrates a rigorous algebraic demonstration that the two populations can never be equal.
The task requires the student to use logarithms to solve an exponential equation in the realistic context of carbon dating, important in archaeology and geology, among other places. Students should be guided to recognize the use of the natural logarithm when the exponential function has the given base of e, as in this problem. Note that the purpose of this task is algebraic in nature -- closely related tasks exist which approach similar problems from numerical or graphical stances.
This problem introduces the method used by scientists to date certain organic material. It is based not on the amount of the Carbon 14 isotope remaining in the sample but rather on the ratio of Carbon 14 to Carbon 12. This ratio decreases, hypothetically, at a constant exponential rate as soon as the organic material has ceased to absorb Carbon 14, that is, as soon as it dies. This problem is intended for instructional purposes only. It provides an interesting and important example of mathematical modeling with an exponential function.
This unit provides Common-Core aligned lessons based for Math 3, English 10, and Biology (NGSS Standards). The subjects are linked by a text on climate change, and they hit the standards of argumentation for English, comparing functions in Math 3, and human effects on environment in Biology.
- Applied Science
- English Language Arts
- Life Science
- Material Type:
- Data Set
- Lesson Plan
- Primary Source
- Teaching/Learning Strategy
- Joanna Schimizzi
- Christine Sheffler
- Rob Leichner
- Theodore Mueller
- Yilmaz Yoruk
- Date Added:
This lesson has students create, compare, and solve linear, quadratic, exponential, and cubic functions based on a primary source from Weather Underground about the melting of the polar ice caps. If the formatting is an issue, contact me at firstname.lastname@example.org for a Google drive link to the lesson plan.
This task gives students an opportunity to work with exponential functions in a real world context involving continuously compounded interest. They will study how the base of the exponential function impacts its growth rate and use logarithms to solve exponential equations.
This lesson unit is intended to help teachers assess how well students are able to interpret exponential and linear functions and in particular to identify and help students who have the following difficulties: translating between descriptive, algebraic and tabular data, and graphical representation of the functions; recognizing how, and why, a quantity changes per unit intervale; and to achieve these goals students work on simple and compound interest problems.
The goal of this task is to develop an understanding of why rational exponents are defined as they are (N-RN.1), however it also raises important issues about distinguishing between linear and exponential behavior (F-LE.1c) and it requires students to create an equation to model a context (A-CED.2
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: Graphite is a mineral with many technological uses and it is perhaps most familiar for its use in writing instruments. At the atomic level, it is made ...
This lesson unit is intended to help teachers assess how well students are able to: articulate verbally the relationships between variables arising in everyday contexts; translate between everyday situations and sketch graphs of relationships between variables; interpret algebraic functions in terms of the contexts in which they arise; and reflect on the domains of everyday functions and in particular whether they should be discrete or continuous.
Students complete this Beer's Law activity in class. Students examine the attenuation of various thicknesses of transparencies. From this activity, students will understand that different substances absorb light differently. This can then be transferred to X-rays to explain that different substances absorb X-rays differently, hence the need for dual-energy analysis. In looking at Beer's Law, students use the properties associated with natural logarithms. After the activity, students complete a series of questions regarding what they observed.
The coffee cooling experiment is a popular example of an exponential model with immediate appeal. The model is realistic and provides a good context for students to practice work with exponential equations.
The purpose of this task is to give students an opportunity to explore various aspects of exponential models (e.g., distinguishing between constant absolute growth and constant relative growth, solving equations using logarithms, applying compound interest formulas) in the context of a real world problem with ties to developing financial literacy skills.
The purpose of this task is to give students experience modeling a real-world example of exponential growth, in a context that provides a vivid illustration of the power of exponential growth, for example the cost of inaction for a year.
These problems give students the opportunity to construct and compare linear, quadratic, and exponential models.