Students examine exponential and logistic growth, identify carrying capacity, distinguish between density-dependent and density-independent limiting factors, apply the population model to data sets, and determine carrying capacity from population data.

In this interactive module, students examine exponential and logistic growth, identify carrying capacity, distinguish between density-dependent and density-independent limiting factors, apply the population model to data sets, and determine carrying capacity from population data.

In this video segment from Cyberchase, Harry plays a game of chess with a young friend and suggests a wager on the game. Harry?s friend uses a story to explain how putting a penny on the first square and then doubling the amount on each square of the chessboard can generate a tremendous amount of money over time. Teaching tips are also provided that discuss frame, focus and follow-up suggestions for using this video in a math lesson.

This assessment will check for understanding of goals NC.M4.AF.3.1, NC.MA.AF.3.2 and NC.MA.AF.3.3.  This is an open-ended assessment that includes changing from expnential to logarithmic form, properties of logarithms, solving logarithmic and exponential equations and applications with exponential and logarithmic functions.

In this video segment from Cyberchase, the CyberSquad is trying to escape from a desert island by crossing a large body of water. They find a special lily pad that was left for them by the Red Warrior. When the lily pad is put in water it doubles and the number of lily pads keeps on doubling. The CyberSquad tries to use this doubling effect to help them create a lily pad bridge to cross the water and escape. Teaching tips are also provided that discuss frame, focus and follow-up suggestions for using this video in a math lesson.

Adapted from mathematicsvisionproject.com’s Material Overview:
The Mathematics Vision Project (MVP) was created as a resource for teachers to implement the Common Core State Standards (CCSS) using a task-based approach that leads to skill and efficiency in mathematics by first developing understanding. The MVP approach develops the Standards of Mathematical Practice through experiential learning. Students engage in mathematical problem solving, guided by skilled teachers, in order to achieve mathematical proficiency: conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive disposition. The MVP authors created a curriculum where students do not learn solely by either “internalizing what a teacher or book says or, on the other hand, solely by inventing mathematics on their own.”
The MVP classroom experience begins by confronting students with an engaging problem and allows them to grapple with solving it. As students’ ideas emerge, take form, and are shared, the teacher deliberately orchestrates the student discussions and explorations toward a focused math goal. Students justify their own thinking while clarifying, describing, comparing, and questioning the thinking of others leading to refined thinking and mathematical fluency. What begin as ideas become concepts that lead to formal, traditional math definitions and properties. Strategies become algorithms that lead to procedures supporting efficiency and consistency. Representations become tools of communication which are formalized as mathematical models. Students learn by doing mathematics.

Students come to see the exponential trend demonstrated through the changing temperatures measured while heating and cooling a beaker of water. This task is accomplished by first appealing to students' real-life heating and cooling experiences, and by showing an example exponential curve. After reviewing the basic principles of heat transfer, students make predictions about the heating and cooling curves of a beaker of tepid water in different environments. During a simple teacher demonstration/experiment, students gather temperature data while a beaker of tepid water cools in an ice water bath, and while it heats up in a hot water bath. They plot the data to create heating and cooling curves, which are recognized as having exponential trends, verifying Newton's result that the change in a sample's temperature is proportional to the difference between the sample's temperature and the temperature of the environment around it. Students apply and explore how their new knowledge may be applied to real-world engineering applications.