In this problem, students use given data points to calculate the average rate of change of a function over a specific interval, foreshadowing the idea of limits and derivatives to students.

Open Middle provides math problems that have a closed beginning, a closed end, and an open middle. This means that there are multiple ways to approach and ultimately solve the problems. Open middle problems generally require a higher Depth of Knowledge than most problems that assess procedural and conceptual understanding.

This task addresses the first part of standard F-BF.3: “Identify the effect on the graph of replacing f(x) by f(x)+k, kf(x), f(kx), and f(x+k) for specific values of k (both positive and negative).” Here, students are required to understand the effect of replacing x with x+k, but this task can also be modified to test or teach function-building skills involving f(x)+k, kf(x), and f(kx) in a similar manner.

This classroom task gives students the opportunity to prove a surprising fact about quadrilaterals: that if we join the midpoints of an arbitrary quadrilateral to form a new quadrilateral, then the new quadrilateral is a parallelogram, even if the original quadrilateral was not.

This task assumes students are familiar with mixing problems. This approach brings out different issues than simply asking students to solve a mixing problem, which they can often set up using patterns rather than thinking about the meaning of each part of the equations.

This Illustrative Mathematics task deals with a rational expression which is built up from operations arising naturally in a context: adding the volumes of the fertilizer and the water, and dividing the volume of the fertilizer by the resulting sum.

This task could serve as an introduction to periodic functions and as a lead-in to sinusoidal functions. By visualizing the height of a plane that is moving along the circumference of a circle several times, students get the idea that output values of the height functions will repeat themselves after each complete revolution. They also connect the situation with key features on the graph, for example they interpret the midline and amplitude of the function as the height of the center of the circle and its radius.

The purpose of this task is to model the population data for London with a variety of different functions. In addition to the linear, quadratic, and exponential models, this task introduces an additional model, namely the logistic model.

A marine survey ship maps ocean depth by using sonar to reflect a sound pulse off the ocean floor. Figure A shows the ship's location at B on the surface of the ocean. The sonar apparatus aboard the ship is capable of emitting sound pulses in an arc measuring from 2 to 30 degrees. In two dimensions this arc is shown within Figure A by , and the emanating sound pulses are displayed by the dashed lines and the solid lines BA and BC.

The goal of this task is to construct and use an exponential model to approximate hard disk storage capacity on personal computers.

Motels and hotels hire people to clean the rooms after each evening's use. Develop a mathematical model for the cleaning schedule and use of cleaning resources. Your model should include consideration of such things as stay-overs, costs, number of rooms, number of rooms per floor, etc. Draft a letter to the manger of a major motel or hotel complex that recommends your model to help them in the management of their operation.

A great deal of preparation must take place before a movie can be filmed. Important sets and scenes need to be identified, resource needs must be calculated, and schedules must be arranged. The issue of the schedule is the focus of the modeling activities. A large studio has contacted your firm, and they wish to have a model to allow for scheduling a movie. You are asked to answer the questions below. You should provide examples and test cases to convince the movie executives that your model is effective and robust.

Mathematical modeling involves two equally important steps - building models based on real world situations and interpreting predictions made by those models back in the real world. This problem places equal emphasis on both steps.

We are at the start of the 2008 U.S. presidential elections, and one important area of debate is sure to be the national debt. As high school students, you have a particular interest in this subject since you are the people who will pay off or at least manage the national debt in the future. The rate at which the national debt changes depends on the difference between federal income (primarily taxes) and federal expenditures. Your first task is to build a model that can be used to help understand the national debt and make forecasts based on different assumptions. As usual, modeling involves a balance between so much complexity that the model may be intractable and so little complexity that it is unrealistic and useless. Your model needs, at the very least, to allow you to consider different tax policies and different expenditure policies.

Honeybees, along with a few other key animals, are critical to human existence on our planet. Along with honey production, these insects provide the vital role of pollination of many trees and plants that provide food for our survival. In 2007, the term Colony Collapse Disorder (CCD) was created to describe the decline of honeybee populations around the world.[1] Bee decline can be attributed to factors such as viruses, pesticides, predators, habitat destruction, and environmental conditions.

In 2014, the world saw the infectious Ebola virus spreading in western Africa. Throughout human history, epidemics have come and gone with some infecting and/or killing thousands and lasting for years and others taking less of a human toll. Some believe these events are just nature’s way of controlling the growth of a species while others think they could be a conspiracy or deliberate act to cause harm. This problem will most likely come down to how to expend (or not expend) scarce resources (doctors, containment facilities, money, research, serums, etc...) to deal with a crisis.

TuvaLabs Data Stories provide resources for teachers to engage students in rich discourse about an interesting topic and then allows students to come to conclusions using mathematical reasoning and tools.

TuvaLabs Data Stories provide resources for teachers to engage students in rich discourse about an interesting topic and then allows students to come to conclusions using mathematical reasoning and tools.

This deceptively simple task asks students to find the domain and range of a function from a given context. The function is linear and if simply looked at from a formulaic point of view, students might find the formula for the line and say that the domain and range are all real numbers. However, in the context of this problem, this answer does not make sense, as the context requires that all input and output values are non-negative integers, and imposes additional restrictions.

The task asks students to identify when two quantitative variables show evidence of a linear association, and to describe the strength and direction of that association.

The purpose of this task is to explore reflections in the context of paper folding.