Putting Math to Work

Type of Unit: Problem Solving

Prior Knowledge

Students should be able to:

Solve problems involving all four operations with rational numbers.
Write ratios and rates.
Write and solve proportions.
Solve problems involving scale.
Write and solve equations to represent problem situations.
Create and interpret maps, graphs, and diagrams.
Use multiple representations (i.e., tables, graphs, and equations) to represent problem situations.
Calculate area and volume.
Solve problems involving linear measurement.

Lesson Flow

Students apply and integrate math concepts they have previously learned to solve mathematical and real-world problems using a variety of strategies. Students have opportunities to explore four real-world situations involving problem solving in a variety of contexts, complete a project of their choice, and work through a series of Gallery problems.

First, students utilize their spatial reasoning and visualization skills to find the least number of cubes needed to construct a structure when given the front and side views. Then, students select a project to complete as they work through this unit to refine their problem-solving skills. Students explore the relationship between flapping frequency, amplitude, and cruising speed to calculate the Strouhal number of a variety of flying and swimming animals. After that, students explore the volume of the Great Lakes, applying strategies for solving volume problems and analyzing diagrams. Next, students graphically represent a virtual journey through the locks of the Welland Canal, estimating the amount of drop through each lock and the distance traveled. Students have a day in class to work on their projects with their group.

Then, students have two days to explore Gallery problems of their choosing. Finally, students present their projects to the class.

Students first create a diagram that represents the distance a ship drops in each of a series of locks. Students create their diagrams based on a video of an actual ship traveling through the locks. Students need to use contextual clues in order to determine the relative drops in each of the locks.Key ConceptsStudents are expected to use the mathematical skills they have acquired in previous lessons or in previous math courses. The lessons in this unit focus on developing and refining problem-solving skills.Students will:Try a variety of strategies to approaching different types of problems.Devise a problem-solving plan and implement their plan systematically.Become aware that problems can be solved in more than one way.See the value of approaching problems in a systematic manner.Communicate their approaches with precision and articulate why their strategies and solutions are reasonable.Make connections between previous learning and real-world problems.Create efficacy and confidence in solving challenging problems in a real-world setting.Goals and Learning ObjectivesRead and interpret maps, graphs, and diagrams.Solve problems that involve linear measurement.Estimate length.Critique a diagram.

Working With Rational Numbers

Type of Unit: Concept

Prior Knowledge

Students should be able to:

Compare and order positive and negative numbers and place them on a number line.
Understand the concepts of opposites absolute value.

Lesson Flow

The unit begins with students using a balloon model to informally explore adding and subtracting integers. With the model, adding or removing heat represents adding or subtracting positive integers, and adding or removing weight represents adding or subtracting negative integers.

Students then move from the balloon model to a number line model for adding and subtracting integers, eventually extending the addition and subtraction rules from integers to all rational numbers. Number lines and multiplication patterns are used to find products of rational numbers. The relationship between multiplication and division is used to understand how to divide rational numbers. Properties of addition are briefly reviewed, then used to prove rules for addition, subtraction, multiplication, and division.

This unit includes problems with real-world contexts, formative assessment lessons, and Gallery problems.

Students learn the metric units engineers use to measure mass, distance (or length) and volume. They make estimations using these units and compare their guesses with actual values. To introduce the concepts, the teacher needs access to a meter stick, a one-liter bottle, a glass container that measures milliliters and a gram scale.

In this activity, students will make go-carts and measure the speed, then change different factors of the experiment and see how the speed changes. In day 1, students will design and build a cart based on a specified set of materials, and then complete several trials to test the cart by rolling it down a ramp. Through discussion and journaling students will share their designs and compare the speeds of carts. In day 2, students will experiment with ways to increase or decrease the speed of their cart.

Students design, build and test model race cars made from simple materials (lifesaver-shaped candies, plastic drinking straws, Popsicle sticks, index cards, tape) as a way to explore independent, dependent and control variables. They measure the changes in distance travelled with the addition of mass to the vehicles. Students also practice the steps of the engineering design process by brainstorming, planning, building, testing, and improving their "mint-mobiles."

This is a PBL lesson where students explore the Physics behind motion of objects.

Students collaborate and to create a video showing the relationship between time, distance, and average velocity.

Graphical and analytical analysis is employed using google sheets..

Definition of perimeter of polygon, list of items needed when calculating perimeter of polygon on coordinate plane; image of polygon on coordinate plane with additional information in hotspots. Detailed example of calculation of perimeter.

This is a PBL lesson where students explore the Physics behind motion of objects. Students will collaborate and come up with a video showing the difference between distance and displacement, speed and velocity and acceleration.

Lesson plan using a Cyberchase activity, students become more familiar and proficient at metric unit measurement by examining measurement and predictions concerning indoor tracks (200 m) and outdoor tracks (400 m). This CYBERCHASE activity is motivated by a Cyberchase For Real segment in which Bianca competes in her first track meet, but is not convinced that the staggered distances of the lanes on the indoor track are all the same distance.

This original resource was a great activity using technology. I would use this as well as have students use the interactive map to find distance between towns/cities they may be more familiar with.

In this design challenge, students will harness the power of the sun to design, construct and evaluate a solar-powered model car of their creation. Students will utilize the design process and select an optimal gear ratio and components for their car.

Students strengthen their communicate skills about measurements by learning the meaning of base units and derived units, including speed one of the most common derived units (distance/time). Working in groups, students measure the time for LEGO MINDSTORMS(TM) NXT robots to move a certain distance. The robots are started and stopped via touch sensors and programmed to display the distance traveled. Using their collected data, students complete a worksheet to calculate the robots' (mean/average) speeds at given motor powers.

Students work as engineers to design and test trebuchets (in this case LEGO® MINDSTORMS® robots) that can launch objects. During the testing stage, they change one variable at a time to study its effect on the outcome of their designs. Specifically, they determine how far objects travel depending on their weights. As students learn about the different components of robot design and the specific function controls, they determine what design features are important for launching objects.

This applet simulates two runners moving along a track and creates a graph of the time-versus-distance relationship of their motion. Students then observe the simulated races as they happen and relate the changing positions of the two runners to dynamic representations that change as the events occur. Students can predict the effects on the graph of changing the starting position or the length of the stride of either runner. They can observe and analyze how a change in one variable, such as length of stride, relates to a change in speed. This computer simulation uses a familiar context that students understand from daily life, and the technology allows them to analyze the relationships in this context deeply because of the ease of manipulating the environment and observing the changes that occur.

The graph displayed shows relative flight distance of birds vs. the number of generations since the birds have been present in urban areas. Each data point represents a different kind of bird, and the line is a linear regression line for this data.

This is a lesson about the vertical dimension of the atmosphere and includes four activities. Activity 1 Introduces concepts related to distance, including length and height and units of measurement. Students are asked to make comparisons of distances. In activity 2, students learn about the vertical profile of the atmosphere. They work with a graph and plot the heights of objects and the layers of the atmosphere: troposphere, stratosphere, mesosphere, thermosphere, and exosphere. In activity 3, students learn about other forms of visual displays using satellite imagery. They compare images of the same weather feature, a hurricane, using two different images from MODIS and CALIPSO. One image is looking down on the hurricane from space, the other looks through the hurricane to display a profile of the hurricane. Activity 4 reinforces the concept of the vertical nature of the atmosphere. Students will take a CALIPSO satellite image that shows a profile of the atmosphere and use this information to plot mountains and clouds on their own graph of the atmosphere. The recommended order for the activities is to complete the first two activities on day one, and the second two activities on day two. Each day will require approximately 1 to 1.5 hours.

Students learn about electric motors and rotational sensors. They learn that motors convert electrical energy to mechanical energy and typically include rotational sensors to enable distance measuring. They also learn the basics about gear trains and gear ratios. Students create a basic program using the LEGO MINDSTORMS(TM) NXT interface to control a motor to move a small robot. Then, through a 10-minute mini-activity, they make measurements and observations to test a LEGO rotation sensor's ability to measure distance in rotations. This prepares them for the associated activity during which they calculate how many wheel rotations are needed to travel a distance. A PowerPoint® presentation, worksheet and pre/post quizzes are provided.