Using the same method for measuring friction that was used in the previous lesson (Discovering Friction), students design and conduct an experiment to determine if weight added incrementally to an object affects the amount of friction encountered when it slides across a flat surface. After graphing the data from their experiments, students can calculate the coefficients of friction between the object and the surface it moved upon, for both static and kinetic friction.

This lesson unit is intended to help assess how well students are able to interpret and use scale drawings to plan a garden layout. This involves using proportional reasoning and metric units.

This lesson unit is intended to help you assess how well students are able to: solve simple problems involving ratio and direct proportion; choose an appropriate sampling method; and collect discrete data and record them using a frequency table.

Students are introduced to the concept of energy conversion, and how energy transfers from one form, place or object to another. They learn that energy transfers can take the form of force, electricity, light, heat and sound and are never without some energy "loss" during the process. Two real-world examples of engineered systems light bulbs and cars are examined in light of the law of conservation of energy to gain an understanding of their energy conversions and inefficiencies/losses. Students' eyes are opened to the examples of energy transfer going on around them every day. Includes two simple teacher demos using a tennis ball and ball bearings. A PowerPoint(TM) presentation and quizzes are provided.

Students learn about kinetic and potential energy, including various types of potential energy: chemical, gravitational, elastic and thermal energy. They identify everyday examples of these energy types, as well as the mechanism of corresponding energy transfers. They learn that energy can be neither created nor destroyed and that relationships exist between a moving object's mass and velocity. Further, the concept that energy can be neither created nor destroyed is reinforced, as students see the pervasiveness of energy transfer among its many different forms. A PowerPoint(TM) presentation and post-quiz are provided.

Students are presented with a guide to rain garden construction in an activity that culminates the unit and pulls together what they have learned and prepared in materials during the three previous associated activities. They learn about the four vertical zones that make up a typical rain garden with the purpose to cultivate natural infiltration of stormwater. Student groups create personal rain gardens planted with native species that can be installed on the school campus, within the surrounding community, or at students' homes to provide a green infrastructure and low-impact development technology solution for areas with poor drainage that often flood during storm events.

Students apply the concepts of conduction, convection and radiation as they work in teams to solve two challenges. One problem requires that they maintain the warm temperature of one soda can filled with water at approximately human body temperature, and the other problem is to cause an identical soda can of warm water to cool as much as possible during the same 30-minute time period. Students design their engineering solutions using only common everyday materials, and test their devices by recording the water temperatures in their two soda cans every five minutes.

This lesson unit is intended to help teachers assess how well students are able to interpret percent increase and decrease, and in particular, to identify and help students who have the following difficulties: translating between percents, decimals, and fractions; representing percent increase and decrease as multiplication; and recognizing the relationship between increases and decreases.

Through multi-trial experiments, students are able to see and measure something that is otherwise invisible to them seeing plants breathe. Student groups are given two small plants of native species and materials to enclose them after watering with colored water. After being enclosed for 5, 10 and 15 minutes, teams collect and measure the condensed water from the plants' "breathing," and then calculate the rates at which the plants breathe. A plant's breath is known as transpiration, which is the flow of water from the ground where it is taken up by roots (plant uptake) and then lost through the leaves. Students plot volume/time data for three different native plant species, determine and compare their transpiration rates to see which had the highest reaction rate and consider how a plant's unique characteristics (leaf surface area, transpiration rate) might figure into engineers' designs for neighborhood stormwater management plans.

With the assistance of a few teacher demonstrations (online animation, using a radiometer and rubbing hands), students review the concept of heat transfer through convection, conduction and radiation. Then they apply an understanding of these ideas as they use wireless temperature probes to investigate the heating capacity of different materials sand and water under heat lamps (or outside in full sunshine). The experiment models how radiant energy drives convection within the atmosphere and oceans, thus producing winds and weather conditions, while giving students the hands-on opportunity to understand the value of remote-sensing capabilities designed by engineers. Students collect and record temperature data on how fast sand and water heat and cool. Then they create multi-line graphs to display and compare their data, and discuss the need for efficient and reliable engineer-designed tools like wireless sensors in real-world applications.

Students use everyday building materials sand, pea gravel, cement and water to create and test pervious pavement. They learn what materials make up a traditional, impervious concrete mix and how pervious pavement mixes differ. Groups are challenged to create their own pervious pavement mixes, experimenting with material ratios to evaluate how infiltration rates change with different mix combinations.

As a weighted plastic egg is dropped into a tub of flour, students see the effect that different heights and masses of the same object have on the overall energy of that object while observing a classic example of potential (stored) energy transferred to kinetic energy (motion). The plastic egg's mass is altered by adding pennies inside it. Because the egg's shape remains constant, and only the mass and height are varied, students can directly visualize how these factors influence the amounts of energy that the eggs carry for each experiment, verified by measurement of the resulting impact craters. Students learn the equations for kinetic and potential energy and then make predictions about the depths of the resulting craters for drops of different masses and heights. They collect and graph their data, comparing it to their predictions, and verifying the relationships described by the equations. This classroom demonstration is also suitable as a small group activity.

Four full-year digital course, built from the ground up and fully-aligned to the Common Core State Standards, for 7th grade Mathematics. Created using research-based approaches to teaching and learning, the Open Access Common Core Course for Mathematics is designed with student-centered learning in mind, including activities for students to develop valuable 21st century skills and academic mindset.

Getting Started

Type of Unit: Introduction

Prior Knowledge

Students should be able to:

Understand ratio concepts and use ratios.
Use ratio and rate reasoning to solve real-world problems.
Identify and use the multiplication property of equality.

Lesson Flow

This unit introduces students to the routines that build a successful classroom math community, and it introduces the basic features of the digital course that students will use throughout the year.

An introductory card sort activity matches students with their partner for the week. Then over the course of the week, students learn about the routines of Opening, Work Time, Ways of Thinking, Apply the Learning (some lessons), Summary of the Math, Reflection, and Exercises. Students learn how to present their work to the class, the importance of students’ taking responsibility for their own learning, and how to effectively participate in the classroom math community.

Students then work on Gallery problems, to further explore the resources and tools and to learn how to organize their work.

The mathematical work of the unit focuses on ratios and rates, including card sort activities in which students identify equivalent ratios and match different representations of an equivalent ratio. Students use the multiplication property of equality to justify solutions to real-world ratio problems.

Discuss the important ways that listeners contribute to mathematical discussions during Ways of Thinking presentations. Then use ratio and rate reasoning to solve a problem about ingredients in a stew.Key ConceptsStudents find the unit rate of a ratio situation.Goals and Learning ObjectivesContribute as listeners during the Ways of Thinking discussion.Understand the concept of a unit rate that is associated with a ratio.Use rate reasoning to solve real-world problems.

Students use ratio cards to find equivalencies and form partnerships for the week. As a class, students discuss and decide on classroom norms.Give each student a ratio card. Instruct students to find a classmate whose card has a ratio that is equivalent to theirs. Classmates with equivalent ratios are now partners for the week. With the class, discuss and decide on classroom norms, or rules. Tell students how to access the application they will use this year.Key ConceptsStudents understand that ratio relationships are multiplicative. They use ratio tables to show ratio relationships.Goals and Learning ObjectivesDistinguish between ratio tables and tables that do no show equivalent ratios.Understand how ratio tables are used to solve ratio problems.Use the basic features of the application.Create and understand the classroom norms.Use mathematical reasoning to justify an answer.

Students discuss classroom routines and expectations, work with partners to present their work matching different representations of a ratio situation, and then prepare math summaries.Introduce classroom routines and expectations prior to the full mathematics lesson. Ask students to discuss how to clearly present their work to their classmates. Model an example of partner work, and then have students work with their partners to match different representations of a ratio situation. Read and discuss a Summary of the Math, and then have students write Reflections about their wonderings.Key ConceptsStudents match a data card with its corresponding ratio, decimal, fraction, percent, and description of the relationship in words. Students construct viable arguments for their matches and critique the reasoning of their partner and other classmates.Goals and Learning ObjectivesDescribe the classroom routines and expectations.Consider how to present work clearly to classmates.Collaborate with a partner.Critique a partner’s reasoning.Connect different representations of a ratio situation.

Proportional Relationships

Type of Unit: Concept

Prior Knowledge

Students should be able to:

Understand what a rate and ratio are.
Make a ratio table.
Make a graph using values from a ratio table.

Lesson Flow

Students start the unit by predicting what will happen in certain situations. They intuitively discover they can predict the situations that are proportional and might have a hard time predicting the ones that are not. In Lessons 2–4, students use the same three situations to explore proportional relationships. Two of the relationships are proportional and one is not. They look at these situations in tables, equations, and graphs. After Lesson 4, students realize a proportional relationship is represented on a graph as a straight line that passes through the origin. In Lesson 5, they look at straight lines that do not represent a proportional relationship. Lesson 6 focuses on the idea of how a proportion that they solved in sixth grade relates to a proportional relationship. They follow that by looking at rates expressed as fractions, finding the unit rate (the constant of proportionality), and then using the constant of proportionality to solve a problem. In Lesson 8, students fine-tune their definition of proportional relationship by looking at situations and determining if they represent proportional relationships and justifying their reasoning. They then apply what they have learned to a situation about flags and stars and extend that thinking to comparing two prices—examining the equations and the graphs. The Putting It Together lesson has them solve two problems and then critique other student work.

Gallery 1 provides students with additional proportional relationship problems.

The second part of the unit works with percents. First, percents are tied to proportional relationships, and then students examine percent situations as formulas, graphs, and tables. They then move to a new context—salary increase—and see the similarities with sales taxes. Next, students explore percent decrease, and then they analyze inaccurate statements involving percents, explaining why the statements are incorrect. Students end this sequence of lessons with a formative assessment that focuses on percent increase and percent decrease and ties it to decimals.

Students have ample opportunities to check, deepen, and apply their understanding of proportional relationships, including percents, with the selection of problems in Gallery 2.

Students analyze the graph of a proportional relationship in order to find the approximate constant of proportionality, to write the related formula, and to create a table of values that lie on the graph.Key ConceptsThe constant of proportionality determines the steepness of the straight-line graph that represents a proportional relationship. The steeper the line is, the greater the constant of proportionality.On the graph of a proportional relationship, the constant of proportionality is the constant ratio of y to x, or the slope of the line.A proportional relationship can be represented in different ways: a ratio table, a graph of a straight line through the origin, or an equation of the form y = kx, where k is the constant of proportionality.Goals and Learning ObjectivesIdentify the constant of proportionality from a graph that represents a proportional relationship.Write a formula for a graph that represents a proportional relationship.Make a table for a graph that represents a proportional relationship.Relate the constant of proportionality to the steepness of a graph that represents a proportional relationship (i.e., the steeper the line is, the greater the constant of proportionality).

Students connect percent to proportional relationships in the context of sales tax.Key ConceptsWhen there is a constant tax percent, the total cost for items purchase—including the price and the tax—is proportional to the price.To find the cost, c , multiply the price of the item, p, by (1 + t), where t is the tax percent, written as a decimal: c = p(1 + t).The constant of proportionality is (1 + t) because of the structure of the situation:c = p + tp = p(1 + t).Because of the distributive property, multiplying the price by (1 + t) means multiplying the price by 1, then multiplying the price by t, and then taking the sum of these products.Goals and Learning ObjectivesFind the total cost in a sales tax situation.Understand that a proportional relationship only exists between the price of an item and the total cost of the item if the sales tax is constant.Find the constant of proportionality in a sales tax situation.Make a graph of an equation showing the relationship between the price of an item and the total amount paid.