Unit 2: Introducing Ratios
Lesson 4: Color Mixtures

This is the second of two lessons that help students make sense of equivalent ratios through physical experiences. In this lesson, students mix different numbers of batches of a recipe for green water by combining blue and yellow water (created ahead of time with food coloring) to see if they produce the same shade of green. They also change the ratio of blue and yellow water to see if it changes the result. The activities here reinforce the idea that scaling a recipe up (or down) requires scaling the amount of each ingredient by the same factor (MP7). Students continue to use discrete diagrams as a tool to represent a situation.

For students who do not see color, the lesson can be adapted by having students make batches of dough with flour and water. 1 cup of flour to 5 tablespoons of water makes a very stiff dough, and 1 cup of flour to 6 tablespoons of water makes a soft (but not sticky) dough. In this case, doubling a recipe yields dough with the same tactile properties, just as doubling a colored-water recipe yields a mixture with the same color. The invariant property is stiffness rather than color. The principle that equivalent ratios yield products that are identical in some important way applies to both types of experiments.

Unit 2: Introducing Ratios
Lesson 5: Defining Equivalent Ratios

Previously, students understood equivalent ratios through physical perception of different batches of recipes. In this lesson, they work with equivalent ratios more abstractly, both in the context of recipes and in the context of abstract ratios of numbers. They understand and articulate that all ratios that are equivalent to a:b can be generated by multiplying both a and b by the same number (MP6).

By connecting concrete quantitative experiences to abstract representations that are independent of a context, students develop their skills in reasoning abstractly and quantitatively (MP2). They continue to use diagrams, words, or a combination of both for their explanations. The goal in subsequent lessons is to develop a general definition of equivalent ratios.

This Nrich activity encourages students to notice and wonder and to engage in mathematical discussions. Students then use graphs to represent the situations.

In this Nrich activity, students explore combinations in a situation with constraints. The combinations are complicated by sets and subsets. The situation uses a Venn Diagram.

Students sort and classify triangles based on similarities. Students are asked to discuss their decisions in this Nrich activity.

This Nrich can be very engaging. It has the ability to introduce pupils to some logical reasoning as well as being solvable by trial and improvement.

This Nrich problem is a good opportunity to encourage children to have a system for finding all possible solutions. It is also an ideal context in which to help children deepen their understanding of factors and multiples in a playful environment.

Unit 7: Rational Numbers
Lesson 5: Using Negative Numbers to Make Sense of Contexts

In this lesson, students are introduced to conventions for using signed numbers to represent money spent and received, as well as inventory gained and lost. While money contexts can be represented without signed numbers, there are many situations that are more efficiently modeled by signed numbers. For example, if a person has $50 in the bank and writes a $20 check, we can represent the balance as 50 - 20. If they had written an $80 check, we can still write the balance as 50-80, as long as we have adopted the convention that negative numbers represent what the person owes the bank (and assuming the bank allows overdrafts). Since students do not operate on signed numbers in this grade, this lesson is simply an introduction to the convention of using signed numbers to represent a change in money or a change in inventory, an important convention in modeling financial situations with mathematics (MP4). In a later lesson, students will be introduced to the idea of an account balance. In grade 7, students will study addition and subtraction of signed numbers and apply those concepts in accounting situations.

The lesson is to be used to review the six addition strategies the first grade class has already learned. The strategies are: Add Zero Facts, Count On Facts, Make 10 Facts, Doubles Facts, Add 10 Facts, and Add 9 Facts. It also touches on the equal sign.

Unit 7: Rational Numbers
Lesson 10: Interpreting Inequalities

In this final lesson on inequalities, students explore situations in which some of the solutions to inequalities do not make sense in the situation’s context. Students learn to think carefully about a situation’s constraints when coming up with reasonable solutions to an inequality. Students also see that inequalities can represent a comparison of two or more unknown quantities.

Unit 7: Rational Numbers
Lesson 11: Points on the Coordinate Plane

In earlier lessons, students extended the number line to include negative numbers. In this lesson, students extend the coordinate axes to expand the coordinate plane. In a previous unit, students worked in the coordinate plane when they examined ratio and other relationships between two quantities with positive values. They now consider an expanded coordinate plane where negative numbers appear on both the vertical and horizontal axes. The crossing axes create the four regions of the coordinate plane, called quadrants. In this first lesson on the coordinate plane, students extend their understanding of the coordinate plane to points with negative coordinates. They gain experience by choosing and plotting points in order to hit targets or to maneuver through mazes in all four quadrants of the coordinate plane.

Unit 7: Rational Numbers
Lesson 12: Constructing the Coordinate Plane

In this lesson, students explore the idea of scaling axes appropriately to accommodate data where coordinates are rational numbers. Students attend to precision as they plan where to place axes on a grid and how to scale them to represent data clearly (MP6). In an optional activity, students practice working with coordinates in all 4 quadrants as they navigate a maze on a coordinate grid. This lesson gives students the opportunity to develop fluency with plotting coordinates in all 4 quadrants and scaling axes to fit data that is essential for the context-driven work over the next few lessons.

This Nrich low threshold high ceiling task is accessible to everyone. It gives children the chance to share the way they picture (visualize) numbers and their methods of counting. One of the key features of this task is that it can be interpreted differently, depending on the image, so that children can decide for themselves whether they are counting individual fruit, cartons of fruit... Therefore there may also be an opportunity for children to develop their estimation skills as well as appreciating different ways of counting.

This Nrich problem encourages children to work together to develop a method for finding a solution which will always work. It will also help to reinforce understanding of odd and even numbers.

Students gain an understanding of the factors that affect wind turbine operation. Following the steps of the engineering design process, engineering teams use simple materials (cardboard and wooden dowels) to build and test their own turbine blade prototypes with the objective of maximizing electrical power output for a hypothetical situation—helping scientists power their electrical devices while doing research on a remote island. Teams explore how blade size, shape, weight and rotation interact to achieve maximal performance, and relate the power generated to energy consumed on a scale that is relevant to them in daily life. A PowerPoint® presentation, worksheet and post-activity test are provided.

Unit 8: Data Sets and Distribution
Lesson 14: Comparing Mean and Median

In this lesson, students investigate whether the mean or the median is a more appropriate measure of the center of a distribution in a given context. They learn that when the distribution is symmetrical, the mean and median have similar values. When a distribution is not symmetrical, however, the mean is often greatly influenced by values that are far from the majority of the data points (even if there is only one unusual value). In this case, the median may be a better choice.

At this point, students may not yet fully understand that the choice of measures of center is not entirely black and white, or that the choice should always be interpreted in the context of the problem (MP2) and should hinge on what insights we seek or questions we would like to answer. This is acceptable at this stage. In upcoming lessons, they will have more opportunities to include these considerations into their decisions about measures of center.

Unit 9: Putting It All Together
Lesson 2: If Our Class Were the World

This lesson is optional. In this lesson, students look at ratios of different populations in the world and determine what their class would be like if its ratios were equivalent (MP1, MP2). In the process, they again work with percentages that are not whole numbers, using knowledge gained in a previous unit. Moreover, the ratios will be “close” to being equivalent because the exact world population is not known and all populations need to be whole numbers (MP6). The activities in this lesson could take anywhere from one to four days, depending on how much time is available and how far the class takes it. Earlier activities are needed for later ones in this lesson. A variant on this activity involves developing, administering, and analyzing a survey: If the school were our class. Students brainstorm some questions they would like to know about the students in their school. Questions might include:

How many people in the school play an instrument?
How many people in the school eat school lunches?
How many people in the school ride the bus to school?
How many people in the school have a cell phone?
How many people in the school plan to attend a four-year college or university?
How many people in the school were born outside of this state?
How many people in the school have traveled outside of this country?
As with all lessons in this unit, all related standards have been addressed in prior units; this lesson provides an optional opportunity to go more deeply and make connections between domains.

Unit 9: Putting It All Together
Lesson 4: How Do We Choose?

This lesson is optional. This is the first of three lessons that explore the mathematics of voting: democratic processes for making decisions. The activities in these lesson build on each other. Doing all of the activities in the three lessons would take more than three class periods—possibly as many as five. It is up to the teacher how much time to spend on this topic. It is not necessary to do the entire set of activities to get some benefit from them, although more connections are made the farther one gets. As with all lessons in this unit, all related standards have been addressed in prior units; this lesson provides an optional opportunity to go more deeply and make connections between domains.

The activities in this lesson are about voting on issues where there are two choices. Students use proportional reasoning concepts and skills developed in grade 6 to compare voting results of two groups, to determine whether an issue wins an election with a supermajority rule, and discover that a few people can determine the results of an election when very few people vote.

Most of the activities use students’ skills from earlier units to reason about ratios and proportional relationships (MP2) in the context of real-world problems (MP4). While some of the activities do not involve much computation, they all require serious thinking. In many activities, students have to make choices of how to assign votes and justify their methods (MP3).

Most importantly, this lesson addresses topics that are important for citizens in a democracy to understand. Teachers may wish to collaborate with a civics or government teacher to learn how the fictional middle-school situations in this lesson relate to real-world elections.

Unit 9: Putting It All Together
Lesson 6: Picking Representatives

This lesson is optional. The five activities in this third lesson on the mathematics on voting return to the situation of an election with two choices. However, rather than directly choosing the result, voters elect representatives, each of whom then casts a single vote for all the people they represent. The activities explore ways to “share” the representatives fairly between groups of people. In the first activity, numbers have been designed so that representatives (or computers) can be shared exactly proportionally between several groups. In later activities, it’s impossible to share representatives fairly; students may use division with decimal quotients or with remainders to try to find the least unfair way. The final activity asks students to gerrymander several districts: to divide it into sections in two ways to influence the final voting result in opposite ways. The mathematics here involves geometric properties of shapes on maps: area and connectedness, as well as some proportional reasoning.

Most of the activities use students’ skills from earlier units to reason about ratios and proportional relationships (MP2) in the context of real-world problems (MP4). While some of the activities do not involve much computation, they all require serious thinking and decision making (MP3).

Most importantly, this lesson addresses topics that are important for citizens in a democracy to understand. Teachers may wish to collaborate with a civics or government teacher to learn how the fictional middle-school situations in this lesson relate to real-world elections.