Unit 3: Unit Rates and Percentages
Lesson 12: Percentages and Tape Diagrams

In previous lessons students used double number lines to reason about percentages. Double number lines show different percentages when a given amount is identified as 100%, and emphasize that percentages are a rate per 100. In this lesson they use tape diagrams. Tape diagrams are useful for seeing the connection between percentages and fractions.

Tape diagrams are useful in solving problems of the form A is B% of C when you are given two of the numbers and must find the third. When reasoning about percentages, it is important to indicate the whole as 100%, just as it is important to indicate the whole when working with fractions (MP6).

PhET provides fun, free, interactive, research-based simulations for math and science. The simulations are written in HTML5, and can be used online or downloaded to your computer. This is free for all students and teachers. There are hundreds of lesson plans and materials using the simulations.

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.

In this activity, students will solve a math problem and explain their thinking using SeeSaw.

This Nrich problem is a good challenge for those children who are competent with calculating using fractions and decimals.

In this Nrich problem students extend their understanding of percentages in a multi-step real world situation.

Place value is a concept that children have a difficult time grasping. The virtual manipulative at Toy Theater helps to teach place value by providing a visual representation for ones, tens, hundreds, thousands, ten thousand, and millions place value.

Students will fill in the blanks with > (greater than), < (less than).

Students will get a better understanding of what place value is and the different ways to express numbers.

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.

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 2: Points on the Number Line

In this second lesson on signed numbers, students learn about opposites​. First they revisit the context of temperature, represented on a vertical number line, extending previous work with interpreting equally spaced divisions to the negative part of the number line. The purpose of this activity is to reestablish the interpretation of distance on the number line in the context of negative numbers. They then create folded number lines to reason about opposites, which are numbers that are on opposite sides of 0 but the same distance from zero. Students will have more practice placing rational numbers of all kinds on the number line in future lessons. In this lesson, it is more important to focus on the concept of opposites than plotting different kinds of rational numbers.

Unit: Area and Surface Area
Lesson 11: Polygons

Students have worked with polygons in earlier grades and throughout this unit. In this lesson, students write a definition that characterizes polygons. There are many different accurate definitions for a polygon. The goal of this lesson is not to find the most succinct definition possible, but to articulate the defining characteristics of a polygon in a way that makes sense to students.

Another key takeaway for this lesson is that the area of any polygon can be found by decomposing it into triangles. The proof that all polygons are triangulable (not a word students need to know) is fairly sophisticated, but students can just take it as a fact for now. In observing and using this fact students look for and make use of structure (MP7).

Unit: Area and Surface Area
Lesson 13: Polyhedra

In this lesson, students learn about polyhedra and their nets. They also study prisms and pyramids as types of polyhedra with certain defining features.

Polyhedra can be thought of as the three-dimensional analog of polygons.

Here are some important aspects of polygons:

They are made out of line segments called edges.
Edges meet at a vertex.
The edges only meet at vertices.
Polygons always enclose a two-dimensional region.
Here is an analogous way to characterize polyhedra:

They are made out of filled-in polygons called faces.
Faces meet at an edge.
The faces only meet at edges.
Polyhedra always enclose a three-dimensional region.
Students do not need to memorize a formal definition of a polyhedron, but help them make sense of nets and surface area.

Unit 7: Rational Numbers
Lesson 1: Positive and Negative Numbers

Students in grade 6 have spent considerable time developing their understanding and fluency with positive numbers. In this lesson, students extend their thinking to negative numbers by exploring temperature and elevation. In these two contexts, zero represents a physical situation (freezing point of water, sea level) and numbers less than zero describe a physical state in the real world. Students abstract temperatures and elevations to positive and negative numbers on a number line (MP2).

Unit 6: Expressions and Equations
Lesson 4: Practice Solving Equations and Representing Situations with Equations

In this lesson, students consolidate their equation writing and solving skills. In the first activity they solve a variety of equations with different structures, and in the second they work to match equations to situations and solve them. Students may choose any strategy to solve equations, including drawing diagrams to reason about unknown quantities, looking at the structure of the equation, or doing the same thing to each side of the equation. They choose efficient tools and strategies for specific problems. This will help students develop flexibility and fluency in writing and solving equations.

In this activity, students will solve a math problem on patterns and explain their thinking using SeeSaw.