Students extend what they learned about solving equations in Grade 6. They learn to solve equations that require them to use both the addition and the multiplication properties of equality. They use what they know about solving equations such as 2x = 6 and x + 3 = 7 to solve equations such as 2x + 3 = 8. They connect solving problems using arithmetic to solving problems using equations. They solve equations containing both positive and negative rational numbers.Key ConceptsAddition property of equality: If a = b, then a + c = b + c.Multiplication property of equality: If a = b, then ac = bc.For any equation, add or subtract the same value from both sides of the equation and the equation will still be true.For any equation, multiply or divide both sides of the equation by the same value and the equation will still be true.In this lesson, students use both properties to solve equations. They then solve equations that contain both positive and negative rational numbers.Goals and Learning ObjectivesSolve equations using both the addition and multiplication properties of equality.Relate solving problems using arithmetic to solving problems using equations.Solve equations containing both positive and negative rational numbers.

Constructions and Angles

Unit Overview

Type of Unit: Concept

Prior Knowledge

Students should be able to:

Use a protractor and ruler.
Identify different types of triangles and quadrilaterals and their characteristics.

Lesson Flow

After an initial exploratory lesson involving a paper folding activity that gets students thinking in general about angles and figures in a context, the unit is divided into two concept development sections. The first section focuses on types of angles—adjacent, supplementary, complementary, and vertical—and how they are manifested in quadrilaterals. The second section looks at triangles and their properties, including the angle sum, and how this affects other figures.

In the first set of conceptual lessons, students explore different types of angles and where the types of angles appear in quadrilaterals. Students fold paper and observe the angles formed, draw given angles, and explore interactive sketches that test many cases. Students use a protractor and ruler to draw parallelograms with given properties. They explore sketches of parallelograms with specific properties, such as perpendicular diagonals. After concluding the investigation of the angle types, students move on to the next set.

In the second set of conceptual development lessons, students focus on triangles. Students again fold paper to create figures and certain angles, such as complementary angles.

Students draw, using a protractor and ruler, other triangles with given properties. Students then explore triangles with certain known and unknown elements, such as the number of given sides and angles. This process starts with paper folding and drawing and continues with exploration of interactive sketches. Students draw conclusions about which cases allow 0, 1, 2, or an infinite number of triangles. In the course of the exploration, students discover that the sum of the measure of the interior angles of a triangle is 180°. They also learn that the sum of the measures of the interior angles of a quadrilateral is 360°. They explore other polygons to find their angle sum and determine if there is a relationship to angle sum of triangles. The exploration concludes with finding the measure of the interior angles of regular polygons and speculating about how this relates to a circle.

Lastly, students solve equations to find unknown angle measures. Using their previous experience, students find the remaining angle measures in a parallelogram when only one angle measure is given. Students also play a game similar to 20 Questions to identify types of triangles and quadrilaterals. Having completed the remaining lessons, students have a four-day Gallery to explore a variety of problems.

The unit ends with a unit assessment.

Students learn more about the characteristics of parallelograms by folding paper and measuring the angles in a parallelogram. Students use a ruler and protractor to draw parallelograms with given properties. Then, students use a ruler and protractor to draw a rectangle.Key ConceptsOpposite angles of a parallelogram are congruent.Consecutive angles of a parallelogram are supplementary.Diagonals of a parallelogram bisect each other.Diagonals of a rectangle are congruent.Goals and Learning ObjectivesAccess prior knowledge of parallelograms.Understand that the sum of angle measures in any quadrilateral is 360°.Understand the relationship of the angles and diagonals in a parallelogram.Understand the relationship of the angles and diagonals in a rectangle.

Students learn to classify triangles by the size of the angles and the measures of the sides.Key ConceptsTriangles are polygons with three sides.Scalene triangles have all sides with a different length and all angles with a different measure.Isosceles triangles have two sides with the same length and two angles with the same measure.Equilateral triangles have all sides with the same length and all angles with the same measure.Acute triangles have all angles with a measure less than 90°.Obtuse triangles have one angle with a measure greater than 90°.Right triangles have one angle with a measure of 90°.ELL: Keep in mind that consistency at the beginning is very important as students begin to learn and apply math vocabulary.Goals and Learning ObjectivesExplore conditions that result in triangles.Identify types of triangles based on the measure of the angles or the measures of the sides.

Students learn how the diagonals of a rhombus are related. They use interactive sketches to learn about the properties of the angles and diagonals of squares, rectangles, rhombuses, parallelograms, and other quadrilaterals.Key ConceptsThe sum of the measures of the angles of all quadrilaterals is 360°.The alternate angles (nonadjacent angles) of rhombuses and parallelograms have the same measure.The measure of the angles of rectangles and squares is 90°.The consecutive angles of parallelograms and rhombuses are supplementary. This applies to squares and rectangles as well.The diagonals of a parallelogram bisect each other.The diagonals of a rectangle are congruent and bisect each other.The diagonals of a rhombus bisect each other and are perpendicular.Goals and Learning ObjectivesMeasure the angles formed by the intersection of the diagonals of a rhombus.Explore the relationships of the angles of different squares, rectangles, rhombuses, parallelograms, and other quadrilaterals.Explore the relationships of the diagonals of different squares, rectangles, rhombuses, parallelograms, and other quadrilaterals.

Lesson OverviewStudents explore different polygons by drawing them, measuring angles, and manipulating interactive sketches to find the angle sum for any polygon. Students also explore the angle measures in regular polygons.Key ConceptsThe angle sum in a triangle is 180°. A quadrilateral can be composed of two triangles, so the angle sum of a quadrilateral is 360°.The number of triangles that compose a polygon is two less than the number of sides (angles). The sum of the interior angles in a polygon with n sides is 180° (n – 2).Goals and Learning ObjectivesFind angle sums in polygons.Generalize to find the angle sum for any polygon.Find interior angle measures for regular polygons. 

Students explore properties of triangles. They fold paper to make a triangle and measure its angles. Students also draw triangles with given conditions.Key ConceptsThe sum of the measures of the angles in a triangle is always 180°.Given different side measures and/or angle measures, some scenarios will create triangles and others will not. Students explore various cases and draw conclusions about which conditions create triangles and why.Goals and Learning ObjectivesDraw triangles with given conditions.Find the sum of the measures of the angles of a triangle.Explore conditions that result in triangles.

Students learn about four types of angles: adjacent, vertical, supplementary, and complementary. They explore the relationships between these types of angles by folding paper, measuring angles with a protractor, and exploring interactive sketches.Key ConceptsAdjacent angles are two angles that share a common vertex and a common side, but do not overlap. Angles 1 and 2 are adjacent angles.Supplementary angles are two angles whose measures have a sum of 180°. Angles 3 and 4 are supplementary angles. Complementary angles are two angles whose measures have a sum of 90°. Angles 5 and 6 are complementary angles. Vertical angles are the opposite angles formed by the intersection of two lines. Vertical angles are congruent. Angles 1 and 2 are vertical angles. Angles 3 and 4 are also vertical angles.Goals and Learning ObjectivesMeasure angles with a protractor and estimate angle measures as greater than or less than 90°.Understand the definition of vertical, adjacent, supplementary, and complementary angles.Explore the relationships between these types of angles.

Gallery OverviewAllow students who have a clear understanding of the content thus far in the unit to work on Gallery problems of their choosing. You can then use this time to provide additional help to students who need review of the unit’s concepts or to assist students who may have fallen behind on work.Problem DescriptionsParallelogram to CubeStudents have a chance to review angle measurements in a parallelogram. Building the cube helps students see the transition from two-dimensional shapes and their relationship to three-dimensional figures.QuadrilateralsStudents investigate the possible quadrilaterals that can be made from any four given side lengths, focusing on those that can’t make a quadrilateral. Students also look at possible parallelograms with two sides given and possible rhombuses with four sides given.DiagonalsStudents further investigate diagonals in quadrilaterals. If the diagonals are perpendicular, is the figure a rhombus?TrapezoidsHow many right angles can a trapezoid have? How many congruent angles or congruent sides can it have? Can its diagonals be perpendicular or congruent? Students investigate possible trapezoids.More AnglesStudents explore three intersecting lines and the combinations of angles.Diagonals and AnglesThe sides of a parallelogram are extended beyond the vertices, and students explore which angles are congruent and which are supplementary. Students also explore the effect diagonals have on interior angles.Exterior AnglesStudents  explore the sum of exterior angles for several polygons and speculate about the results.Angles and SidesStudents explore the relationship between angles and sides in a triangle and discover that the longest side is opposite the largest angle, and the shortest side is opposite the smallest angle (and congruent sides are opposite congruent angles).Ratios and AnglesStudents explore the ratios of the legs of a right triangle to the angles in the triangle. Students see that there is a unique ratio for each angle, and vice versa. This is an informal look at trigonometry.Find the AngleStudents solve equations to find angle measures in polygons.TessellationsStudents explore quadrilateral tessellations and why they tessellate. Students also explore tessellations of pentagons and other polygons.

Students solve for missing angle measures by applying what they have learned about types of angles and the angle measures of polygons. Students do a pre-assessment at the end of the lesson.Key ConceptsThere are many defining characteristics for angles, triangles, quadrilaterals, and polygons. Students have discovered these properties throughout this unit and have investigated why they are true. These characteristics and properties will be looked at more formally in high school geometry.Goals and Learning ObjectivesSolve for missing angle measures in polygons.

Students critique and revise their work from the Self Check after receiving feedback. Students then take a quiz to review the goals of the unit.Key ConceptsStudents reflect on their work and apply what they've learned about the characteristics of geometric figures.Goals and Learning ObjectivesCritique and revise work on the Self Check.Apply skills learned in the unit.Understand the relationship of angles:Created by intersecting lines.Found in quadrilaterals, triangles, and polygons.

Students discuss what they know about shapes and their characteristics through a paper-folding activity that results in a parallelogram.Key ConceptsQuadrilaterals and triangles are classified by their different characteristics; the types of angles and sides define the shapes. While students are familiar with some of the characteristics of these shapes, they begin to explore other aspects of theses figures. Students review what they know about these shapes so far.Goals and Learning ObjectivesReview characteristics that describe quadrilaterals and triangles.Discuss what students know about these shapes.Explore other aspects of these shapes.

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.

Allow students who have a clear understanding of the content thus far in the unit to work on Gallery problems of their choosing. You can then use this time to provide additional help to students who need review of the unit's concepts or to assist students who may have fallen behind on work.Gallery OverviewRepresent a Math ProblemExplore ways to represent math problems—make an equation using a double number line, a ratio table, and a tape diagram. Then solve a problem comparing the prices of different types of nails using one of the representations.Research Ratios and RatesResearch how ratios and rates are related to proportional relationships. Watch the video tutorials and explore the other resources.Louisiana PurchaseSolve a problem about the Louisiana Purchase.

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.

Review the multiplication property of equality. Demonstrate the use of “ask myself” questions to understand a problem before solving it. Have students discuss the strategies that they can use when they feel stuck on a problem. Direct partners to solve a problem using a ratio table and equations, and then justify their solution in a presentation using the multiplication property of equality. Have each student write a Summary of the Mathematics in the lesson and work together to create a classroom summary.Key ConceptsStudents use the multiplication property of equality to justify their solution to a ratio problem.Goals and Learning ObjectivesBefore starting to work on a problem, make sense of the problem by using “ask myself” questions.Persevere in solving a problem even when feeling stuck.Solve a ratio problem using two different strategies.Link arithmetic and algebraic methods to solve a ratio problem.Use the multiplication property of equality to solve ratio problems

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.

Review the ways classroom habits and routines can strengthen students’ mathematical character. Explain what a Gallery is and how to choose a Gallery problem to solve. Direct students to choose one of three Gallery problems that introduce the unit’s technology resources. The three Gallery problems combine working with ratios and rates with the application resources available with this unit.Key ConceptsStudents understand that a Gallery gives them a choice of problems to solve. Students think about the features of the problems to use when choosing a problem. Students know how to work on a Gallery problem and present a solution.Goals and Learning ObjectivesKnow how to choose a problem from a Gallery.

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.