Students use estimation or other methods to place the decimal points in products and quotients. They review the algorithms for the four basic decimal operations and solve multistep word problems involving decimal operations.Key ConceptsThe algorithms for whole-number operations can be extended to decimal operations. Students learned the algorithms for decimal operations in Grade 5. By the end of Grade 6, they should be fluent with these operations.For decimal addition and subtraction, once the decimal points of the addends are aligned (which aligns like place values), the algorithms are the same as for whole numbers. The decimal point in the sum or difference goes directly below the decimal point in the numbers that were added or subtracted.For decimal multiplication and division, one method is to ignore the decimal points and apply the whole-number algorithms. Then use estimation or some other method to place the decimal point in the answer.Goals and Learning ObjectivesReview and practice the algorithms for all four decimal operations.Solve real-world problems involving decimal operations.

Students explore methods for dividing a whole number by a fraction.Key ConceptsIn earlier grades, students learned to think of a whole number division problem, such as 8 ÷ 4, in terms of two types of equal groups.Divisor as the Number of Groups Divide 8 into 4 equal groups and find the size of each group.Divisor as the Group Size Divide 8 into groups of 4 and find the number of groups.To divide a fraction by a whole number in Lesson 2, students used the first interpretation. For example, to find 89 ÷ 4, they divided 8 ninths into 4 equal groups and found that there were 2 ninths in each group.To divide a whole number by a fraction, the second interpretation is most helpful. For example, to find 3 ÷ 34, we find the number of groups of 3 fourths in 3 wholes. The diagram in the Opening shows that there are 4 groups, so 3 ÷ 34 = 4.Just as with whole number division, the quotient when a whole number is divided by a fraction is not always a whole number. Below is a model for 2 ÷ 35. The model shows that there are 3 groups of 3 fifths in 2 wholes plus 13 of another group (13 of a group of 3 fifths is 1 fifth). Therefore, 2 ÷ 35 = 313. Notice that once we have divided the 2 wholes into fifths, we are finding the number of groups of 3 fifths in 10 fifths. This is simply 10 ÷ 3.These models can help explain that the “multiply by the reciprocal” method of dividing a whole number by a fraction works. To find 2 ÷ 35, we can multiply 2 by 5 to find the total number of fifths in 2 and then divide the result (10) by 3 to find the number of groups of 3 of these fifths in 2. So, 2÷35=2×53=2×53.ELL: Encourage students to verbalize their explanations. To help students gain confidence and increase their understanding, allow those that share the same language of origin to speak in small groups using their prefered language.Goals and Learning ObjectivesUse models and other methods to divide a whole number by a fraction.

Getting Started

Type of Unit: Introduction

Prior Knowledge

Students should be able to:

Solve and write numerical equations for whole number addition, subtraction, multiplication, and division problems.
Use parentheses to evaluate numerical expressions.
Identify and use the properties of operations.

Lesson Flow

In this unit, students are introduced to the rituals and routines that build a successful classroom math community and they are introduced to the basic features of the digital course that they 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 lesson routines: Opening, Work Time, Ways of Thinking, Apply the Learning, Summary of the Math, and Reflection. Students learn how to present their work to the class, the importance of 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 program’s technology resources and tools and learn how to organize their work.

The mathematical work of the unit focuses on numerical expressions, including card sort activities in which students identify equivalent expressions and match an expression card to a word card that describes its meaning. Students use the properties of operations to identify equivalent expressions and to find unknown values in equations.

Students participate in an icebreaker activity, finding a classmate whose card contains an expression equivalent to the expression on their own card. The resulting student pairs will be partners for this unit. Students spend time exploring the digital course. They learn new symbols for multiplication and detect possible errors in evaluating numeric expressions. The class discusses and decides upon norms for math class.Key ConceptsStudents evaluate numerical expressions and identify equivalent expressions. They explore why the order of operations affects calculation results and how to use parentheses to clearly describe the order of the operations.Goals and Learning ObjectivesEvaluate numerical expressions.Understand the reason for the order of operations and how to use parentheses in numerical expressions.Use the basic features of the application.Create and understand the classroom norms.Use mathematical reasoning to justify an answer.PreparationPrint out the Expressions Icebreaker cards. Select the number of pairs of Partner 1 and Partner 2 cards needed for your class. Shuffle the cards before distributing to students.Write on the board or chart paper: Find a classmate whose card has an expression that is equivalent to the expression on your card.Choose a hand signal or phrase for common activities, such as putting technology away and focusing on the teacher.

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.Gallery DescriptionRepresent a Math ProblemStudents explore the number line tool and the double number line tool. They use the number line tool to solve a problem about the weights of a cheetah and a fisher cat.Research ExpressionsStudents learn the difference between numerical expressions and variable expressions. They watch video tutorials, review worked examples, use the Glossary, and explore other resources.Fish TankStudents create diagrams and use text and images as they solve a problem about the size of a fish tank.

Students are introduced to classroom routines and expectations, and complete a full mathematics lesson. The class discusses how to clearly present work to classmates. Partner work is modeled, and partners then work to match numerical expressions to corresponding word descriptions. Students read and discuss a summary of the math in the lesson, and then write a reflection about their thoughts.Key ConceptsStudents match a numerical expression to its corresponding description in words. Students interpret parentheses and brackets in numerical expressions and they construct viable arguments and critique the reasoning of others. Students learn to use the exponent 2 to represent squaring.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 a numerical expression to its corresponding word description.Learn to use an exponent of 2 to represent squaring.

Students are introduced to classroom routines and expectations, and complete a full mathematics lesson. The class discusses how to clearly present work to classmates. Partner work is modeled, and partners then work to match numerical expressions to corresponding word descriptions. Students read and discuss a summary of the math in the lesson, and then write a reflection about their thoughts.Key ConceptsStudents match a numerical expression to its corresponding description in words. Students interpret parentheses and brackets in numerical expressions and they construct viable arguments and critique the reasoning of others. Students learn to use the exponent 2 to represent squaring.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 a numerical expression to its corresponding word description.Learn to use an exponent of 2 to represent squaring.

Students review the ways classroom habits and routines can strengthen their mathematical character. Students learn what a Gallery is and how to choose a Gallery problem to work on. They then choose one of three Gallery problems that introduce the unit’s technology resources. The three Gallery problems combine working with expressions with the resources available with this unit.Key ConceptsUnderstand that a Gallery gives students a choice of several problems. Understand what to consider when choosing a problem. Know how to work on a Gallery problem and how to present work on gallery problems.Goals and Learning ObjectivesKnow how to choose a problem from a Gallery.

The class reviews the properties of operations. The use of “ask myself” questions to make sense of problems and persevere is modeled. Students review things to do when they feel stuck on a problem. Finally, students use the properties of operations to evaluate expressions.Key ConceptsStudents use the properties of operations to justify whether two expressions are equivalent.Goals and Learning ObjectivesTo start to work on a problem, make sense of the problem by using “ask myself” questions.Persevere in solving a problem even when feeling stuck.Use the properties of operations to evaluate expressions.

Students discuss as a class the important ways that listeners contribute to mathematical discussions during Ways of Thinking presentations. Students then use the properties of operations to find the value of each fruit used in equations.Key ConceptsStudents use the properties of operations to find the value of each fruit used in different equations. By considering several equations, students can match each of the 10 fruits to the whole numbers 0 through 9. This work helps students see why representing unknown numbers with letters is useful.Goals and Learning ObjectivesContribute as listeners during the Ways of Thinking discussion.Identify the whole numbers that make an equation true.Use the properties of operations, when appropriate, to justify which whole numbers represent unknown values.

Putting Math to Work

Type of Unit: Problem Solving

Prior Knowledge

Students should be able to:

Solve problems with rational numbers using all four operations.
Write ratios and rates.
Use a rate table to solve problems.
Write and solve proportions.
Use multiple representations (e.g., tables, graphs, and equations) to display data.
Identify the variables in a problem situation (i.e., dependent and independent variables).
Write formulas to show the relationship between two variables, and use these formulas to solve for a problem situation.
Draw and interpret graphs that show the relationship between two variables.
Describe graphs that show proportional relationships, and use these graphs to make predictions.
Interpret word problems, and organize information.
Graph in all quadrants of the coordinate plane.

Lesson Flow

As a class, students use problem-solving steps to work through a problem about lightning. In the next lesson, they use the same problem-solving steps to solve a similar problem about lightning. The lightning problems use both rational numbers and rates. Students then choose a topic for a math project. Next, they solve two problems about gummy bears using the problem-solving steps. They then have 3 days of Gallery problems to test their problem-solving skills solo or with a partner. Encourage students to work on at least one problem individually so they can better prepare for a testing situation. The unit ends with project presentations and a short unit test.

In this lesson and the next, student groups make their presentations, provide feedback about other students' presentations, and get evaluated on their listening skills.Key ConceptsIn this culminating event, students must present their project plan and solution to the class. The presentation allows students to explain their problem-solving plan, to communicate their reasoning, and to construct a viable argument about a mathematical problem. Students also listen to other project presentations and provide feedback to the presenters. Listeners have the opportunity to critique the mathematical reasoning of others.Goals and Learning ObjectivesPresent project to the class.Give feedback on other project presentations.Exhibit good listening skills.

Student groups make their presentations, provide feedback for other students' presentations, and get evaluated on their listening skills.Key ConceptsIn this culminating event, students must present their project plan and solution to the class. The presentation allows students to explain their problem-solving plan, to communicate their reasoning, and to construct a viable argument about a mathematical problem. Students also listen to other project presentations and provide feedback to the presenters. Listeners have the opportunity to critique the mathematical reasoning of others.Goals and Learning ObjectivesPresent project to the class.Give feedback on other project presentations.Exhibit good listening skills.Reflect on the problem-solving and project development processes.

Students choose a project idea and a partner or group. They write a proposal for the project.Key ConceptsProjects engage students in the applications of mathematics. It is important for students to apply mathematical ways of thinking to solve rich problems. Students are more motivated to understand mathematical concepts if they are engaged in solving a problem of their own choosing. In this lesson, students are challenged to identify an interesting mathematical problem and to choose a partner or a group to work with collaboratively on solving that problem. Students gain valuable skills in problem solving, reasoning, and communicating mathematical ideas with others.Goals and Learning ObjectivesIdentify a project idea.Identify a partner or group to work collaboratively on a math project.

Students work in a whole-class setting, independently, and with partners to design and implement a problem-solving plan based on the mathematical concepts of rates and multiple representations (e.g., tables, equations, and graphs). They analyze a rule of thumb and use this relationship to calculate the distance in miles from a viewer's vantage point to lightning.Key ConceptsThroughout this unit, students are encouraged to apply the mathematical concepts they have learned over the course of this year to new settings. Help students develop and refine these problem-solving skills:Creating a problem-solving plan and implementing the plan systematicallyPersevering through challenging problems to find solutionsRecalling prior knowledge and applying that knowledge to new situationsMaking connections between previous learning and real-world problemsCommunicating their approaches with precision and articulating why their strategies and solutions are reasonableCreating efficacy and confidence in solving challenging problems in the real worldGoals and Learning ObjectivesCreate and implement a problem-solving plan.Organize and interpret data presented in a problem situation.Analyze the relationship between two variables.Create a rate table to organize data and make predictions.Apply the relationship between the variables to write a mathematical formula and use the formula to solve problems.Create a graph to display proportional relationships, and use this graph to make predictions.Articulate strategies, thought processes, and approaches to solving a problem, and defend why the solution is reasonable.

Lesson OverviewAllow students who have a clear understanding of the content in the unit to work on Gallery problems of their choosing. You can 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.GalleryMap ReadingStudents will examine a street map, interpret the street map coordinates, and relate the street map to an x-y coordinate grid.Burning CandlesStudents will explore how the height of a burning candle changes with time. In one case, they will get information by watching a video of candles burning. In another case, they will be given data with which they can calculate a candle's height at any given time.More DirectionsStudents are given instructions on what path to take to get to a friend's house. But these instructions are incomplete because a starting point is not given. Their task is to figure out which starting points will make these instructions work.LeaguesStudents will determine how the length of a sports league's playing season is affected by increasing the number of teams from 22 to 30. They will use the number of times a game can be played each week and the present length of the season to solve the problem.TrafficStudents will figure out the distance between cars on a highway and how many cars there are along a 1-mile stretch of the highway. They will use the speed of the cars and how often a car passes a certain point to solve the problem.Leaky FaucetStudents will watch a video of a sink filling with water. From that, they will determine how long it will take for the sink to fill up completely.Frog JumpStudents are shown a drawing of yellow and green frogs arranged in a straight line. The yellow frogs are on the left, and the green frogs are on the right. Their task is to figure out how the yellow and green frogs can switch positions if they can move only by leaping according to certain rules.

Students design and work on their projects in class. They review the project rubric and, as a class, add criteria relevant to their specific projects.Key ConceptsThroughout this unit, students are encouraged to apply the mathematical concepts they have learned over the course of this year to new settings. Help students develop and refine these problem-solving skills:Creating a problem-solving plan and implementing their plan systematicallyPersevering through challenging problems to find solutionsRecalling prior knowledge and applying that knowledge to new situationsMaking connections between previous learning and real-world problemsCommunicating their approaches with precision and articulating why their strategies and solutions are reasonableCreating efficacy and confidence in solving challenging problems in a real worldGoals and Learning ObjectivesCreate and implement a problem-solving plan.Organize and interpret data presented in a problem situation.Analyze the relationship between two variables.Use ratios.Write and solve proportions.Create rate tables to organize data and make predictions.Use multiple representations—including tables, graphs, and equations—to organize and communicate data.Articulate strategies, thought processes, and approaches to solving a problem and defend why the solution is reasonable.

During this two-day lesson, students work with a partner to create and implement a problem-solving plan based on the mathematical concepts of rates, ratios, and proportionality. Students analyze the relationship between different-sized gummy bears to solve problems involving size and price.Key ConceptsThroughout this unit, students are encouraged to apply the mathematical concepts they have learned over the course of this year to new settings. Help students develop and refine these problem-solving skills:Creating a problem solving plan and implementing their plan systematicallyPersevering through challenging problems to find solutionsRecalling prior knowledge and applying that knowledge to new situationsMaking connections between previous learning and real-world problemsCommunicating their approaches with precision and articulating why their strategies and solutions are reasonableCreating efficacy and confidence in solving challenging problems in a real worldGoals and Learning ObjectivesCreate and implement a problem-solving plan.Organize and interpret data presented in a problem situation.Analyze the relationship between two variables.Use ratios.Write and solve proportions.Create rate tables to organize data and make predictions.Use multiple representations—including tables, graphs, and equations—to organize and communicate data.Articulate strategies, thought processes, and approaches to solving a problem, and defend why the solution is reasonable.

During this two-day lesson, students work with a partner to create and implement a problem-solving plan based on the mathematical concepts of rates, ratios, and proportionality. Students analyze the relationship between different-sized gummy bears to solve problems involving size and price.Key ConceptsThroughout this unit, students are encouraged to apply the mathematical concepts they have learned over the course of this year to new settings. Helping students develop and refine these problem solving skills:Creating a problem solving plan and implementing their plan systematicallyPersevering through challenging problems to find solutionsRecalling prior knowledge and applying that knowledge to new situationsMaking connections between previous learning and real-world problemsCommunicating their approaches with precision and articulating why their strategies and solutions are reasonableCreating efficacy and confidence in solving challenging problems in a real worldGoals and Learning ObjectivesCreate and implement a problem-solving plan.Organize and interpret data presented in a problem situation.Analyze the relationship between two variables.Use ratios.Write and solve proportions.Create rate tables to organize data and make predictionsUse multiple representations—including tables, graphs, and equations—to organize and communicate data.Articulate strategies, thought processes, and approaches to solving a problem and defend why the solution is reasonable.

Students create and implement a problem-solving plan to solve another problem involving the relationship between the sound of thunder and the distance of the lightning.Key ConceptsThroughout this unit, students are encouraged to apply the mathematical concepts they have learned over the course of this year to new settings. Help students develop and refine these problem-solving skills:Creating a problem-solving plan and implementing their plan systematicallyPersevering through challenging problems to find solutionsRecalling prior knowledge and applying that knowledge to new situationsMaking connections between previous learning and real-world problemsCommunicating their approaches with precision and articulating why their strategies and solutions are reasonableCreating efficacy and confidence in solving challenging problems in a real worldGoals and Learning ObjectivesCreate and implement a problem-solving plan.Organize and interpret data presented in a problem situation.Analyze the relationship between two variables.Create a rate table to organize data and make predictions.Apply the relationship between the variables to write a mathematical formula and use the formula to solve problems.Create a graph to display proportional relationships and use this graph to make predictions.Articulate strategies, thought processes, and approaches to solving a problem and defend why the solution is reasonable.