Students critique the work of other students and revise their own work based on feedback from the teacher and peers.Key ConceptsConcepts from previous lessons are integrated into this unit task: rewriting expressions, using parentheses, and using the distributive property. Students apply their knowledge, review their work, and make revisions based on feedback from you and their peers. This process creates a deeper understanding of the concepts.Goals and Learning ObjectivesApply knowledge of expressions to correct the work of other students.Track and review the choice of strategy when problem solving.

Students use a geometric model to investigate common factors and the greatest common factor of two numbers.Key ConceptsA geometric model can be used to investigate common factors. When congruent squares fit exactly along the edge of a rectangular grid, the side length of the square is a factor of the side length of the rectangular grid. The greatest common factor (GCF) is the largest square that fits exactly along both the length and the width of the rectangular grid. For example, given a 6-centimeter × 8-centimeter rectangular grid, four 2-centimeter squares will fit exactly along the length without any gaps or overlaps. So, 2 is a factor of 8. Three 2-centimeter squares will fit exactly along the width, so 2 is a factor of 6. Since the 2-centimeter square is the largest square that will fit along both the length and the width exactly, 2 is the greatest common factor of 6 and 8. Common factors are all of the factors that are shared by two or more numbers.The greatest common factor is the greatest number that is a factor shared by two or more numbers.Goals and Learning ObjectivesUse a geometric model to understand greatest common factor.Find the greatest common factor of two whole numbers equal to or less than 100.

Students express the lengths of trains as algebraic expressions and then substitute numbers for letters to find the actual lengths of the trains.Key ConceptsAn algebraic expression can be written to represent a problem situation. More than one algebraic expression may represent the same problem situation. These algebraic expressions have the same value and are equivalent.To evaluate an algebraic expression, a specific value for each variable is substituted in the expression, and then all the calculations are completed using the order of operations to get a single value.Goals and Learning ObjectivesEvaluate expressions for the given values of the variables.

Students write an expression for the length of a train, using variables to represent the lengths of the different types of cars.Key ConceptsA numerical expression consists of a number or numbers connected by the arithmetic operations of addition, subtraction, multiplication, division, and exponentiation.An algebraic expression uses letters to represent numbers.An algebraic expression can be written to represent a problem situation. Sometimes more than one algebraic expression may represent the same problem situation. These algebraic expressions have the same value and are equivalent.The properties of operations can be used to make long algebraic expressions shorter:The commutative property of addition states that changing the order of the addends does not change the end result:a + b = b + a.The associative property of addition states that changing the grouping of the addends does not change the end result:(a + b) + c = a + (b + c).The distributive property of multiplication over addition states that multiplying a sum by a number gives the same result as multiplying each addend by the number and then adding the products together:a(b + c) = ab + ac.Goals and Learning ObjectivesWrite algebraic expressions that describe lengths of freight trains.Use properties of operations to shorten those expressions.

Students represent problem situations using expressions and then evaluate the expressions for the given values of the variables.Key ConceptsAn algebraic expression can be written to represent a problem situation.To evaluate an algebraic expression, a specific value for each variable is substituted in the expression, and then all the calculations are completed using the order of operations to get a single value.Goals and Learning ObjectivesDevelop fluency in writing expressions to represent situations and in evaluating the expressions for given values.

Fractions and Decimals

Type of Unit: Concept

Prior Knowledge

Students should be able to:

Multiply and divide whole numbers and decimals.
Multiply a fraction by a whole number.
Multiply a fraction by another fraction.
Write fractions in equivalent forms, including converting between improper fractions and mixed numbers.
Understand the meaning and structure of decimal numbers.

Lesson Flow

This unit extends students’ learning from Grade 5 about operations with fractions and decimals.

The first lesson informally introduces the idea of dividing a fraction by a fraction. Students are challenged to figure out how many times a 14-cup measuring cup must be filled to measure the ingredients in a recipe. Students use a variety of methods, including adding 14 repeatedly until the sum is the desired amount, and drawing a model. In Lesson 2, students focus on dividing a fraction by a whole number. They make a model of the fraction—an area model, bar model, number line, or some other model—and then divide the model into whole numbers of groups. Students also work without a model by looking at the inverse relationship between division and multiplication. Students explore methods for dividing a whole number by a fraction in Lesson 3, for dividing a fraction by a unit fraction in Lesson 4, and for dividing a fraction by another fraction in Lesson 6. Students examine several methods and models for solving such problems, and use models to solve similar problems.

Students apply their learning to real-world contexts in Lesson 6 as they solve word problems that require dividing and multiplying mixed numbers. Lesson 7 is a Gallery lesson in which students choose from a number of problems that reinforce their learning from the previous lessons.

Students review the standard long-division algorithm for dividing whole numbers in Lesson 8. They discuss the different ways that an answer to a whole number division problem can be expressed (as a whole number plus a remainder, as a mixed number, or as a decimal). Students then solve a series of real-world problems that require the same whole number division operation, but have different answers because of how the remainder is interpreted.

Students focus on decimal operations in Lessons 9 and 10. In Lesson 9, they review addition, subtraction, multiplication, and division with decimals. They solve decimal problems using mental math, and then work on a card sort activity in which they must match problems with diagram and solution cards. In Lesson 10, students review the algorithms for the four basic decimal operations, and use estimation or other methods to place the decimal points in products and quotients. They solve multistep word problems involving decimal operations.

In Lesson 11, students explore whether multiplication always results in a greater number and whether division always results in a smaller number. They work on a Self Check problem in which they apply what they have learned to a real-world problem. Students consolidate their learning in Lesson 12 by critiquing and improving their work on the Self Check problem from the previous lesson. The unit ends with a second set of Gallery problems that students complete over two lessons.

Students determine how many times they would need to fill a quarter cup to measure the ingredients in a recipe.Key ConceptsThis lesson informally introduces the idea of dividing by a fraction. Students must figure out how many times a quarter cup must be filled to measure the ingredients in a recipe. This involves dividing each amount by 14. Here are some methods students might use:Add 14 repeatedly until the sum is the desired amount. Count the number of 14s that were added.Start with the amount in the recipe. Subtract 14 repeatedly until the difference is 0. Count the 14s that were subtracted.Draw a model (e.g., a bar or a number line model) to represent the amount in the recipe. Divide it into fourths and count the number of fourths.Goals and Learning ObjectivesLearn how to divide by a fraction.

Students solve decimal multiplication and division problems related to the basic fact 3 × 7 = 21.Students match cards that represent word problems, visual models, and numerical solutions to problems that include the numbers 0.8 and 0.2 for all four operations.Key ConceptsNo new mathematics is introduced in this lesson. Students apply their knowledge about decimal operations.Goals and Learning ObjectivesUse reasoning and mental math to solve problems.Solve word problems involving simple addition, subtraction, multiplication, and division with decimals.

Students explore methods of dividing a fraction by a unit fraction.Key ConceptsIn this lesson and in Lesson 5, students explore dividing a fraction by a fraction.In this lesson, we focus on the case in which the divisor is a unit fraction. Understanding this case makes it easier to see why we can divide by a fraction by multiplying by its reciprocal. For example, finding 34÷15 means finding the number of fifths in 34. In this lesson, students will see that this is 34 × 5.Students learn and apply several methods for dividing a fraction by a unit fraction, such as 23÷14.Model 23. Change the model and the fractions in the problem to twelfths: 812÷312. Then find the number of groups of 3 twelfths in 8 twelfths. This is the same as finding 8 ÷ 3.Reason that since there are 4 fourths in 1, there must be 23 × 4 fourths in 23. This is the same as using the multiplicative inverse.Rewrite both fractions so they have a common denominator: 23÷14=812÷312. The answer is the quotient of the numerators. This is the numerical analog to modeling.Goals and Learning ObjectivesUse models and other methods to divide fractions by unit fractions

Students use models and the idea of dividing as making equal groups to divide a fraction by a whole number.SWD: Some students with disabilities will benefit from a preview of the goals in each lesson. Students can highlight the critical features or concepts in order to help them pay close attention to salient information.Key ConceptsWhen we divide a whole number by a whole number n, we can think of making n equal groups and finding the size of each group. We can think about dividing a fraction by a whole number in the same way.8 ÷ 4 = 2 When we make 4 equal groups, there are 2 wholes in each group.89÷4=29  When we make 4 equal groups, there are 2 ninths in each group.When the given fraction cannot be divided into equal groups of unit fractions, we can break each unit fraction part into smaller parts to form an equivalent fraction.34 ÷ 6 = ?     68 ÷ 6 = ?     68 ÷ 6 = 18  Students see that, in general, we can divide a fraction by a whole number by dividing the numerator by the whole number. Note that this is consistent with the “multiply by the reciprocal” method.ab÷n=a÷nb=anb=an×1b=an×b=ab×1nGoals and Learning ObjectivesUse models to divide a fraction by a whole number.Learn general methods for dividing a fraction by a whole number without using a model. 

Students solve word problems that require dividing and multiplying with fractions and mixed numbers.Key ConceptsStudents apply their knowledge about multiplying and dividing fractions to solve word problems. This includes applying the general methods for dividing fractions learned in previous lessons:Rewrite the dividend and the divisor so they have a common denominator. The answer to the original division will be the quotient of the numerators.Multiply the dividend by the reciprocal of the divisor.Goals and Learning ObjectivesApply knowledge of fraction multiplication and division to solve word problems.

Gallery 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.Gallery DescriptionStew RecipeStudents use fraction operations to help Molly figure out if she has enough potatoes to make stew for all the guests at her party.Multiply or Divide?Students match descriptions of situations to multiplication and division situations.Card SortStudents find the diagram, expression, and answer that match given word problems.Complex FractionsStudents learn about complex fractions and how they are useful for dividing fractions.

Gallery 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.Gallery DescriptionTiling a FloorStudents determine which size tiles are cheaper to use to tile a floor with given dimensions.Adam's HomeworkStudents find and correct an error in a whole number division problem.Then and NowStudents solve comparison problems involving census data from 1940 and 2010.Graphical MultiplicationGiven points m and p on a number line, students must locate m × p.When Does Zero Matter?Students must determine how the placement of 0 affects the value of a number.

Students explore whether multiplying by a number always results in a greater number. Students explore whether dividing by a number always results in a smaller number.Key ConceptsIn early grades, students learn that multiplication represents the total when several equal groups are combined. For this reason, some students think that multiplying always “makes things bigger.” In this lesson, students will investigate the case where a number is multiplied by a factor less than 1.Students are introduced to division in early grades in the context of dividing a group into smaller, equal groups. In whole number situations like these, the quotient is smaller than the starting number. For this reason, some students think that dividing always “makes things smaller.” In this lesson, students will investigate the case where a number is divided by a divisor less than 1.Goals and Learning ObjectivesDetermine when multiplying a number by a factor gives a result greater than the number and when it gives a result less than the number.Determine when dividing a number by a divisor gives a result greater than the number and when it gives a result less than the number.

Students critique and improve their work on the Self Check.Key ConceptsNo new concepts are introduced in this lesson. To solve the problems in the Self Check, students use fraction division and operations with decimals.Goals and Learning ObjectivesUse knowledge of fraction division and decimal operations to solve problems.

Students review the standard long-division algorithm and discuss the different ways the answer to a whole-number division problem can be expressed (as a whole number plus a remainder, as a mixed number, or as a decimal).Students solve a series of real-world problems that require the same whole number division operation, but have different answers because of how the remainder is interpreted.Key ConceptsStudents have been dividing multidigit whole numbers since Grade 4. By the end of Grade 6, they are expected to be fluent with the standard long-division algorithm. In this lesson, this algorithm is reviewed along with the various ways of expressing the answer to a long division problem. Students will have more opportunities to practice the algorithm in the Exercises.Goals and Learning ObjectivesReview and practice the standard long-division algorithm.Answer a real-world word problem that involves division in a way that makes sense in the context of the problem.

Students explore methods of dividing a fraction by a fraction.Key ConceptsStudents extend what they learned in Lesson 4 to divide a fraction by any fraction. Students are presented with two general methods for dividing fractions:Rewrite the dividend and the divisor so they have a common denominator. The answer to the original division will be the quotient of the numerators.Multiply the dividend by the reciprocal of the divisor.These two methods will work for all cases, including cases in which one or both of the numbers in the division is a fraction or whole number.Goals and Learning ObjectivesUse models and other methods to divide fractions by fractions.

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.