5th Grade - Gateway 1

The materials reviewed for Open Up Resources K–5 Math Grade 5 meet expectations for assessing grade-level content and, if applicable, content from earlier grades. 

The curriculum is divided into eight units, and each unit contains a written End-of-Unit Assessment for individual student completion. The Unit 8 Assessment is an End-of-Course Assessment and includes problems from across the grade. Examples from End-of-Unit Assessments include: 

• Unit 2, Fractions as Quotients and Fraction Multiplication, End-of-Unit Assessment, Problem 3, “There are 8 ounces of pasta in the package. Jada cooks \frac{2}{3} of the pasta. How many ounces of pasta did Jada cook? A. 2\frac{2}{3}, B.5\frac{1}{3}, C. 7\frac{1}{3}, D, 12.” (5.NF.4a, 5.NF.6)

• Unit 3, Multiplying and Dividing Fractions, End-of-Unit Assessment, Problem 4, “440 meters is \frac{1}{4} of the way around the race track. How far is it around the whole race track? Explain or show your reasoning.” (5.NF.7b, 5.NF.7c)

• Unit 5, Place Value Patterns and Decimal Operations, End-of-Unit Assessment, Problem 3, “What is 1.357 rounded to the nearest hundredth? What about to the nearest tenth? To the nearest whole number? Explain or show your reasoning.” (5.NBT.4)

• Unit 7, Shapes on the Coordinate Plane, End-of-Unit Assessment, Problem 3, “Fill in each blank with the correct word, ‘sometimes,’ ‘always,’ or ‘never.’ a. A parallelogram is ___ a rhombus. b. A rhombus is ___ a parallelogram. c. A rectangle is ___ a rhombus. d. A quadrilateral with a 35 degree angle is ___ a rectangle.” (5.G.3, 5.G.4)

• Unit 8, Putting It All Together, End-of-Course Assessment and Resources, Problem 1, “Select all expressions that represent the volume of this rectangular prism in cubic units. a. 5\times4\times3, b. (3\times4)+4 , c. 5\times(4+3) , d. 3\times20, e. 4\times15.” A prism is shown next to the problem. (5.MD.5a, 5.MD.5b, 5.OA.A)

The instructional materials reviewed for Open Up Resources K–5 Math Grade 5 meet expectations for the materials giving all students extensive work with grade-level problems to meet the full intent of grade-level standards.

The instructional materials provide extensive work in Grade 5 as students engage with all CCSSM standards within a consistent daily lesson structure. Per the Grade 5 Course Guide, “A typical lesson has four phases: a Warm-up, one or more instructional activities, the lesson synthesis, a Cool-down.” Examples of extensive work include:

• Unit 2, Fractions as Quotients and Fraction Multiplication, Section C, Lessons 12 and 16; Unit 3, Multiplying and Dividing Fractions, Section C, Lesson 19; and Unit 8, Putting It All Together, Section C, Lesson 13 engage students in extensive work with 5.NF.4 (Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction). Unit 2, Lesson 12, Decompose Area, Activity 1, Student Work Time, students find the area of a rectangle with a whole number side length and a side length that is a mixed number. Student Facing, “a. Noah’s garden is 5 yards by 6\frac{1}{4} yards. Draw a diagram of Noah’s garden on the grid. b. ​Priya’s garden is 6 yards by 5\frac{1}{4} yards. Draw a diagram of Priya’s garden on the grid. c. Whose garden covers a larger area? Be prepared to explain your reasoning.” Unit 2, Lesson 16, Estimate Products, Activity 2, Student Work Time,  students reason about the value of products by rounding either the whole number or mixed number factors and multiplying. Student Facing, “1. Write a whole number product that is slightly less than, slightly greater than, or about equal to the value of 7\times12\frac{8}{9}. a slightly less b. slightly greater c. just right. 2. Write a whole number product that is slightly less than, slightly greater than, or about equal to the value of 9\times4\frac{2}{29}. a. slightly less b. slightly greater c. just right. 3. a. Without calculating, use the numbers 2, 3, 5, 6, and 7, to complete the expression with a value close to 20. (An equation model for multiplying a whole number by a mixed number is provided.) b. Explain how you know your expression represents a value close to 20.” Unit 3, Lesson 19, Fraction Games, Warm-up: Estimation Exploration: Multiply Fractions, Student Work Time, students develop strategies for finding the product of a fraction and a mixed number. Student Facing “28\times2\frac{8}{9} Record an estimate that is: too high, about right, too low.” Unit 8, Lesson 13, Multiply Fractions Game Day, Activity 1, Student Work Time, students practice multiplying fractions. Student Facing, “a. Use the directions to play Fraction Multiplication Compare with your partner. Spin the spinner. Write the number you spun in one of the empty boxes. Once you write a number, you cannot change it. Player two spins and writes the number on their game board. Continue taking turns until all four blank boxes are filled. Multiply your fractions. The player with the greatest product wins. Play again. b. What strategy do you use to decide where to write the numbers?”

• Unit 5, Place Value Patterns and Decimal Operations, Section B, Lessons 11, 12, and 13 engage students in extensive work with 5.NBT.7 (Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used). Lesson 11, Make Sense of Decimal Addition, Activity 2, Student Work Time, students use place value understanding to add decimals. Student Facing, “Directions: a. Play one round of Target Numbers. Partner A, Start at 0. Roll the number cube. Choose whether to add that number of tenths or hundredths to your starting number. Write an equation to represent the sum. Take turns until you’ve played 6 rounds. Each round, the sum from the previous equations becomes the starting number in the new equation. The partner to get a sum closest to 1 without going over wins. b. Describe a move that you could have made differently to change the outcome of the game.” Lesson 12, Estimate and Add, Cool-down, students build on knowledge of the standard algorithm for addition from a prior unit. Student Facing, “Find the value of 3.45+21.6. Explain or show your reasoning.” Lesson 13, Analyze Addition Mistakes, Activity 1, Launch and Student Work Time, students analyze a common error when using the standard algorithm to add decimals. In Launch, “‘Solve the first problem on your own.’ 2–3 minutes: independent work time. ‘Now, work on the second problem on your own for a few minutes, and then talk to your partner about it.’” In Student Work Time, Student Facing, “a. Find the value of 621.45+72.3. Explain or show your reasoning. b. Elena and Andre found the value of 621.45+72.3. Who do you agree with? Explain or show your reasoning.” Work is shown for Elana and Andre, showing that Elena lined digits up incorrectly as she wrote the problem vertically.

• Unit 6, More Decimal and Fraction Operations, Section A, Lessons 5 and 7 engage students in extensive work with 5.MD.1 (Convert among different-sized measurement units within a given measurement system…). Lesson 5, Multi-step Conversion Problems: Metric Length, Activity 2, Student Work Time. Student Facing, students convert between meters and kilometers to decide which of two measurements is larger, “a. Use the table to find the total distance Tyler ran during the week. Explain or show your reasoning. A table with columns for day and distance (km) is shown: Monday 8.5, Tuesday 6.25, Wednesday 10.3, Thursday 5.75, Friday 9.25. b. Use the table to find the total distance Clare ran during the week. Show your reasoning. A table with columns for day and distance (m) is shown. Monday 5,400, Tuesday 7,500, Wednesday 8,250, Thursday 6,750, Friday 7,250. c. Who ran farther, Clare or Tyler? How much farther? Explain or show your reasoning.” Lesson 7, Multi-step Conversion Problems: Customary Length, Activity 2, Student Work Time, students solve multi-step conversion problems using customary length units. Student Facing, “a. A rectangular field is 90 yards long and 42\frac{1}{4} yards wide. Priya says that 6 laps around the field is more than a mile. Do you agree with Priya? Explain or show your reasoning. b. A different rectangular field is 408\frac{1}{2} feet long and 240\frac{1}{4} feet wide. How many laps around this field would Priya need to run if she wants to run at least 2 miles?”

The instructional materials provide opportunities for all students to engage with the full intent of Grade 5 standards through a consistent lesson structure. According to the Grade 5 Course Guide, “The first event in every lesson is a Warm-up. Every Warm-up is an Activity Narrative. The Warm-up invites all students to engage in the mathematics of the lesson… After the Warm-up, lessons consist of a sequence of one to three instructional activities. The activities are the heart of the mathematical experience and make up the majority of the time spent in class… After the activities for the day, students should take time to synthesize what they have learned. This portion of class should take 5-10 minutes before students start working on the Cool-down…The Cool-down task is to be given to students at the end of the lesson. Students are meant to work on the Cool-down for about 5 minutes independently and turn it in. The Cool-down serves as a brief formative assessment to determine whether students understood the lesson.” Examples of meeting the full intent include:

• Unit 1, Finding Volume, Section C, Lessons 10 and 12; Unit 4, Wrapping Up Multiplication and Division with Multi-Digit Numbers, Section A, Lesson 9; and Unit 8, Putting It All Together, Section B, Lesson 6 engage students in the full intent of 5.MD.5 (Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume). Unit 1, Lesson 10, Represent Volume with Expressions, Activity 1, Student Work Time, students find volume of figures in different ways. Student Facing “a. Write an expression to represent the volume of the figure in unit cubes. b. Compare expressions with your partner.  How are they the same?  How are they different? c. If they are the same, try to find another way to represent the volume.” An image of a rectangular prism is provided. Unit 1, Lesson 12, Lots and Lots of Garbage, Activity 1,  Student Work Time, students find different ways to arrange 60 shipping containers. Student Facing, “a. Find at least 5 different ways to arrange 60 containers. Represent each arrangement with an expression. b. Create a visual display to show which is the best arrangement for shipping the 3,300 tons of garbage.” Unit 4, Lesson 9, The Birds, Cool-down, students calculate volume. Student Facing, “To make a birdhouse for a screech owl, the recommended area of the floor is 8 inches by 8 inches and the recommended height is 12 inches to 15 inches. What is the recommended range of volumes for a screech owl birdhouse? Explain or show your thinking.” Unit 8, Lesson 6, Revisit Volume, Activity 1, Student Work Time, students revisit the concept that volume is the number of unit cubes required to fill a space without gaps or overlaps. Student Facing, “A company packages 126 sugar cubes in each box. The box is a rectangular prism. a. What are some possible ways they could pack the cubes? b. How would you choose to pack the cubes? Explain or show your reasoning. c. The side lengths of the box are about 1\frac{7}{8} inches by 3\frac{3}{4} inches by 4\frac{3}{8} inches. What can we say about how the sugar cubes are packed?”

• Unit 5, Place Value Patterns and Decimal Operations, Section A, Lessons 2, 5, and 7 engage students in the full intent of 5.NBT.3 (Read, write, and compare decimals to thousandths). Lesson 2, Thousandths on Grids and in Words, Activity 2, Student Work Time, students consider different ways to name a decimal shown on a hundredths grid. Student Facing, “Several students look at the diagram and describe the shaded region in different ways. Who do you agree with? Why? A. Jada says it’s ‘15 hundredths.’ B. Priya says it’s ‘150 thousandths.’ C. Tyler says it’s ‘15 thousandths.’ D. Diego says it’s ‘1 tenth and 5 hundredths.’ E. Mai says it’s ‘1 tenth and half of a tenth.’” Lesson 5, Compare Decimals, Cool-down, students use place value understanding to compare decimals. Student Facing, “Lin threw the frisbee 5.09 meters. Andre threw the frisbee 5.1 meters. Who threw the frisbee farther? Explain or show your reasoning.” Lesson 7, Round Doubloons, Activity 2, Student Work Time, students examine numbers in different situations and decide if they are exact or approximate. Student Facing, “Decide if each quantity is exact or an estimate. Be prepared to explain your reasoning. a. There are 14 pencils on the desk. b. The population of Los Angeles is 12,400,000. c. It's 2.4 miles from the school to the park. d. The runner finished the race in 19.78 seconds.”

• Unit 7, Shapes on the Coordinate Plane, Section B, Lessons 4, 5, 6, and 7 engage students in the full intent of 5.G.4 (Classify two-dimensional figures in a hierarchy based on properties). Lesson 4, Sort Quadrilaterals, Activity 2, Student Work Time, students determine appropriate categories as they sort quadrilaterals. Student facing, “Your teacher will give you a set of cards. a. Sort all of the quadrilateral cards in a way that makes sense to you. Name the categories in your sort. b. Sort the quadrilateral cards in a different way and name each of the categories in your new sort. Lesson 5, Trapezoids, Warm-up, Student Work Time, students share what they know about and how they can represent trapezoids. Student facing, “What do you know about trapezoids?” Lesson 6, Hierarchy of Quadrilaterals, Activity 2, Student Work Time, students determine if quadrilaterals are squares, rhombuses, rectangles, or parallelograms. Student facing, “a. Draw 3 different quadrilaterals on the grid, making sure at least one of them is a parallelogram. b. For each of your quadrilaterals determine if it is a: square, rhombus, rectangle, parallelogram. Explain or show your reasoning. c. Draw a rhombus that is not a square. Explain or show how you know it is a rhombus but not a square. d. Draw a rhombus that is a square. Explain or show how you know it is a rhombus and a square. e. Diego says that it is impossible to draw a square that is not a rhombus. Do you agree with him? Explain or show your reasoning.” Lesson 7, Rectangles and Squares, Activity 1, Student Work Time, students deepen their understanding of the quadrilateral hierarchy as they recognize specific attributes. Student facing, “Spread out your shape cards so you and your partner can see all of them. Work together to find a shape that fits each clue. If you don’t think it is possible to find that shape, explain why. You can only use each shape one time. a. Find a quadrilateral that is not a parallelogram. b. Find a rhombus that is also a square. c. Find a rhombus that is not a square. d. Find a trapezoid that is not a rectangle. e. Find a rectangle that is not a square. f. Find a parallelogram that is not a rectangle. g. Find a square that is not a rectangle.”

The materials reviewed for Open Up Resources K–5 Math Grade 5 meet expectations that, when implemented as designed, the majority of the materials address the major clusters of each grade. The instructional materials devote at least 65% of instructional time to the major clusters of the grade: 

• The approximate number of units devoted to major work of the grade (including assessments and supporting work connected to the major work) is 7 out of 8, approximately 88%.

• The number of lessons devoted to major work of the grade (including assessments and supporting work connected to the major work) is 138 out of 156, approximately 88%. The total number of lessons devoted to major work of the grade include: 130 lessons plus 8 assessments for a total of 138 lessons.

• The number of days devoted to major work of the grade (including assessments and supporting work connected to the major work) is 145 out of 151, approximately 96%.

A lesson-level analysis is most representative of the instructional materials as the lessons include major work, supporting work connected to major work, and the assessments embedded within each unit. As a result, approximately 88% of the instructional materials focus on major work of the grade.

The materials reviewed for Open Up Resources K–5 Math Grade 5 meet expectations that supporting content enhances focus and coherence simultaneously by engaging students in the major work of the grade. 

Materials are designed so supporting standards/clusters are connected to the major standards/ clusters of the grade. These connections are listed for teachers on a document titled “Lessons and Standards” found within the Course Guide tab for each unit. Connections are also listed on a document titled “Scope and Sequence”. Examples of connections include:

• Unit 1, Finding Volume, Section C, Lesson 9, Activity 2, Student Work Time, connects the supporting work of 5.OA.2 (Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them) to the major work of 5.MD.5c (Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real world problems). Students find the volume of composite figures. Student Facing states, “1. Explain how each expression represents the volume of the figure. Show your thinking. Organize it so it can be followed by others. a. ((2\times3)\times4)+((3\times3)\times2). b. (5\times6)+(3\times4). 2. How does each expression represent the volume of the prism? Explain or show your thinking. Organize it so it can be followed by others. a. (5\times8\times6)+(5\times4\times9) cubic inches. b. (5\times4\times3)+(5\times12\times6) cubic inches.”

• Unit 2, Fractions as Quotients and Fraction Multiplication, Section C, Lesson 13, Warm-up, Student Work Time, connects the supporting work of 5.OA.1 (Use parenthesis, brackets, or braces in numerical expressions, and evaluate expressions with these symbols) to the major work of 5.NF.4 (Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction). Students multiply a whole number and a fraction as they solve problems with grouping symbols. Student Facing states, “Find the value of each expression mentally. 5\times(7+4), (5\times7)+(5\times4), (5\times7)+(5\times\frac{1}{4}), (5\times7)-(5\times\frac{1}{4}).”

• Unit 6, More Decimal and Fraction Operations, Section B, Lesson 14, Activity 1, Student Work Time, connects the supporting work of 5.MD.2 (Make a line plot to display a data set of measurements in fractions of a unit [\frac{1}{2}, \frac{1}{4}, \frac{1}{8}], Use operations on fractions for this grade to solve problems involving information presented in line plots) to the major work of 5.NF.A (Use equivalent fractions as a strategy to add and subtract fractions). Students make a line plot and then analyze the data to solve problems using operations with fractions. A spinner with the fractions \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{5}{8} is provided. Student Facing states, “a. Play Sums of Fractions with your partner. Take turns with your partner. Spin the spinner twice. Add the two fractions. Record the sum on the line plot. Play the game until you and your partner together have 12 data points. b. How did you know where to plot the sums of eighths? c. What is the difference between your highest and lowest number? d. What do you notice about the data you collected?”

The instructional materials for Open Up Resources K–5 Math Grade 5 meet expectations for including problems and activities that serve to connect two or more clusters in a domain or two or more domains in a grade. 

Materials are coherent and consistent with the Standards. These connections can be listed for teachers in one or more of the four phases of a typical lesson: instructional activities, lesson synthesis, or Cool-down. Examples of connections include:

• Unit 4, Wrapping Up Multiplication and Division with Multi-Digit Numbers, Section B, Lesson 16, Activity 2, Student Work Time and Activity Synthesis, connects the major work of 5.NBT.B (Perform operations with multi-digit whole numbers and with decimals to hundredths) to the major work of 5.NF.B (Apply and extend previous understandings of multiplication and division to multiply and divide fractions). Students consider the most precise estimate for a fractional length, connecting division to what they know about fractions. In Student Work Time, Student Facing states, “Han said that each person will get about 25\frac{1}{4} feet of noodle. Do you agree with Han? Explain or show your reasoning.” The problem context states that 400 people equally shared a 10,119 foot noodle. Activity Synthesis states, “Display: 25\frac{119}{400}. ‘What does 25\frac{119}{400} mean in this situation?’ (Each person gets 25 feet of the noodle and then the 119 feet leftover would be divided into 400 equal pieces.) Display: 25\frac{1}{4} ’Why is Han's estimate reasonable?’ (Because is \frac{119}{400} really close to \frac{100}{400} and \frac{100}{400}=\frac{1}{4}) ‘Do you think they actually measured and cut the noodle into equal pieces when they served it?’ (No, because it would take too long and be too difficult. Yes, because if long noodles represent long life they probably want to serve the noodle soup with sections that are one piece of the original noodle.).”

• Unit 5, Place Value Patterns and Decimal Operations, Section C, Lesson 21, , Student Work Time and Activity Synthesis, connects the major work of 5.NBT.A (Understand the place value system) and 5.NBT.B (Perform operations with multi-digit whole numbers and with decimals to hundredths). Students apply what they know about area and multiplication of decimals to a situation where the side length of the rectangle are decimals. In Student Work Time, Student Facing states, “Central Park is a large park in Manhattan. It is about 3.85 kilometers long and 0.79 km wide. What is the area of Central Park? Record an estimate that is: too low, about right, too high.” Activity Synthesis states, “Invite students to share their estimates. ‘How do you know the area is greater than 2 square kilometers?’ (I know that 3 x 0.7 is 21 tenths or 2.1 and it’s more than that.) ‘How do you know the area is less than 3.2 square kilometers?’ (I know 3.85 is less than 4 and 0.79 is less than 0.8. Then 4\times0.8 is 32 tenths or 3.2.).”

• Unit 7, Shapes on the Coordinate Plane, Section C, Lesson 11, , Student Work Time, connects the supporting work of 5.OA.B (Analyze patterns and relationships) to the supporting work of 5.G.A (Graph points on the coordinate plane to solve real-world and mathematical problems). Students look for patterns in points plotted on a coordinate grid. Student Facing states, “What do you notice? What do you wonder?” Students see a coordinate grid with points plotted in the first quadrant. Student Response includes, “Students may notice: The points are scattered, There are 4 points labeled A - D, Points B and D are on the same horizontal line, The numbers on the vertical and horizontal axis skip count by two, Some points are not on the vertices of the grid. Students may wonder: What do the points represent? Can we connect the points? If we connect the points, what shape will it make?”

• Unit 8, Putting It All Together, Lesson 8, Activity 1, Launch and Student Workt Time, connects the major work of 5.MD.C (Geometric measurement: Understand concepts of volume and relate volume to multiplication and to addition) to the major work of 5.NBT.B (Perform operations with multi-digit whole numbers and with decimals to hundredths). Students apply what they know about multiplication and division to solve problems involving the volume of the Radio Flyer, a rectangular prism. The Launch states, “Groups of 2. Display: 27 feet long, 13 feet wide, 2 feet deep. ‘These are the approximate dimensions of the actual Radio Flyer. How do they compare to the estimates you made in the previous lesson?’ (We were close for the length and depth but the actual wagon is wider than what we guessed.). ‘Imagine the wagon was being filled with sand. Would you want to buy large bags of sand or small bags of sand? Why?’ (I would want large bags because it would take fewer of them.).” In Student Work Time, Student Facing states, “The Radio Flyer wagon is 27 feet long, 13 feet wide and 2 feet deep. a. A 150-pound bag of sand will fill about 9 cubic feet. How many bags of sand will it take to fill the wagon with sand? b. A 150-pound bag of sand costs about $12. About how much will it cost to fill the wagon with sand? Explain or show your reasoning. c. How many pounds of sand does the Radio Flyer hold when it is full? Explain or show your reasoning.”

The instructional materials reviewed for Open Up Resources K–5 Math Grade 5 meet expectations that content from future grades is identified and related to grade-level work, and materials relate grade-level concepts explicitly to prior knowledge from earlier grades. 

Prior and Future connections are identified within materials in the Course Guide, Scope and Sequence Section, within the Dependency Diagrams which are shown in Unit Dependency Diagram, and Section Dependency Diagram. An arrow indicates the prior section that contains content most directly designed to support or build toward the content in the current section. While future connections are all embedded within the Scope and Sequence, descriptions of prior connections are also found within the Preparation tab for specific lessons and within the notes for specific parts of lessons. 

Examples of connections to future grades include:

• Course Guide, Scope and Sequence, Unit 3, Multiplying and Dividing Fractions, Section B: Fraction Division, Section Learning Goals connect 5.NF.7 (Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions) to the work of interpreting and computing quotients of fractions in 6.NS.1. The section states, “Students may notice that to find 5\div\frac{1}{2}, they can multiply 5 by 2 because there are 2 halves in each of the 5 wholes. It is not essential, however, that students generalize division of fractions at this point, as they will do so in grade 6.”

• Course Guide, Scope and Sequence, Unit 4, Wrapping Up Multiplication and Division with Multi-Digit Numbers, Section B: Multi-digit Division Using partial Quotients, Section Learning Goals connect 5.NBT.6 (Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models) to the work of dividing multi-digit numbers using the standard algorithm in 6.NS.2. The section states, “Students see that some decompositions may be more helpful than others for finding whole-number quotients. They use this insight to make sense of algorithms using partial quotients that are more complex. Note that use of the standard algorithm for division is not an expectation in grade 5, but students can begin to develop the conceptual understanding needed to do so. The algorithms using partial quotients seen here are based on place value, which will allow students to make sense of the logic of the standard algorithm they’ll learn in grade 6.”

• Course Guide, Scope and Sequence, Unit 5, Place Value Patterns and Decimal Operations, Unit Learning Goals connect the work of 5.NBT.7 (Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used) to work with operations with decimals in Grade 6. Lesson Narrative states, “Students then apply their understanding of decimals and of whole-number operations to add, subtract, multiply, and divide decimal numbers to the hundredths, using strategies based on place value and the properties of operations. They see that the reasoning strategies and algorithms they used to operate on whole numbers are also applicable to decimals. For example, addition and subtraction can be done by attending to the place value of the digits in the numbers, and multiplication and division can still be understood in terms of equal-size groups. In grade 6, students will build on the work here to reach the expectation to fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.”

Examples of connections to prior knowledge include:

• Unit 1, Finding Volume, Section A, Lesson 1, Preparation connects 5.MD.3 (Recognize volume as an attribute of solid figures and understand concepts of volume measurement) to the work with concepts of area from Grades 3 and 4. Lesson Narrative states, “In previous grades, students learned that they can count the number of square tiles that cover a plane shape without gaps or overlaps to find the area of the shape. In this lesson, students explore the concept of volume as they build and compare objects made of cubes. Students learn that objects can have different shapes but still take up the same amount of space and that we call this amount an object’s volume.”

• Unit 5, Place Value Patterns and Decimal Operations, Section A, Lesson 1, Preparation connects 5.NBT.A (Understand the place value system) to work with decimal fractions from 4.NF.C. Lesson Narrative states, “In grade 4, students studied decimal fractions with denominators 10 and 100. They represented tenths and hundredths with hundredths grids, number lines, and decimal notation. In this lesson students make sense of representations of tenths, hundredths, and thousandths with hundredths grid diagrams, fractions, and decimals. They also see relationships between these values, namely that a tenth of a tenth is a hundredth and a tenth of a hundredth is a thousandth. Students may use informal language to describe the relationship between decimals (for example, to get from 0.01 to .001 you add a zero in front of the one.) This language supports students in sharing their developing understanding. Teachers should ask questions to help students develop more precise language to describe base-ten representations (for example, what does the extra 0 you wrote in .001 represent?). They will have many opportunities to develop this understanding in upcoming lessons.”

• Unit 7, Shapes on the Coordinate Plane, Section B, Lesson 4, Warm-up connects 5.G.3 (Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category) to work with classifying two-dimensional shapes from Grade 4. Narrative states, “The purpose of this What Do You Know About ___? is for students to share what they know about and how they can represent quadrilaterals. In previous courses students have drawn and described squares, rectangles, and rhombuses and they will revisit and classify all of these shapes over the next several lessons.”