5th Grade - Gateway 1

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 5 meet expectations for assessing grade-level content and if applicable, content from earlier grades. The materials for Grade 5 are divided into eight units, and each unit contains a written End-of-Unit Assessment. Additionally, the Unit 8 Assessment is an End-of-Course Assessment, and it includes problems from the entire grade level. Examples of End-of-Unit Assessments include:

• Unit 1, Finding Volume, End-of-Unit Assessment, Problem 4, students “Find the volume of a rectangular prism with the given side lengths. 1.The length is 2 units, the width is 5 units, and the height is 7 units. 2.The base has an area of 200 square inches and the height is 6 inches.” (5.MD.5)

• Unit 3, Multiplying and Dividing Fractions, End of Unit Assessment, Problem 6, “An apple weighs \frac{1}{2} pound. Diego cuts the apple into 4 equal pieces. How many pounds does each piece of the apple weigh? Explain your reasoning.” (5.NF.7)

• Unit 6, More Decimal and Fraction Operations, End of Unit Assessment, Problem 5, “Han’s backpack weighs \frac{5}{4} as much as Lin’s backpack. Clare’s backpack weighs \frac{2}{3} as much as Lin’s backpack. Whose backpack weighs the most? Whose backpack weighs the least? Explain or show how you know.” (5.NF.5)

• 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. 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)

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 5 meet expectations for giving all students extensive work with grade-level problems to meet the full intent of grade-level standards. The materials provide extensive work with and opportunities for students to engage in the full intent of Grade 5 standards by including in every lesson a Warm Up, one to three instructional activities, and Lesson Synthesis. Within Grade 5, students engage with all CCSS standards.

Examples of extensive work include:

• Unit 1, Finding Volume, Lesson 4, Using Layers to Determine Volume, Lesson 10, Represent Volume with Expressions, and Unit 2, Fractions as quotients and Fraction Multiplication, Lesson, Lesson 8, Divide to Multiply Non-unit Fractions, and Unit 4, Wrapping Up Multiplication and Division with Multi-Digit Numbers, Lesson 3, Partial Products in Algorithms,   provide students with extensive work with 5.OA.2 (Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them). In Lesson 4, Using Layers to Determine Volume, Activity 2, students interpret multiplication expressions that represent the volume of an illustrated rectangular prism with dimensions 4, 5, 6, “1. Explain or show how the expression  5\times24 represents the volume of this rectangular prism., 2. Explain or show how the expression 6\times20 represents the volume of this rectangular prism., 3. Find a different way to calculate the volume of this rectangular prism. Explain or show your thinking., 4. Write an expression to represent the way you calculated the volume.” In Lesson 10, Activity 1, students write expressions to represent the volume of an illustrated prism that has an L-shaped base, “1. Write an expression to represent the volume of the figure in unit cubes.” In Unit 2, Fractions as Quotients and Fraction Multiplication, Lesson 8, Divide to Multiply Non-unit Fractions, students determine if expressions are equivalent during the warm up, “Decide if each statement is true or false. Be prepared to explain your reasoning. 2\times(\frac{1}{3}\times6)=\frac{2}{3}\times6, 2\times(\frac{1}{3}\times6)=2\times(6\div3), \frac{2}{3}\times6=2\times(\frac{1}{4}\times6).” In Unit 4, Wrapping Up Multiplication and Division with Multi-Digit Numbers, Lesson 3, Partial Products in Algorithms, Activity 1, students use different illustrated area models to represent multi-digit multiplication with different expressions, “1. Take turns picking out a set of expressions that are equal to 245\times35 when added together. Use the diagrams if they are helpful., 2. Explain how you know the sum of your expressions is equal to 245\times35., 3. What is the value of 245\times35? Explain or show your reasoning.” In Unit 5, Place Value Patterns and Decimal Operations, Lesson 17, Multiply Decimals and Whole Numbers, Activity 1, students create visual models of different expressions using 100s grids, “Find the value of each expression in a way that makes sense to you. Explain or show your reasoning. Use the grids, if needed., 1. 2times0.7, 2. 2\times0.08, 3. 2\times0.78. In Lesson 23, Activity 2, students relate a multiplication and division expression to a single diagram. The provided diagram is a 100s grid with 2 columns colored blue, followed by 2 columns colored orange, repeated for 10 columns. “2. This is the diagram and explanation Tyler used to justify why 12\div0.2=60. ‘$$12\div0.2=60$$ There are 5 groups of 0.2 in 1 and there are 12 so that is 12 groups of 5.’ Explain how the expression 12\times(0.1\div0.2) relates to Tyler’s reasoning.” In Unit 7, Lesson 12, Represent Problems on the Coordinate Grid, students decide if expressions are equivalent during the warm up, “Decide if each statement is true or false. Be prepared to explain your reasoning., (2\times10)+(3\times5)=(3\times10)+(1\times5), (3\times25)+(5\times5)=8\times25, (4\times25)+(10\times5)=(2\times25)+(10\times10).”

• Unit 5, Place Value Patterns and Decimal Operations, Lessons 5, 6, and 9 engage students in extensive work of 5.NBT.3b (Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.) In Lesson 5, Compare Decimals, Activity 1: Faster and Faster, students compare decimals to hundredths place. “1. Diego and Jada were competing to see who could throw the frisbee further. Diego threw the frisbee 5.10 meters. Jada threw the frisbee 5.01 meters. Who threw the frisbee further? Be prepared to explain your thinking. 2. Tyler and Han were competing to see who could swim the length of the pool faster. Tyler swam the length of the pool in 35.15 seconds. Han swam the length of the pool in 35.30 seconds. Who swam the length of the pool faster? Be prepared to explain your thinking.” In Lesson 6, Compare Decimals on a Number Line, Activity 2, Label and Compare Decimals, students accurately label the marks for decimals through the thousandths place on a number line. Students are given 3 number lines to label the thousandths place. “1. Label the tick marks on each number line. 2. Which of the number lines would you use to compare 0.534 and 0.537? Explain or show your reasoning.” In Lesson 9, Ordering Decimals, Warm Up: True or False: Decimal Inequalities, students compare pairs of decimal numbers and explain their reasoning. Students “decide whether each statement is true or false. Be prepared to explain your reasoning. 0.909>0.91; 4.1<4.100; 0.99<0.999$.”

• Unit 6, More Decimal and Fraction Operations, Lesson 5, Multi-step Conversion Problems: Metric Length, students engage with extensive work with 5.MD.1 (Convert like measurement units within a given measurement system). In Activity 2: Who Ran Farther?, Student Task Statements, students convert between meters and kilometers to decide which of two measurements is larger. “1. Use the table to find the total distance Tyler ran during the week. Explain or show your reasoning. Column labels include day and distance (km) with the following data: Monday 8.5, Tuesday 6.25, Wednesday 10.3, Thursday 5.75, Friday 9.25. 2. Use the table to find the total distance Clare ran during the week. Show your reasoning. Column labels include day and distance (m) with the following data: Monday 5,400, Tuesday 7,500, Wednesday 8,250, Thursday 6,750, Friday 7,250. 3. Who ran farther, Clare or Tyler? How much farther? Explain or show your reasoning.” 

Examples of full intent include:

• Unit 5, Place Value Patterns and Decimal Operations, Lessons 7, Round Doubloons, and Lesson 8, Round Decimals, provides the opportunity for students to engage with the full intent of standard 5.NBT.4 (Use place value understanding to round decimals to any place). Lesson 7, Round Doubloons, Activity 1: Gold Doubloons, students round to the nearest tenth and hundredth. “Display the image. “This is a doubloon. What do you notice? What do you wonder?” Problem 4, “Use the number lines to find which hundredth of a gram the doubloon weights are each closest to.” Lesson 8, Round Decimals, Activity 2: Which Number is Closest?, students round a decimal number to the nearest whole number, tenth, and hundredth. Problem 2, “Round 4.158 to the nearest whole number, tenth, and hundredth.” 

• Unit 6, More Decimal and Fraction Operations, Lesson 12, Solve Problems, provides students with opportunities to meet the full intent of 5.NF.2 (Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers.) In Activity 1, students are shown a recipe for salad dressing that uses \frac{3}{4} cup of olive oil, and answer questions requiring subtraction of fractions with unlike denominators and assessing the reasonableness of an estimate, “1. Priya has \frac{2}{3} cup of olive oil. She is going to borrow some more from her neighbor. How much olive oil does she need to borrow to have enough to make the dressing?, 2. 1 tablespoon is equal to \frac{1}{16} of a cup. Priya decides that 1 tablespoon of olive oil is close enough to what she needs to borrow from her neighbor. Do you agree with Priya? Explain or show your reasoning.” In the Cool Down, students are required to add fractions with unlike denominators, “2. On Monday, Andre hiked \frac{3}{4} mile in the morning and 1\frac{1}{3} miles in the afternoon. How far did Andre hike on Monday? Explain or show your reasoning.”

• Unit 7, Shapes on the Coordinate Plane, Lesson 2, Points on the Coordinate Grid, and Lesson 3, Plot More Points, engage students with the full intent of 5.G.1 (Graph points on the coordinate plane to solve real-world and mathematical problems). In Lesson 2, Points on the Coordinate Grid, Activity 2: Plot and Label Points, students write ordered pairs of numbers to represent points in the coordinate plane and to plot points with given coordinates. Student Task Statements, a coordinate plane and three points are provided for students. “1. List the coordinates for each point. 2. Plot points D, E, F on the same grid. D (6,4), E (2,5), F (8,3).” In Lesson 3, Plot More Points, Cool-down, Missing Coordinate,“Here is a grid with some points labeled. Plot and label the points (3,0), (0,2) and (3,2). Explain or show your reasoning.”

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 5 meet expectations that, when implemented as designed, the majority of the materials address the major clusters of the grade. The instructional materials devote at least 65 percent of instructional time to the major clusters of the grade:

• The approximate number of units devoted to the major work of the grade (including assessments and supporting work connected to 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 major work) is 137 lessons out of 156 lessons, approximately 88%. The total number of lessons include 129 lessons plus 8 assessments for a total of 137 lessons. 

• The number of days devoted to major work of the grade (including assessments and supporting work connected to major work) is 143.5 days out of 164 days, approximately 88%.

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

The materials reviewed for Imagine Learning Illustrative Mathematics 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 with supporting standards/clusters connected to the major standards/clusters of the grade. These connections are listed for teachers on a document titled, “Pacing Guide and Dependency Diagram” found on the Course Guide tab for each Unit. Teacher Notes also provide the explicit standards listed within the lessons. Examples of connections include:

• Unit 2, Fractions as Quotients and Fraction Multiplication, Lesson 13, Area and Properties of Operations, Warm-up: Number Talk: Parentheses, 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 Task Statements, “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 4, Wrapping Up Multiplication and Division with Multi-Digit Numbers, Lesson 11, Different Partial Quotients, Activity 2: Division Expressions, Student Task Statement, 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.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). Students rewrite division expressions to find their values. “1. Choose a set of expressions that, when added together, is equal to 308\div14. Not all expressions will be used. 2. Explain to your partner how you know that your cards represent a sum that is equal to 308\div14. 3. Choose one of the sets of expressions whose sum is equal to 308\div14 and use it to find the value of 308\div14.”

• Unit 6, More Decimals and Fraction Operations, Lesson 4, Metric Conversion and Multiplication by Powers of 10, Activity 1: Long Jump, Javelin Throw, and Shot Put connects the supporting work of 5.MD.1 (Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real-world problems) to the major work of 5.NBT.2 (Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10). Student Task Statement, “Below are some results Jackie Joyner-Kersee recorded in different events in 1988. Complete the table. (Table has the following columns with the following information in the rows: Event- long jump, javelin throw, shot put; centimeters- 727, 4,566, and 1,580; meters- (blank). (Students are to convert centimeters to meters)” “What pattern do you notice? 727\div100=7.27; 4,566\div100=45.66; 1,580\div100=15.80”

The materials reviewed for Imagine Learning Illustrative Mathematics 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. Examples of connections between major work to major work and/or supporting work to supporting work throughout the materials, when appropriate, include:

• Unit 4, Wrapping Up Multiplication and Division with Multi-digit Numbers, Lesson 16, World’s Record Noodle Soup connects the major work of 5.NF.B (Apply and extend previous understandings of multiplication and division to multiply and divide fractions) to the major work of 5.NBT.B (Perform operations with multi-digit whole numbers and with decimals to hundredths). In Activity 2, Han’s Estimate, students apply understanding of multiplication and division and perform multiplication and division with multi-digit whole and fractions. Lesson Synthesis, “Today, we solved problems about a real life context. We also discussed solutions that were mixed numbers. In what ways did we use division today?”

• Unit 5, Place Value Patterns and Decimal Operations, Lesson 21, Multiply More Decimals, Activity 1, connects the major work of 5.NBT.A (Understand The Place Value System) with the major work of 5.NBT.B (Perform Operations With Multi-Digit Whole Numbers And With Decimals To Hundredths). Students explain equivalence among expressions using both decimals and whole numbers. Student Task Statement, “1. Explain or show why each pair of expressions is equivalent., a. 7.2\times5.3 and (72\times53)\times0.01, b. 6.5\times2.8 and (65\times28)\div100, c. 31\times0.44 and (31\times44)\times\frac{1}{100}, 2. Find the value of the products in the previous problem.”

• Unit 7, Shapes on the Coordinate Plane, Lesson 13, Perimeter and Area of Rectangles, Activity 1, 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 draw rectangles on the coordinate plane. Student Task Statement, “1. Jada drew a rectangle with a perimeter of 12 centimeters. What could the length and width of Jada’s rectangle be? Use the table to record your answer. 2. Plot the length and width of each rectangle on the coordinate grid. 3. If Jada drew a square, how long and wide was it? 4. If Jada’s rectangle was 2.5 cm long, how wide was it? Plot this point on the coordinate grid. 5. If Jada’s rectangle was 3.25 cm long, how wide was it? Plot this point on the coordinate grid.” 

• Unit 8, Putting it All Together, Lesson 14, Notice and Wonder, Activity 3, Design Your Notice and Wonder, Part 2, connects the major work of 5.NBT (Numbers and Operations In Base Ten) with the major work of 5.NF (Number and Operations - Fractions). Students find meaningful mathematics in images or photographs. Students are encouraged to make connections to a mathematical topic they have become familiar with. “1. Find an image that you find interesting and would encourage your classmates to notice and wonder about a mathematical topic you have learned this year. 2. Fill in the possible things students might notice and wonder about your image.”

The materials reviewed for Imagine Learning Illustrative Mathematics 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. 

The Section Dependency Chart explores the Unit sections relating to future grades. The Section Dependency Chart states, “arrow indicates the prior section that contains content most directly designed to support or build toward the content in the current section.” 

Examples of connections to future grades include:

• Unit 3, Multiplying and Dividing Fractions, Section B, Fraction Division, Section summary connects 5.NF.7 (Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.) with work done in grade 6. “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.”

• Unit 4, Wrapping Up Multiplication and Division with Multi-digit Numbers, Section B, Multi-digit Division Using Partial Quotients, connects 5.NBT.5 (Fluently multiply multi-digit whole numbers using the standard algorithm.), 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.), 5.NF.3 (Interpret a fraction as division of the numerator by the denominator (\frac{a}{b}=a\div b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem), 5.OA.2 (Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them.) with work done in grade 6. “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.”

• Unit 5, Finding Volume, Section C Narrative: Volume of Solid Figures connects 5.NBT.2 (Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.) and 5.NBT.5 (Fluently multiply multi-digit whole numbers using the standard algorithm) to work in grade 6. About this Section, “The work reminds students that they can decompose multi-digit factors by place value to find their product, paving the way toward the standard algorithm for multiplication in a later unit.”

Examples of connections to prior knowledge include:

• Unit 1, Finding Volume, Lesson 3, Volumes of Prism Drawings, Warm-up: Number Talk: Multiplication connects 4.MD.5 (Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume) to the work in grade 3, “In grade 3, students learned how to find the area of a rectangle by tiling and found that multiplying the side lengths yields the same result (3.MD.7).”

• Unit 2, Fractions as Quotients and Fractions as Multiplication, Section B: Fractions of Whole Numbers, Section Narrative, “In grade 4, students saw that a non-unit fraction can be expressed as a product of a whole number and a unit fraction, or a whole number and a non-unit fraction with the same denominator. For instance, \frac{8}{3} can be expressed as 8\times\frac{1}{3}, as 4\times\frac{2}{3}, or as 2\times\frac{4}{3}. In the previous section, students interpreted a fraction like \frac{8}{3} as a quotient: 8\div3. This section allows students to connect these two interpretations of \frac{8}{3} and relate 8\frac{1}{3} and 8\div3.”

• Unit 6, More Decimal and Fraction Operations, Unit Overview, Full Unit Narrative, “In this unit, students deepen their understanding of place-value relationships of numbers in base ten, unit conversion, operations on fractions with unlike denominators, and multiplicative comparison. The work here builds on several important ideas from grade 4. In grade 4, students learned the value of each digit in a whole number is 10 times the value of the same digit in a place to its right. Here, they extend that insight to include decimals to the thousandths. Students recognize that the value of each digit in a place (including decimal places) is \frac{1}{10} the value of the same digit in the place to its left.”