2nd Grade - Gateway 1

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 2 meet expectations for assessing grade-level content and if applicable, content from earlier grades. The materials for Grade 2 are divided into nine units, and each unit contains a written End-of-Unit Assessment. Additionally, the Unit 9 Assessment is an End-of-Course Assessment, and it includes problems from the entire grade level. Examples of End-of-Unit Assessments include: 

• Unit 2, Adding and Subtracting within 100, End-of-Unit Assessment, Problem 2, “Jada has 40 stickers. She gets 13 more stickers. How many stickers does she have? Jada gave 15 stickers to Noah. How many stickers does Jada have now? Show your thinking using drawings, numbers, words.” (2.NBT.5, 2.OA.1)

• Unit 6, Geometry, Time, and Money, End-of-Unit Assessment, Problem 4, “1. Split the circle into 4 equal parts. 2. Explain why 4 fourths of the circle is the whole circle.” (2.G.3)

• Unit 7, Adding and Subtracting within 1,000, End-of-Unit Assessment, Problem 6, students “Find the value of each difference. Show your thinking. Use base-ten blocks if it helps. a. 528-315, b. 471-124, c. 600-594.” (2.NBT.7) 

• Unit 8: Equal Groups, End-of-Unit Assessment, Problem 4, “For each number, decide whether the number is even or odd. Write each even number as the sum of 2 equal addends. 1. 6, 2. 11, 3. 14.” (2.OA.3)

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 2 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 2 standards by including in every lesson a Warm Up, one to three instructional activities, and Lesson Synthesis. Within Grade 2, students engage with all CCSS standards.

Examples of extensive work include:

• Unit 3, Measuring Length, Lessons 4 and 7, engage students in extensive work with 2.MD.3 (Estimate lengths using units of inches, feet, centimeters, and meters). In Lesson 4, Measure and Estimate in Centimeters, Cool-down, students estimate the length of a picture of a pencil in centimeters. “1. Estimate: I think the length of the pencil is about ___cm.” In Lesson 4, Measure and Estimate in Centimeters, Activity 1: Estimate Length in Centimeters, students are provided with different objects and are asked to estimate their lengths in centimeters. “Now look at the objects I gave each group and think about how long they are. Record your estimates on the recording sheet on your own.” In Lesson 7, Center Day 1, Activity 1: Introduce Estimate and Measure, students estimate and measure the length of objects in centimeters, inches, and feet. “Choose an object to measure and hold it up for students to see. What tool would you use to measure this object? Before measuring the object, you and your partner will both estimate the length of the object using the unit you chose. Your estimates do not have to be the same. Record your estimate.”

• In Unit 4, Addition and Subtraction on the Number Line, Lesson 2, Features of a Number Line, and Unit 5 Numbers to 1000, Lesson 1, How Do We Compose a Hundred? students engage with extensive work with 2.NBT.2 (Count within 1000; skip-count by 5s, 10s, and 100s)., as students choral count from 0-100 by 5s. Teacher Guidance states, “‘Count by 5, starting at 0.’ Record as students count. Stop counting and recording at 100.” In Lesson 3, Unlabeled Tick Marks, Cool Down Problem 1, students are provided a number line that begins at 15, includes large tick marks in intervals of 5 and small tick marks in intervals of 1. The numbers 15, 20 and 45 are included on the number line, and there are blanks below the other multiples of 5. “‘Complete each number line by filling in the missing labels with the number the tick mark represents.’ Part b, ‘Locate and label 37 on the number line.’” Problem 2 is similar, but includes blanks only below multiples of ten. Students are asked to locate and label 35 on the number line. In Unit 5, Lesson 1, How Do We Compose a Hundred?, students choral count from 0 to 300 by 10. The Teacher Guidance instructs the teacher to say, “‘Count by 10, starting at 0.’ Record as students count. ‘Record 10 numbers in each row. Then start a new row directly below. Stop counting and recording at 300.’” In Unit 6, Geometry, Time, and Money, Lesson 12, Count by 5 to Tell Time, students count by 5s on a clock. The Warm Up includes an image of an analog clock with the labels 5, 10, 15, etc. shown outside the clock adjacent to 1, 2, 3, etc. In Unit 7, Add and Subtract within 1,000, Lesson 1: Compare, Count on, and Count Back, students count by 100s. In Activity 2, students complete number lines. “‘1. Fill in the missing numbers. Does this number line show counting on by 10 or counting on by 100?’ The problem includes an image of a number line with labels at 502, 702, and 902, and blanks at 602, 802. Problems 2 and 3 are similar, but include different images on the number lines. On the Cool Down, students count by 100. Problem 2, ‘Complete the list of numbers to show counting on by 100. 552, ___, ___, 852, 952 Explain how you know your list shows counting on by 100 and not counting on by 10.’” 

• Unit 6, Geometry, Time, and Money, Lesson 11, Tell Time with Halves and Quarters, and Lesson 12, Count by 5 to Tell Time, engages students in extensive work with 2.MD.7 (Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m.) In Lesson 11, Tell Time with Halves and Quarters, Activity 2: Card Sort: Halves and Quarters, students use their understanding of halves and quarters of circles to match clock faces, partitioned circles, and the phrases “quarter past,” “half past,” and “quarter till.”  Student Task Statements, “Problem 1. Find matching sets of cards. Each set should have 3 cards. Be prepared to explain why they match. Problem 2. Write the time shown on each clock using the words half past, quarter past, or quarter till.” In Lesson 12, Count by 5 to Tell Time, Activity 1: Count by 5 on the Clock, students count by fives and tell time on an analog clock. The Teacher Guide, “Display the image of the clock that shows 4:30 with the minutes labeled on the outside. ‘Tell your partner 2 ways to read this time.’ (4:30 or half past 4) 30 seconds: partner discussion ‘How could you prove that the time is 4:30?’ (Each tick mark shows one minute. You could count by 1 to 30. You can count the minutes by 5. Start at the 12 and count by 5 for each number until you get to 6. 5, 10, 15, 20, 25, 30. So it is 4:30.)’ Display the image of the clock that shows 4:15. ‘What time does this clock show?’ (4:15 or quarter past 4). ‘When telling time, we can count by 5 to determine how many minutes have passed since the hour.’ Use a clock to demonstrate starting at 4:00 and moving the minute hand to the 1, 2, then 3, as you say, “4:00, 4:05, 4:10, 4:15. Give each group a set of cards. ‘You are going to continue counting by 5 to tell time. Take turns telling the time on your cards. Work together to put the cards in order based on the times they show.’” Student Task Statements, “1. Discuss 2 ways to read the time on this clock. 2. A clock is shown with the time 4:15. What time does this clock show? 3. Read the time on each clock card with your partner. Put the clocks in order based on the times they show.”

Examples of full intent include:

• Unit 3, Measuring Length, Lesson 4, Measure and Estimate in Centimeters, Lesson 5, Measure in Meters, Lesson 8, What is an Inch, and Lesson 9, From Feet to Inches meet the full intent of 2.MD.3 (Estimate lengths using units of inches, feet, centimeters, and meters.) Lesson 4, Measure and Estimate in Centimeters, Activity 1, students make estimates of objects lengths in centimeters, beginning with a notebook. The Teacher Guidance instructs the teacher to show the notebook next to a 10 cm tool, “‘Let’s look at another image of the object.’ Display the image or hold a folder next to a 10-centimeter tool.” After launching the activity, the teacher instructs students to estimate the length of different objects, “‘Now look at the objects I gave each group and think about how long they are. Record your estimates on the recording sheet on your own. When you and your partner finish, compare your estimates and explain why you think they are ‘about right.’” In Activity 2, students measure the objects and compare their estimates to the measurements. In Lesson 5, Measure in Meters, students estimate the length of an object in meters during the Cool Down, “Noah held a gecko at the zoo. The length of the gecko fit in his hands. He measured it and said it was about 13 meters long. Do you think his  measurement is correct? Why or why not?” In Lesson 8, What is an Inch?, Activity 2, students estimate the length of the sides of different shapes that are pictured in the materials, “1. Here is a rectangle. How long is the long side of the rectangle in inches? Estimate: ___ Measure the long side of the rectangle. Actual length: ___.” Problem 2 follows the same structure but includes a square, while Problem 3 is about a triangle. Finally, in Lesson 9, From Feet to Inches, Activity 2, students estimate lengths in feet, “Estimate the length of objects around the room. Say if you will measure in inches or feet.” The Lesson 9 Cool Down also provides an opportunity to estimate length in feet, “Tyler told Han that a great white shark is about 16 inches long, but Han disagrees. Han believes it would be about 16 feet long. Who do you agree with? Explain.”

• Unit 5, Numbers to 1000, Lesson 4, Write Three-digit Numbers, and Lesson 6, Represent Numbers in Different Ways, engages students with the full intent of 2.NBT.3 (Read and write numbers to 1000 using base-ten numerals, number names, and expanded form.) In Lesson 4, Write Three-digit Numbers, Activity 1: Place Value Riddles, students write the number of hundreds, tens, and ones, and represent the value as a three-digit number. The Teacher Guide, “Give students access to base-ten blocks. ‘I have 4 hundreds, 3 ones, and 2 tens. Which of these shows the total value written as a three-digit number? Explain how you know.’ Display 432, 234, 423. ‘You are going to solve number riddles using base-ten blocks.’” Student Task Statements, “Solve each riddle and write the three-digit number. Use the table to help you organize the digits. 1. I have 2 ones, 7 tens, and 6 hundreds. 2. I have 3 ones, 5 tens, and 2 hundreds. 3. I have 7 hundreds, 5 ones, and 3 tens. 4. I have 5 hundreds, no tens, and 9 ones. 5. I have 4 ones, 6 tens, and 3 hundreds. 6. I have 8 tens, 1 hundred, and no ones.” Lesson 5, Expanded Form of Numbers, Cool-down: Three-digit Numbers in Expanded Form, “1. Represent the number 375 as the sum of hundreds, tens, and ones. Expanded form: 2. Represent 200+40+7 as a three-digit number. Three-digit number:” Lesson 6, Represent Numbers in Different Ways, Activity 1: Numbers as Words, students use words to represent three-digit numbers. The Teacher Guide, “Display the anchor chart that shows the different forms of 253. Complete the chart together. “This number has ____ hundreds, ____ tens, and ____ ones.’ (2, 5, 3) ‘The expanded form of this number is ___. The three-digit number is ___. These other forms can help us think about writing a number using number names. What is this number?’ (two hundred fifty-three) Write the number name as the students say two hundred fifty-three. ‘Fifty-three has a hyphen because numbers with tens and ones representing 21 through 99 use a hyphen to show the 2 parts of a two-digit number.’” Activity 2: Represent the Numbers, Student Task Statements, “Represent the number on your poster. Be sure to represent the number using: a three-digit number, a base-ten diagram, expanded form, words.” Cool-down: Words and Other Ways, “1. Represent 147 with words. 2. Represent 147 in one other way.”

• Unit 8, Equal Groups, Lesson 9 and Lesson 10, students engage in the full intent of 2.OA.4 (Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends). In Lesson 9, A Sum of Equal Addends, Activity 1: Sums of Rows and Sums of Columns, students recognize that an expression with equal addends can represent the sum of the number of objects in each row or the sum of the number of objects in each column. “‘Mai and Diego represented the number of objects in the same array with different expressions. Diego wrote 2+2+2+2+2+2. Mai wrote 6+6. Who do you agree with? Work with a partner to decide who you agree with and be prepared to explain your reasoning.’” In Lesson 10, Write Expressions and Equations to Represent Arrays, Activity 1: Build Arrays and Write Equations, students write equations that represent the number of objects in the rows or columns of an array. “‘First, you will arrange counters to make an array. Then you will write an equation that has equal addends. There are 2 equations that match each array. To find the total number of counters, you can use any method that makes sense to you.’”

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 2 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 9, approximately 78%.

• The number of lessons devoted to major work of the grade (including assessments and supporting work connected to major work) is 122 lessons out of 155 lessons, approximately 79%. The total number of lessons include 114 lessons plus 8 assessments for a total of 122 lessons. 

• The number of days devoted to major work of the grade (including assessments and supporting work connected to major work) is 131 days out of 163 days, approximately 80%.

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 79% of the instructional materials focus on major work of the grade.

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 2 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 3, Measuring Length, Lesson 15, Create Line Plots, Activity 1: Measure and Plot Pencil Lengths connects the supporting work of 2.MD.9 (Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by making repeated measurements of the same object. Show the measurements by making a line plot, where the horizontal scale is marked off in whole- number units) to the major work of 2.MD.1 (Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes.) Student Task Statements, the teacher provides the groups with 10-12 pencils of various lengths, “1. Measure the pencils in centimeters. Work with a partner and check each other’s measurements. Record each measurement in the table. 2. Create a line plot to represent the lengths of all the pencils in your group.”

• Unit 6, Geometry, Time, and Money, Lesson 18, Money Problems, Activity 1: Shop for School Supplies, connects the supporting work of 2.MD.8 (Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and ¢ symbols appropriately) to the major work of 2.OA.1 (Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem). Students engage in the task, “Lin has these coins: 2 quarters, 3 dimes, 1 nickel. a. How much money does Lin have for supplies? b. If Lin buys an eraser, how much money will she have left? Show your thinking.”

• Unit 8, Equal Groups, Lesson 5, Patterns with Even and Odd Numbers, Cool-down: Odd One Out, connects the supporting work of 2.OA.3 (Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends.) to the major work of 2.OA.2 (Fluently add and subtract within 20 using mental strategies.) Student Task Statements, “1. Elena has 8 counters. Does she have an even or odd number of counters? Explain or show your reasoning. 2. Without adding, explain which one of these expressions represents an odd number. A. 4+4, B. 8+1, C. 8+2.”

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 2 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 3, Measuring Lengths, Lesson 11, Saree Silk Stories, Necklaces and Bracelets connects the major work of 2.MD.B (Relate addition and subtraction to length) to the major work of 2.NBT.B (Use Place Value Understanding and Properties of Operations to Add and Subtract). In Cool-down: More Saree Ribbon, students solve subtraction problems within 100 with the unknown in all positions. Student Task Statements, “Priya had a piece of ribbon that was 74 inches long. She cut off 17 in. How long is Priya’s ribbon now? Show your thinking. Use a diagram if it helps. Don’t forget the unit in your answer.” 

• Unit 4, Addition and Subtraction on the Number Line, Lesson 2, Features of a Number Line connects the major work of 2.MD.B (Relate addition and subtraction to length) to the major work of 2.NBT.A (Understand place value). In Activity 2: Analyze Number Lines, students analyze number lines to determine whether they represent numbers within 10 as lengths from 0. Student Task Statements, “1. How should Jada revise her number line? 2. How should Andre revise his number line? 3. Elena’s number line. How should Elena revise her number line? 4. Fill in the numbers to create your own number line.” 

• Unit 8, Equal Groups, Lesson 12, Partition Rectangles Into Squares connects the supporting work of 2.OA.C (Work with equal groups of objects to gain foundations for multiplication) to the supporting work of 2.G.A (Reason with shapes and their attributes). In Activity 1, How Many Squares?, students partition rectangles into rows and columns. Teaching Notes, “Use 8 tiles to make a rectangle. Your tiles should be in 2 rows now, draw lines in the rectangle to show the squares. It should have the same number of equal-size squares as the rectangle you made out of tiles. You may use a ruler if it helps you.”

• Unit 9, Putting It All Together, Lesson 3, Measure on a Map, Activity 2, connects the major work of 2.MD.A (Measure and estimate lengths in standard units) with the major work of 2.MD.B (Relate addition and subtraction to length). Students use maps to answer measurement questions. Student Task Statements, “Use your map and the stories from the previous activity to answer the questions. Represent each story with an equation with a symbol for the unknown length. 1. How much longer is the total length of Diego’s trip than the total length of Lin’s trip? 2. How much longer is the total length of Diego’s trip than the total length of Noah’s trip? 3. How much shorter is the total length of Noah’s trip than the total length of Lin’s trip?”

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 2 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, Measuring Length, Section B, Customary Measurement, Section narrative, connects work with 2.MD.1 (Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes.), 2.MD.2 (Measure the length of an object twice, using length units of different lengths for the two measurements; describe how the two measurements relate to the size of the unit chosen.), 2.MD.3 (Estimate lengths using units of inches, feet, centimeters, and meters.), 2.MD.4 (Measure to determine how much longer one object is than another, expressing the length difference in terms of a standard length unit.), 2.MD.5 (Use addition and subtraction within 100 to solve word problems involving lengths that are given in the same units, e.g., by using drawings (such as drawings of rulers) and equations with a symbol for the unknown number to represent the problem.), 2.NBT.5 (Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.), 2.OA.2 (Fluently add and subtract within 20 using mental strategies) to work done in grade 3. “As in the previous section, students make choices about the tool to use based on the length of the object being measured (MP5) and measure the length of the same object in both feet and inches. They begin to generalize that when they use a longer length unit, fewer of those units are needed to span the full length of the object. This understanding is a foundation for their work with fractions in grade 3 and beyond.”

• Unit 6, Geometry, Time, and Money, Lesson 9: You Ate the Whole Thing, About this lesson, connects 2.G.3 (Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape) and 2.NBT.2 (Count within 1000; skip-count by 5s, 10s, and 100s) to the work in grade 3. “In this lesson, students continue to practice partitioning circles and describe halves, thirds, and quarters of circles using the language a half of, a third of, and a quarter of to describe a piece of the shape. They also use this language to describe the whole shape as a number of equal pieces. Students recognize that a whole shape can be described as 2 halves, 3 thirds, or 4 fourths. This understanding is the foundation for students' work with a whole and fraction equivalency in grade 3.”

• Unit 8, Equal Groups, Lesson 9, Activity 3: Add It All Up connects 2.OA.3 (Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends) and 2.OA.4 (Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends) to work in grade 3. Activity 3, “The purpose of this activity is for students to determine the total number of objects in an array and match expressions to arrays by paying attention to the number of objects in each row and the number of objects in each column. For example, students recognize that 3 rows with 4 in each row would be 4+4+4. The arrays in this task provide students opportunities to compare different ways an array could be decomposed to find the total number of objects” when students compare the different ways they find the total number of objects in the array to expressions that use equal addends to represent the sums of rows or sums of columns. (3.OA.1)

Examples of connections to prior knowledge include:

• Unit 1, Adding, Subtracting, and Working with Data, Lesson 1, Add and Subtract Within 10, Warm-up: Notice and Wonder, A Picture of Shapes connects 2.G.1 (Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces…) to the work in grade 1, “Students develop fluency with addition and subtraction within 10 in grade 1. This lesson provides an opportunity for formative assessment of students' fluency within 10, including recognizing sums with a value of 10 (1.OA.6).”

• Unit 4, Addition and Subtraction on the Number Line, Lesson 4, Compare Numbers on a Number Line, About this lesson, connects 2.MD.6 (Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2, ..., and represent whole-number sums and differences within 100 on a number line diagram.), 2.NBT.5 (Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.) to work done in grade 1. “In this lesson, students recognize that as you move to the right on the number line, numbers increase in value because they are a greater distance from 0. Students also use the relative position of numbers and generalize that a number that is greater than a given number if it is farther to the right on the number line. To demonstrate this understanding, students compare numbers within 100 (a skill from grade 1) and use the number line to explain their comparison (MP7).”

• Unit 5, Numbers to 1,000, Lesson 1, How Do We Compose a Hundred?, About this lesson, connects 2.NBT.1 (Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones) and 2.NBT.2 (Count within 1000; skip-count by 5s, 10s, and 100s) to the work in grade 1. “In grade 1, students were introduced to a ten as a unit made of 10 ones. They used that understanding to represent two-digit numbers and add within 100. Students used connecting cubes to make and break apart two-digit numbers. In previous units in grade 2, students used the words compose and decompose as they made and broke apart tens when they added and subtracted within 100.”