2nd Grade - Gateway 1

The materials reviewed for Eureka Math² Grade 2 meet expectations for assessing grade-level content and, if applicable, content from earlier grades. 

The curriculum is divided into six modules and each includes a Module Assessment. Examples of grade-level items from Module Assessments include:

• Module 1, Module Assessment, Part 1, Item 16, “Read: The truck is 27 cm long. The car is 21 cm long. How much shorter is the car than the truck? Draw: (there is space for students to draw) Write: The car is ___ cm shorter than the truck.” (2.MD.5)

• Module 2, Module Assessment, Item 11, “Read: 90 toy cars are in a box. Ling takes some out. Now 48 toy cars are in the box. How many toy cars does Ling take out? Draw: (there is space for students to draw) Write: (there is space for students to write their response).” (2.OA.1)

• Module 3, Module Assessment, Item 6, “Ling and Tim both draw a hexagon. Ling says Tim’s shape is not a hexagon. Is Ling correct? Tell how you know.” Ling and Tim’s shapes are pictured for students. (2.G.1)

• Module 4, Module Assessment, Item 13, Tim’s Way for finding the solution is shown, including representation of 10s and 1s and some trading.“Tim finds 34+23+17+20=95 Look at Tim’s work. Show a different way.” (2.NBT.6)

The materials reviewed for Eureka Math² Grade 2 meet expectations for giving all students extensive work with grade-level problems to meet the full intent of grade-level standards. 

According to the Grade 2 Implementation Guide, “Each lesson is structured in four sections: Fluency, Launch, Learn, and Land. Lessons are designed for one 60-minute instructional period. Fluency provides distributed practice with previously learned material. It is designed to prepare students for new learning by activating prior knowledge and bridging small learning gaps. Launch creates an accessible entry point to the day’s learning through activities that build context and often create productive struggle that leads to a need for the learning that follows. Every Launch ends with a transition statement that sets the goal for the day’s learning. Learn presents new learning related to the lesson objective, usually through a series of instructional segments. This lesson component takes most of the instructional time. Suggested facilitation styles vary and may include direct instruction, guided instruction, group work, partner activities, interactive video, and digital elements. The Problem Set, an opportunity for independent practice, is included in Learn. Land helps you facilitate a brief discussion to close the lesson and provides students with an opportunity to complete the Exit Ticket. In the Debrief portion of Land, suggested questions, including key questions related to the objective, help students synthesize the day’s learning. The Exit Ticket provides a window into what students understand so that you can make instructional decisions.”

Instructional materials engage all students in extensive work with grade-level problems through the consistent lesson structure. Examples include:

• Module 1, Lessons 25 through 27 engage students in extensive work with 2.NBT.3 (Read and write numbers to 1000 using base-ten numerals, number names, and expanded form).  Lesson 25: Write three-digit numbers in unit form and show the value that each digit represents, Learn, students express numbers in unit form and show the value of each digit. “Gather students and place 2 hundreds bundles, 4 tens bundles, and 3 individual sticks in the boxes. ‘How many of each unit do you see–from largest to smallest?’ (2 hundreds 4 tens 3 ones) ‘What number does that represent?’ (243) ‘When a number is written by using only digits and no units, it is called standard form. It is the standard, or most common, way to show numbers.’ Write the following term and example: Standard form: 243, Show 243 with place value cards. Pull the cards apart to show the value that each digit represents. Push them back together so students see how the values comprise one number. Then have students do the same. Hold up 2 hundreds bundles. ‘Which of your cards shows this number of sticks?’ (Holds up 200 card) Hold up 4 tens bundles. ‘Which of your cards shows this number of sticks?’ (Holds up 40 card) ‘Which has a greater value, 2 hundreds or 4 tens?’ (2 hundreds) Tell me the number of each unit. (Point to each box.) (2 hundreds 4 tens 3 ones) Write the following term and example: Unit form: 2 hundreds 4 tens 3 ones ‘Numbers can also be written with their unit. This written notation is called unit form. What if we had 4 tens 3 ones 2 hundreds? What number does that represent?’ (It’s still 243.) Rearrange the boxes so that students see that 3 ones 4 tens 2 hundreds represents the same total. ‘When numbers are written in unit form, we can rearrange the order without changing the value.’ Invite students to think–pair– share about why unit form can be rearranged without changing the value but the digits in standard form cannot. ‘Unit form shows the unit, or value, of each digit, so it does not matter how you arrange them. Unit form is like the place value bundles–each bundle shows the value. If you rearrange the digits in standard form, you change the value. I know 432 is not the same as 243.’ Repeat the process with the following suggested sequence: 351, 252, 104. Model each number in the boxes as students do the following: Represent each number with whole number place value cards. Whisper the number in standard form. Whisper the number in unit form to a partner.” Lesson 26: Write base-ten numbers in expanded form, Land, students write base-ten numbers in expanded form. “Initiate a class discussion by using the following prompts. Encourage students to restate their classmates’ responses in their own words. Refer students to problem 4 on their Problem Set. ‘What is the same and different about the equations?’ (In the first one, the units were in order from greatest to least, but in the second one the units were all mixed up. Even though the units are in a different order, the total is still the same, 257.) ‘When we are writing in expanded form, does the order of the units matter? Does it affect the total value?’ (No, as long as the number of hundreds, tens, and ones doesn’t change, you can write the parts in any order. When you are adding, the order of the parts doesn’t change the total.) ‘You have discovered that expanded form is another way to represent a number, and the order of the units does not change the total value.’”

• Module 1, Lessons 6-8 and Module 5, Lesson 9 engage students in extensive 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). Module 1, Lesson 6: Make a 10 cm ruler and measure objects, Learn, Measure Objects, students use their 10 cm rulers to measure objects. “Partner students and have them use their 10 cm rulers to measure five classroom objects, such as an eraser, a glue stick, a paper clip, a pair of scissors, and a crayon. Tell students to choose objects that are shorter than their rulers. Have students record their measurements in their student books.” Module 1, Lesson 7: Measure lengths and relate 10 cm and 1 cm, Learn, students combine tools to measure an object that is longer than 10 cm. Module 1, Lesson 8: Make a meter stick and measure with various tools, Learn, students measure with Meters and Centimeters and discuss different ways to measure an object and reason about units. “Gather students on the rug with the student-created meter sticks. Invite students to think–pair–share to relate 1, 10, and 100 cm. ‘How many 1 cm cubes are the same length as the meter stick? How can you be sure?’ (100. We used ten 10 cm rulers, so we can count the rulers by ten to be sure there are 100 cm.) ‘Use a double-sided meter stick (numberless side) to confirm. Point to each unit of 10 as students chorally count.’ (10, 20, 30, … , 80, 90, 100) Remind students about the rug they started to measure in the previous lesson segment. ‘Now that we have this longer length unit, the meter, let’s use it to measure the rug.’ Have students chorally count by hundreds as you lay several double-sided meter sticks (numberless side) end to end to measure the rug. ‘1 meter is how many centimeters?’ (100 cm) ‘2 meters is how many centimeters?’ (200 cm) Pause when the remaining length is less than a meter. Have students think–pair–share about how to finish measuring the rug.” Module 5, Lesson 9: Use an inch ruler and a yard stick to estimate and measure the length of various objects, Learn, students use benchmarks to estimate and measure by using an appropriate tool. “‘Let’s practice selecting a unit and measuring the length of an object.’ Direct students to a classroom bulletin board. ‘Which unit is the long side of the bulletin board closest to: 1 inch, 1 foot, or 1 yard?’ (1 yard) ‘Let’s use a benchmark to estimate how long it is. We know the length of the table is 1 yard. About how many table lengths do you think would fit on the long side of the bulletin board? Picture it in your mind.’ Direct students to give a thumbs-up once they’ve pictured how many table lengths would fit and they have an estimate. ‘If 1 table length is about 1 yard, about how many yards do you think the bulletin board is?’ (About 3 yards, About 2 yards) Partner students and distribute a double-sided meter stick to each pair. Have students turn and talk about the different units on the meter stick. ‘How long is 1 yard in inches?’ (36 inches) ‘How many inches are shown on the meter stick?’ (It stops labeling inches at 36 but there are still more inches.) ‘Why do you think it stops labeling at 36 inches? (Because 36 inches is a yard. Because it’s a meter stick and a meter is longer than a yard.) ‘This tool is called a meter stick because from end to end it is 1 meter, or about 39 inches. To measure in yards, we would have to mark and move forward from the 36-inch mark instead of from the end of the meter stick. That method of measuring would be harder and less accurate. Let’s use the rulers we made to create a yardstick so we can more accurately measure in yards. Each ruler is 1 foot, or 12 inches, and 1 yard is 3 feet, or 36 inches. We can put 3 rulers together to make a yard stick.’ Distribute the additional student-created rulers and tape. Then model putting together three student-created rulers to make a yard stick. ‘Now we can use our yard sticks with the mark-and-move-forward technique to measure the bulletin board.’ Model measuring the bulletin board by using the mark-and-move forward technique with a student- created yard stick. Record the measurement of the bulletin board by using both the word yard and the abbreviation, yd. Help students connect the abbreviation yd to the word yard. Direct students’ attention to the short side of an easel. ‘Which unit is the short side of an easel closest to: 1 inch, 1 foot, or 1 yard?’ After that the students measure lengths to the nearest inch, foot, and yard. Direct students to work in centers and rotate to different centers approximately every 4 minutes. The centers are numbered in order from simple to complex: Center 1: Students measure to the nearest inch. Center 2: Students measure to the nearest foot. Center 3: Students measure to the nearest yard. Center 4: Students decide whether to measure in inches, feet, or yards. Clarify the task by stating that students will measure actual objects in the classroom and not the pictures of the objects on their paper. Direct students to go to their assigned centers with their books and begin their measurement work. Have students rotate to the next center on your signal. After students have visited all the centers, direct students to return to their seats. ‘What do you notice about the objects at center 1?’ (They are all measured in inches.) ‘What do you notice about the objects at center 2?’ (They are all measured in feet.) ‘What do you notice about the objects at center 3?’ (All the objects are measured in yards.) Invite students to think–pair–share about how they decided which unit to use when measuring at center 4. (I thought about each benchmark and asked myself, ‘Is the length of this object more like the paper clip, the whiteboard, or the table?’ For all the objects less than 1 foot, I measured in inches. If the objects are longer than 1 foot but shorter than 1 yard, I measured in feet. If the objects are more than 1 yard, I measured in yards.)”

• Module 3, Lessons 10, 11, and 13 engage students in extensive work with 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). Lesson 11: Partition circles and rectangles into equal parts, and describe those parts as halves, thirds, or fourths, Launch, students reason about partitioning a whole into thirds to solve a sharing problem. Three images are shown: a square cut in half vertically, a square cut in half horizontally, and a square cut in half diagonally. “Present the following problem and use the Math Chat routine to engage students in mathematical discourse. ‘Imani, Ming, and Zoey want to share a small rectangular cake. Show two ways they can cut the cake to share it equally.’ Give students 1 minute of silent work time to draw a model. Have students give a silent signal to indicate they are finished. Have students discuss their thinking with a partner. Circulate and listen as they talk. Identify a few students to share their thinking. Purposefully choose work that allows for rich discussion about connections between strategies, such as vertically and horizontally partitioning the cake. Then facilitate a class discussion. Invite students to share their thinking with the whole group and record their reasoning. Transition to the next segment by framing the work. ‘Today, we will partition, count, and describe different units of a whole.’” Lesson 13: Recognize that equal parts of an identical rectangle can be different shapes, Learn, students cut apart and compare halves of the same whole to determine that equal parts can have different shapes. “Distribute one paper square to each student. Pair students and designate each student as partner A or partner B. Direct partner A to fold their square vertically and partner B to fold their square diagonally. Demonstrate lining up the corners so the squares fold into 2 equal parts. Invite students to turn and talk to compare the similarities and differences between the two different ways to make halves. Direct students to cut along the fold line. ‘What do you notice about the 2 parts you cut from each square?’ (My 2 parts are rectangles, but my partner’s parts are triangles.) Direct students to place 1 rectangular half on top of one triangular half. Then invite students to think–pair–share about whether the 2 parts are equal. (I don’t think they are equal, because one is a rectangle and one is a triangle. I don’t think they are equal, because they don’t match up. I think they are equal because they both are 1 half from the same-size square.) Demonstrate folding the triangle in half twice. Then cut along the folds so there are four separate triangles. Direct students to repeat the steps with 1 triangular half. Direct students to arrange the 4 triangular pieces on top of the rectangle so they cover the whole rectangle. Invite students to think–pair–share about whether their thinking has changed about the 2 parts being equal. (I think they are equal halves. When we moved the pieces of the triangle, they covered the same amount of space. The pieces from the triangular half cover the rectangular half without any gaps or overlaps, so the halves must be the same size. They take up the same amount of space.) ‘Equal parts from the same whole can be different shapes. We have 1 half that is a triangle and 1 half that is a rectangle, but they are still equal parts. They take up the same amount of space.’”

The instructional materials provide opportunities for all students to engage with the full intent of all Grade 2 standards through a consistent lesson structure. Examples include:

• Module 1, Lesson 20, and Module 6, Lesson 1 engage students with the full intent of 2.NBT.2 (Count within 1000; skip-count by 5s, 10s, and 100s). Module 1, Lesson 20: Count and bundle ones, tens, and hundreds to 1,000, Learn, students count and bundle 10 tens as 1 hundred to develop place value understanding. “‘Now, we can count by tens. Let’s show that with our fingers first. Give each finger a value of 10. Count with me.’ (Count the math way, beginning with the right pinkie. Students begin with their left pinkie. 10, 20, 30, … , 100)  At 100, loudly clap hands together and lace fingers, as students do the same. ‘10 tens can be bundled to make 1 hundred. Hundreds are the next larger place value unit after tens. What do 10 tens make?’ (1 hundred) Demonstrate how to count and bundle tens as students count chorally the math way. In a later portion of Learn, students count and bundle 10 hundreds as 1 thousand to develop place value understanding. “‘How many ones did it take to make a ten?’ (10 ones) ‘How many tens did it take to make a hundred?’ (10 tens) Invite students to think–pair–share about how many hundreds they think it will take to make a thousand. (10 hundreds, because the numbers always go 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, and then we get a new unit. It will take 10 hundreds, because we always make one bigger group out of 10 smaller groups.) ‘10 hundreds can be bundled to make 1 thousand. Thousands are the next larger place value unit after hundreds.’ Direct students to confirm by counting by hundreds the math way and clasp hands when they reach 1 thousand. Invite groups, one at a time, to place their hundreds in a central location as the class counts by hundreds. (100, 200, 300, … , 1,000) When the class reaches 1,000, bundle the new unit. ‘Ten hundreds make the next larger unit, a thousand.’ (Gesture to the bundle of the new unit, 1,000.) Point to each additional hundred and guide students to count. (1 thousand 1 hundred, 1 thousand 2 hundreds) Demonstrate how to draw ones and bundles of tens and hundreds for the Problem Set.” Module 6, Lesson 1: Compose equal groups and write repeated addition equations, Fluency, students write the time to the nearest 5 minutes and use picture clues to distinguish between a.m. and p.m. to build fluency with time from module 3. “Display the picture of the blank clock. ‘Let’s count by 5 minutes around the clock together.’ Point to the numbers on the clock as students count by fives from 0 to 60. (0, 5, 10, … , 60) Display the picture of the boy waking up and the clock that shows 7:00.”

• Module 3, Lessons 14, 17 and 18 engage students with the full intent of 2.MD.7 (Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m). Lesson 14: Distinguish between a.m. and p.m., Learn, students determine the difference between a.m. and p.m. “Display the timeline. ‘Look at this timeline of a day. What do you notice?’ (I notice that a whole day is 24 hours. I notice that the day is split into 2 equal halves. I notice that a.m. takes up half of the day and p.m. takes up the other half of the day.) ‘The day is divided into two equal parts: a.m. and p.m. Both are 12 hours long. The morning hours, referred to as a.m., begin at midnight, when we are typically sleeping, and end at noon, the middle of the day, around lunch. The afternoon and evening, from noon to midnight, is referred to as p.m. We typically think of a.m. as the morning. However, a.m. starts at midnight when we are sleeping, so it is dark outside for part of the a.m. (Display the timeline with Dark outside labeled for early a.m.) In most places, it starts getting dark again in the evenings when we go to bed and fall asleep.” Lesson 18: Tell time to the nearest five minutes, Learn, students apply their ability to count by fives to tell time on a clock by five minutes intervals. The teacher uses a demonstration clock. “‘What unit does each tick mark on the clock represent?’ (Each tick mark represents 1 minute.) ‘How many minutes are there between each number on the clock? (Gesture to the numbers on the clock.)’ (5 minutes) ‘Count by 5 minutes with me.’ (0 minutes, 5 minutes, 10 minutes, 15 minutes, … , 60 minutes) ‘How many minutes are in 1 hour?’ (60 minutes) ‘When the minute hand gets to the 12, 60 minutes have elapsed, or gone by. Then a new hour begins.’ Show 7:00 on the demonstration clock. ‘Watch what happens to the hour hand as I move the minute hand.’ Show 7:05 on the demonstration clock. ‘How many minutes have elapsed since seven o’clock?’ (5 minutes) ‘We say this time as “seven oh five,” and we write it like this. (Write 7:05.) What do you notice about the hour hand?’ (It moved away from the 7 a little bit.) Move the minute hand around the clock, asking students to write the time on their whiteboards at each 5-minute interval (7:10, 7:15, 7:20, … , 7:55). Highlight the position of the hour hand relative to the minute hand. Stop at 7:30. ‘How many minutes until the next hour?’ (30 minutes) ‘What do you notice about the hour hand?’ (It’s halfway between the 7 and 8.) Stop at 7:55. ‘How many minutes until the next hour?’ (5 minutes) ‘What do you notice about the hour hand?’ (It’s really close to the 8. It’s almost touching it.) Show 8:00 on the demonstration clock. ‘What time is it now?’ (8:00) ‘Where is the hour hand?’ (It is directly on the 8.) Direct students to write the time they see. ’True or false: This time is written as 7:60.’ (False) Invite students to think–pair–share about why the time is not written as 7:60. (It’s eight o’clock now, so the hour should be an 8. No minutes in the eight o’clock hour have gone by, so it starts at zero again. As soon as the minute hand hits 60 minutes, the hour changes and the minutes restart at zero.) ‘When 60 minutes have passed, the minutes are renamed as 1 hour. The hour changes, and the minutes go back to zero to show that a new hour has started.’ Invite students to turn and talk about why we never see 60 minutes displayed on a digital clock.”

• Module 6, Lessons 14, 15, and 16 engage students with the full intent 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). Lesson 14: Relate doubles to even numbers and write equations to express the sums, Land, students engage in discourse as they relate doubles to even numbers and write mathematical equations to express sums. “Use the following prompts to initiate a class discussion. Encourage students to restate their classmates’ responses in their own words. Direct students to problems 6–11 on the Problem Set. ‘Did you answer even or not even for each problem? Why?’ (Even, because every time we double a number, the sum is an even number.) ‘How do you determine whether a number is even or not even?’ (I ask myself if each number is the sum of a doubles fact. If it is, I know the number is even. I skip-count by twos starting at 0. If I say the number while I count, I know it’s even.) ‘How about the number 23? Is it even or not even? Why?’ (I don’t think it’s even because I know 10+10=20, so that means 11+11=22. I know 23 is 1 more than 22. It’s not the sum of a doubles fact, so I know it’s not even. No, 23 is not even. I counted by twos and skipped 23. I said 20, 22, 24.)” Lesson 16: Use rectangular arrays to investigate combinations of even and odd numbers, Learn, students find the sum of two even addends. “Pair students and designate one student as partner A and the other as partner B. Direct partner A to make 2 rows of 3 and partner B to make 2 rows of 4. ‘Partner A, how many tiles do you have?’ (6 tiles) Record 6. ‘Is 6 even or odd?’ (6 is even.) Record an E under the 6 to show that it is an even number. ‘Partner B, how many tiles do you have?’ (8 tiles) Record 8. ‘Is 8 even or odd?’ (8 is even.) Record an E under the 8 to show that it is an even number. ‘What happens when we add an even number with an even number?’ Record a plus sign between the 6 and the 8. Invite students to think–pair–share about whether they think the sum will be an even number or an odd number. (I think the sum will be even because both numbers are even and right now all the tiles have a partner. I think the sum will be even because we are just putting 6 and 8 together. We’re not adding any additional tiles or taking any away. Each tile will continue to have a partner.) Direct students to slide their arrays together so they touch in the middle. ‘What is the total number of tiles now?’ (The total number of tiles is 14.) Record 14 as the sum of 6 + 8. ‘Is 14 even or odd?’ (14 is even.) Label 14 with an E. ‘How do you know 14 is even?’ (I know 14 is even because it is the total of a doubles fact. 14 is even because it has a 4 in the ones place. 14 is even because each object has a partner. I say 14 when I skip-count by twos starting at 0, so it must be even.) ‘So, an even number plus an even number is … (Pause.)’ (An even number) Fill in Even next to Even + Even on the chart.”

The materials reviewed for Eureka Math² Grade 2 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 percent of instructional time to the major clusters of the grade. 

• The number of modules devoted to major work of the grade (including assessments and supporting work connected to the major work) is 4.5 out of 6, approximately 75%.

• The number of lessons devoted to major work of the grade (including supporting work connected to the major work) is 113.25 out of 139, approximately 81%. 

• The number of days devoted to major work of the grade (including assessments and supporting work connected to the major work) Is 118.25 out of 146, approximately 81%.

A lesson-level analysis is most representative of the instructional materials as the lessons include major work and supporting work connected to major work. As a result, approximatley 81% of the instructional materials focus on major work of the grade.

The materials reviewed for Eureka Math² 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 so supporting standards are connected to the major work standards and teachers can locate these connections on a tab called, “Achievement Descriptors and Standards” within lessons. Examples include:

• Module 1, Topic A, Lesson 3: Use information presented in a bar graph to solve put together and take apart problems, Land, Exit Ticket, connects the supporting work of 2.MD.10 (Draw a picture graph and a bar graph [with single-unit scale] to represent a data set with up to four categories. Solve simple put- together, take-apart, and compare problems using information presented in a bar graph) 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). Students see a bar graph labeled “Prizes” as they answer questions. “1. How many prizes are there in all? (18) Write a number sentence. (3+4+6+5=18) 2. Take away 2 of each prize. What is the new total? (10)” 

• Module 5, Topic A, Lesson 4: Solve one- and two-step word problems to find the total value of a group of bills, Launch, 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.NBT.2 (Count within 1000; skip-count by 5s, 10s, and 100s). “Pair students. Distribute one set of dollar bills to each student pair. Work with a partner to organize and count the money in your wallet. Allow students 3 to 4 minutes to work. ‘How did you organize and count your bills?’ (We put all the one-dollar bills together, all the five-dollar bills together, all the ten-dollar bills together, all the twenty-dollar bills together and all the hundred-dollar bills together. We skip-counted by each bill’s value to find the total value of each group. Then we added all five values together. We made groups of ten. Then we put 10 tens together and made 1 hundred. We lined up the bills from greatest value to least value, and then skip-counted by each value until we counted all the bills.) ‘What is the total value of the bills?’ (513) Invite students to think–pair–share about which strategy is the most efficient and why. (I think making groups of tens and then of hundreds is most efficient because you can find the total quickly without doing much addition. Grouping bills into like units is most efficient because you can skip-count easily. I think organizing the bills from greatest value to least value and then skip-counting is efficient because the last number is the total. You don’t have to add.) Transition to the next segment by framing the work.”

• Module 5, Topic C, Lesson 15: Use measurement data to create a line plot, Learn, Collect, Organize and Plot Data, 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). Each student measures their pencil, then the data for all pencils is used to create a line plot. “Have students choose a pencil that is the length they like to write with best. Invite students to think–pair–share about how they can find which pencil lengths most of the class likes to write with. (We can measure each person’s pencil and make a chart with tally marks. Each person can measure their pencil and record the measurement on a sticky note.) ‘We know that graphs can help us answer some questions about data, or information. Let’s use a ruler to measure each person’s pencil to the nearest inch and record the measurements.’ Gather students with their pencils and direct them to the chart with the inch ruler attached and with the long and short pencil lengths plotted. Invite students to check their estimates against the actual measurements of the long and short pencil. Hold up an unsharpened pencil. ‘How many inches long do you think this pencil is?’ (6 inches, 7 inches) ‘Let’s measure to find the actual measurement.’ Place the unsharpened pencil at the endpoint of the ruler. Measure the length of the unsharpened pencil. ‘To measure the pencil to the nearest inch, we need to decide which tick mark the end of the pencil is closest to.’ (It is between 7 inches and 8 inches, but I think it is closer to 8 inches. I think it looks like it is closer to 7 inches—it is right in the middle of the 7 and the 8.) ‘When a measurement is exactly halfway between two numbers or more than halfway, we say it is closest to the next unit. So, even though the pencil is an equal distance from the 7 and the 8, we say that the pencil is about 8 inches long. I am going to place an X on the grid line above the 8 on the ruler.’ Place an X above the 8-inch tick mark on the ruler. ‘If the pencil is less than halfway, we say it is closer to 7, and we would plot the length as 7 inches.’ Direct students to estimate the length of their pencil. ‘Did anyone estimate that their pencil is about 1 inch long?’(Yes.) ‘Let’s find the actual measurement and plot, or record, our data by placing an X above the tick mark on the ruler. The ruler is the number line for the line plot. The grid paper helps us plot our data in straight vertical lines. When the X’s are the same size, it is easier to read our data.’ Invite students to measure the actual length of their pencils and record the measurement on the grid paper above the ruler. Then direct students to the completed data set. ‘We just organized our data into a line plot. A line plot is a graph with measurement data organized above a number line. How is a ruler similar to or different from a number line?’ (The ruler and the number line both have numbers in order. The units are equally spaced on a ruler and on a number line.) ‘What does each X represent?’ (Each X represents the length of 1 pencil.) Remove the ruler from below the data. ‘How many of our pencils are less than 8 inches?’ (The ruler isn’t there anymore, so we don’t know.) ‘Let’s draw a number line, or a scale, to show each unit. This way even if the ruler isn’t there, we can see the length of each pencil.’ Replace the ruler and use it as a guide to make a number line. Draw a tick mark and write the inches until all measurements are represented on the line plot. Remove the ruler. ‘What do you notice about all the tick marks?’ (They are equally spaced just like the tick marks on the ruler. They are all labeled with numbers. No numbers are skipped, even if there isn’t a pencil with that length.) ‘Since the ruler is no longer here, we need to label the line plot so that we know the unit we used to measure the pencils. What unit did we use to measure our pencils?’ (Inches) Write: Length (inches) under the scale. ‘Does the line plot tell us what we measured in inches?’ (No.) ‘Let’s add a title to the line plot.’ Invite students to turn and talk about a title for the graph. Write Pencil Lengths as the title. Invite students to turn and talk about how the scale was created for the line plot and how the data were plotted. Invite students to think–pair–share about why having a scale, a title, and labels is important. (It helps us know what we measured and what unit we used to measure it. We need the scale to know the exact measurements.) ‘Now that our data are organized, we can ask and answer questions about it.’ Use the following prompts to facilitate a discussion about the data: ‘How many pencils are 5 inches long? How many pencils are more than 5 inches long? How many pencils are less than 5 inches long? What pencil length occurs most often, or is the most common? What pencil length occurs least often, or is the least common? What is the total number of pencils we measured? ‘Does the line plot tell us whose pencils are 6 inches long?’ (No, there are no names on the line plot. We only know that 6 pencils are 6 inches long.) Invite students to think–pair–share about what other questions they can use this graph to answer. (We can see how many pencils we measured. We can see which pencil length is the most common. We can answer how many questions, like, How many pencils are 2 inches long? We can see how many pencils are closer in length to the long pencil and how many are closer in length to the short pencil.)”

• Module 6, Topic B, Lesson 5: Compose arrays with rows and columns and use a repeated count to find the total, Learn, Unequal Groups, 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) to the major work of 2.NBT.2 (Count within 1000; skip-count by 5s, 10s, and 100s). Students use skip-counting to test their ideas and articulate why unequal groups cannot be organized into an array. “Present the following statement: ‘Unequal groups can be rearranged into rows or columns to make an array.’ Use the Always Sometimes Never routine to engage students in constructing meaning and discussing their ideas. Give students 1 minute of silent think time to evaluate whether the statement is always, sometimes, or never true. Have students discuss their thinking with a partner. Circulate and listen as they talk. Identify a few students to share their thinking. Then facilitate a class discussion. Invite students to share their thinking with the whole group. Encourage them to provide examples and nonexamples to support their claim.”  Later in the Debrief portion of the lesson, students compose arrays with rows and columns and use a repeated count to find the total. “‘Is this an array? Why?’ (No, it is not an array because the rows are not equal groups. The columns aren’t either. There are some squares missing. It is not an array because the tiles are not organized in equal groups.) ‘Is it useful to organize objects into an array? How?’ (Yes, organizing objects into an array helps because you can skip-count to find the total. Yes, organizing objects into an array helps you see the number of groups and the number of objects in each group. Yes, organizing objects into an array helps you write a repeated addition equation to find the total.)”

The instructional materials reviewed for Eureka Math² 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. 

Grade 2 lessons are coherent and consistent with the Standards and teachers can locate standard connections on a tab called, “Achievement Descriptors and Standards” within lessons. Examples include:

• Module 1, Topic E, Lesson 22: Use counting strategies to solve add to with change unknown word problems, Learn, Represent and Solve an Add to with Change Unknown Word Problem, connects the major work of 2.OA.A (Represent and solve problems involving addition and subtraction) to the major work 2.NBT.A (Understand place value). Students apply place value understanding to solve an add to with change unknown word problem. “Direct students to the problem in their books. Chorally read the problem with the class: ‘Ming biked 64 miles. He wants to bike 100 miles. How many more miles should Ming bike?’ Direct the class to use the Read–Draw–Write process to solve the problem. Invite students to think–pair–share about what they could draw to represent the problem. (We can draw 6 bundles of tens and 4 ones to show 64. We can draw a tape diagram to show 64 as the part and 100 as the total.) Circulate and observe as students work. Select a few students to share their work. Look for examples to highlight multiple solution strategies, including using place value units to count on. As students share their strategies, ask them to explain their rationale. The student work samples shown demonstrate several possible solution strategies. Consider asking these or similar questions to connect the representations: ‘Where do you see 64 in each representation? 36? The total? When did you change units? Why? Where do you see a benchmark number? How does your drawing match the situation?’”

• Module 3, Topic D, Lesson 16: Use a clock to tell time to the half hour or quarter hour, Learn, Decompose a Clock into Four Quarters, connects the supporting work of 2.MD.C (Work with time and money) with the supporting work of 2.G.A (Reason with shapes and their attributes). Students partition a clock into halves and then into fourths to relate fractions to time. “Direct students to remove the clock from their books. Cut out the circle along the dotted line as students do the same. Have students locate the top of the clock ensuring that the write-on lines are oriented correctly. Circulate and assist as needed. Fold the circle in half vertically and have students do the same. ‘How many equal parts do you see?’ (2 equal parts) Trace along the folded line to show the 2 halves and write a 12 at one end of the line (the top) and a 6 at the other end of the line (the bottom). Direct students to do the same. Refold the clock in half along the line, then fold it in half again horizontally, to make fourths. Direct students to do the same. Prompt students to unfold their clocks. ‘How many equal parts do you see now?’ (4) ‘What unit is 4 equal parts?’ (Fourths, or quarters) ‘How do you know there are quarters?’ (I know there are quarters because the whole is partitioned into 4 equal parts.) Invite students to think–pair–share about how halves became quarters. (We split each half in half. Now each half has 2 equal parts, so the whole has 4 equal parts. I know 4 equal parts of a whole are called fourths, or quarters. Each equal part was split into 2 parts, so now there are 4 equal parts) Direct students to trace along the second fold line and label the 3 and the 9 on the clock as you do the same. Guide students to cut out and attach the clock hands with a brad fastener. ‘Let’s show different times on our clocks. The hands on your clocks don’t move like the demonstration clock, so you need to put the hour and the minute hands in the right place.’ Show 5:00 on your clock. Circulate and give students immediate feedback as they work. Repeat the process with the following sequence: half past 5:00, quarter to 6:00, seven o’clock, quarter to 8:00, half past 9:00, half past 7:00, quarter past 8:00.”

• Module 5, Topic A, Lesson 1: Organize, count, and represent a collection of coins, Learn, Organize, Count and Record, connects the supporting work of 2.MD.C (Work with time and money) to the supporting work of 2.MD.D (Represent and interpret data). Students use a collection of coins, organize the coins in some way to count, and determine how much money is represented. “Partner students and distribute a collection to each pair. Invite partners to work together to estimate the value of the coins in their collection. Direct them to write their estimates on the recording sheet. Encourage partners to discuss how they will organize their collection before they count. Invite partners to select organizing tools, with the understanding that they may exchange tools as they refine their plans. Ask partners to begin counting their collection. Circulate and notice how students engage in the following behaviors: Organizing: Strategies may include sorting by coin type or value, making tens, making a dollar, and writing expressions or equations. Counting: Students may repeatedly add to find the total or skip-count by using unit form or standard form. Some students may use less efficient counting strategies, such as counting by ones. Recording: Recordings may include drawings, numbers, expressions, equations, and written explanations. Circulate and use questions and prompts such as the following to assess and advance student thinking: ‘Show and tell me what you did. How can you organize your collection to make it easier to count? How does the way you organized your collection make it easier to count? How can making a ten or a hundred help you find the total? How did you keep track of what you already counted and what you still needed to count? How close was your estimate to your actual count? How can you count this in a way that challenges you?’ Select three pairs of students to share their work in the next segment. Look for samples that demonstrate the following strategies: Grouping like units (e.g., grouping all pennies [ones] together and all dimes [tens] together), Starting the count with coins that have the greatest value, Composing a ten or a hundred, As partners share, consider displaying their work alongside their counting collections so students can see the written representation that corresponds to each counting collection. Collect written representations as informal assessment after the lesson.” 

• Module 5, Topic C, Lesson 16: Create a line plot to represent data and ask and answer questions, Learn: Measure Student Height and Plot Data, connects the major work of 2.MD.A (Measure and estimate lengths in standard units) to the major work of 2.NBT.B (Use place value understanding and properties of operations to add and subtract). Students plot their height on a line plot, measure themselves with a yard stick and use addition to convert yards to inches. “‘Let’s use yard sticks to create a scale for our vertical line plot.’ Direct students to the yard sticks taped to the door. ‘Do you recall at the beginning of the year when you put yourselves in a line by height order? Let’s do that again, but this time we will measure your height in inches instead of centimeters.’ Direct students to order themselves in a straight line from shortest to tallest. Measure the shortest and tallest students first to establish the range. Have one student stand next to the yard sticks and guide the other student to mark an X at their height. Then have students switch roles. Continue measuring pairs of students’ heights. When finished, reposition the paper so there is space to create a scale between the yard sticks and the X’s. Draw a vertical line beside the X’s that extends beyond the first and last measured heights. ‘How many inches compose 1 yard?’ (36 inches)  Invite students to mentally find the total number of inches for each row of X’s. ‘Our first measurement is 11 more inches than 1 yard. What is 11 more than 36?’ (47) Make a tick mark and write the number of the first measured height. ‘Our number line starts at 0, but do we need to write every number from 0 to the first measurement on our scale?’ (No.) Mark 0 on the number line and then draw two slashes //. ‘The slashes show that we are skipping all of the numbers from 0 to the length of our first height measurement.’ Complete the scale and continue to record each measurement as students find the total for the remaining data.”

The materials reviewed for Eureka Math² 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.

Content from future grades is identified within materials and related to grade-level work. These references are consistently included within Topic and Module Overviews and less commonly found within teacher notes at the lesson level. Examples include:

• Module 1: Place Value Concepts Through Metric Measurement and Data \cdot Place Value, Counting, and Comparing Within 1,000, Module Overview Part 1, After This Module, connects 2.MD.A (Measure and estimate lengths in standard units) and 2.NBT.A (Understand place value) to work done in Grade 3, Modules 2 and 5. “Grade 3 Module 2: Students estimate and measure weight and liquid volume. They explore the relationship between place value units by reasoning that there are 1,000 grams in 1 kilogram and 1,000 milliliters in 1 liter. Students apply their understanding of metric measurement as they represent word problems with a tape diagram and solve flexibly. In addition, students use their understanding of the number line to read vertical measurement scales. Finally, students represent data in scaled bar graphs and solve problems related to graphs. Grade 3 Module 5: Students use the interval from 0 to 1 on the number line as the whole. They iterate fraction tiles to partition a number line into fractional units. Students count unit fractions and relate the placement of a fraction on the number line to its distance from 0. Then students apply their understanding of fractions on the number line to rulers and to the creation of line plots.” (3.MD.A, 3.MD.B, 3.NF.A)

• Module 5, Topic B: Use Customary Units to Measure and Estimate Length, Topic Overview, connects 2.MD.A (Measure and estimate lengths in standard units) and 2.MD.B (Relate addition and subtraction to length) to partitioning from 0 to 1 in Grade 3. “Now that students have a solid conceptual understanding of length, they use a number line to represent distances, for example, the distance a rocket travels. Students apply their knowledge of a ruler to a number line, where they refer to the space between each tick mark as an interval. By using the distance between points and their ability to skip-count by fives and tens, students identify unknown numbers on a number line with a given interval. They reason about how the size of each interval affects the number of intervals, discovering that the more intervals a length is divided into, the smaller each interval is. This lays the foundation for partitioning the interval from 0 to 1 into equal parts in grade 3.” (3.NF.A, 3.MD.B)

• Module 6, Topic A, Lesson 4: Represent equal groups with a tape diagram, Launch, connects 2.OA.C (Work with equal groups of objects to gain foundations for multiplication) to work with multiplication in Grade 3. “This Launch segment is intended to gently guide students to using a more abstract representation of equal groups, the tape diagram. Both equal groups and tape diagrams are acceptable representations and are used throughout the module. In grade 3, students draw tape diagrams to represent multiplication and division problems.” (3.OA.A)

Materials relate grade-level concepts from Grade 2 explicitly to prior knowledge. These references can be found consistently within Topic and Module Overviews and less commonly within teacher notes at the lesson level. Examples include:

• Module 1, Topic B: Metric Measurement and Concepts About the Ruler, connects 2.MD.A (Measure and estimate lengths in standard units) to previous work from Grade 1. “Metric measurement is intrinsically related to place value understanding, as both systems include units of ones, tens, hundreds, and thousands. In topic B, students extend their grade 1 understanding of measurement skills and concepts. They begin this work with centimeter cubes, laying multiple cubes end to end to create their own numberless ruler. Through this concrete experience, students discover concepts about the ruler, including that no gaps or overlaps should appear between length units and the length units should be the same size. Students come to see that they are counting the number of length units, rather than tick marks, from the zero point. They also develop a proportional mental image of a unit of one.” (1.MD.A)

• Module 2: Addition and Subtraction Within 200, Module Overview, connects 2.OA.1 (Represent and solve problems involving addition and subtraction) and 2.OA.2 (Add and subtract within 20) to the work from Grade 1, Module 6, Part 2. “Grade 1 students deepen their problem- solving skills as they use tape diagrams and drawings to represent and solve more complex problems within 20, which include start unknown problem types. Students build on their work in module 5 by extending addition strategies to larger numbers within 100. The focus is on making easier problems by decomposing one or both addends. Students may add like units, add tens then ones, or vice versa, or make the next ten. They use various tools and recording methods, such as number bonds, the number path, and the arrow way to support their strategy work.” (1.OA.A)

• Module 3, Topic A, Lesson 2: Use attributes to identify, build, and describe two-dimensional shapes, Learn, Teacher Note, connects 2.G.A (Reason with shapes and their attributes) to previous work from earlier grades. “The terms triangle and hexagon were formalized in kindergarten and quadrilateral and pentagon were introduced in grade 1. This lesson formally defines all four terms for students and expects students not only to visually identify the shapes but to classify and draw the polygons based on attributes.” (K.G.A, 1.G.A)