The materials reviewed for Everyday Mathematics 4 Grade 2 meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings.

All units begin with a Unit Organizer, Planning for Rich Math Instruction. This component indicates where conceptual understanding is emphasized within each lesson of the Unit. The Focus portion of each lesson introduces new content, designed to help teachers build their students’ conceptual understanding through exploration, engagement, and discussion. The materials include problems that develop conceptual understanding throughout the grade level, especially where called for in the standards. Examples include:

• Lesson 1-4, Class Number Scroll, Focus: Exploring Patterns on the Number Grid, students describe and explore patterns on a number grid. Students are expected to observe such things as every other number on the grid is odd and all of the numbers in the far right column end in a 0. Next, the teacher leads the students in using the chart to skip count by 2s, 5s, and 10s. “Lead children in counting by 2s in unison starting at 0. Point to the numbers on the Number-Grid Poster as you count. Ask: What patterns do the count-by-2 numbers make on the grid?” This activity supports conceptual understanding of 2.NBT.2, “Count within 1000, skip-count by 5s, 10s, and 100s.”

• Lesson 2-5, The Near-Doubles Strategy, Focus: Discussing the Near-Doubles Strategy, the teacher leads a discussion about how using a near-doubles fact can help find the answer to a more difficult problem. “Display problems $5+7=?$ and $6+4=?$ and tell partners to use their knowledge of doubles facts to solve them. After a few minutes, invite volunteers to share their strategies. These may include adding or subtracting 2 from a double (for example, $5 + 7 = 5 + 5 + 2 = 12$) or ‘sharing’ (for example, taking 1 from 7 and giving it to 5 to make the double $6 + 6$).” This activity supports the conceptual understanding of 2.OA.2, “Fluently add and subtract within 20 using mental strategies.”

• Lesson 4-4, Numeration and Place Value, Focus: Reviewing Values of Digits, students examine base-10 blocks. “Display a cube, a long, and a flat. Remind children that these are called base-10 blocks. Hold up a cube. Say: This is a base-10 cube. It represents 1. Then hold up a long and say: This is a long. It represents 10. Ask children to explain why a long represents 10. Hold up a flat and say: This is a flat. It represents 100. Ask children to explain why a flat represents 100.” This class discussion supports students in developing a conceptual understanding of 2.NBT.1, “Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones.”

• Lesson 5-5, Exploring Arrays, Time, and Shapes, Focus: Introducing Arrays, students make arrays and write number models to represent them. “Explain that the arrangement of dots in rows and columns, as shown in the Math Message, is called an array. All of the rows have the same number of dots, and all of the columns have the same number of dots. Ask: How many rows of dots are there? How many dots are in each row? How many dots are there in all?” This activity supports conceptual understanding 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.”

• Lesson 9-5, Reviewing Place Value, Focus: Comparing Multi-digit Numbers, the teacher leads a discussion about expanding three-digit numbers. “How can we use the expanded form of each number to help us compare them? Sample answer: First we can look at the hundreds. Both numbers have 2 hundreds, or 200, so next, we look at the tens. We see that 292 has 9 tens, or 90, but 289 has 8 tens, or 80. So we know 292 is larger.” Then the teacher asks, “Do we need to look at the ones?” Students may answer, “No. The tens told us that 292 is larger.” The teacher repeats this activity with other pairs of 3-digit numbers as needed. This activity supports conceptual understanding of 2.NBT.4, “Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record results of comparisons.”

Home Links and Games provide opportunities for students to independently demonstrate conceptual understanding throughout the grade. Examples include:

• Lesson 1-8, My Reference Book, Quarters and Math Boxes, Math Journal 1, Problem 3, students make multiple representations of 25 cents. “Show two ways to make 25 cents. Use [circle with a P inside, circle with an N inside, circle with a D inside].” This practice activity supports conceptual understanding of 2.MD.8, “Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies using $ and ¢ symbols appropriately.”

• Lesson 2-8, Exploring Addition Tools, Odd and Even Patterns, and Shapes, Math Journal 1, Problem 1, students use number lines provided to find sums. “Show $62 + 10$ on the number line below.” Then students record their answers. This practice activity supports the conceptual understanding of 2.NBT.1, “Add and subtract within 1000 using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method.”

• Lesson 4-5, Using Place Value to Compare Numbers, Math Journal 1, Problem 2, students write numbers in expanded form to compare them. “Write each number in expanded form. Then write < or > in the box to compare the two numbers.” Students compare 42 and 48, then write $40+2$ and $40+8$ so they can use the ones to make a comparison. This practice activity supports conceptual understanding of 2.NBT.3, “Read and write numbers to 1000 using base-10 numerals, number names, and expanded form.”

The materials reviewed for Everyday Mathematics 4 Grade 2 meet expectations for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency.

All units begin with a Unit Organizer, Planning for Rich Math Instruction. This component indicates where procedural skill and fluency exercises are identified within each lesson of the Unit. The Mental Math Fluency exercises found at the beginning of each lesson develop fluency with basic facts and other skills that need to be automatic while engaging learners. The Practice portion of the lesson provides ongoing practice of skills from past lessons and units through activities and games. Examples include:

• Lesson 2-3, Doubles and Combinations of 10, Focus: Using Double Ten Frames, students explore strategies to find dot totals on double ten frames. “Flash each Quick Look Card for 2 to 3 seconds before removing it from view or covering it. Always allow a second look and follow up by asking children both what they saw and how they saw it. Asking such questions will allow a variety of strategies to emerge. Encourage children to share multiple strategies but focus attention on those that involve doubles and combinations of 10.” This activity provides an opportunity for students to develop fluency in 2.OA.2, “Fluently add and subtract within 20 using mental strategies.”

• Lesson 5-7, Open Number Lines, Focus: Using Open Number Lines, students use mental strategies to solve addition number stories and record their thinking on open number lines. “Pose the following number story and have children draw open number lines on their slates to record their thinking: Peter has 64 blocks in his toy box and 30 blocks on the table. How many blocks does he have in all?” This activity provides an opportunity for students to develop fluency of 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.”

• Lesson 7-3, Playing Basketball Addition, Focus: Introducing Basketball Addition, students are introduced to a new game. “Basketball Addition is played by two teams of 3 to 5 players each. The number of points scored by each player in each half is determined by rolling one 20-sided polyhedral die or by rolling three 6-sided dice and using their sum. The team that scores the most points wins the game.” This activity provides an opportunity for students to develop fluency of 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.”

• Routine 3: Attendance Routine, students use data from the attendance chart to tell number stories. “Today, 23 children are here. If 2 more children arrive, how many will be present? There are 24 children here today. If 6 go to the library, how many are left?” This activity provides continuous opportunities for students to develop fluency of 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.”

Math Boxes, Home Links, Games, and Daily Routines provide opportunities for students to independently demonstrate procedural skills and fluency throughout the grade. Examples include:

• Lesson 2-4, The Making-10 Strategy, Practice: Math Journal 1, students apply the making 10 strategy. “Use the double ten frames to make 10. Then find the sum. Write the combination of 10 that helped.” In Question 1, students are shown 9 and 5. The suggested answer is, “The combination of 10 that helped: $9+1=10$ and fact $9+5=10$.” This activity provides an opportunity for students to independently demonstrate the procedural skill of 2.OA.2, “Fluently add and subtract within 20 using mental strategies.”

• Lesson 4-11, Matching Facts with Strategies, Measuring a Path, Exploring Arrays, Practice: Math Journal 1, students match subtraction facts to strategies used to solve. “In small groups children discuss their reasoning for their pairings, focusing especially on differences in how they matched facts and strategies.” This activity provides an opportunity for students to independently demonstrate the procedural skill of 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.”

• Lesson 5-7, Open Number Lines, Practice: Math Journal 2, Problem 2, students use Open Number Lines to record their thinking when adding two 2-digit numbers. “You build the second tower with 37 red blocks and 32 blue blocks. How many blocks did you use?” This activity provides an opportunity for students to independently demonstrate the procedural skill of 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.”

• Lesson 9-7, Expand-And-Trade Subtraction, Part 2, Practice; Math Journal 2, Problem 3, students are given a subtraction problem and find a ballpark estimate, write each number in expanded form, and solve, “” This activity provides an opportunity for students to independently demonstrate the procedural skill of 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.”

The materials reviewed for Everyday Mathematics 4 Grade 3 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics.

Materials include multiple routine and non-routine applications of mathematics throughout the grade level. Focus activities introduce new content, provide routine exercises, review recent learning, and provide challenging problem-solving tasks that help build conceptual understanding, procedural skill and fluency, and application of mathematics. Open Response lessons provide challenging problems that involve more than one strategy or solution. Home-Links relate to the Focus activity and provide informal mathematics activities for students to do at home. Examples of routine and non-routine applications of the mathematics include:

• Lesson 3-9, Going-Back-Through-10 Strategy for Subtraction, Focus: Going Back Through 10, Math Journal 1, Problem 4, students use a subtraction strategy to solve problems. “Make up and solve your own subtraction story.” Students apply their understanding 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.”

• Lesson 5-3, Counting Up With Money, Focus: Making Change, Math Journal 2, students count up from the cost of an item to the amount paid using Pine School’s Fruit and Vegetable Sale. “I am buying an orange. I give the clerk two dimes. How much change should the clerk give back?” Students apply their understanding of 2.MD.8, “Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and ¢ symbols appropriately. Example: If you have 2 dimes and 3 pennies, how many cents do you have?”

• Lesson 6-4, Animal Number Stories, Focus: Drawing a Bar Graph, Math Journal 2, Problem 4, students draw a bar graph and use the information to write their own number story. “Use the data in the bar graph to write your own number story.” Students apply their understanding 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.”

• Lesson 8-8, Equal-Groups and Array Number Stories, Home Link, Problem 2, “Find the total number of objects in each picture. Then write a number model.” Students are shown a picture of a muffin tin. “There are ____ muffin cups. Number Model ____.” This activity provides the opportunity for students to apply their understanding 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.”

Materials provide opportunities for students to independently demonstrate multiple routine and non-routine applications of the mathematics throughout the grade level. Independent Problem Solving provides “additional opportunities for children to apply the content they have learned during the section to solve non-routine problems independently. These problems often feature: applying math in the real world, multiple representations, drawing information or data from pictures, tables, or graphs, and opportunities for children to choose tools to support their problem solving.” Examples of independent demonstration of routine and non-routine applications of the mathematics include:

• Independent Problem Solving 1a, “to be used after Lesson 1-6”, Problem 1, students use mental strategies to add and subtract. “a. Sarah used a calculator to check her brother’s math homework. When she tried to enter 15 into the calculator she found that the 5 key was broken. Write at least 3 number sentences that show how Sarah can show 15 on the calculator without using the 5 key.” This activity provides the opportunity for students to independently demonstrate an understanding of 2.OA.2, “Fluently add and subtract within 20 using mental strategies.”

• Independent Problem Solving 1b, “to be used after Lesson 1-11”, Problem 1, students draw combinations of coins to show 35 cents. “Juan’s mom gave him and his sister Maria each 35¢. They noticed they each had a different combination of coins. Use P, N, D, Q. Draw possible coins for Juan and Maria.” This activity provides the opportunity for students to independently demonstrate understanding of 2.MD.8, “Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and ¢ symbols appropriately.”

• Independent Problem Solving 5b, “to be used after Lesson 5-10”, Problem 1, students use a table with temperatures of 3 cities at 6:00 am and 6:00 pm to write and solve number stories. “Use the information in the table above to write a number story about the temperature change in one of the cities/towns. Solve your number story. Show your work below.” This activity provides the opportunity for students to independently demonstrate an understanding 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.”

• Independent Problem Solving 7a, “to be used after Lesson 7-1”, Problem 1, students draw possible combinations of coins to show 35 cents. “a. Ellen has saved $26. She needs$80 to buy a new gaming headset. Ellen knows she could subtract to find out how much more money she needs, but the subtraction key on her calculator is broken. How might she use her broken calculator to get from 26 to 80? b. Ellen needs ___ more to buy the headset. c. What other tool(s) could Ellen use to start at 26 and end at 80? d. Explain how Ellen might use the tool(s) you list above.” This activity provides the opportunity for students to independently demonstrate an understanding 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.”

The materials reviewed for Everyday Mathematics 4 Grade 2 meet expectations in that the three aspects of rigor are not always treated together and are not always treated separately. There is a balance of the three aspects of rigor within the grade.

All three aspects of rigor are present independently throughout the grade. Examples where materials attend to conceptual understanding, procedural skill and fluency, or application include:

• Lesson 2-2, Addition Number Stories, Focus: Creating and Solving Addition Number Stories, students make up and solve number stories, “1. Display the story or draw a picture that illustrates the story but doesn't suggest a solution strategy. 2. Draw an empty unit box below the story. 3. Have children write a label in the unit box and share how they would answer the question in the story. 4. Ask a volunteer to write a number model for the story. 5. Ask another volunteer to explain how the numbers in the number model connect to the story.” This activity develops a conceptual understanding 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.”

• Lesson 3-2, Subtraction From Addition: Think Addition, Focus: Generating Related Addition and Subtraction Facts, students generate related addition and subtraction facts based on dominoes, “Display a domino with 5 dots on one side and 4 dots on the other, Help children discover the addition facts and subtraction facts it shows. Ask: Which addition facts describe the domino? If needed, remind children about using the turn-around rule to generate related addition facts. Which subtraction facts describe this domino?” Students repeat the activity with other dominos. This activity develops the procedural skill of 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.”

• Lesson 7-8, Representing Data: Arm Spans, Focus: Making a Line Plot of Arm Span Data, students measure their arm spans, enter the data in a frequency table, and draw a line plot. “Discuss children’s completed line plots. Ask: What does it mean when there are a lot of Xs above a number? Which arm span is the most common? What do you know about the numbers that have no Xs above them? How many children have an arm span of 51 inches? Of 46 inches?” This activity provides an opportunity for students to apply their understanding 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.”

Multiple aspects of rigor are engaged simultaneously to develop students’ mathematical understanding of a single unit of study throughout the materials. Examples include:

• Lesson 4-7, Playing Target, Focus: Playing Target to 50, students add and subtract using base-10 blocks, “Players take turns. When it is your turn, do the following: Turn over 2 cards. You may either use one card to make a 1-digit number or both cards to make a 2-digit number. Model your number with base-10 blocks. Put these blocks just below your Target Game Mat but not on the mat. You now have two choices: Choice 1: Add all of the base-10 blocks below the mat to the blocks already on your Target Game Mat. Choice 2: Subtract all of the blocks below the mat from the blocks already on your Target Game Mat. If you decide to subtract, you may first have to make exchanges on the mat.” The first player to have a mat with a value of exactly 50 wins. Students develop conceptual understanding of 2.NBT.1, “Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones,” and procedural skills with 2.NBT.7, “Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.”

• Lesson 4-11, Matching Facts with Strategies, Measuring a Path, Exploring Arrays, Focus: Exploration A: Matching Facts with Strategies, students match subtraction facts to possible solution strategies. “Children independently match subtraction facts from Math Masters, page 110 to strategies they could use to solve them on journal page 89. In small groups children discuss their reasoning for their pairings, focusing especially on differences in how they matched facts and strategies.” Students develop procedural skills with 2.OA.2, “Fluently add and subtract within 20 using mental strategies,” and conceptual understanding of 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.”

• Lesson 6-4, Animal Number Stories, Focus: Solving Silly Animal Stories and Writing Silly Animal Stories, students solve number stories comparing the heights and lengths of various animals. “Have children share the names and lengths of the longest animal and the shortest animal.” Then students write and solve their own number stories, “Children write two number stories. In each story, they compare or add the lengths in feet of two animals from journal page 146.” Students develop procedural skills with 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,” and application 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.”

The materials reviewed for Everyday Mathematics 4 Grade 3 meet expectations for supporting the intentional development of MP1: Make sense of problems and persevere in solving them; and MP2: Reason abstractly and quantitatively, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

Each lesson targets one to three MPs. Math Practices are identified for teachers in several places: Pathway to Mastery Correlation to the Mathematical Processes and Practices, Focus, Student Math Journals, Student Reference Book, Independent Problem Solving Masters, and Practice. Materials refer to the Mathematical Practices as GMPs (Goals for Mathematical Practice). 

Materials provide intentional development of MP1 to meet its full intent in connection to grade-level content. Students make sense of problems and persevere in solving them as they work with the support of the teacher and independently throughout the units. Examples include:

• Lesson 3-2, Subtraction From Addition: Think Addition, Focus: Summarizing Subtraction Strategies, students consider different strategies to solve problems involving subtraction. “Have partners discuss the following questions: If you want to solve a subtraction fact that you don’t know, what strategies could you use? Pose a few subtraction facts to provide a context, such as $12-6$, $15-8$, and $16-7$. Encourage children to refer to their My Subtraction Fact Strategies Table on journal page 48. As volunteers share their ideas, record their strategies on the Class Data Pad and lead a class discussion about the similarities and the differences between the strategies.” 

• Lesson 3-4, Playing Salute! Focus: Introducing and Playing Salute! students use various addition strategies and determine if answers make sense while playing a game to practice adding and finding missing addends. “Circulate among the groups as they play. Whenever possible, encourage children to reflect on and discuss strategies they think would have been more efficient for a given round. Have them resolve any discrepancies between their answers.”

• Lesson 5-2, Using Coins to Buy Things, Focus: Reviewing Money Equivalencies, students analyze and make sense of money problems and as they find equivalent coin combinations. “Display a nickel. Ask: What is this called? How much is it worth? Write nickel and 5 cents on the Class Data Pad. Ask: How much are 2 nickels worth? Repeat with a penny, a dime, a quarter, and a $1 bill.”

• Independent Problem Solving 5b, “to be used after Lesson 5-10”, Problem 2, students solve a word problem in two different ways. “Hai had some action figures. His friend Amalia gave him 9 more. Now Hai has 24 action figures. How many action figures did Hai have to start with? a) Solve the problem in two different ways. Use drawings, words, or both to show your thinking in the space below.”

Materials provide intentional development of MP2 to meet its full intent in connection to grade-level content. Students reason abstractly and quantitatively as they work with the support of the teacher and independently throughout the units. Examples include:

• Lesson 2-4, The Making-10 Strategy, Focus: Exploring the Making-10 Strategy, students use the making-10 strategy to determine the number of dots in ten frames and record symbolically.  “Remind children that Quick Look activities help us think about addition strategies. Flash Quick Look cards 95 and 98 in sequence. Have children write words or number sentences on their slates to record how they figured out the total number of dots on each card.” 

• Lesson 8-7, Partitioning Rectangles, Part 2, Focus: Partitioning Strategies, students explain number representations when partitioning rectangles into same-size squares and discuss connections between rows and columns. “Ask children to share their strategies for determining how many squares are needed to cover the rectangle. Some children may have visually estimated how many squares will fit in one row and one column, while others may have used their fingers or marks on paper to help them estimate. Ask: How were you able to make sure that your squares were the same size?”

• Lesson 9-5, Reviewing Place Value, Focus: Comparing Multi-Digit Numbers, Math Journal 2, students attend to the meaning of quantities as they compare numbers after writing in expanded form, “Write each number in expanded form. Then write < or > in the box to compare the two numbers.” For example, in Problem 5, students compare the numbers 1,583 and 1,221.

• Independent Problem Solving 3a, “to be used after Lesson 3-4”, Problem 2, students write a word problem to go with an equation. “Write a number story for the number sentence $4=8-4$.”

The materials reviewed for Everyday Mathematics 4 Grade 2 meet expectations for supporting the intentional development of MP3: Construct viable arguments and critique the reasoning of others, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

Each lesson targets one to three MPs. Math Practices are identified for teachers in several places: Pathway to Mastery Correlation to the Mathematical Processes and Practices, Focus, Student Math Journals, Student Reference Book, Independent Problem Solving Masters, and Practice. Materials refer to the Mathematical Practices as GMPs (Goals for Mathematical Practice). 

Materials provide support for the intentional development of MP3 by providing opportunities for students to construct viable arguments in connection to grade-level content. Examples include:

• Lesson 3-8, Using Doubles to Subtract, Home Link, students construct viable arguments as they explain their strategy to someone at home. “Look at the missing addend in each Fact Triangle. Tell someone at home how to use doubles to help find it. Explain how you found the missing addend.”

• Lesson 8-4, (Day 2): Drawing and Reasoning About Quadrilaterals, Focus: Setting Expectations, students construct viable arguments as they discuss solutions to the Open Response Problem. “Review the open response problem from Day 1. Ask: What do you think a complete answer to this problem needs to include? Sample answer: It needs drawings of gardens for Juan and Linda and an explanation for why the circled shape has the attributes for Linda’s plan.”

• Independent Problem Solving 2b, “to be used after Lesson 2-10”, Problem 2, students construct viable arguments as they look for patterns in the sums of even and odd numbers. “Alice wonders if the sum of dots on a domino with an even number of dots on one side and an odd number of dots on the other side will always be even, odd, or could be either. a) Draw at least 1 different even/odd domino and see if you can help Alice figure this out. Use drawings or words or both to show Alice what you think and why.”

Materials provide support for the intentional development of MP3 by providing opportunities for students to critique the reasoning of others in connection to grade-level content. Examples include:

• Unit 3, More Fact Strategies, Open Response Assessment, A Subtraction Strategy, students critique the reasoning of others as they explain and use subtraction strategies. “Grace solved $12-7$ this way: ‘I started at 12 and took away 2 to get to 10. Then I took away 5 more. I ended up at 5. So $12-7=5$.’ Grace solved $13-4$ this way: ‘I started at 13 and took away 3 to get to 10. Then I took away 1 more. I ended up at 9. So $13-4=9$.’ Show and explain how to use Grace’s subtraction strategy to solve $14-8$.”

• Lesson 4-7, Playing Target, Practice: Practicing Place-Value Concepts, Math Journal 1, Problem 8, students critique the reasoning of others as they analyze a provided student strategy. “Marta wrote 24 to describe the number shown by these base-10 blocks: Do you agree with Marta? Explain your answer.”

• Independent Problem Solving 6b, “to be used after Lesson 6-8”, Problem 2, students critique the reasoning of another student’s addition strategy. “Louisa is checking supplies for a party. She counts 27 small cups and 34 large cups. She wants to know how many cups she has in all. Her strategy to solve $27+34=$ ___ is: $24+30=57$, $57+4=61$, $27+34=61$. Do you agree that Louisa’s strategy works? Use drawings or words or both to explain why or why not.”

The materials reviewed for Everyday Mathematics 4 Grade 2 meet expectations for supporting the intentional development of MP4: Model with mathematics; and MP5: Use appropriate tools strategically, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

Each lesson targets one to three MPs. Math Practices are identified for teachers in several places: Pathway to Mastery Correlation to the Mathematical Processes and Practices, Focus, Student Math Journals, Student Reference Book, Independent Problem Solving Masters, and Practice. Materials refer to the Mathematical Practices as GMPs (Goals for Mathematical Practice). 

Materials provide intentional development of MP4 to meet its full intent in connection to grade-level content. Students model with mathematics to solve real-world problems, identify important quantities to make sense of relationships, and represent them mathematically as they work with the support of the teacher and independently throughout the units. Examples include:

• Lesson 9-3, Focus: Solving the Open Response Problem, students use the math they know to divide muffins equally among two and four children, describing each child’s share. “Distribute Math Masters, pages 254-255 to all children. Read the problem as a class and ask partners to discuss what the problem asks them to do. Encourage children to refer to the Equal Shares posters and use fraction vocabulary like that on the poster as they talk about and write responses to the problem. Review the terms one-half, two-halves, one-fourth, and four-fourths. Tell children that an important part of the task is to write how much muffin is in one child’s share. Circulate as children work. Ask children to explain their drawings and descriptions of one child’s share and encourage them to add details to clarify their responses. You may also want to make notes about children’s strategies. Ask: How did you show your work and thinking for this problem? Did you use words, symbols, or anything else?”

• Independent Problem Solving 6a, “to be used after Lesson 6-5”, Problems 1 and 2, students model the situation with an appropriate representation as they solve word problems involving addition. “1. Asia has 24 tickets. She was happy to have 4 more tickets than last year. She wants to get a bracelet, a whistle, and a pencil. How many tickets will Asia have left over? Use drawings or words or both to show your thinking. 2. Mario has 2 tickets fewer than Asia. He wants to use all his tickets. What items could he buy at the carnival?”

• Independent Problem Solving 8b, “to be used after Lesson 8-7”, Problems 1 and 2, students use the math they know to solve problems and everyday situations to find the total number of objects arranged in an array. “1. The principal asked Koke to set up the gym for the science fair. Each student needs the same amount of space for their project. Koke drew a picture of the gym floor and partitioned it into same-sized squares. Each square can hold one project. Before he counted the total number of squares, he accidentally spilled milk on his drawing. Look at Koke’s drawing. Help him figure out how many squares there are in all. 2. Use words to explain how you found the total number of squares in Koke’s drawing.”

Materials provide intentional development of MP5 to meet its full intent in connection to grade-level content. Students use appropriate tools strategically as they work with the support of the teacher and independently throughout the units. Examples include:

• Lesson 9-11, Multiples of 10 and 5, Focus: Math Message, students choose and use appropriate tools and strategies as they solve a word problem. “You have 6 boxes of markers with 10 markers in each box. How many markers do you have in all? Talk to a partner about how you could solve this problem using each of the following tools: a number line, a number grid, and base-10 blocks. Then solve the problem.” 

• Independent Problem Solving 7b, “to be used after Lesson 7-7”, Problem 1, students solve problems that involve measuring and estimating lengths using any tool or strategy. “Two partnerships are each measuring the length of a wall. Jack and Tessa used two yardsticks. Tessa placed the edge of one yardstick at the beginning of the wall. Jack placed the second yardstick at the end of the Tessa’s yardstick. Then Tessa picked up her yardstick and placed it at the end of Jack’s yardstick. They did this over and over until they reached the end of the wall. Does Jack and Tessa’s measuring strategy work? Why or why not?”

• Independent Problem Solving 8b, “to be used after Lesson 8-7”, Problem 3, students choose tools and strategies to solve a word problem. “Mike’s mom told him to clean his closet. He wants to neatly arrange his shoes in his closet. He has 8 pairs of shoes. (Think: How many shoes are in a pair?) Show 2 ways Mike can arrange his pairs of shoes in equal rows. Use counters, drawings, or any other tool to help.”

The materials reviewed for Everyday Mathematics 4 Grade 2 meet expectations for supporting the intentional development of MP7: Look for and make use of structure; and MP8: Look for and express regularity in repeated reasoning, for students, in connection to grade-level content standards, as expected by the mathematical practice standards. 

Each lesson targets one to three MPs. Math Practices are identified for teachers in several places: Pathway to Mastery Correlation to the Mathematical Processes and Practices, Focus, Student Math Journals, Student Reference Book, Independent Problem Solving Masters, and Practice. Materials refer to the Mathematical Practices as GMPs (Goals for Mathematical Practice). 

Materials provide intentional development of MP7 to meet its full intent in connection to grade-level content. Students look for and make use of structure throughout the units as they describe, and make use of patterns within problem-solving as they work with the support of the teacher and independently throughout the units. Examples include

• Lesson 7-2, Four of More Addends, Focus: Counting Pencils, students look for and explain the structure within mathematical representations as they discuss strategies for solving a number story with three addends. “Were any of the three ways easier for you to find the sum? Why? Does it make a difference in what order the three numbers are added? Why? If no child mentions the turn-around rule, ask: How do you think this is related to the turn-around rule for addition?”

• Independent Problem Solving 2a, “to be used after Lesson 2-3”, Problem 2, students analyze a problem and look for more than one approach as they fluently add and subtract within 20. “Jamal said he could always tell how many pennies Jason had in his fist by looking at the number of pennies on the plate. He said he could use a chart to write all the combinations of 15. Help Jamal complete the chart he started. a. Do you agree that Jason’s chart will help him with all the combinations? b. Explain your thinking.”

• Independent Problem Solving 9b, “to be used after Lesson 9-9”, Problem 1, students analyze a problem and look for more than one approach as they solve a money problem in two different ways. “Maya purchased items from Moran’s Market. She spent exactly $50. Show two different combinations of items she might have bought. Show your work below.” Students are given a picture of a table with items and their prices.

Materials provide intentional development of MP8 to meet its full intent in connection to grade-level content. Students look for and express regularity in repeated reasoning throughout the units to make generalizations and build a deeper understanding of grade level math concepts as they work with the support of the teacher and independently throughout the units. Examples include:

• Lesson 3-6, -0 and -1 Fact Strategies and Subtraction Top-It, Focus: Discussing the -0 and -1 Strategies, students create methods for solving -0 and -1 facts. “Ask children to copy the facts onto their slates and solve. Prompt them to describe how to find the answers to -0 facts and -1 facts. Encourage children to look at the facts to find patterns and determine rules for solving these types of facts. Sample answers: If 0 is subtracted from any number, that number does not change. If 1 is subtracted from any number, the result is the next smaller number.”

• Lesson 9-10, Connecting Doubles Facts, Even Numbers, and Equal Groups, Focus: Connecting Doubles and Equal Groups, students notice repeated calculations as they use doubles facts to solve equal-groups stories. “After all 10 possible arrays have been recorded, have children examine the list. Ask: What patterns do you notice? Referring to the two lists of possible number models, discuss the idea that when children need to find the total number of objects in 2 equal groups (or multiply by 2), they can use addition doubles. Ask: How can we use doubles facts to help us solve number stories about 2 equal groups?” 

• Independent Problem Solving 3a, “to be used after Lesson 3-4”, Problem 1, students evaluate the reasonableness of answers and thinking as they add and subtract. “a. Mrs. Foy asked her class to share anything they noticed about $4+3$ and $3+4$. Akilah noticed that both number sentences equal 7 and figured out that the order of the addends does not matter. Dakota said that this also works for subtraction: If $4-3$ is 1, then $3-4$ is 1 as well. Do you agree with Dakota? b. Use drawings or words or both to show how you know.”

The materials reviewed for Everyday Mathematics 4 Grade 2 meet expectations for providing teacher guidance with useful annotations and suggestions for how to enact the student materials and ancillary materials, with specific attention to engaging students in order to guide their mathematical development.

Materials provide comprehensive guidance that will assist teachers in presenting the student and ancillary materials. Examples include:

• Teacher's Lesson Guide, Welcome to Everyday Mathematics, explains how the program is presented. “Throughout Everyday Mathematics, emphasis is placed on problem solving in everyday situations and mathematical contexts; an instructional design that revisits topics regularly to ensure depth of knowledge and long-term learning; distributed practice through games and other daily activities; teaching that supports “productive struggle” and maintains high cognitive demand; and lessons and activities that engage all children and make mathematics fun!”

• Implementation Guide, Guiding Principles for the Design and Development of Everyday Mathematics, explains the foundational principles. “The foundational principles that guide Everyday Mathematics development address what children know when they come to school, how they learn best, what they should learn, and the role of problem-solving and assessment in the curriculum.”

• Unit 7, Whole Number Operations and Measurement and Data, Lesson Organizer, Coherence, 2.NBT.6, provides an overview of content and expectations for the unit. “In Grade 1, children applied their knowledge of addition strategies to solve number stories involving three whole numbers whose sum is less than 20. In Unit 9, children will choose at least three items to purchase for $100. In Grade 3, children will add and subtract numbers using strategies and algorithms.”

Materials include sufficient and useful annotations and suggestions that are presented within the context of the specific learning objectives. Examples include:

• Implementation Guide, Everyday Mathematics Instructional Design, “Lesson Structure and Features include; Lesson Opener, Mental Math and Fluency, Daily Routines, Math Message, Math Message Follow-Up, Assessment Check-In, Summarize, Practice, Math Boxes, and Home-Links.”

• Lesson 1-6, Equivalent Names for Numbers, Focus: Assessment Check-in, teacher guidance supports students to find equivalent number names. “Expect that most children will find at least one equivalent name each for Problems 1-5 on journal page 4. Some children may find more than one name for each; others may use more than one operation. For children who struggle to find one name for each problem, have them count out the number of counters that is the same as the ‘show’ number, divide the counters into two groups, and write an addition number model to represent the groups. Be sure that the number model does not contain the number of the broken key.”

• Lesson 4-7, Playing Target, Focus: Making Exchanges, Adjusting the Activity, teacher guidance supports struggling students. “For children who struggle to remember the values of the base-10 blocks, provide a card showing pictures of the base-10 blocks and their values.”

• Lesson 8-10, Playing Array Concentration, Focus: Math Message Follow-Up, teacher guidance connects students' prior knowledge to new concepts. “Remind children that when they arrange things in equal rows, they are making arrays. Ask volunteers to share their arrays and number models. If children wrote multiplication number models, ask them to suggest addition models as well.”

The materials reviewed for Everyday Mathematics 4 Grade 2 meet expectations for containing adult-level explanations and examples of the more complex grade-level concepts and concepts beyond the current grade so that teachers can improve their own knowledge of the subject.

Each Unit Organizer Coherence table provides adult-level explanations and examples of complex grade/course-level concepts so teachers can improve their content knowledge. Professional Development side notes within Lessons support teachers in building knowledge of key mathematical concepts. Examples include:

• Lesson 1-9, Even and Odd Number Patterns, Focus: Identifying Even and Odd Numbers, Professional Development, teacher guidance clarifies the pairing strategy for even numbers. “The pairing definition for even numbers does not apply to the number 0 because 0 objects cannot be paired. In later grades, children will learn that a number is even if dividing it by 2 yields a remainder of 0. Using this definition, 0 is an even number because $0\div2$ has a remainder of 0.”

• Lesson 4-9, The Inch, Focus: Sharing Strategies, Professional Development, teacher guidance explains the transition from iterating units to standard measurement. “The 12-inch (foot-long) ruler is introduced in this lesson. Some children may not yet see the ruler as composed of a series of inch-long intervals. The activities in this lesson are designed to show children that the length of an object must be the same number of inches, whether it is measured with 1-inch long blocks or with a ruler-a potentially unfamiliar concept to children.”

• Lesson 5-8, Change-to-More Number Stories, Focus: Introducing the Change Diagram, Professional Development, teacher guidance explains how number models are used beyond the grade. “In Everyday Mathematics, children use number models to represent situations and summarize relationships among quantities. In first through fifth grade, number models are used to clarify the quantitative relationships in a problem. Writing number models may help some children decide how to solve a problem, but more importantly, doing so helps them learn the mathematical-symbol system. Translating a word problem into a number model that can be manipulated to find an answer comes later when children begin to learn formal algebra.”

• Lesson 6-3, Interpreting Number Stories, Focus: Sharing Strategies, Professional Development, explains the connection between previous lessons and this lesson. “Until now, lessons have focused on one type of number story at a time. For example, all the problems in Lesson 6-2 were comparison stories, and the comparison diagram was the only diagram used. In this lesson, children are asked to categorize addition and subtraction number stories and then solve them. Do not force any number story into a particular mold. There may be several ways to interpret a problem.”

• Lesson 8-7, Partitioning Rectangles, Part 2, Focus: Partitioning Strategies, Professional Development, supports teachers with concepts for work beyond the grade. “Work with partitioning in Grade 2 lays the foundation for area measurement in Grade 3. In Grade 2 children develop the ability to visualize a rectangle as a collection of squares arranged in a row-by-column structure. This structure is important because it allows children to see the one-to-one correspondence between the number of squares in a row or a column and the number of units of measurement in the rectangle’s sides.”

• Lesson 9-10, Connecting Doubles Facts, Even Numbers, and Equal Groups, Focus: Connecting Doubles and Equal Groups, Professional Development, supports teachers with concepts for work beyond the grade. “This lesson focuses on finding the total number of objects in two equal groups and expressing an even number of objects as the sum of the number of objects in two equal groups. These are expectations for second grade but also build an understanding of how doubles addition facts relate to multiplying by 2. Children are exposed to multiplication number models in this lesson to build readiness for solving 2s multiplication facts early in third grade.”

The materials reviewed for Everyday Mathematics 4 Grade 2 meet expectations for including standards correlation information that explains the role of the standards in the context of the overall series. 

Correlation information is present for the mathematics standards addressed throughout the grade level/series and can be found in several places, including the Correlations to the Standards for Mathematics, Unit Organizers, Pathway to Mastery, and within each lesson. Examples include:

• Grade 2 Math, Correlation to the Standards for Mathematics Chart includes a table with each lesson and aligned grade-level standards. Teachers can easily identify a lesson when each grade-level standard will be addressed. 

• Mastery Expectations, 2.NBT.2, “First Quarter: Count by 1s to at least 120; skip count by 5s using a calculator, and skip count by 10s to at least 200. Second Quarter: Count by 1s within 500; skip count by 5s and 10s past 200; count by 100 to 900. Third Quarter: Count within 1000; skip-count by 5s, 10s, and 100s. Fourth Quarter; Ongoing practice and application.”

• Lesson 3-6, -0 and -1 Fact Strategies and Subtraction Top-It, standards identified in the Focus are 2.OA.2, 2.NBT.5, 2.NBT.9, and standards identified in the Practice are 2.OA.2, 2.NBT.3, and 2.NBT.5. Lessons contain a consistent structure that includes an Overview, Before You Begin, Vocabulary, Warm-Up, Focus, Assessment Check-In, Practice, Math Boxes, and Home-Link. This provides an additional place to reference standards within each lesson.

Each Unit Organizer Coherence table includes an overview of content standards addressed within the unit as well as a narrative outlining relevant prior and future content connections for teachers. Examples include:

• Unit 4, Place Value Measurement, Unit 4 Organizer, Coherence, 2.NBT.1a, includes an overview of how the content in Kindergarten builds from previous grades and extends to future grades. “In Grade 1, children used base-10 blocks to learn about place value of 2-digit numbers through a variety of hands-on activities. In Grade 3, children will use place value understanding to round whole numbers to the nearest 10 or 100.”

• Unit 6, Whole Number Operations and Number Stories, Unit 6 Organizer, Coherence, 2.OA.1, includes an overview of how the content in Kindergarten builds from previous grades and extends to future grades. In Grade 1, children modeled and solved number stories within 20 of all different types, with the position of the unknown varying. In Grade 3, children will solve one-and two-step number stories using all operations.”

• Unit 8, Geometry and Arrays, Unit 8 Organizer, Coherence, 2.G.1, includes an overview of how the content in Kindergarten builds from previous grades and extends to future grades. “In Grade 1, children began to differentiate between defining and non-defining attributes as they sorted attribute blocks and defined a rectangle by its attributes. Such work extended informal Kindergarten discussions of attributes and activities that involved building and drawing shapes. In Grade 3, children will compare and classify polygons based on attributes and will explore the relationships between categories of quadrilaterals.”

The materials reviewed for Everyday Mathematics 4 Grade 2 meet expectations for providing explanations of the instructional approaches of the program and identification of the research-based strategies. 

Instructional approaches to the program are described within the Teacher’s Lesson Guide. Examples include:

• Teacher’s Lesson Guide, Welcome to Everyday Mathematics, The University of Chicago School Mathematics Project (UCSMP) describes the five areas of the Everyday Mathematics 4 classroom. “Problem solving in everyday situations and mathematical contexts; an instructional design that revisits topics regularly to ensure depth of knowledge and long-term learning; distributed practice through routines, games, and other activities; teaching that supports ‘productive struggle’ and maintains high cognitive demand; and lessons and activities that engage all children and make mathematics fun!” 

• Teacher’s Lesson Guide, About Everyday Mathematics, An Investment in How Your Children Learn, The Everyday Mathematics Difference, includes the mission of the program as well as a description of the core beliefs. “Decades of research show that children who use Everyday Mathematics develop deeper conceptual understanding and greater depth of knowledge than students using other programs. They develop powerful, life-long habits of mind such as perseverance, creative thinking, and the ability to express and defend their reasoning.”

• Teacher’s Lesson Guide, About Everyday Mathematics, A Commitment to Educational Equality, outlines the student learning experience. “Everyday Mathematics was founded on the principle that every child can and should learn challenging, interesting, and useful mathematics. The program is designed to ensure that each of your children develops positive attitudes about math and powerful habits of mind that will carry them through college, career, and beyond. Provide Multiple Pathways to Learning, Create a System for Differentiation in Your Classroom, Access Quality Materials, Use Data to Drive Your Instruction, and Build and Maintain Strong Home-School Connections.”

• Teacher’s Lesson Guide, About Everyday Mathematics, Problem-based Instruction, approach to teaching skills helps to outline how to teach a lesson. “Everyday Mathematics builds problem solving into every lesson. Problem solving is in everything they do. Warm-up Activity: Lessons begin with a quick, scaffolded Mental Math and Fluency exercise. Daily Routines: Reinforce and apply concepts and skills with daily activities. Math Message: Engage in high cognitive demand problem-solving activities that encourage productive struggle. Focus Activities: Introduce new content with group problem solving activities and classroom discussion. Summarize: Discuss and make connections to the themes of the focus activity. Practice Activities: Lessons end with a spiraled review of content from past lessons.” 

• Teacher’s Lesson Guide, Everyday Mathematics in Your Classroom, The Everyday Mathematics Lesson, outlines the design of lessons. “Lessons are designed to help teachers facilitate instruction and engineered to accommodate flexible group models. The three-part, activity-driven lesson structure helps you easily incorporate research-based instructional methods into your daily instruction. Embedded Rigor and Spiraled Instruction: Each lesson weaves new content with the practice of content introduced in earlier lessons. The structure of the lessons ensures that your instruction includes all elements of rigor in equal measure with problem solving at the heart of everything you do.”

Preparing for the Module provides a Research into Practice section citing and describing research-based strategies in each unit. Examples include:

• Implementation Guide, Everyday Mathematics & the Common Core State Standards, 1.1.1 Rigor, “The Publishers’ Criteria, a companion document to the Common Core State Standards, defines rigor as the pursuit, with equal intensity, of conceptual understanding, procedural skill and fluency, and applications (National Governors Association [NGA] Center for Best Practices & Council of Chief State School Officers [CCSSO], 2013, p. 3).

• Implementation Guide, Differentiating Instruction with Everyday Mathematics, Differentiation Strategies in Everyday Mathematics, 10.3.3, Effective Differentiation Maintains the Cognitive Demand of the Mathematics, “Researchers broadly categorize mathematical tasks into two categories; low cognitive demand tasks, and high cognitive demand tasks. While the discussion of cognitive demand in mathematics lessons is discussed widely, see Sten, M.K., Grover, B.W. & Henningsen, M. (1996) for an introduction to the concept of high and low cognitive demand tasks.”

• Implementation Guide, Open Response and Re-Engagement, 6.1 Overview, “Research conducted by the Mathematics Assessment Collaborative has demonstrated that the use of complex open response problems “significantly enhances student achievement both on standardized multiple-choice achievement tests and on more complex performance-based assessments” (Paek & Foster, 2012, p. 11).”

• The University of Chicago School Mathematics Project provides Efficient Research on third party studies. For example:

  • Use of Student Constructed Number Stories in a Reform-Based Curriculum.

  • An Action-Based Research Study on How Using Manipulatives Will Increase Student’s Achievement in Mathematics.

  • Differentiating Instruction to Close the Achievement Gap for Special Education Students Using Everyday Math.

  • Implementing a Curriculum Innovation with Sustainability: A Case Study from Upstate New York.

  • Mental Computation of Students in a Reform-Based Mathematics Curriculum.

  • ARC Center Tri-State Achievement Study.

  • Teacher-Initiated Differentiation.

  • The Impact of Two Standards-Based Mathematics Curricula on Student Achievement in Massachusetts.

The materials reviewed for Everyday Mathematics 4 Grade 2 meet expectations for providing a comprehensive list of supplies needed to support instructional activities. 

A year-long list of materials needed is provided in the Teacher’s Lesson Guide, Getting to Know Your Classroom Resource Package, Manipulative Kits, and eToolkit. “The table below lists the materials that are used on a regular basis throughout Second Grade Everyday Mathematics.” Each unit includes a Materials Overview section outlining supplies needed for each lesson within the unit. Additionally, specific lessons include notes about supplies needed to support instructional activities, found in the overview of the lesson under Materials. Examples include:

• Lesson 2-1, Grouping by 10s, Overview, Materials, Math Message, “Math Masters, pp. G11-G13; scissors; envelope, paper clip, or rubber band”

• Lesson 4-1, Clocks and Telling Time, Focus: Math Message, “Display two clock faces as shown in the margin. Which clock shows 4:30? Explain to a partner how you know. Use the words minute hand and hour hand.”

• Unit 5, Addition and Subtraction, Unit 5 Organizer, Unit 5 Materials, each lesson has materials under the following categories: Math Master, Activity Cards, Manipulative Kit, and Other Materials. For example, Lesson 5-3, materials listed, Math Masters: “pp.121; 123-125”, Activity Card: “66-67”, Manipulative Kit; “toolkit coins; per group: 4 each of number cards 0-10; per partnership: two 6-sided dice”, Other Materials: “slate; scissors.” 

• Unit 7, Whole Number Operations and Measurement and Data, Unit 7 Organizer, Unit 7 Materials, each lesson has materials under the following categories: Math Master, Activity Cards, Manipulative Kit, and Other Materials. For example, Lesson 7-8, materials listed, Math Masters: “pp.155 (1 copy per partnership); 202-206”, Activity Card: “94-95”, Manipulative Kit; “tape measure; per partnership; on 6-sided die”, Other Materials: “slate; stick-on notes; stick-on notes from Math Message.”

The materials reviewed for Everyday Mathematics 4 Grade 2 meet expectations for providing strategies and supports for students in special populations to support their regular and active participation in learning grade-level mathematics.

Materials regularly provide strategies, supports, and resources for students in special populations to help them access grade-level mathematics. Implementation Guide, Differentiating Instruction with Everyday Mathematics, 10.1 Differentiating Instruction in Everyday Mathematics: For Whom?, “Everyday Mathematics lessons offer specific differentiation advice for four groups of learners. Students Who Need More Scaffolding, Advance Learners, Beginning English Language Learners, and Intermediate and Advanced English Language Learners.” Differentiation Lesson Activities notes in each lesson provide extended suggestions for working with diverse learners. Supplementary Activities in each lesson include Readiness, Enrichment, Extra Practice, and English Language Learner. 

For example, the supplementary activities of Unit 1, Establishing Routines, Lesson 1, include:

• Readiness, “For experience reading and comparing numbers, children put number cards in order. Write a sequence of numbers (such as 0-10) on index cards. Children work together to put the cards in order.”

• Enrichment, “To explore whole numbers represented as lengths from 0 on a number line, children complete number-line puzzles.”

• Extra Practice, “For additional practice with counting, children fill in missing numbers on a number line. Depending on children’s ability to read and order numbers, write one or more 1-, 2-, or 3-digit numbers on the number lines on Math Masters, page 3. Children review their work with a partner and explain how they found the missing numbers.”

• English Language Learner, Beginning ELL, “The Total Physical Response (TPR) technique allows beginning English language learners to participate in lesson activities even with minimal English language proficiency. To teach children actions they can use to demonstrate understanding, display a number line and then model actions as you say the following: Point to the number that comes before 5. Circle the number that comes after 6. Use your finger to hop on the number line from 7 to 10. After demonstrating several times with other numbers, have children respond to the same directions without your modeling.”

The materials reviewed for Everyday Mathematics 4 Grade 2 meet expectations for providing extensions and/or opportunities for students to engage with grade-level mathematics at higher levels of complexity.

Materials provide multiple opportunities for advanced students to investigate the grade-level content at a higher level of complexity rather than doing more assignments. The Implementation Guide, Differentiation Instructions with Everyday Mathematics, 10.4 Working with Advanced Learners, “Nearly all Everyday Mathematics lessons include a set of high cognitive demand tasks with mathematical challenges that can be extended. Every regular lesson includes recommended enrichment activities related to the lesson content on the Differentiation Options page opposite the Lesson Opener Everyday Mathematics lessons incorporate varied grouping configurations which enables the kind of flexibility that is helpful when advanced learners in heterogeneous classrooms. Progress Check lessons include suggestions for extending assessment items for advanced learners and additional Challenge problems.” The 2-day Open Response and Re-Engagement lesson rubrics provide guidance for students in Exceeding Expectations. Examples include:

• Lesson 3-5, Subtraction Strategies: Counting Up and Counting Back, Enrichment, “To extend their understanding of subtraction, partners find differences between two 2-digit numbers by playing The Number-Grid Difference Game. On each turn, players mark two numbers on a number grid with counters and then find the difference.”

• Unit 4, Place Value and Measurement, Challenge, Problem 2, an analog clock is pictured with the time 12:10,“Simon thinks the time says 2:00. Is he correct? Explain how you know.”

• Lesson 7-2, Four or More Addends (Day 1), Focus: Solving the Open Response Problem, Adjusting the Activity, “If children quickly solve the problem and write a complete explanation, ask them to check their work by using a second strategy to solve the problem.”

The materials reviewed for Everyday Mathematics 4 Grade 2 meet expectations for providing strategies and supports for students who read, write, and/or speak in a language other than English to regularly participate in learning grade-level mathematics.

The Teacher’s Lesson Guide and ConnectED Teacher Center include guidance for the teacher in meeting the needs of English Language Learners. There are specific suggestions for making anchor charts or explaining new vocabulary. The Implementation Guide, English Language Learners, Everyday Mathematics addresses the needs of three groups of ELL based on their English language proficiency (beginning, emerging, and advanced), “Beginning English language learners fall into Entering (level 1) and Emerging (level 2) proficiencies. This group is typically within the first year of learning English; students' basic communication skills with everyday language are in their early development. These students require the most intensive language-related accommodations in order to access the mathematics in most lessons. Intermediate and Advanced English learners represent Levels 3, 4, and 5 (Developing, Expanding, and Bridging) in the English language proficiencies identified above. Students in this category are typically in their second to fourth year of learning English. They may be proficient with basic communications skills in English and able to carry on everyday conversations, but they are still developing proficiency with more cognitively demanding academic language of the mathematics class.” The ConnectED Teacher Center offers extended suggestions for working with diverse learners including English Language Learners. The Teacher’s Lesson Guide provides supplementary activities for beginning English Language Learners, Intermediate, and Advanced English Language Learners. In every lesson, there are Differentiation Support suggestions, English Language Learner for Beginning ELL located on the Differentiation Options Page and Focus section. Examples include:

• Lesson 1-4, Class Number Scroll, Differentiation Options, English Language Learner, Beginning ELL, “Introduce the work pattern by showing examples of simple patterns and examples that are not patterns, using materials such as pattern blocks, classroom objects, and strings of numbers. Point to examples of a pattern and say: This is a pattern. Point to the non-examples and say: This is not a pattern. As you point to examples and non-examples, ask yes/no questions. For example: Is this a pattern?”

• Lesson 6-6, Recording Addition Strategies, Differentiation Options, English Language Learner, Beginning ELL, “The expression ballpark estimate is a familiar usage in American English. Provide context for this term for English language learners by displaying visuals of a ballpark. Introduce the expressions in the ballpark and out of the ballpark. Gesture to one of the visuals to demonstrate what happens when a home run is hit out of the ballpark. Connect that to an estimate that is far away from the actual answer, or out of the ballpark. Estimates close to the actual answer are in the ballpark (such as a ground ball hit in the infield) and are therefore called ballpark estimates.”

• Lesson 9-7, Expand-and-Trade Subtractions, Part 2, Differentiation Options, English Language Learner, Beginning ELL, “Use role play to review the term trade. Give a child 10 pennies and place a dime in front of you on the table. Say: I need to trade. Please give me your 10 pennies, and I will give you my dime. Have children practice with equivalent amounts in pennies, nickels and dimes. As they trade, encourage them to use sentence frames like the following: I need to trade. Please give me ___, and I will give you ____.”

• The online Student Center and Student Reference Book use sound to reduce language barriers to support English language learners. Students click on the audio icon, and the sound is provided. Questions are read aloud, visual models are provided, and examples and sound definitions of mathematical terms are provided. 

• The Differentiation Support ebook available online contains Meeting Language Demands providing suggestions addressing student language demands for each lesson. Vocabulary for the lesson and suggested strategies for assessing English language learners’ understanding of particularly important words needed for accessing the lesson are provided.

The materials reviewed for Everyday Mathematics 4 Grade 2 meet expectations for providing manipulatives, both virtual and physical, that are accurate representations of the mathematical objects they represent and, when appropriate, are connected to written methods.

The materials consistently include suggestions and/or links, within the lesson notes, for virtual and physical manipulatives that support the understanding of grade-level math concepts. Examples include: 

• Lesson 4-4, Numeration and Place Value, Focus: Matching Numbers to Base-10 Block Representations, materials reference use of base-10 blocks. “Display 200 + 40 + 8. Ask children to show the number using base-10 blocks.”

• Lesson 8-5, Attributes of 3-Dimensional Shapes, Focus: Describing Cubes, materials reference use of 3-Dimensional shapes. “Distribute a centimeter cube to each partnership. Have children share with their partners what they notice about the cube. After a few minutes, bring the class together to discuss children’s observations.”

• Lesson 9-1, Creating and Naming Equal Parts, Focus: Math Message, materials reference use of fraction squares. “Take 8 paper squares. Two children want to share a sandwich equally. Fold a paper square to show how to divide the sandwich into 2 equal shares. Draw a line on the fold. Talk with a partner. Did you both fold the square the same way?”