The materials reviewed for Everyday Mathematics 4 Grade 1 meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific content 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 1-6, Comparing Numbers, Focus: Comparing and Ordering Numbers, students use number cards from 1-15 to compare numbers and order sets of numbers. Students mix their cards up, draw two cards, and use the class number line to decide which number is larger. Students are encouraged to, “Use comparative language such as ‘8 is larger than 2, and 12 is smaller than 15.” This activity supports conceptual understanding of 1.NBT.3, “Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <.”

• Lesson 3-1, Parts-and-Total Number Stories, Focus: Introducing Domino Addition, students represent the number of dots on dominoes with parts-and-total diagrams. For example, for a domino with 3 and 5 dots students write Part: 3, Part: 5, and Total: 8 on the diagram in their math journal and then write the corresponding number sentence. “Complete a parts-and-total diagram for the domino. Point out that one part of the domino has 3 dots and the other part has 5 dots - so the domino has 8 dots in all.” This activity supports conceptual understanding of 1.OA.6, “Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing unknowns in all positions.”

• Lesson 5-1, Introducing Place Value, Focus: Exchanging Base-10 Blocks, students represent numbers using base-ten blocks. “Display 1 long and 15 cubes on your mat. Ask: What number is shown? Discuss how children can be sure. Demonstrate the following two exchanges, or trades, starting with 1 long and 15 cubes each time: Trade the long for 10 cubes, then count the total number of cubes to get to 25. Trade 10 cubes for 1 long and place the long in the tens column. As you trade, emphasize that you are making another ten (1 long) from 10 ones (10 cubes). There are now 2 longs and 5 cubes. Ask: What number is shown? What does the 2 represent? What does the 5 represent?” This activity supports conceptual understanding of 1.NBT.2, “Understand that the two digits of a two-digit number represent amounts of tens and ones.” 

• Lesson 7-6, Exploring Attributes, Fractions, and Salute! Focus: Making an Attribute Train, one student brings an attribute block to the front of the room and is designated the conductor. They choose a child to join the train and that student must bring a block to add to the train that differs from the conductor’s block in only one way, “For example, a child with a large, thick, blue triangle may choose a child with a large, thin, blue triangle, or a large, thick blue square. Each child who joins the train chooses the next child.” This activity supports conceptual understanding of 1.G.1, “Distinguish between defining attributes versus non-defining attributes, build and draw shapes to possess defining attributes.”

• Lesson 8-11, Focus: Adding and Subtracting 10 Mentally, students apply strategies for adding or subtracting 10. “Discuss how children found their answers to the Math Message (35 cents). Emphasize strategies, including visualizing a number grid and moving down a row from 25 to 35, thinking about adding a long to base-10 blocks representing 25, and adding 1 to the tens digit of 25.” This activity supports conceptual understanding of 1.NBT.5, “Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count to explain the reasoning used.”

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

• Lesson 4-3, More Length Measurement, Math Journal, students independently practice measuring objects such as their desktop using one new pencil as the unit. Then students draw a picture or write the name of the measured object and record its measurement, “Draw pictures or write the names of 4 objects. Measure each object with a new pencil. Record your answer.” This activity supports the conceptual understanding of 1.MD.2, “Express the length of an object as a whole number of length units, by laying multiple copies of a shorter object end to end; understand that the length measurement of an object is the number of same-size length units that span it with no gaps or overlaps.”

• Lesson 5-8, Exploring Base-10 Exchanges, Lengths, and Path Measurement, Focus: Introducing Base-10 Exchange, students play with a partner practicing exchanging ones for tens after rolling the die to determine how many base-10 blocks to get. The first student to get 10 longs wins, “As children play, be sure to ask questions that reinforce place-value concepts.” This activity provides practice of conceptual understanding of 1.NBT.2, “Understand that the two digits of a two-digit number represent amounts of tens and ones.”

• Lesson 8-2, Halves, Math Journal, Problem 2, students partition pictures of pancakes and crackers in equal shares and explain how they know the share is equal. “Show how to share 1 cracker between 2 people.” This activity supports the conceptual understanding of 1.G.3, “Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and a quarter of.”

The materials reviewed for Everyday Mathematics 4 Grade 1 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 2-1, Introducing the Strategy Wall, Focus: Introducing Roll and Total, students take turns with a partner to roll one numeral die and one dot die. “This encourages children to start with the numeral die and count on the number of dots, thus moving beyond counting all to find sums.” Students should recognize that it is more efficient to count on from the larger number being added. This activity provides an opportunity for students to develop fluency of 1.OA.5, “Relate counting to addition and subtraction.”

• Lesson 3-3, Exploring Counting, Matching Pairs, and Ordering by Length, Focus: Counting Large Numbers of Pennies, groups of students are given a container of at least 50 pennies, estimate the total, then count and record the number of pennies. “Encourage children to work together to count the pennies and to check each other’s counts. Then have them record their group’s counting strategy in the journal using pictures or words.” This activity provides an opportunity for students to develop fluency of 1.NBT.1, “Count to 120, starting at any number less than 120.”

• Lesson 3-6, Counting to Add and Subtract, Focus: Introducing Addition on the Number Line, students represent and make sense of word problems by drawing hops on a number line. “Cynthia had 8 model cars. She got 3 more model cars. How many model cars does Cynthia have now?” Teachers encourage children to represent and make sense of the problem by drawing hops on the first number line. This activity provides an opportunity for students to develop fluency of 1.OA.6, “Add and subtract within 20, demonstrating fluency for addition and subtraction within 10.”

• Routine 6: Math Any Time Routine, the class daily schedule can be used to practice time concepts on individual student clocks. “How long does lunch last? It is 9:00 a.m. now, how long will it be until we start math? Which activity takes the shortest amount of time today?” This activity provides a continuous opportunity for students to develop fluency of 1.MD.3, “Tell and write time in hours and half-hours using analog and digital clocks.”

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 2-3, More Decomposing Numbers with 10, Home Link, Question 1, students find pairs of numbers that add to 10. “Count up by 1s. 7, 8, 9, ___, ___, ___, ___, ___, ___, ___, ___.” This provides an opportunity for students to independently demonstrate the procedural skill of 1.NBT.1, “Count to 120 starting at any number less than 120.”

• Lesson 8-1, Building Shapes with Defining Attributes, Practice: Math Journal 2, Question 4, students write fact families from fact triangles (11, 6, and 5) dominos (8 and 6). “How can $9+3=12$ help you solve $12-9$?” This activity provides an opportunity for students to independently demonstrate the procedural skill of 1.OA.6, “Add and subtract within 20, demonstrating fluency for addition and subtraction within 10.”

• Lesson 8-2, Halves, Practice: Math Journal 2, Question 4, students practice addition and subtraction. “Use your number grid. Add. $17+10=$ ____, $17+20=$ ____, $17+30=$ ____.” This activity provides an opportunity for students to independently demonstrate the procedural skill of 1.NBT.4, “Add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10.”

• Lesson 9-4, Exploring Broken Calculators, Fractions, and Facts, Practice: Math Journal 2, students complete a “My Facts Inventory Record.” Students are given addition facts such as $10+2$, then check a box if they know it or don’t know it, then explain how they can figure it out, “Encourage them to use more sophisticated and efficient strategies when applicable (for example, substituting near doubles or making 10 for counting on) for remaining facts that they do not know.” This activity provides an opportunity for students to independently demonstrate the procedural skill of 1.OA.6, “Add and subtract within 20, demonstrating fluency for addition and subtraction within 10.”

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 3-2, Number Story Strategies, Practice: Modeling Number Stories, Home-Link, students write number models to solve stories. Problem 1, “Walt was at the carnival. He had 8 carnival tickets and 2 pens. He traded 4 tickets for 1 more pen. How many tickets does Walt have now? How many pens does Walt have now?” This activity provides the opportunity for students to apply their understanding of 1.OA.1, “Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions.”

• Lesson 4-4, Measuring a Marker, Focus: Solving the Open Response Problem, students work with a partner to analyze measurement strategies. Math Masters, p. 97, Problem 2, “Here is how four children used blocks to measure the length of a marker. Who made the best measurement? Write a note to convince a friend that you are right.” Students are given 4 pictures of makers with blocks pictured above each marker to show how each one was measured. Students apply their understanding of 1.MD.2, “Express the length of an object as a whole number of length units, by laying multiple copies of a shorter object (the length unit) end to end; understand that the length measurement of an object is the number of same-size length units that span it with no gaps or overlaps.”

• Lesson 4-10, Adding Three Numbers, Focus: Adding Three Numbers, students add three numbers from School Supply Cards. “Our class has 7 pencils, 4 pens, and 3 crayons. How many writing tools do we have in all?” Students apply their understanding of 1.OA.2, “Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20.”

• Lesson 5-10, Comparison Number Stories, Focus: Finding How Much More or Less, students solve comparison number stories using diagrams. “Alberto has 12 cents. June has 7 cents. Who has more money? How much more money?” Students apply their understanding of 1.OA.1, “Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions.”

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 2a, “to be used after Lesson 2-5”, Problem 1, students make combinations of 8. “A dog park has 8 dogs in it. Some dogs are large and some are small. How many of each could there be? Draw a picture to show one way you could have 8 dogs.” This activity provides the opportunity for students to independently demonstrate an understanding of 1.OA.1, “Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions.”

• Independent Problem Solving 3b, “to be used after Lesson 3-7”, Problem 1, students solve a routine word problem. “Eisa had 13 pencils. She lost 6 pencils. How many pencils does Eisa have now? Use pictures or words to show your thinking. Use pictures or words to show a different way you could solve the number story.” This activity provides the opportunity for students to independently demonstrate an understanding of 1.OA.1, “Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.”

• Independent Problem Solving 4b, “to be used after Lesson 4-10”, Problem 2, students use addition strategies to solve a problem. “Kierre has 6 red grapes, 4 green grapes, and 6 purple grapes. How many grapes does Kierre have in all? Which numbers did you add first? Why did you add these first? What could you add first instead?” This activity provides the opportunity for students to independently demonstrate an understanding of 1.OA.1, “Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.”

• Independent Problem Solving 9b, “to be used after Lesson 9-8”, Problem 2, students compare weights of items by using place value strategies. “Look at your Animal Cards. Choose a pair of animals whose total weight is heavier than Pair 1 but lighter than Pair 2. Write them in the middle section below. Pair 1: A flamingo and an octopus. My Pair. Pair 2. A penguin and a starfish. Write or show how you decided.” This activity provides the opportunity for students to independently demonstrate an understanding of 1.NB.3, “Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <.”

The materials reviewed for Everyday Mathematics 4 Grade 1 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 skills, and fluency, or application include:

• Lesson 3-8, Skip Counting to Add and Subtract, Focus: Using Number-Grid Counting to Add and Subtract, students solve number stories by skip counting on a number grid, “Tell children the following number story: Justin’s teacher gives out stickers for good behavior. At the beginning of the day, Justin had 0 stickers. During the day, he earned 10 stickers. How many stickers does Justin have at the end of the day? Have children share solutions to the problem.” This activity provides the opportunity to apply an understanding of 1.OA.3, “Apply properties of operations as strategies to add and subtract.”

• Lesson 4-8, Combinations of 10, Focus: Fact Strategy Review, students discuss addition facts and note which facts they know well on a Facts Inventory Record, “Explain that addition facts are two numbers from 0 to 10 and their sum. Point out that children can use the strategies on the Strategy Wall to solve addition facts. Record children’s ideas about adding 0 on the Strategy Wall.” This activity develops the procedural skill of 1.OA.6, “Add and subtract within 20, demonstrating fluency for addition and subtraction within 10.”

• Lesson 5-2, Digits and Place Value, Focus: Investigating Base-10 Block Patterns, students make place-value exchanges and discuss various ways to represent numbers, “Tell children that today they will explore patterns in numbers using both base-10 blocks and calculators. Display the following with your Tens-and-Ones Mat as children follow along on their own mats. 1. Display 9 cubes in the ones column. 2. Add 1 cube to the ones column. 3. Exchange the 10 cubes for 1 long and put it in the tens column. This activity develops a conceptual understanding of 1.NBT.2, “Understand that the two digits of a two-digit number represent amounts of tens and ones.”

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

• Lesson 2-1, introducing the Strategy Wall, Focus: Introducing the Turn-Around Rule, students discover and define the turn-around rule for addition. “Solve. Show your work. Use drawings, numbers, or words. 1. On Monday, Ellie found 2 pennies. On Tuesday, Ellie found 3 pennies. How many pennies did Ellie find in all? 2. On Monday, Tommy found 3 pennies. On Tuesday, Tommy found 2 pennies. How many pennies did Tommy find in all?” After solving, students discuss with a partner. “Help children recognize that both problems involve putting together two parts, which is called adding. They should also observe that both problems have the same numbers and the same total, or sum, but the order of the numbers being added is different.” Students develop the procedural skills of 1.OA.6, “Add and subtract within 20, demonstrating fluency for addition and subtraction within 10,” and application of 1.OA.1, “Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions.”

• Lesson 6-6, Introducing Making 10, Focus: Math Message and Developing the Making-10 Strategy, students first show how they represent numbers on a double ten frame, then they represent the making-10 strategy, “How could you show 15 on a double ten frame? How about 18? 19? Record your thinking on your slate. Display a double ten frame. Have children use their own double ten frames and counters to share their thinking. Although children may represent the numbers in the Math Message in a variety of ways, emphasize filling one ten frame with ten and then repeating the remaining ones on the second ten frame. Generalize this idea by asking: How can you represent teen numbers?” Students develop conceptual understanding of 1.NBT.2, “Understand that two digits of a two-digit number represent amounts of tens and ones,” and procedural skill with 1.NBT.1, “Count to 120, starting at any number less than 120.”

• Lesson 8-8, time to the Half Hour, Focus: Math Message and Introducing Time to the Half Hour, students are introduced to half-past an hour and shade half of a clock face, “Draw the clock face, and shade half of the clock. How much time has passed if the minute hand begins at 12 and goes through all of the shaded parts? Children share their drawings. Compare the different representations and discuss how both pieces of the clock must be equal to be divided in half. Ask: What would you name the part of the clock that you shaded? Tell children that today they will use what they know about halves to learn more about telling time.” Students develop conceptual understanding of 1.G.3, “Partition circles and rectangles into two and four equal shares,” and procedural skills with 1.MD.3, “Tell and write time in hours and half-hours using analog and digital clocks.”

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 2-8, Change-to-More number Stories, Focus: Introducing Change-to-More Diagrams, students use change-to-more diagrams to add and determine if their answers make sense. “List some of the children’s predictions and have them explain their thinking. Ask the class whether the predictions seem reasonable. For example, predicting a sum that is smaller than one of the numbers being added would not be reasonable.” 

• Lesson 6-9, Understanding Equivalence, Focus: Illustrating Equivalence, students monitor their progress as they learn the word equivalent and that numbers can have different names like people or things. “Ask children to list the addition facts that have sums of 7. Each fact contains an equivalent name for 7: $4+3$, $2+5$, $1+6$, $0+7=7$ and so on.” “Ask children if they can think of another way to show 7 with the counters.”

• Lesson 8-9, Review: Data, Focus: Math Message, students analyze and make sense of problems when shown a tally chart with missing tallies for one of three categories. “Mr. Chan’s class took a survey to figure out which pet is the favorite. They made a tally chart, but the tallies for Turtle got erased. If 19 children voted, how many voted for Turtle? What do you need to do to solve this problem? Record your answer, and explain how you found it.” 

• Independent Problem Solving 8a, “to be used after Lesson 8-14”, Problem 1, students use a diagram to make sense of a word problem. “Javon and Sam are sharing a pizza. Javon eats 1-half of the pizza. Sam eats 1-fourth of the pizza. Draw and label the pizza to show how much each friend eats.” Students are given a picture of a circle. “If another friend joins them, name what part of the pizza she can eat. Use pictures, words, or both to show how you figured it out.”

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:

• Routine 3, Extending the Attendance Routine, students attend to the meaning of quantities when using a display of class attendance using a parts-and-total diagram. “When you introduce this diagram, ask children what numbers should go in each space in the diagram. Have the Attendance Helper explain which numbers represent the parts and the total each day.”

• Lesson 1-7, Organizing Data in a Tally Chart, Focus: Introducing Tally Marks, students represent situations symbolically when learning to use tally marks to create mathematical representations of numbers. “Have children draw tally marks on their slates to represent different numbers. Say to children: Show the number 10 using tally marks. The number 11. The number 18. As you call out a number, point to it on the Class Number Line or hold up the appropriate number card.”

• Lesson 6-4, Introducing Near Doubles, Focus: Developing the Near-Doubles Strategy, students demonstrate understanding of mathematical representations when shown two “Quick Look” cards for 2-3 seconds that have arrangements of dots. The first card is arranged as a doubles fact, the second shows dots arranged as that doubles fact plus one. “Ask children to share what they saw and how they saw it.”

• Independent Problem Solving 2b, “to be used after Lesson 2-11”, Problem 2, students create a word problem to represent an equation. “Write a number story to go with this equation. Solve. You may wish to draw a picture to help.” Students are given the equation “___ + 4 + 10.”

The materials reviewed for Everyday Mathematics 4 Grade 1 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 2-5, 10 Apples, Focus: Find the Missing Day, students construct viable arguments as they discuss strategies for determining whether any days of the week are missing from a journal page. “Are any days missing? Which days are missing? Tell your partner how you know.”

• Lesson 4-4, Measuring a Marker, Focus: Making Arguments, Math Journal 1, Problem 1, students construct viable arguments as they share strategies and discuss which ribbon they believe is longer. “Have children share their answers for Problem 1. Ask volunteers to explain how they decided which ribbon was longer. Sample answer: The top one looked longer. I could fit more paper clips along the top ribbon. I used a string to show how long one ribbon was, then lined the string up with the other ribbon.”

• Independent Problem Solving 4a, “to be used after Lesson 4-4”, Problem 2, students construct viable arguments about measurement tools. “Katelyn used a paperclip to measure the length of her journal. Now she wants to measure one side of the school building. Would using her paperclip make sense to measure one side of the school building? Why or why not?”

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:

• Lesson 7-10, Addition Facts: “What’s My Rule?” Practice: Math Boxes, Problem 5, students critique the reasoning of others when analyzing the expression 3 + 4. “Raol wants to show (picture of 7 longs) in the box. Is that right or wrong? Explain. Sample answer: That’s wrong because that shows 7 tens and (picture of 7 cubes) is 7 ones.”

• Unit 9, Two-Digit Addition and Subtraction and Review, Open Response Assessment, students critique the reasoning of others as they analyze another student’s work in a number-grid. “Deena filled in the number-grid puzzle. Finish the puzzle and find her mistake. Cross out her mistake and write the correct number. Explain how you used patterns in the number grid to correct Deena’s mistake.”

• Independent Problem Solving 1a, “to be used after Lesson 1-5”, Problem 2, students critique an argument about the number of shoes being worn by classmates. “How many shoes are there in class today? How do you know? Yasi says she can count by 2s to find the number of shoes. Do you agree? Explain.”

The materials reviewed for Everyday Mathematics 4 Grade 1 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 2-11, Finding Unknowns, Focus: Modeling Unknowns, students model the situation with an appropriate representation as the teacher is encouraged to, “Remind children about their recent work with number models. Tell them that today they will write more number models to represent number stories. Jackie dropped 3 pennies in a cup. Then she dropped 4 more pennies in the cup. She dropped 7 pennies in all. Display a change-to-more diagram and have children help you complete it. Have children suggest a representative number model and record it.” 

• Independent Problem Solving 1b, “to be used after Lesson 1-10”, Problem 1, students model the situation as they write a number sentence to represent a picture. “Use the picture to write a number story.” Students are given a picture of a rack with 3 shelves containing various balls (basketballs, soccer balls, footballs, etc.). “Solve your number story. Use pictures or numbers to show your thinking.”

• Independent Problem Solving 8a, “to be used after Lesson 8-4”, Problem 2, students model the situation with an appropriate representation and use an appropriate strategy as they reason about shapes and their attributes. “a. Cole’s family is also sharing a pizza. Cole suggests cutting the pizza as shown below. Is this a fair way to share the pizza between the four people in Cole’s family? Use words and drawings to show your thinking. b. For dessert Cole’s family shares a giant brownie. Show two ways they could share the brownie fairly.”

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 5-12, Adding Animal Weights, Focus: Solving the Open Response Problem, students choose appropriate tools and strategies as they solve an open-response problem. “Ask children to name some other tools they have used to add. Make a list of the tools on the board. Be sure number grids, number lines, and base-10 blocks are on the list. Tell children that they will choose one of the tools from the class list and use it to add animal weights.”

• Independent Problem Solving 6b, “to be used after Lesson 6-11”, Problem 1, students choose and use appropriate tools and strategies as they compare numbers using place value. “Use your toolkit coins, base-10 blocks, number grid, or another tool to help you. You want to buy a notebook that costs 70 cents. Find the total money in cents for each choice below. Draw an X on the choices that are not enough money to buy the notebook.”

• Independent Problem Solving 9b, “to be used after Lesson 9-8”, Problem 1, students choose appropriate tools and strategies as they compare weights. “Use your Animal Cards for these problems. Use base-10 blocks, a number grid, drawings, or another tool to help you. A pair of first-graders and a pair of animals are going to ride together in wagons. Circle the pair that weighs more. Two first-grade girls. A beaver and a koala. Write or show how you decided. Write a number model with < or > to compare the total weight of the pair of first-graders and the total weight of the pair of animals.”

The materials reviewed for Everyday Mathematics 4 Grade 1 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 7-3, Relating Special Addition and Subtraction Facts, Focus: Think Addition to Subtract with Combinations of 10, students analyze a problem and look for more than one approach. “Point out that understanding combinations of 10 can also be useful for games such as Fishing for 10. For example, if they are holding a 3, they may think 3 + ___ = 10 or 10 - 3 = ___. Therefore, thinking of addition might help them solve problems involving decomposing a 10.” 

• Lesson 9-9, Review: Place Value, Focus: Making Up and Solving Number-Grid Puzzles, students look for patterns or structures when using place value understanding to create and solve number-grid puzzles. “Have children fold Math Masters, page TA43 into four equal parts. Invite children to make up number-grid puzzles that their partners can solve using their understanding of tens and ones patterns on the number grid. Children draw around some of the grid cells on one part of the sheet to make a puzzle piece and then write a 2-digit number in one of the cells. They trade puzzles with their partners who then fill in all of the missing numbers.”

• Independent Problem Solving 4b, “to be used after Lesson 4-10”, Problem 1, students look for and explain the structure within mathematical representations as they solve addition problems. “Think of an addition doubles fact about this picture of an alligator. Write the fact below. ___ + ___ = ___. Write a story to fit the doubles fact you wrote.”

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 2-6, More Counting On to Add, Focus: Introducing High Roller, students describe and explain the “counting on” strategy and the “turn-around” rule in their own words. “High Roller makes use of counting on and the turn-around rule. Refer to the Strategy Wall and introduce High Roller as practice for both strategies (assuming counting on has already been listed). Have children restate the turn-around rule and counting on strategy in their own words. If counting on has not yet emerged, display two dot dice, one with 5, the other with 3, and ask children how to find the sum. Have them share the strategies, and highlight counting on (starting with 5 and counting on, 6, 7, 8) when it emerges. Have children practice counting on with a second pair of dice (showing 6 and 4).”

• Lesson 3-6, Counting to Add and Subtract, Focus: Introducing Addition on the Number Line, students notice repeated calculations and make generalizations/create shortcuts as they solve addition problems. “Next, ask children to draw hops on the third number line to solve $2+9$. Select children to show counting on from 2 and counting on from 9. Discuss which strategy is more efficient. Help children generalize that counting on from the larger number, or 9, is faster because there are fewer numbers to count.” 

• Independent Problem Solving 7a, “to be used after Lesson 7-4”, Problem 2, students use repeated reasoning when they define attributes of rectangles. “Find and draw 2 different things that are not shaped like rectangles. What are they? Use what you know about rectangles to explain why your shapes are not rectangles.”

The materials reviewed for Everyday Mathematics 4 Grade 1 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 4, Length and Addition Fact, Lesson Organizer, Coherence, 1. MD.2, provides an overview of content and expectations for the unit. “In Kindergarten, children identified measurable attributes of objects, including length. They also used direct comparison to determine which of the two objects is longer. In Unit 5, children will be presented with more complicated paths, made up of several straight lines joined together, and will use an iteration of a single paper clip to determine the total length. In Grade 2, children will measure a single object using two different units of measure and discuss why the length measurements differed based on the chosen unit.”

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 4-11, 10 More, 10 Less, Focus: Assessment Check-In, teacher guidance supports students in identifying 10 more or less than a given number. “Observe whether children can find 10 more and 10 less than a given number while playing What’s Your Way? Expect most children to do this using a number grid. Have children who struggle discuss patterns they see as they count by 10s on a number grid. Have those who exceed expectations complete the Enrichment activity for this lesson. Children practice finding 10 more and 10 less throughout Grade 1.”

• Lesson 6-6, Introducing Making 10, Focus: Developing the Making-10 Strategy, Math Message Follow-Up, teacher guidance connects students' prior knowledge to new concepts. “DIsplay a double ten frame (Math Masters, p. TA19). Have children use their own double ten frames (Math Journal 1, Activity Sheet 3) and counters to share their thinking. Although children may represent the numbers in the Math Message in a variety of ways, emphasize filling one ten frame with ten and then representing the remaining ones on the second ten frame.”

• Lesson 8-4, Sharing Paper Squares (Day 2), Focus: Solving the Open Response Problem, Common Misconception, teacher guidance addressed common misconceptions as students decompose shapes. “Children sometimes do not understand that decomposing shapes into a larger number of equal shapes creates smaller shares. In this problem, for example, children may say that the boys get more because there are more boys (4 pieces is more than 2 pieces). As you review children’s work, plan to address this misconception through the re-engagement discussion.”

The materials reviewed for Everyday Mathematics 4 Grade 1 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 4-1, Introducing Length Measurement, Focus: Introduction Length Comparison, Professional Development, teacher guidance explains the concept of indirect comparison. “Making a direct comparison of two objects involves clearly observing and comparing the aspects of those objects. In the first part of the activity, children place the objects next to each other to compare their lengths directly. In the second part of the activity, they cannot directly compare two edges of the same box, so they use an intermediary (such as string, strips of paper, paper clips, or inches) to measure each object. Then they compare the results. This is called indirect comparison.”

• Lesson 5-11, Two-Digit Addition and Subtraction, Focus: Adding Animal Weights, Professional Development, supports teachers with concepts for work beyond the grade. “The sample strategies listed in this lesson demonstrate children’s understanding of one or more of the following: place value, properties of operations, and the relationship between addition and subtraction. Although these are not the only possible strategies, they are listed here to help you connect your class’s work to 1.NBT.4 and 1.NBT.6. Children are not expected to use these phrases.”

• Lesson 7-4, More Subtraction Fact Strategies, Focus: Comparing Counting Up and Counting Back to Subtract, Professional Development, teacher guidance explains the strategy. “Children were informally introduced to counting up as a subtraction strategy in Unit 2. It is a powerful subtraction fact strategy because it makes use of the relationship between addition and subtraction. 1.OA.4 Counting up to subtract is an accessible strategy that improves children's accuracy. It also transfers well to multi-digit computation.”

• Lesson 7-8, Finding Unknowns: “What’s My Rule?”, Focus: Introducing “What’s My Rule?”, Professional Development, teacher guidance explains a routine that works with number patterns. “The Everyday Mathematics routine, “What’s My Rule?”, provides practice with number patterns and arithmetic facts as well as a format for thinking about relationships between pairs of numbers in a function. A function machine is a diagram or metaphor that indicates the input and output numbers in “What’s My Rule?” tables are related.”

• Lesson 8-2, Halves, Focus: Naming Shares, Note, supports teachers with concepts for work beyond the grade. “Although it may come up in discussion, avoid using formal fraction notation, such as $\frac{1}{2}$, to name one or all of the shares. The focus in Grade 1 is using words and pictures to describe equal shares. Fractional notation will be introduced in later grades.”

• Lesson 9-5, Vending Machine Addition and Subtraction, Focus: Adding 2-Digit Vending Machine Prices, Professional Development, teacher guidance clarifies sample strategies. “The sample strategies listed in this lesson demonstrate children’s understanding of one or more of the following: place value, properties of operations, and the relationship between addition and subtraction. These are not the only possible strategies. They are listed as they are to help you connect your class’s work to 1.NBT.4 and 1.NBT.6. Do not expect children to use these phrases.”

The materials reviewed for Everyday Mathematics 4 Grade 1 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 1 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, 1.OA.4, “First Quarter: Understand that some addition strategies can be used to solve subtraction problems. For example, think ‘What do I need to add to 7 in order to get 10?’ Second Quarter: Understand that a difference can be found with both subtraction and addition. Third Quarter: Understand subtraction as an unknown addend problem. For example, subtract 10-8 by finding the number that makes 10 when added to 8. Fourth Quarter: Ongoing practice and application.”

• Lesson 2-7, Labeling Counts, standards identified in the Focus and Practice are identified as 1.OA.5, 1.NBT.1. 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 1, Counting, Unit 1 Organizer, Coherence, 1.NBT.3, includes an overview of how the content in Kindergarten builds from previous grades and extends to future grades. “In Kindergarten, children learned to compare numbers less than or equal to 10 when expressed in numeral form. In Grade 2, children will use their understanding of place value to compare numbers up to 1000.”

• Unit 5, Place Value Comparisons, Unit 5 Organizer, Coherence, 1.NBT.4, includes an overview of how the content in Kindergarten builds from previous grades and extends to future grades. “In Kindergarten, children developed an understanding of teen numbers as 10 ones and some further ones. In Grade 2, they will expand upon those strategies to make sense of addition and subtraction within 1000.”

• Unit 9, Two-Digit Addition and Subtraction and Review, Unit 9 Organizer, Coherence, 1.MD.2, includes an overview of how the content in Kindergarten builds from previous grades and extends to future grades. “In Unit 5, children used iteration of a single paper clip to determine lengths of pathways. This work built upon Kindergarten experiences identifying measurable attributes of objects, including length. In Grade 2, children will learn to use tools such as yardsticks and measuring tapes to measure lengths with standard units.”

The materials reviewed for Everyday Mathematics 4 Grade 1 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 children 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 1 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 First 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:

• Unit 4, Length and Addition Facts, Unit 4 Organizer, Unit 4 Materials, each lesson has materials under the following categories: Math Masters, Activity Cards, Manipulative Kit, and Other Materials. For example, Lesson 4-8, materials listed, Math Masters: “pp.28 (optional); 111-113; G26 (optional); G27”, Activity Card: “50”, Manipulative Kit; “Quick Look Cards 60, 64, 69, 84, 86, 88, 90, 92, 93; Ten Frame; per partnership; 2 dot dice, 10 pennies, 10 counters”, Other Materials: “slate; Ten Frame; timer or stopwatch; per partnership; plastic cup.”

• Lesson 5-11, Two-Digit Addition and Subtraction, Overview, Materials: Math Message, “number line, number grid base-10 blocks, counters (optional).”

• Lesson 7-5, Attributes of Shapes, Math Message, “Choose a block. Record everything you notice about your block.” Note: “Place a set of attribute blocks in a box or basket near the Math Message. Because of the differences among attribute block sets, the activities in this lesson are generic. Adapt the activities to the blocks that you are using.”

• Unit 8, Geometry, Unit 8 Organizer, Unit 8 Materials, each lesson has materials under the following categories: Math Masters, Activity Cards, Manipulative Kit, and Other Materials. For example, Lesson 8-6, materials listed, Math Masters: “pp.240-241; TA4; TA40; G58 (optional)”, Activity Card: “99”, Manipulative Kit; “base-10 blocks; 3-dimensional blocks; pattern blocks”, Other Materials: “slate; ball; can; box; everyday objects; paper bag, scissors, glue.”

The materials reviewed for Everyday Mathematics 4 Grade 1 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 7, Subtraction Fact Strategies and Attributes of Shapes, Lesson 7, include:

• Readiness, “To provide additional experience with attributes of shapes, have children use the Shape Sorting Cards prepared from Math Masters, page 203. Ask children to find three shapes that have one or more attributes in common. Then have them determine a rule that tells the commonality. Then have children choose three more shapes with at least one attribute in common and suggest a rule that tells what those shapes have in common. Repeat as time permits.”

• Enrichment, “To extend children's understandings of defining and nondefining attributes, have them build composite polygons with pattern blocks. Children discuss the attributes of polygons and composite polygons. They determine whether shapes are the same after being rotated.”

• Extra Practice, “To provide practice comparing attributes, have children play Attribute Train. For detailed instructions, see Lesson 7-6. Observe: Which children change only one attribute when placing the next shape? Discuss: Which attribute did you change when you chose this shape? What can you do if you do not know what shape to play next?”

• English Language Learner, Beginning ELL, “Prepare children for discussing shapes by looking at examples. Display pairs of shapes with one differing attribute. Ask yes/no questions about color, shape, and size to identify how the shapes are the same and different. For example: Are all the shapes squares? Are all the shapes the same color? Are they the same shape? Use the terms in common and same to summarize answers. Repeat with the words different and not the same.”

The materials reviewed for Everyday Mathematics 4 Grade 1 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:

• Unit 6, Addition Fact Strategies, Challenge, “Fill the name-collection box for 11 using these rules: The names must use either addition or subtraction. Each Name can only use two numbers between 1 and 20. For example, $1+10=11$ and $12-1=11$. $1+10$ and $12-1$ are already in the box. With these rules, there are nineteen different names for the number 11. Finish writing ALL nineteen names for 11 in the name-collection box below.”

• Lesson 7-1, Fact Families, Focus: Writing Fact Families, Enrichment, “To further explore finding unknowns in number sentences and recording fact families, have children write facts and complete their partners’ fact families.”

• Lesson 9-1, Review: Measurement, Focus: Measuring with Rulers, Adjusting the Activity, “Children who are already familiar with standard rulers may choose to write the numerals where units meet on their paper-clip rulers. Connect this representation to a number line by making sure that they write 1 at the end (rather than the beginning) of the first unit.”

The materials reviewed for Everyday Mathematics 4 Grade 1 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 1-5, 1 More, 1 Less, Differentiation Options, English Language Learner, Beginning ELL, “Show a picture of a bunny and demonstrate how it hops. Tell children that a bunny is a baby rabbit and that in English many animals have different names for their babies. Mention other examples such as dogs/puppies and cats/kittens. Encourage children to use the new words for their own number stories.”

• Lesson 4-9, More Combinations of 10, Differentiation Options, English Language Learner, Beginning ELL, “Children may associate fishing with catching real fish. They should learn that this word can also apply to asking for something or trying to get something. Display a common object, such as a card, and say: May I have that card? I want that card. I would like that card. I am fishing for that card. This will help children understand the word fishing in the context of asking for something. Have them use sentence frames, such as ‘I am fishing for ___’ to practice asking for objects, and later, for numbers.”

• Lesson 8-1, Building shapes with Defining Attributes, Differentiation Options, English Language Learner, Beginning ELL, “To review defining attributes, have children play a guessing game with yes or no questions. One child secretly puts an attribute block inside a bag. The guesser asks questions, such as: Does it have vertices? Does it have three sides? Is it a ___? The first player may reach into the bag to touch the block while answering questions. Demonstrate one round with a child, with you as the guesser. Then reverse roles so children can practice using the terms to ask questions. You may wish to play as a class or form partnerships.”

• 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 1 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 4-5, Exploring Data, Shapes, and Base-10 Blocks, Focus: Exploration B: Geoboard Shapes with Defining Attributes, materials reference use of geoboards. “Children make shapes with specified attributes using rubber bands on a geoboard. They record their shapes on Math Masters, page TA16 or TA17.”

• Lesson 5-1, Introducing Place Value, Focus: Naming Numbers with Base-10 Blocks, materials reference use of base-10 blocks. “Explain that today children will learn more about tens and ones. Provide each child with a Tens-and-Ones Math (Math Journal 1, Activity Sheet 4) and display a demonstration Tens-and-Ones Math (Math Masters, page TA20). Have the class use base-10 blocks to represent numbers.”

• Lesson 8-6, 3-Dimensional Shapes, Focus: Describing 3-Dimensional Shapes, materials reference use of solid shapes. “Divide children into 6 groups, one group for each of the following shapes: cube, non-cube rectangular prism, sphere cylinder, pyramid, and cone. Provide each group with blocks or everyday objects of their shape. After children examine the 3-dimensional shapes, encourage them to match their objects to representations on journal page 170.”