The materials reviewed for Everyday Mathematics 4 Grade 4 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 clusters. Examples include: 

• Lesson 1-2, Place-Value Concepts, Focus: Math Message, students compare the numbers 46,385 and 463,850 using place value. “Ask students to respond on their slates to the following questions about 46,385 using the place-value chart on Math Masters, page 2 for reference. Which digit is in the hundreds place? What is the value of the digit? Which digit is in the ones place? What is the value of the digit? Which digit is in the ten-thousands place? What is the value of the digit?” Later, students compare and order numbers, “Pose the following problem: Which number is larger, 47,899 or 48,908? Ask: How can we use expanded form to tell which is larger?” Students develop a conceptual understanding of 4.NBT.2, “Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form.”

• Lesson 3-1, Equal Sharing and Equivalence, Focus: Math Message, students model equal-sharing situations and examine equivalent names for those models. “Two brothers go to lunch and share three 8-inch pizzas equally. How much pizza does each brother get? Have students share the drawings they used to model and solve the problem.” Within the Math Journal activity, students practice using visual representations of fractions and equal sharing to include subdividing “leftover” pieces to produce fair shares, “Use drawings to help you solve the problems. Solve each problem in more than one way. Show your work. 1. Three friends shared 4 chicken quesadillas equally. How many quesadillas did each friend get?” Students develop conceptual understanding of 4.NF.A, “Extend understanding of fraction equivalence and ordering.”

• Lesson 4-3, Partitioning Rectangles, Focus: Partitioning Rectangles to Multiply, Math Journal 1, Problems 1 and 2, students partition rectangles to multiply. “Maya wants to lay tile on a floor that is 8 feet wide by 24 feet long. The tiles she wants to use are 1 square foot each. How many tiles will Maya need? Draw a picture to represent Maya’s floor. Explain how you figure out how many tiles Maya needs.” Students develop a conceptual understanding of 4.NBT.5, “Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations.”

• Lesson 5-4, Adding Mixed Numbers, Focus: Math Message, students use fraction circles to solve, “Use fraction circles on the Number-Line Poster to solve this problem on your slate. $2\frac{1}{4}=\frac{?}{4}$.” Students develop a conceptual understanding of 4.NF.1, “Explain why a fraction $\frac{a}{b}$ is equivalent to a fraction $\frac{(n\times a)}{(n\times b)}$ by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size.”

• Lesson 6-3, Strategies for Division, Focus: Math Message, students use multiples to solve division problems. “Mariana is in charge of seating students for an assembly. Each table seats 6. Seventy-eight students will attend the assembly. How many tables will Mariana need to seat all of the students?” In the Teacher's Lesson Guide, “Invite students to share strategies for solving the problem, discussing the various steps they take. Emphasize the following strategies: Representing the problem concretely, subtracting groups of 6 from 78, and finding multiples of 6. Tell students that today they will use multiples to help find the answers to division problems more efficiently. Pose two more division problems for the class to try. Guide a discussion of how students make sense of the problem and think through solving the problem.” Students develop a conceptual understanding of 4.NBT.6, “Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, properties of operations, and/or the relationship between multiplication and division.”

Home Links, Math Boxes, and Practice provide opportunities for students to independently demonstrate conceptual understanding throughout the grade. Examples include:

• Lesson 3-6, Comparing Fractions, Home Link, Problem 1, students compare fractions to solve number stories. “Tenisha and Christa were each reading the same book. Tenisha said she was $\frac{3}{4}$ of the way done with it, and Christa said she was $\frac{6}{8}$ of the way finished. Who has read more, or have they read the same amount? How do you know?” Students independently demonstrate conceptual understanding of 4.NF.A, “Generalize place value understanding for multi-digit whole numbers.”

• Lesson 4-6, Introducing Partial-Products Multiplication, Focus: Math Message, students use the partial-products multiplication strategy to extend their conceptual understanding of multiplication and place value. “Helen wants to paint the sidewalk for her block party. She needs to know the area of the sidewalk so she’ll know how much paint to buy. The sidewalk is 5 feet wide and 660 feet long. What is the area of Helen’s sidewalk? ___square feet. 1. Draw a picture to represent Helen’s sidewalk. 2. Show how you figured out the area of the sidewalk.” Students independently demonstrate conceptual understanding of 4.NBT.5, “Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations.”

• Lesson 7-6, Open Response Three Fruit Salad, Focus: Making Exact Numbers of Whole, Math Journal 2, Problem 1, students use tools of their choice to solve. “You may choose tools such as fraction circles or the Number line Poster to help you solve the problems. Does this fraction make an exact number of wholes? Explain or show why or why not for each?” Problem 2, “What number of eighths makes 5 wholes? Show how you know.” Students independently demonstrate conceptual understanding of 4.NF.3, “Understand a fraction $\frac{a}{b}$ with $a>1$ as a sum of fractions $\frac{1}{b}$.”

The materials reviewed for Everyday Mathematics 4 Grade 4 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 3-4, An Equivalent Fraction Rule, Practice: Math Journal 1, Problem 1, students use the four operations to solve number stories. “Each day a company delivers newspapers to the town of Wayland. It has 158 customers on the north side of town, and 237 customers on the south side. The company receives 900 newspapers to deliver. How many will be leftover?” Students develop procedural skills and fluency of 4.NBT.4, “Fluently add and subtract multi-digit whole numbers using the standard algorithm.”

• Lesson 3-8, Modeling Tenths with Fraction Circles, Warm-Up: Mental Math and Fluency, teachers state numbers in expanded form and students write the numbers in standard form on their slates. “Level 1: 5 [100s] + 8 [1s]; 4 [1,000s] + 3 [100s] + 7 [1s]; 2 [1,000s] + 6 [100s] + 9 [10s]. Level 2: 9 [10,000s] + 5 [1,000s] + 6 [10s]; 5 [10,000s] + 8 [1s]; 1 [10,000s] + 5 [1,000s] + 2 [10s] + 7 [1s]. Level 3: 2 [100,000] + 9 [100s] + 6 [10s]; 6 [100,000s] + 8 [1,000s] + 2 [10s] + 4 [1s]; 8 [1,000,000s] + 7 [100,000] + 3 [10s].” Students develop procedural skills and fluency of 4.NBT.2, “Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form.”

• Lesson 6-13, Extending Understandings of Whole-Number Multiplication, Warm-Up: Mental Math and Fluency, students add and subtract fractions with like denominators, “, $\frac{5}{8}+\frac{2}{8}=$, $\frac{2}{3}+\frac{2}{3}=$, $1\frac{3}{5}+2\frac{1}{5}=$.” Students develop procedural skills and fluency of 4.NF.3d, “Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators.”

• Lesson 8-10, Fractions and Liquid Measures, Practice: Math Journal 2, Problem 7, students divide larger numbers and interpret remainders. “The school purchased 1,245 new fiction books for the third, fourth, and fifth grade classrooms at Portland South School. There are 3 classrooms at each grade level. a. Can the school divide the books evenly among the classrooms? Why or why not? b. What would be a fair way to divide the books among the classrooms?” Students develop procedural skills and fluency of 4.OA.3, “Solve multi-step word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted.”

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 5-6, Queen Arlene’s Dilemma, Home Link, Problem 3, students solve four addition or subtraction problems using the standard algorithm, “.” Students independently demonstrate procedural skill and fluency of 4.NBT.4, “Fluently add and subtract multi-digit whole numbers using the standard algorithm.”

• Lesson 6-10, Using a Half-Circle Protractor, Home Link, Problem 9, students practice adding or subtracting multi-digit numbers vertically using the standard algorithm, “.” Students independently demonstrate procedural skill and fluency of 4.NBT.4, “Fluently add and subtract multi-digit whole numbers using the standard algorithm.”

• Subtraction Target Practice, game, Student Reference Book, students can play independently or with a partner to practice subtraction skills. “Directions: 1. Shuffle the cards and place the deck number-side down on the table. Each player starts at 250. 2. Players take turns. Each player has 5 turns in a game. When it is your turn, do the following: Turn 1: Turn over the top 2 cards and make a 2-digit number (You may place the cards in either order). Subtract this number from 250 on scratch paper. Check the answer on a calculator. Turns 2-5: Take 2 cards and make a 2-digit number. Subtract this number from the result obtained in your previous subtraction problem. Check the answer on a calculator. 3. The player whose final result is closest to 0, without going below 0, is the winner. If there is only 1 player, the object of the game is to get as close to 0 as possible, without going below 0.” Students independently demonstrate procedural skill and fluency of 4.NBT.4, “Fluently add and subtract multi-digit whole numbers using the standard algorithm.”

• Multiplication Wrestling, online game, students multiply 2-digit numbers, “Directions: Try to get the highest score you can. During each round, arrange your four number cards into the largest 2-digit numbers you can and use those numbers to make your ‘teams.’ Find your teams’ partial products and then the total product. Each time you get a larger total, it will become your high score!” Students independently demonstrate procedural skill and fluency of 4.NBT.1, “Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right” and 4.NBT.5, “Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations.”

The materials reviewed for Everyday Mathematics 4 Grade 4 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 1-8, Cracking the Muffin Code, Focus: Solving the Open Response Problem, Problem 1, students use patterns to decipher codes in real-world problems. “Marcus takes orders at the Marvelous Muffin Market. The muffins are packed into boxes that hold 27, 9, 3, or 1 muffins. Marcus always fills the largest box first, uses the fewest number of boxes possible, and always sends boxes that are full. When a customer asks for muffins, Marcus fills out an order form. Hints. For an order of 5 muffins, Marcus writes: _ _12. For an order of 19 muffins, Marcus writes: _201. For an order of 34 muffins, Marcus writes: 1021. For an order of 32 muffins, what would Marcus write on the order form? Explain or show how you know.” This activity provides the opportunity for students to apply their understanding of 4.OA.5, “Generate a number or shape pattern that follows a given rule.”

• Lesson 4-2, Making Reasonable Estimates for Products, Home-Link, Problem 3, students solve multiplication problems and use estimates. “There are 30 Major League Baseball (MLB) teams and 32 National Football League (NFL) teams. The expanded roster for MLB teams is 40 players and it is 53 for NFL teams. How many more players are in the NFL than the MLB?” This activity provides the opportunity for students to apply their understanding of 4.OA.3, “Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted.”

• Lesson 5-3, Adding Fractions, Focus: Solving Fraction Number Stories, Problem 1, students solve fraction addition number stories. “Ryan and his 3 sisters painted the walls of their family room. Ryan used $\frac{2}{3}$ of a can of paint. Each one of his sisters used $\frac{1}{3}$ of the same-size can. How much paint did they use all together?” This activity provides the opportunity for students to apply their understanding of 4.NF.3d, “Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem.”

• Lesson 8-3, Pattern Block Angles, Focus: Solving the Open Response Problem, Problem 3, students find measures of pattern-block angles. “Julie and Penny solved the problem below in different ways. What is the measure of an angle of a yellow hexagon? Draw a picture and explain how you know. Julie’s Solution: I know that the measure of the hexagon’s angle is 120°. The measure of the white rhombus’s small angle is 30° Four angles measuring 30° fit inside the hexagon’s angle. So, $30\degree+30\degree+30\degree+30\degree=120\degree$. Penny’s Solution: I know that the measure of the hexagon’s angle is 120° because the measure of the square’s angle is $90\degree$ and the measure of the white rhombus’s small angle is $30\degree$. So, $90\degree+30\degree=120\degree$. Who is correct, Julie, Perry, or both? Write a note to another student explaining your thinking on the back of this page.” This activity provides the opportunity for students to apply their understanding of  4.MD.7, “Recognize angle measure as additive.” 

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 1-7”, Problem 2, students solve word problems involving multiplicative comparison. “Gabrielle has a babysitting job that pays $6 per hour. On Saturdays and Sundays, she babysits 2 hours each day. With the money she earns, she wants to buy a video game for $59. How many weekends will Gabrielle have to babysit to have enough money to buy the game? Explain how you found your answer.” This activity provides the opportunity for students to independently demonstrate an understanding of 4.OA.2, “Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison.”

• Independent Problem Solving 2b, “to be used after Lesson 2-12”, Problem 2, students use multiplication to solve word problems. “Two fourth grade classes are keeping track of the books they read in October. Every time they read 10 books, they add a square to the diagrams they are making. These diagrams show how many books the classes read after two weeks. a. Peter says Mr. Smith’s class read twice as many books as Dr. Caswell’s class. Explain to Peter why he’s wrong. b. Ms. Liebman’s class read twice as many books as Dr. Caswell’s and Mr. Smith’s classes read together. Draw a diagram to show how many books Ms. Liebman’s class read. c. Write a number model that shows how the number of books Mr. Liebman’s class read compares to the number of books read by Dr. Caswell’s and Mr. Smith’s classes.” This activity provides the opportunity for students to independently demonstrate an understanding of  4.OA.2, “Multiply or divide to solve word problems involving multiplicative comparisons.”

• Independent Problem Solving 4a, “to be used after Lesson 4-6”, Problem 2, students solve a multi-step word problem and assess the reasonableness of their answer. “Claire buys 7 dozen donuts for Math Night. There is a sign in the donut shop that says: Donuts Buy 12, get 2 free. a. How many free donuts will Claire get? b. She expects 110 people to attend Math Night. Do you think Claire will have enough donuts for the meeting? c. Explain your reasoning.” This activity provides the opportunity for students to independently demonstrate an understanding of  4.OA.3, “Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.”

• Independent Problem Solving 8a, “to be used after Lesson 8-4”, Problem 2, students multiply fractions to solve word problems. “Mrs. Murkowski’s art class is making bead necklaces. a. Alex is planning to make a 36-bead necklace that has $\frac{1}{6}$ red beads, $\frac{1}{2}$ green beads, and $\frac{1}{3}$ blue beads. How many of each color will he need? b. Charlotte wants to use 12 red beads to make a necklace that is $\frac{2}{3}$ red. How many beads will she need in all? Explain how you solved this problem. Charlie wants to make a 12-bead necklace with $\frac{1}{2}$ red beads, $\frac{1}{4}$green beads, and $\frac{1}{3}$ blue beads. What would you tell Charlie? d. Design your own 24-bead necklace. Color the necklace below to show your design. What fraction of each color bead does your necklace have?” This activity provides the opportunity for students to independently demonstrate an understanding of  4.NF.4, “Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.”

The materials reviewed for Everyday Mathematics 4 Grade 4 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 Grade 7. Examples where materials attend to conceptual understanding, procedural skill and fluency, or application include:

• Lesson 1-5, More, 1 Less, Home-Link, Problem 1, students estimate solutions to multi-step number stories. “On the walk home from school, Meg stopped at the library for 22 minutes and at her grandmother’s house for 38 minutes. She spent 17 minutes walking. She left at 3:00 and was supposed to be home by 4:00. Did Meg make it home on time? How did you get your answer?” Students engage in application of 4.OA.3, “Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.”

• Lesson 4-6, Introducing Partial-Products Multiplication, Focus: Introducing Partial-Products Multiplication, Math Journal 1, Problem 3, students partition a rectangle and use the partial-products multiplication strategy to solve number stories. “The mayor wants to beautify part of the highway by planting marigolds. She wants to plant 4 marigolds along every foot of highway for an entire mile or 5,280 feet. How many marigolds will she need? Draw a partitioned rectangle to represent the problem. Then use partial-products multiplication to record your work in a similar way.” Students extend their conceptual understanding of 4.NBT.5, “Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.” 

• Independent Problem Solving 3b, “to be used after Lesson 3-12”, Problem 2, students fluently add and subtract multi-digit whole numbers to calculate the total cost of a remodeling project. “Keith has saved $9,000. He wants to remodel his house. He plans to spend $2,567 for his kitchen, $1,189 for the bathroom, paint three rooms for $148 each and purchase a living room sofa for $1,799. a. What is the total cost of his remodeling project? b. Write a number model to show the cost of Keith’s remodeling project.c. How much of the $9,000 Keith saved will be left after his remodeling? d. Do you think Keith should do the remodeling project? Why or why not?” Students develop procedural skills and fluency of 4.NBT.4, “Fluently add and subtract multi-digit whole numbers using the standard algorithm.”

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

• Lesson 3-1, Equal Sharing and Equivalence, Home-Link, Problem 1, students generate equivalent fractions and solve equal sharing number stories. “Four friends shared 5 pizzas equally. How much pizza did each friend get?” Students engage with conceptual understanding and application of 4.OA.4, “Find all factor pairs for a whole number in the range 1-100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1-100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1-100 is prime or composite.”

• Lesson 4-8, Money Number Stories, Home-Link, Problem 4, students solve multi-step number stories involving money. “If the cashier only has $10 and $1 bills, what are two ways he could make Mr. Russo’s change?” Students engage with procedural skill and application of 4.MD.2, “Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale.”

• Lesson 5-3, Adding Fractions, Focus: Solving Fraction Addition Number Stories, students solve number stories with fractions. “After running $\frac{3}{4}$ of a mile, Marisa stopped for a drink of water. Then she ran another $\frac{3}{4}$ of a mile. How far did she run in all? What is the whole? Display ‘1 mile’ in a whole box. Does each of the fractions in this problem refer to the same whole? Will your answer be more or less than 1 mile? Encourage strategies such as the following: Use fraction names: just as 3 dogs + 3 dogs = 6 dogs, 3 fourths + 3 fourths = 6 fourths. The unit is fourths. Think about $\frac{3}{4}$ as the sum of unit fractions: $\frac{1}{4}+\frac{1}{4}+\frac{1}{4}$. Or more simply with equations: $\frac{(3+3)}{4}=\frac{6}{4}$. Use the Number-Line Poster: Place a finger on $\frac{3}{4}$. Then, beginning at $\frac{3}{4}$, count up $\frac{3}{4}(\frac{1}{4}+\frac{1}{4}+\frac{1}{4})$ to $\frac{6}{4}$.” Students develop all three aspects of rigor simultaneously of 4.NF.3a, “Understand addition and subtraction of fractions as joining and separating parts referring to the same whole.”, 4.NF.3b, “Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation.” and 4.NF.3d, “Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators.”

The materials reviewed for Everyday Mathematics 4 Grade 4 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 1-7, U.S. Traditional Addition, Focus, Introducing U.S. Traditional Addition, Math Journal 1, Problem 7, students analyze and make sense of addition problems. “There are 279 boys and 347 girls at a school assembly. How many students are at the assembly?”

• Lesson 5-1, Fraction Decomposition, Focus: Practicing Decomposing Fractions, Student Math Journal, Problem 3, students use a variety of strategies as they write equations and shade parts of circles. “Decompose $\frac{1}{2}$ into a sum of fractions with the same denominator in three different ways. Record each decomposition with an equation and justify each one by shading the parts of the circle.”

• Lesson 8-1, Extending Multistep Number Stories, Focus: Cracking a Number Story Code, Math Journal 2, Problem 6, students use a variety of strategies to solve multistep number stories. “In April some astronomy experts from the local science museum visited the school and offered to show interested students the constellations inside their special star globe. The globe could hold 8 students at a time. There were 12 Kindergarteners, 17 first graders, 25 second graders, 28 third graders, 23 fourth graders, and 39 fifth graders lined up to go inside the star globe. How many groups of students went in the globe?” Teacher’s Lesson Guide, Summarize, “Discuss which problem students found most challenging and why. Have them share different ways they tried to make sense of the most difficult problems.” 

• Independent Problem Solving 4a, “to be used after Lesson 4-6”, Problem 1, students make sense of the information presented in word problems involving time. “Thursdays, Saraha spends 360 minutes in school. He has two 90-minute classes. The other classes are 45 minutes. a. How many 45-minute classes does he have on Thursday? b. How many hours does he spend in school on Thursday? c. On the other days of the week, Saraha spends 315 minutes in school. How much longer does Saraha spend in school on Thursday compared to the other days of the week? d. Why do you think Saraha spends more time in school on Thursdays?”

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 2-8, Multiplicative Comparisons, Focus: Creating and Interpreting Statements and Equations, Math Journal 1, Problem 5, students represent situations symbolically to represent an unknown in an equation, solve the unknown, and then interpret number stories in context. “Sally is 21 years old. Tonya is 3 times as old as Sally. How old is Tonya? a. Equation with unknown. b. Answer: ___ years old.” Problem 6, “Write a comparison number story using the equation $8\star5=40$.” 

• Lesson 3-2, Fraction Circles and Equivalence, Home Link, Problem 1, students understand the relationships between problem scenarios and mathematical representations as they represent equivalent fractions. “Divide into 4 equal parts. Shade $\frac{1}{4}$.” Problem 4, “Create your own. Divide into equal parts and shade a portion. Record the amount you shaded.” Empty circles are provided for students. 

• Lesson 8-2, Real-Life Angle Measures as Additive, Practice: Solving Number Stories, Math Journal 1, Problem 4, students represent situations symbolically as they multiply fractions by whole numbers. “Officer Wells drove back and forth between Northbrook and Deerfield once every day for a few days. At the end of that time, he had driven $40\frac{8}{10}$ miles. How many days did he do this? Number model with unknown: ___. Answer: ___ days.”

• Independent Problem Solving 6b, “to be used after Lesson 6-13”, Problem 1, students consider units involved in problem solving and attend to the meaning of quantities as they find unknown measurements. “Main Street, Pine Street, and Davis Street come together to make a three-way intersection. a. What is the measure of ∠a? b. Explain how you found the measure of ∠a. West Street, East Street, and South Street also come together to make a three-way intersection. What is the measure of ∠d? d. How did you find the measure of ∠d? Design your own street intersection. Write an angle problem about your intersection.”

The materials reviewed for Everyday Mathematics 4 Grade 4 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 3-7, Fractions and Decimals, Focus: Math Message, students justify their strategies and thinking when they compare two fractions. “Which fraction is smaller: $\frac{3}{8}$ or $\frac{1}{5}$? Or are they equivalent? Record your answer on your slate using one of the symbols >, =, or <. Be prepared to justify your conclusion.” 

• Lesson 5-3, Fraction and Mixed-Number Computation; Measurement, Practice: Reviewing Decimal Concepts, Math Journal 2, Problem 5b, students construct viable arguments as they order and compare decimals. “Cassie said, ‘I think 10.6 is less than 10.06 since it doesn’t have any hundredths.’ Is she correct? Explain your answer.”

• Independent Problem Solving 6a, “to be used after Lesson 6-8”, Problem 2, students critique the reasoning of others as they decompose fractions into a sum of fractions. “Sasha and Rosetta are making smoothies with frozen berries, bananas, yogurt, and apple juice. The recipe calls for $1\frac{3}{8}$ cups of apple juice. Sasha adds juice from an open bottle, but it’s only $\frac{7}{8}$ cup. Rosetta opens a new bottle and tells Sasha that she needs to add $\frac{1}{2}$ cup more apple juice. Sasha disagrees and says she needs to add $\frac{4}{8}$ cup more apple juice. a. Explain who’s right 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:

• Lesson 8-9, More Fractions Multiplication Number Stories, Focus, Solving an Area Problem with Multiplication, Math Journal 2, students construct a mathematical argument when they solve a number story by multiplying a whole number by a fraction. “Ella bought 3 yards of fabric from a bolt that is 45 inches wide. She said, “I have 135 square inches of fabric for my project.” Do you agree with Ella? Explain why or why not?”

• Independent Problem Solving 2a, “to be used after Lesson 2-15”, Problem 1, students construct mathematical arguments as they use multiplicative comparisons to multiply problems. “The Catbird Pet Store sells only birds and cats. One day there were 18 animal legs in the store. a. How many cats and birds might there have been? b. Write a number model that fits your answer. c. Keri-Anne says there could have been 8 birds. Do you agree? Why or why not?”

• Independent Problem Solving 7a, “to be used after Lesson 7-7”, Problem 2, students critique the reasoning of others as they multiply fractions with a whole number. “Mr. Apple’s students have solved many problems such as $7\star\frac{1}{2}$ and $15\star\frac{1}{4}$, so one day he asks them to make some conjectures about multiplying unit fractions by whole numbers. He reminds them that the Student Reference Book defines a conjecture as ‘a statement that is thought to be true based on information or mathematical thinking.’ a. Jeremiah’s conjecture is that whenever you multiply a unit fraction by a whole number, the product is always greater than 1. Jeremiah’s example is $5\star\frac{1}{4}=\frac{5}{4}$. Does Jeremiah’s example fit his conjecture? Do you think Jeremiah’s conjecture is true? Why or why not?”

The materials reviewed for Everyday Mathematics 4 Grade 4 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 6-11, Angle Measures as Additive, Focus: Finding Unknown Angle Measures, Math Journal 2, students model with mathematics as they solve liquid measurement problems with fractions using drawings, measurement scales, and equations. Students must determine if the statement is true or false and explain it. Students analyze a punch recipe and use the recipe to determine if statements are true or false and provide an explanation. Problem 4, “There is more than twice as much orange juice as apple juice in the recipe.” Problem 7, “The combined amount of juice in the recipe is $1\frac{3}{4}$ cups more than the amount of soda.”

• Independent Problem Solving 2b, “to be used after Lesson 2-12”, Problem 1, students use the math they know to solve problems and everyday situations as they draw sketches of a rectangle to find the area. “Miguel’s old tomato garden was a rectangle 4 feet by 6 feet. But he wanted to grow more tomatoes, so he decided to make his garden bigger. He decided to make all the sides of his new garden twice as long. Draw sketches of Miguel’s old garden and his new garden. Label the lengths of the sides of both gardens. a. Miguel used chicken wire to make a fence around his old garden to keep the rabbits out. He can use the chicken wire from his old garden for his new garden, but it won’t be enough. How much more chicken wire will he need for his new garden? b. Miguel grew 6 tomato plants in his old garden. How many do you think he can grow in his new garden? c. Why?”

• Independent Problem Solving 5a, “to be used after Lesson 5-4”, Problem 2, students model the situation with an appropriate representation and use an appropriate strategy as they use fraction operations to convert measurements. “Abel wants to make a frame for a photograph he’s giving to his grandmother as a present. The photograph is 5 inches by 8 inches. Abel plans to glue 12 craft sticks side by side on a piece of cardboard and then put the picture on top of the craft sticks. His craft sticks are 6 inches long and $\frac{3}{4}$ inch wide. He wants to leave $\frac{1}{2}$ of an inch of the craft sticks uncovered all around the edges of the photograph.Will Abel’s plan work? Explain why or why not.”

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

• Lesson 7-6, (Day 2): Three-Fruit Salad, Focus: Solving the Open Response, students create recipes using fraction addition and multiplication using any tool or strategy. “The school cook asks you to create recipes for Three-Fruit Salad. Follow these rules: Each recipe must use exactly 3 different fruits. The combined weight of the fruit for one recipe must be exactly 5 pounds. Make up two recipes that follow the rules. Show that each recipe weighs 5 pounds by using tools such as fraction circles, fraction number lines, drawings, or number models. Use multiplication when possible.”

• Independent Problem Solving 3a, “to be used after Lesson 3-6”, Problem 2, students choose and use appropriate tools as they show fraction equivalence. “Mrs. John had a pizza party. She ordered three large pizzas, one cheese, one sausage, and one pepperoni. The cheese pizza was cut into 8 slices, the sausage into 12 slices, and the pepperoni into 6 slices. At the end of the party, 4 slices of cheese, 7 slices of sausage, and 4 slices of pepperoni were left over. Choose and use an appropriate tool to help you solve this problem. a. Which pizza did Mrs. John and her guests eat the most of? b. Explain your answer and describe the tool you used.”

• Independent Problem Solving 6b, “to be used after Lesson 6-13”, Problem 2, students solve problems involving addition of fractions using any tool or strategy. “Marsha’s class is writing problems to fit the equation $2\frac{3}{4}+z=10$. This is Marsha’s problem: Nick spent 10 hours practicing drums during the school week and then $2\frac{3}{4}$hours more on the weekend. How many hours did he practice in all? a. Do you think Marsha’s problem fits $2\frac{3}{4}+z=10$? Why or why not? b. Write and solve your own problem that fits $2\frac{3}{4}+z=10$.”

The materials reviewed for Everyday Mathematics 4 Grade 4 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 6-1, Extended Division Facts, Focus: Finding Patterns in Extended Division Facts, students make use of the structure as they explore patterns in division facts. Teachers support a discussion around patterns in extended division facts. “Tell students that, as with multiplication, extended facts can help solve division problems. Display the problem 35 divided by 7, 350 divided by 7, and 3,500 divided by 7. Have partners discuss strategies for solving. If no one mentions it, guide students in a discussion of using basic facts and knowledge of place value to solve extended division facts: Identify the basic fact, solve the basic fact, note place value.” Math Journal 2, Problem 5, students are provided a triangle with the numbers 4,800 and 6, $\star$, / inside, “Write a basic division fact and extended division fact for each Fact Triangle.” Problem 7, “What strategy did you use to solve Problem 5?” 

• Lesson 7-9, Generating and Identifying Patterns, Focus: Math Message, Math Journal 2, students look for and explain the structure within mathematical representations as they build arrays representing rectangular numbers. “Use centimeter cubes to build the following arrays: 1-by-2, 2-by-3, 3-by-4. Be prepared to discuss any patterns you notice.” Teacher’s Lesson Guide, “Ask volunteers to share any patterns they noticed. Expect the following: The arrays are rectangles. Each array has an even number of cubes. Each array adds 1 row and 1 column. Ask: What rule can we use to find any square number?”

• Independent Problem Solving 2a, “to be used after Lesson 2-5”, students look for patterns or structures to make generalizations and solve problems as they use factors and multiples. “A class is playing Buzz. (See page 252 of your Student Reference Book for the game directions.) The STOP number is 42. They play with no mistakes and when they reach 42, it’s a BUZZ. As they play the game, they say BUZZ a total of 9 times. a. What could the BUZZ number have been? b. If the BUZZ number was even, what number was it? c. Explain how you solved this problem.”

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 1-12, Angles, Triangles, and Quadrilaterals, Focus: Readiness, Pattern-Block Sort, Math Journal 2, Problem 3, students evaluate the reasonableness of their answers and thinking to sort and create a rule for their pattern blocks. “Label one sheet of paper: These fit the rule. Label another sheet of paper: These do NOT fit the rule. Sort the pattern blocks (hexagons, trapezoid, square, triangle, 2 rhombuses) according to the rules given below. Then use the shapes marked “PB” on your Geometry Template to record the results of your sort. 4 sides and all sides the same length.” Problem 5, “Make up your own rule. Sort the pattern blocks according to your rule. Record your rule and the pattern blocks that fit your rule on the back of this page.”

• Lesson 7-9, Generating and Identifying Patterns, Extra Practice: Trading Cards, Math Journal 2, Problem 2, students notice repeated calculations to understand algorithms and make generalizations about patterns. “a. Fill in the first two lines of the chart using your answers from above. Find the pattern and use it to fill in the rest of the chart. Then answer the questions below. People: 2, 3, 4, 5, 6, 7. Cards: ___, ___, ___, 20, 30, ___. Equation: $2\star(2-1)=2$, $3\star(3-1)=6$, ___, $5\star(5-1)=20$, ___, ___. b. What rule describes the pattern in the chart?”

• Independent Problem Solving 3a, “to be used after Lesson 3-6”, Problem 2, students describe and explain another students’ method. “Zeynep says she can always make a fraction smaller than any fraction you give her. a. What is Zeynep’s rule for making a smaller fraction? b. Does Zeynep’s rule always work? Why or why not? c. Use Zeynep’s rule to complete the table. Add some fractions of your own.”

The materials reviewed for Everyday Mathematics 4 Grade 4 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 students 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, Multi-Digit Multiplication, Organizer, Coherence, provides an overview of content and expectations for the unit. “In Unit 1, students learn U.S. traditional addition. In Units 2-4, students practice multi-digit addition and subtraction with whole numbers in a variety of contexts. In Grade 3, students learn a variety of methods for multidigit addition and subtraction. Beginning in Unit 6, students use multidigit addition and subtraction with whole numbers as they do partial-quotients division. In Grade 5, students will learn how to multiply using U.S. traditional multiplication.”

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 1-13, Finding Perimeters of Squares and Rectangles, Common Misconception teacher guidance addresses common misconceptions as students use formulas for finding the perimeter of a rectangle. “Watch for students who think they have found the answer by adding only the 2 labeled sides. Suggest that they label the lengths of the other two sides. Remind these students that even though all sides may not be labeled, all sides do need to be included in the calculation.”

• Lesson 5-8, Subtracting Mixed Numbers, Focus: Assessment Check-In, teacher guidance supports students in solving mixed number subtraction problems. “Observe students completing journal page 171. Expect most to be able to solve Problems 1 and 2 using a strategy. Encourage students who struggle to work through each step by modeling the mixed numbers, taking pieces away, then renaming them using fraction circles. For students who complete all parts successfully, suggest that they write a number story and illustrate how to solve it in two ways.”

• Lesson 8-7, More Decimal Number Stories, Focus: Solving a Perimeter Problem with Simple Decimals, Math Message Follow-Up, teacher guidance connects students' prior knowledge to new concepts. “Before students share solution strategies, guide a discussion about decimal and fraction equivalence by displaying two-name collection boxes, one for 5.6 and one for 3.8, and completing them as a class. Remind students that sometimes one form of a number is easier to work with than another when solving problems. Sometimes the “easier” form is different for different people. This is why more than one way to solve a problem is encouraged. It is important to remember that mixed numbers and decimals are simply different ways to name the same number and that numbers can be combined in different ways to find an answer.”

The materials reviewed for Everyday Mathematics 4 Grade 4 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 1-10, U.S. Customary Units of Length, Professional Development, supports teachers with concepts for work beyond the grade. “The Grade 4 standard 4.MD.1 expects students to express measurements in a larger unit in terms of a smaller unit. Thus, students may only understand this relationship as moving from larger to smaller. The Grade 5 standards expect that students will explore the relationship in both directions. It is important that students are exposed to converting measurement units from smaller to larger in Grade 4, so as to have a general understanding of the relationship. Do not expect that your students will master the concept. It will not be assessed in Grade 4.”

• Unit 2, Multiplication and Geometry, Unit 2 Organizer, 4.G.2, supports teachers with concepts for work beyond the grade. “Links to the Future: In Grade 5, students use properties of triangles to create a triangle hierarchy.”

• Lesson 4-13, Lattice Multiplication, Professional Development, explains using the lattice multiplication. “Lattice multiplication is a simple alternative to traditional algorithms that deal with whole numbers. It focuses on place value and the organization of a multiplication problem. The lattice method breaks down the numbers into place values, allowing students to work with smaller numbers while solving a multi-digit multiplication problem. Students are not required to learn this method, but they should be encouraged to try. Having choices among methods is important. Given those choices, most students will select ones that work best for them.”

• Lesson 5-10, Rotations and Iterating Angles, Professional Development, explains understanding angles and protractor use. “For students to accurately use a protractor to measure angles, they must first understand what they are measuring. The attribute of angle size is a source of confusion for most students. Using a nonstandard unit like a wedge to measure an angle helps students see that measuring the size of an angle is the same as measuring any other attribute: iterating unit angles fills the spread between an angle’s rays, just as iterating unit lengths fills a given length.”

• Unit 7, Multiplication of a Fraction by a Whole Number; Measurement, Unit 7 Organizer, 4.OA.3, provides support with explanations and examples of the more complex grade/course-level concepts. “Links to the Past: In Grade 3, students solve two-step word problems involving all four operations, using a letter for the unknown quantity. They estimate to assess the reasonableness of results.”

• Lesson 8-7, More Decimal Number Stories, Professional Development, supports teachers with concepts for work beyond the grade. “Even though formal operations with decimals are not a Grade 4 mathematics topic in Indiana Academic Standards, the work in this lesson meets 4.MD.2, which specifies problems involving distances and simple decimals and provides a foundation for more formal work with decimal operations in Grade 5. The link between different representations of numbers, especially fractions and decimals, is a key concept for success with decimal computation in fifth grade.”

The materials reviewed for Everyday Mathematics 4 Grade 4 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:

• 4th Grade 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. 

• 4th Grade Math, Unit 3, Fractions and Decimals, Organizer, Contents Lesson Map outlines lessons, aligned standards, and the lesson overview for each lesson. This is present for all units and allows teachers to identify targeted standards for any lesson.

• Lesson 6-5, (Day 2): Fruit Baskets, Core Standards identified are 4.NF.7, 4.OA.3, and 4.NBT.6. Lessons contain a consistent structure that includes an Overview, Before You Begin, Vocabulary, Warm-Up, Focus, Assessment Check-In, Practice, Minute Math, Math Boxes, and Home-Link. This provides an additional place to reference standards, and language of the standard, within each lesson.

• Mastery Expectations, 4.NBT.3, “First Quarter: Round numbers through the hundred thousands to the thousands place or larger. Second Quarter: Use place value understanding to round multi-digit whole numbers to any place. Third Quarter: Ongoing practice and application. Fourth Quarter: Ongoing practice and application.” Mastery is expected in the Second Quarter. 

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 3, Fractions and Decimals, Organizer, Coherence, includes an overview of how the content in 4th grade builds from previous grades and extends to future grades. “In Grade 3, students use visual models, such as fraction circles, drawings, and number lines to compare fractions. They learn that comparisons are possible only if the whole is the same size. In Grade 5, students apply formal strategies to add and subtract fractions and use estimates to help them assess the reasonableness of their answers.” 

• Unit 5, Fraction and Mixed-Number Computation; Measurement, Organizer, Coherence, includes an overview of how the content in 4th grade builds from previous grades and extends to future grades. “In Grade 3, students used fraction strips, number lines, and fraction circles to explore fractions that are equivalent to whole numbers and to convert whole numbers into fraction equivalents. In Grade 5, students develop formal strategies to add and subtract fractions and mixed numbers.”

• Unit 8, Fraction Operations; Applications, Organizer, Coherence, includes an overview of how the content in 4th grade builds from previous grades and extends to future grades. “In Grade 3, students use concrete and visual representations to explore fractions as equal parts of the whole. In Grade 5, students develop formal strategies, including using equivalent fractions with common denominators, to add and subtract fractions and mixed numbers.”

The materials reviewed for Everyday Mathematics 4 Grade 4 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, a distributed practice through 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 students 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 student can and should learn challenging, interesting, and useful mathematics. The program is designed to ensure that each of your students 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:

  • A Study to Explore How Gardner’s Multiple Intelligences Are Represented in Fourth Grade Everyday Mathematics Curriculum in the State of Texas.

  • 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.

  • Achievement Results for Second and Third Graders Using the Standards-Based Curriculum Everyday Mathematics.

  • The Relationship between Third and Fourth Grade Everyday Mathematics Assessment and Performance on the New Jersey Assessment of Skills and Knowledge in Fourth Grade (NJASK/4).

  • The Impact of a Reform-Based Elementary Mathematics Textbook on Students’ Fractional Number Sense.

  • A Study of the Effects of Everyday Mathematics on Student Achievement of Third, Fourth, and Fifth-grade students in a Large North Texas Urban School District.

The materials reviewed for Everyday Mathematics 4 Grade 4 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 Fourth 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 1, Place Value; Multidigit Addition and Subtraction, Unit 1 Organizer, Unit 1 Materials, teachers need, “number cards 0-9 (4 of each); tape measure; yardstick; 12-inch ruler; place-value tool; calculator in lesson 10.” 

• Lesson 1-10, U.S. Customary Units of Length, Math Message, “Display the measurement scales on journal page 24, along with the yardstick and 12-inch ruler.” Focus: Measuring Lengths in U.S. Customary Units, “Show students the yardstick and 12-inch ruler to remind them of the relative sizes of yards, feet, and inches.”

• Unit 4, Multi-Digit Multiplication, Unit 4 Organizer, Unit 4 Materials, teachers need, “graduated cylinder; beakers (optional); fraction circles or number line; eyedropper; containers; 1-liter pitcher or beaker; water; calculator; paper in lesson 4.” 

• Lesson 7-3, A Fraction as a Multiple of a Unit Fraction, Overview, Materials, “slate; fraction circles; Math Journal 2, pp. 234-235; Student Reference Book p. 267; Math Masters p. G35; number cards 0-9 (4 of each) or one 10-sided die labeled 0-9; Math Journal 2, p. 236; half-circle protractor; Math Masters, p. 266.” Math Message, “Use centimeter cubes and grid paper to show your thinking.”

The materials reviewed for Everyday Mathematics 4 Grade 4 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 5, Fraction and Mixed-Number Computation; Measurement, Lesson 8, include:

• Readiness, “To explore mixed-number subtraction concepts, students decompose mixed numbers in multiple ways. Display the mixed number $2\frac{2}{4}$ and work with students to decompose it using fraction circles. For example, show 1 whole and six-fourths using fraction circles (1 red and 6 yellows). Display the equation $1+\frac{6}{4}=2\frac{2}{4}$. Have students find another solution and record the appropriate equation. Repeat with other mixed numbers. Suggestions: $2\frac{3}{5}$, $3\frac{1}{2}$.”

• Enrichment, “To further explore mixed-number subtraction, students solve number stories involving unlike denominators. Note: As students have not been introduced to computing with unlike denominators beyond 10 and 100, consider doing a few examples as a group to help students understand that they can use the Equivalent Fractions Rule to find equivalent fractions in order to subtract.”

• Extra Practice, “To practice mixed-number subtraction, students complete Frames-and-Arrows diagrams. Create problems to meet the needs of individual students or have them create and solve their own problems.”

• English Language Learner, Beginning ELL, “Building understanding of like using the terms same and alike. Show pairs of objects and model comparing them by pointing to like attributes and using think-alouds: ___ is like ___. They have the same ___, ___, and ___ are alike because ___. Give students various objects and direct them to find the two objects that are alike, like each other, or the same. Encourage students to repeat statements in which the terms are used.”

The materials reviewed for Everyday Mathematics 4 Grade 4 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 3, Fractions and Decimals, Challenge, “Use three different strategies to show how you know that $\frac{3}{5}$ is greater than $\frac{4}{10}$.”

• Lesson 6-13, Extending Understandings of Whole Number Multiplication, Enrichment, “To explore finding an unknown in a number sentence that involves multiplying a fraction by a whole number, students use a variety of structures. They fill in multiplication/division diagrams, draw representations, and write multiplication equations to solve the problems.” Teachers are provided guidance to help advanced learners solve missing groups number stories.”

• Lesson 8-2, Real-Life Angle Measures as Additive, Enrichment, “To extend their understanding of the additive nature of angle measures, students determine the measure of the angles of fraction circle pieces. They explore sums of angle measures of various combinations of fraction circle pieces.”

The materials reviewed for Everyday Mathematics 4 Grade 4 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 1-9, U.S. Traditional Subtraction, English Language Learner Beginning ELL, “Demonstrate the meaning of columns on a grid, using up and down gestures as you say the term. Have students trace columns from top to button with their fingers. Show examples of columns and non-examples (rows) on the grid, asking yes/no questions like: Is this a column? Choose a number from the top of the grid and tell students to point to that column and trace it up and down.”

• Lesson 5-12, Creating Symmetric Figures, Differentiation Options, English Language Learner Beginning ELL, “To scaffold students’ understanding of terms used in the lesson, including line, fold, horizontal, vertical, and mirror image, provide vocabulary cards for each term with the corresponding illustration. Use Total Physical Response prompts to model each term, directing students to find classroom examples that help illustrate each of these terms. For example, point to a vertical line as you say the term, and then ask students to point to another vertical line.” 

• Lesson 6-7, Partial-Quotients Division, Part 2, Differentiating Lesson Activities, Using Partial-Quotients Division, “Encourage students to use academic and content terms as they discuss their strategies. Provide a word bank that includes the terms at least, multiply, divisor, dividend, quotient, partial, estimate, and reasonable. Post sentence frames that encourage the use of the terms to describe the order of the steps they followed and that incorporate terms such as first, then, also, in addition, and finally. For example: First I ___ Then I ___ I also ___ In addition ___ Finally I ___. Encourage them to use Easy Multiples + and its guide in conjunction with the word bank and sentence frames.”

• 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 4 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 2-6, (Day 2): Little and Big, Focus: Solving the Open Response Problem, materials reference the use of paper clips. “How many paper clips tall is Little? Is Big’s height greater than or less than Littles’ height in dog treats? In paper clips? How do the paper clips line up with the dog treats?”

• Lesson 3-2, Fraction Circles and Equivalence, Focus: Starting a Collection of Fraction Names, materials reference use of fraction circles. “Ask students to cover the region in Problem 2 on Math Masters, page 100 with fraction circle pieces. They may use only one color at a time and should try all of the colors.” 

• Lesson 6-9, Measuring Angles, Focus: Making an Angle Measurer, materials reference using an angle measurer and protractor. “Tell students that today they will make a tool for measuring angles. Their tool will be similar to a full-circle protractor, but will only have larger intervals marked.”