The materials reviewed for Everyday Mathematics 4 Grade 3 meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings. 

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

• Lesson 2-12, Exploring Fraction Circles, Liquid Volume, and Area, Practice: Playing Division Arrays, Student Reference Book, students play Division Arrays to practice division by grouping counters equally. “Players take turns. When it is your turn, draw a card and take the number of counters shown on the card. You will use the counters to make an array. Roll the die. The number on the die is the number of equal rows you must have in your array.” Students develop a conceptual understanding of 3.OA.2, “Interpret whole-number quotients of whole numbers.”

• Lesson 4-9, Number Sentences for Area of Rectangles, Focus: Math Message, students find the area of a rectangle. “A cloud is partly covering this rectangle. Find the area of the whole rectangle. Tell a partner how you found the area. Then listen to how your partner found the area. Be ready to share your partner’s ideas.” Students develop conceptual understanding of 3.MD.C, “Geometric measurement: understand concepts of area and relate area to multiplication and to addition.” 

• Lesson 5-11, Multiplication Facts Strategies: Break-Apart Strategy, Focus: Breaking Apart Factors to Solve Facts, Math Journal 1, Problem 1, students decompose factors to solve multiplication problems. “You have a rectangular garden that is 7 feet wide and 8 feet long. You decide to plant flowers in one section and vegetables in another. Sketch at least two different ways you can partition, or divide your garden into two rectangular sections. Label the side lengths of each of your new rectangles. Write a number model using easier helper facts for one of your ways. $7\times8=$ ___ $\times$ ___ $+$ ___ $\times$ ___.” Students develop a conceptual understanding of 3.OA.1, “Interpret products of whole numbers.”

• Lesson 7-5, Fractions on a Number Line, Part 1, Focus: Math Journal 1, Problem 4, students identify fractions greater than one on a number line. “Circle all the fractions greater than 1 on the number lines on pages 232 and 233. What do you notice about fractions greater than 1?” Students develop conceptual understanding of 3.NF, “Develop an understanding of fractions as numbers.”

• Lesson 7-12, Fraction of Collection, Focus: Naming Fraction of Collections, students name fractions of collections using counters. “Direct children to make collections and name fractions of those collections. For example: There are 4 pennies in $\frac{1}{2}$ of the pile. Show me the whole pile. There are 8 crayons in 1 box. How many crayons are in 2 boxes? In $1\frac{1}{2}$ boxes?” Students develop a conceptual understanding of 3.OA.2, “Interpret whole-number quotients of whole numbers.”

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

• Lesson 2-6, Equal Groups, Home Link, students solve problems involving multiples of equal groups by using strategies like repeated addition and skip counting. “Solve. Show your thinking using drawings, words, or number models. A pack of Brilliant Color Markers contains 5 markers. Each pack costs $2. 1. If you buy 6 packs, how many markers will you have?” Students independently demonstrate conceptual understanding of 3.OA.1, “Interpret products of whole numbers.”

• Lesson 7-4, Fraction Strips, Home Link, Problem 1, students shade fraction strips to represent given fractions. “Shade each rectangle to match the fraction below it. ‘’” Students independently demonstrate conceptual understanding of 3.NF.3, “Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.” 

• Lesson 8-4, Setting Up Chairs, Home Link, Problem 1, students make conjectures and arguments to explain why an arrangement of marching band members is best. “There are 24 members in the school band. The band director wants them to march in rows with the same number of band members in each row. Find two different ways that the band members can be arranged. Draw a sketch that shows each arrangement.” Students independently demonstrate conceptual understanding of 3.OA.2, “Interpret whole-number quotients of whole numbers.”

The materials reviewed for Everyday Mathematics 4 Grade 3 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 2-11, Framers and Arrows, Warm-Up: Mental Math and Fluency, students focus on basic fact families. “Pose each basic fact without an answer. Have children write out the rest of the fact family, including the answers, on their slates: $6+4$, $2\times8$, $8+5$, $5\times4$, $9+7$, $5\times9$.” Students develop procedural skills and fluency of 3.OA.7, “Fluently multiply and divide within 100,” and 3.NBT.2, “Fluently add and subtract within 1,000.”

• Lesson 3-3, Partial-Sums Addition, Focus: Adding with Partial Sums, students add by expanding addends. “Display $145+322$ in the vertical form. Ask: What is the expanded form of each addend?” Students develop procedural skills and fluency of 3.NBT.2, “Fluently add and subtract within 1,000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.”

• Lesson 5-9, Multiplication Facts Strategies: Near Squares, Focus: Math Message, students multiply and divide within 100. “Kali knows $7\times7=49$. How could she use $7\times7$ as a helper fact to figure out $8\times7$?” Students develop procedural skills and fluency of 3.OA.7, “Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division.”

• Lesson 6-2, Playing Baseball Multiplication, Focus: Introducing Baseball Multiplication, students are introduced to the game to build multiplication fact fluency. “Tell children that they will practice multiplication facts while playing Baseball Multiplication. Players solve multiplication facts to move counters around the bases and score runs.” Students develop procedural skills and fluency of 3.OA.7, “Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division.”

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

• Lesson 2-8, Picturing Division, Practice: Math Journal 1, Problem 1, students add fluently using strategies or the standard algorithm. “Scientists counted 91 eggs in 2 clutches of python eggs. If 1 python clutch has 52 eggs, how many are in the other clutch? You may draw a diagram or picture.” Students independently demonstrate procedural skill and fluency of 3.NBT.2, “Fluently add and subtract within 1000.”

• Lesson 7-6, Fractions on a Number Line, Part 2, Practice: Student Reference Book, students practice multiplication facts by playing Baseball Multiplication. “Pitching and Batting: Members of the team not at bat take turns ‘pitching’. They roll the dice to get two factors. Players on the ‘batting’ team take turns multiplying the two factors and saying the product.” Students independently demonstrate procedural skill and fluency of 3.OA.7, “Fluently multiply and divide within 100.”

• Facts Workshop, online game, students add and subtract to create fact families. Students are shown a domino that has 2 dots on one side and 3 dots on the other side. Students are asked to select facts that are part of that fact family (i.e. $5-3=2$, $5-2=3$, $3+2=5$). Students independently demonstrate procedural skill and fluency of 3.NBT.2, “Fluently add and subtract within 1000.”

• Division Arrays, online game, students multiply and divide and interpret whole-number quotients. Students can play with a partner or against the computer, “Players take turns making arrays. During each turn, a player is given a total number of counters and numbers of rows, then uses them to build an array. The player earns points equal to the number of counters in one row of the array.” Students develop procedural skills and fluency of 3.OA.7, “Fluently multiply and divide within 100” and 3.OA.2, “Interpret whole-number quotients of whole numbers.”

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 4-7, Area and Perimeter, Home-Link, Problem 2, students solve problems involving perimeter. “Your pace is the length of one of your steps. Find the perimeter, in paces, of your bedroom. Walk along each side and count the number of paces. The perimeter of my bedroom is about ___ paces.” Problem 3, “Which room in your home has the largest perimeter? Use your estimating skills to help you decide. The ___ has the largest perimeter. Its perimeter is about ___ paces.” This activity provides the opportunity for students to apply their understanding of 3.MD.8, “Solve real world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters.”

• Lesson 6-9, Writing Number Stories, Focus: Solving the Open Response Problem, students use the four operations to write and solve real-world problems. “Write a two-step number story to fit the number sentence below. $12-(4\times2)=4$.” This activity provides the opportunity for students to apply their understanding of 3.OA.8, “Solve two-step word problems using the four operations.”

• Lesson 7-3, Number Stories with Measures, Focus: Solving Number Stories with Measures, Problem 3, students solve number stories that involve time in a real-world problem. “Lena has a doctor’s appointment at 8:45 A.M. It takes her 25 minutes to drive to her doctor’s office. How many minutes early will Lena be if she leaves at 8:00 A.M.?” This activity provides the opportunity for students to apply their understanding of 3.MD.1, “Tell and write time to the nearest minute and measure time intervals in minutes. Solve word problems involving addition and subtraction of time intervals in minutes, e.g., by representing the problem on a number line diagram.”

• Lesson 7-8, Finding Rules for Comparing Fractions, Focus: Solving the Open Response Problem, Problem 1, students write rules for ordering fractions. “Think about these fractions: $\frac{1}{6}$, $\frac{1}{8}$, $\frac{1}{10}$, $\frac{1}{3}$, $\frac{1}{4}$, $\frac{1}{2}$, $\frac{1}{5}$. Write the fractions in order from least to greatest: What Patterns do you notice? How are these fractions the same or different?” Problem 3, “Write a rule for ordering fractions with the same numerator.” This activity provides the opportunity for students to apply their understanding of 3.NF.3, “Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.”

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

• Independent Problem Solving 1a, “to be used after Lesson 1-8”, Problem 2, students write their own number story problems and represent their story in their own way. “Write a number story about this number sentence: $?=2\times7$. Then draw a picture that represents your number story and show how you solved it.” This activity provides the opportunity for students to independently demonstrate understanding of  3.OA.3, “Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities.”

• Independent Problem Solving 2b, “to be used after Lesson 2-9”, Problem 1, students use multiplication and division to determine the rows of apples. “a. Maria picked 18 apples for her aunt’s fruit market. She wants to display her apples in an array that has 3 rows. Show how Maria can display the apples. How many apples are in each row? Write a number model to describe Maria’s array. b. Her aunt asked Maria to change her display so that the array has 9 apples in each row. Show Maria’s new array. How many rows of apples are in Maria’s new display? Write a number model to describe this array.” This activity provides the opportunity for students to independently demonstrate understanding of  3.OA.3, “Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities.”

• Independent Problem Solving 4b, “to be used after Lesson 4-10”, Problem 1, students use multiplication and addition strategies to find an area. “For a school project, Keisha has to figure out the area of her back porch. She sketched this model of her porch on grid paper, but her baby brother spilled juice that covered part of her model. Help Keisha figure out the area of her back porch. The area of Keisha’s back porch is ___ square feet. Explain how you figured out the area of Keisha’s back porch.” This activity provides the opportunity for students to independently demonstrate understanding of 3.MD.7, “Relate area to the operation of multiplication and addition.” 

• Independent Problem Solving 5b, “to be used after Lesson 5-11”, Problem 1, students create and write situations for area problems. “Kim is helping his dad tile the floor of the bathroom. The floor is 7 feet by 9 feet. Kim’s dad said that each box of tile will cover 10 square feet. To figure out how much tile they need, Kim sketched this rectangle. Use words and numbers to explain how Kim can use this sketch to figure out how many boxes of tile he and his dad need for the bathroom floor. Kim and his dad need to buy ___ boxes of tile.” This activity provides the opportunity for students to independently demonstrate understanding of 3.MD.7, “Relate area to the operation of multiplication and addition.”

The materials reviewed for Everyday Mathematics 4 Grade 3 meet expectations in that the three aspects of rigor are not always treated together and are not always treated separately. There is a balance of the three aspects of rigor within the grade.

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

• Lesson 2-1, Extended Facts: Addition and Subtraction, Focus: Math Message, students solve multi-digit addition and subtraction problems. “Solve. Record your answers on your slate. Think about the patterns that help you solve each set. $9-7=?$; $90-70=?$; $900-700=?$; $?=7+9$; $?=70+90$; $?=700+900$.” Students develop procedural skills and fluency of 3.NBT.2, “Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.”

• Lesson 3-10, The Commutative Property of Multiplication, Focus: Math Message, students solve problems using the commutative property. “You have 8 apples for sale and want to display them in an array. How many different ways can you arrange them? Make sketches on paper to show your thinking.” Students extend their conceptual understanding of 3.OA.1, “Interpret products of whole numbers” and 3.OA.3, “Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities.”

• Lesson 8-6, Sharing Money, Home-Link, Problem 1, students engage in application as they interpret whole-number quotients by using multiplication and division. “Four friends share $76. They have seven $10 bills and six $1 bills. They can go to the bank to get smaller bills. Use numbers or pictures to show how you solved the problem. Answer: Each friend gets a total of $___.” Students engage in application of 3.OA.2, “Interpret whole-number quotients of whole numbers.”, 3.OA.3, “Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities.”, and 3.OA.7, “Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division.”

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

• Lesson 4-6, Perimeter, Focus: Solving Perimeter Number Stories, Math Journal 1, Problem 2, students solve real-world perimeter problems, “Mr. Lopez wants to put a fence around his rectangular vegetable garden. The longer sides are 14 feet long and the shorter sides are $9\frac{1}{2}$ feet long. How much fencing should Mr. Lopez buy? You may sketch a picture. Number Model: ____. Mr. Lopez should buy ____ feet of fencing.” Students develop all three aspects of rigor simultaneously of 3.MD.8, “Solve real-world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters.”

• Lesson 5-4, Recognize Helper Facts, Practice: Solving Two-Step Number Stories, Math Journal 2, Problem 1, students solve problems involving two steps. “Savannah earns $5 selling lemonade. Jessica earns double the amount of money that Savannah earns. How much money do they have together?” Students engage with procedural skills and application of 3.OA.8, “Solve two-step word problems using the four operations. 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 7-11, Fractions in Number Stories, Home-Link, Problem 3, students solve number stories with fractions. “Solve these number stories. Show your answer as a fraction. You may draw pictures to show your work. Nora rode her bike $\frac{2}{2}$ of a block. Brady rode his bike $\frac{4}{4}$ of the same block. Compare the distances each child rode. What do you notice? Explain your answer.” Students develop conceptual understanding and procedural skills and fluency of 3.NF.1, “Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction $\frac{a}{b}$ as the quantity formed by a parts of size $\frac{1}{b}$.”

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 2-3, More Number Stories, Focus: Organizing Number Story Information, students consider different strategies to solve number stories. “There are 43 children in the soccer club and 25 children in the science club. How many fewer children are in the science club?” The teacher asks, “How will you organize the information from the story? What do you know already?”

• Lesson 4-1, Measuring with a Ruler, Practice: Math Boxes, Problems 3 and 5, students analyze and make sense of subtraction word problems. Problems 3 and 5, “3. An alligator clutch had 82 eggs. 19 eggs did not hatch. How many eggs did hatch? 5. What strategy could you use to check your answer to Problem 3?”

• Lesson 6-6, Multiplication and Division Diagrams, Focus: Representing and Solving Number Stories, Math Journal 2, Problem 2, students reflect on their problem solving strategies. “There are 48 third graders. The gym teacher groups them into teams of 6. How many teams are there? When most children have finished, bring them together to discuss how they used diagrams to organize the information in each problem and write a number model to represent the story.” 

• Independent Problem Solving 9a, “to be used after Lesson 9-4”, Problem 1, students analyze and make sense of the information presented in problems that involve adding time intervals. “Alicia and Jeremy will visit the City Aquarium with their grandpa next week. They plan on leaving their house at 8:15 a.m. It will take them 45 minutes to ride the bus to the aquarium. When they leave the aquarium, they plan on going out to lunch for 1 hour and then taking the 45-minute bus ride home. They need to be back home by 2:15 p.m. The rest of their time will be spent at the aquarium. How much time will they spend at the aquarium altogether? Use words or drawings or both to explain 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:

• Lesson 1-7, Scaled Bar Graphs, Focus: Organizing and Representing Data, Math Journal 1, students consider units involved in a problem and attend to the meaning of quantities as they organize and represent data in bar graphs. “1. How many last names are there? 2. Use the data you collected to make a tally chart for the last names in your class. Add rows as needed. 3. Look at the data in your tally chart. Write at least three things you know from looking at the data.” 

• Lesson 5-7, Patterns in Products, Practice: Finding Clock Fractions, students understand the relationships between problem scenarios and mathematical representations as they connect clocks with fraction circle pieces. “Have children connect fractions of circles with fractions of hours by completing journal page 176, make sure toolkit clocks and fraction circle pieces are available for children to model the problems.” Math Journal 2, “Use your fraction circle pieces and toolkit clock to answer the questions. 1. On Monday, Isaac worked on his science project for 30 minutes. Shade 30 minutes on the clock. What time did he start? Draw hour and minute hands on the clock to show the time Isaac stopped working. What time did he stop? 2. What fraction of the clock did you shade? What fraction of an hour is that?”

• Lesson 9-5, Multi-Digit Multiplication, Focus: Math Message, Problems 1 and 2, students consider units involved in a problem and attend to the meaning of quantities as they partition a rectangle garden. “Jonah’s garden is a rectangle with 16 rows of plants. He wants to plant two sections: one with 10 rows of carrots and the other with 6 rows of beans. Partition the rectangle and label the sections with carrots and beans to show how Jonah could plant his garden. Jonah can plant 9 seeds in each row. How many seeds can he plant all together? Show your work.”

• Independent Problem Solving 3b, “to be used after Lesson 3-13”, Problem 1, students understand the relationships between numbers as they find a target number. “Ashley and Tyrese were playing Name That Number. Here are their cards: 6, 2, 9, 3, 10, Target Number 12. They wanted to write as many different equivalent names as they could for the target number 12. In the space below, use the numbers on the cards to write equivalent names for 12 using +, -, , and .”

The materials reviewed for Everyday Mathematics 4 Grade 3 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 6-10, More Operations, Focus: Exploring Order of Operations, Math Journal 2, Problem 5, students construct a viable argument when they explain to their partner why they picked their answer. “Circle the answer that makes the number sentence true. $2\times(4+3\times2)=?$. a. 28, b. 20, c. 14. Explain to a partner why you picked your answer.”

• Lesson 8-4, Setting Up Chairs, Focus, Math Message, Math Journal 2, students use clues to construct mathematical arguments. “Ms. Soto is setting up chairs for Math Night. Her room cannot fit more than 35 chairs. She places the same number of chairs in each row. As she sets up the chairs, she makes up a problem for her class with these clues: Clue A: When there are 2 chairs in each row, there is 1 leftover chair. Clue B: When there are 3 chairs in each row, there is 1 leftover chair. Clue C: When there are 4 chairs in each row, there is still 1 leftover chair. Clue D: When there are 5 chairs in each row, there are no leftover chairs. Use the clues to figure out how many chairs Ms. Soto set up. Joi, one of Ms. Soto’s students, makes a conjecture that Ms. Soto set up 13 chairs. Work with your partner and use the clues to make a mathematical argument for or against Joi’s conjecture. You may draw pictures or use counters to show your thinking. Explain your reasoning.”

• Independent Problem Solving 3a, “to be used after Lesson 3-6”, Problem 1, students justify their strategies and thinking as students use estimation strategies to solve problems. “Sarah estimated that the difference between the mass of a soccer ball and the mass of a softball was about 200 grams. How do you think Sarah made her estimate? When Sarah figured out the exact difference, she got 361 grams. How does Sarah’s estimate help her realize that her answer of 361 is not reasonable?”

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 7-2, Fractions, Home-Link, Problem 1, students critique the reasoning of others as they determine if two fractions are equivalent. “Nash chose these two cards in a round of Fraction Memory.” One card shows $\frac{5}{6}$ shaded and the other card shows $\frac{6}{8}$ shaded. “Nash says that these cards show equivalent fractions. Do you agree or disagree? Explain.”

• Independent Problem Solving 1a, “to be used after Lesson 1-8”, Problem 1, students critique the reasoning of others and construct mathematical arguments as they solve problems using multiplication strategies. “Grayson’s mother had 3 bags of oranges. Each bag had 5 oranges. She asked Grayson to figure out the total number of oranges in her 3 bags. Grayson said she had 11 oranges in all because $3+5+3=11$. Do you agree or disagree with Grayson? Use words, drawings, and numbers to show your thinking.”

• Independent Problem Solving 5a, “to be used after Lesson 5-4”, Problem 1, students construct mathematical arguments and critique the reasoning of others as they reason about equivalent fractions. “Alia’s mom baked 2 same-sized pizzas. She gave Alia $\frac{4}{8}$ of one pizza. She gave Alia’s friend, Blanca, $\frac{3}{6}$of the other pizza. Alia said she got more than Blanca because 4 slices are more than 3 slices. Do you agree? Show your thinking with words or drawings.”

The materials reviewed for Everyday Mathematics 4 Grade 3 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, Number Models for Two-Step Number Stories, Focus: Writing Number Models, Math Journal 2, Problem 1, students use the math they know to solve problems and everyday situations as they represent multistep number stories using drawings and number models with unknowns. “Ronald bought 2 packs of crackers. There are 5 crackers in each pack. He ate some crackers. Now Ronald has 7 crackers. How many crackers did he eat?”

• Independent Problem Solving 2a, “to be used after Lesson 2-5”, Problem 1, students model with mathematics as they multiply and divide to solve number stories. “Use information from the poster below to solve each problem. Show your work and write number models to keep track of your thinking. Carter bought 3 boxes of mini-stock cars. He shared half of his cars with his brother. How many cars did Carter give to his brother?” 

• Independent Problem Solving 8b, “to be used after Lesson 8-6”, Problem 2, students model the situation with an appropriate representation and use an appropriate strategy as they write their own equal sharing money story. “Write your own equal sharing money number story. Write a number model with a letter for the unknown quantity to model the problem. Solve your story. Use words, numbers, or drawings to show your thinking.”

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 7-11, Fractions in Number Stories, Focus: Solving Fraction Number Stories, Math Journal 2, Problem 2, students solve fraction number stories as they choose appropriate tools and strategies. “Use fraction circles, fraction strips, number lines, or pictures to help solve the number stories. Make sketches to show how you solved. Kaden makes 2 cups of salsa for the party. The 6 guests share the salsa equally. Write a fraction that shows how much each guest eats.”

• Independent Problem Solving 1b, “to be used after Lesson 1-13”, Problem 2, students choose appropriate tools and strategies as they add numbers. “Look at Terry’s and Justin’s record sheets from Problem 1. The person with the larger total sum of rounded numbers wins. Is Terry or Justin in the lead after 3 turns? Show how you know. Explain what you could do to make the other person the winner after 3 turns.”

• Independent Problem Solving 9b, “to be used after Lesson 9-6”, Problem 2, students use tools and strategies to solve problems using multiplication of 10s. “Jaloni’s grandma and the other 3rd grade room parents planned an end-of-year party for the primary grade classrooms. They baked 240 blueberry muffins and had to pack them into boxes that each held 16 muffins. Jaloni’s grandma asked him to figure out how many boxes the room parents need to equally pack all the muffins. He wants to use his calculator to solve, but the + and the keys are both broken. Help Jaloni find a way to use his broken calculator to solve the problem. a. Show or tell how to use Jaloni’s broken calculator to find the number of boxes the room parents need to pack the muffins. b. Show or tell another way for Jaloni to use his broken calculator to solve the problem.”

The materials reviewed for Everyday Mathematics 4 Grade 3 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 2-1, Extended Facts: Addition and Subtraction, Focus: Extending Combinations, Math Journal 1, Problem 4, students make use of the structure as they analyze how to solve addition and subtraction problems. “$14-9=?$, $24-9=?$, $54-9=?$” Problem 5, “Explain how you used a basic fact to help you solve Problem 4.”

• Lesson 6-10, Order of Operations, Focus: Exploring Order of Operations, Math Journal 2, Problem 4, students look for patterns or structures to make generalizations and solve problems using the order of operations. “Use the order of operations to solve each number sentence below. Show your work. To check your work, use a calculator that follows the order of operations. Rules for the Order of Operations. 1. Do operations inside parentheses first. Follow rules 2 and 3 when computing inside parentheses. 2. Then multiply or divide. In order, from left to right. 3. Finally add or subtract, in order, from left to right. $6+4\div2=$ ___.” Teacher’s Lesson Guide, “Ask each group to figure out how each calculator solved the problem.”

• Independent Problem Solving 5b, “to be used after Lesson 5-11”, Problem 2, students look for and explain the structure within rectilinear figures by decomposing them to find the area. “Abdul and Isabella are painting a wall mural that is 8 feet by 6 feet. They need to find the area of the mural so they can buy enough white paint to cover the background of the entire mural. Abdul said he could find the area by sketching an 8-by-6 rectangle and breaking 8 into easier-to-multiply factors 4 and 4. Isabella said she could sketch the same rectangle, but break 6 into 5 and 1. Abdul said that both strategies will work. Do you agree? Explain your thinking using words, numbers, and drawings.”

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

• Lesson 3-10, The Commutative Property of Multiplication, Focus: Introducing the Turn-Around Rule for Multiplication, Math Journal 1, Problem 1, students notice repeated calculations to understand algorithms and look for patterns when they generate pairs of facts and arrays. “Roll a die twice to get 2 factors. Sketch an array using those 2 factors and record a number sentence to match. Switch the factors and record an array and number sentence to match. What do you notice about each pair of arrays?” Teacher’s Lesson Guide, “Bring the class together to share their examples. Demonstrate or have a child turn an array pair for the class to see. Ask: What do you notice when you turn the arrays? Point to the number sentences and ask, If you switch the factors, will you always get the same product? Explain.”

• Lesson 7-7, Comparing Fractions, Focus: Using Benchmarks to Compare Fractions, Math Journal 2, Problem 2, students evaluate the reasonableness of their answers and thinking to develop a general rule for comparing fractions. “Choose two fractions that are less than $\frac{1}{2}$. Of these two fractions, which one is closer to 0? How do you know? Write a number sentence that compares your two fractions. Use <, >, or =.”

• Lesson 8-3, Factors of Counting Numbers, Focus: Recognizing Factor Pairs, Math Journal 2, Problem 6, students use repeated reasoning and make generalizations about factors and multiples. “The Kim family is serving dinner for 24 people. Mrs. Kim could have 1 table with 24 people or 2 tables with 12 people at each. What are some other ways Mrs. Kim could seat 24 people in equal groups at different numbers of tables? Is 1 in a factor pair for every counting number?”

The materials reviewed for Everyday Mathematics 4 Grade 3 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 assists 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 3, Operations, Organizer, Coherence, provides an overview of content and expectations for the unit. “In Unit 2, children represented multiplication number stories with arrays and recorded a number model to match. In Grade 2, children used addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns. They wrote an equation to express the total as a sum of equal addends. In Unit 5, children will use helper facts, doubling, and near-squares to solve for unknown products. In Grade 4, children will use multiplicative comparison statements to interpret a multiplication equation.”

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 2-2, Number Stories, Focus: Assessment Check-In, teacher guidance supports students in writing number stories. “Expect most children to make sense of and solve Problems 1-4 and to write number models with question marks for the unknowns. For children who struggle to make sense of and solve the problems, ask the guiding questions from the lesson. If children struggle to write a number model, suggest that they use a situation diagram or draw a picture to help organize the information from the story.”

• Lesson 4-3, Exploring Measures of Distance and Comparisons of Mass, Focus: Measuring Around Objects, Math Message, teacher guidance connects students' prior knowledge to new concepts. “Have children compare their measurements with a partner and think about whether they make sense. Invite volunteers to share how they measured and how they know whether their measures make sense. Expect responses to include that the distance around someone’s head is more than the distance around someone’s wrist because a head is bigger around than a wrist. Ask: Which measuring tools did you choose and why? When might it be useful to know these measurements?”

• Lesson 7-8, (Day 2): Finding Rules for Comparing Fractions, Common Misconception, teacher guidance addresses common misconceptions as students write rules for ordering fractions. “Watch for children who assume fractions with smaller denominators are smaller in size (or that fractions with larger denominators are larger in size). Encourage them to model two fractions, such as $\frac{1}{3}$ and $\frac{1}{4}$ using fraction circles. Ask: Would you get more pizza if you shared a pizza with three friends or four friends? Why?”

The materials reviewed for Everyday Mathematics 4 Grade 3 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 1-4, Number Lines and Rounding, Professional Development, explains connections between rounding and estimating. “Rounding can help with estimation. This lesson introduces a common rounding method that involves rounding to the nearest 10 or 100. The traditional version of this algorithm involves rounding up if the digit to the right of the target place is 5 or greater, and rounding down if the digit is less than 5.”

• Unit 3, Operations, Unit 3 Organizer, 3.G.1, supports teachers with concepts for work beyond the grade. “Links to the Future: In Grade 4, children will solve multistep number stories using all four operations and interpret the remainders in division number stories. They will explore different estimation strategies.”

• Unit 4, Measurement and Geometry, Unit 4 Organizer, 3.OA.8, provides support with explanations and examples of the more complex grade/course-level concepts. “Links to the Past: In Grade 2, children focused on specific attributes such as the number of sides, angles, and faces. They explored parallel sides.”

• Lesson 5-1, Exploring Equal Parts, Fractions of Different Wholes, and Area, Professional Development, explains concepts for work beyond the grade. “When working with fraction circles, many children may incorrectly think that the red circle is always the whole. For example, a pink piece may be the whole with a yellow piece representing 1-half. Children worked with other pieces as the whole in Lesson 2-12, Exploration A. Flexibility with the whole is important for solving real-world problems in which the size of the whole varies. It also lays a foundation for more complicated computation in later grades.”

• Lesson 6-4, Fact Power and Beat the Calculator, Professional Development, explains connections between multiplication and fluency. “The first five units of Third Grade Everyday Mathematics include many fact strategies to help children develop fluency with multiplication facts. By now, many children will already be automatic with beginning facts, such as 2s, 5s, and 10s, and squares- that is, they are able to recall these facts from memory or use a strategy automatically and instantaneously. For the rest of the year, meaningful practice through games, activities, and fact-extensions exercises will encourage children to progress beyond fluency to automaticity with basic multiplication facts.”

• Lesson 7-1, Liquid Volume, Professional Development, supports teachers with concepts for work beyond the grade. “In Everyday Mathematics, children begin by comparing liquid volume informally, followed by estimating in liters and finally estimating and measuring using liters and milliliters. As with other forms of measurement, such as length and mass, the need for more precise measurement motivates children's use of small, standard units. This progression done interactively, helps children understand volume concretely before exploring more abstract volume concepts in later grades.”

The materials reviewed for Everyday Mathematics 4 Grade 3 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:

• 3rd 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. 

• 3rd Grade Math, Unit 2, Number Stories and Arrays, 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 6-3, Taking Inventory of Known Fact Strategies, Core Standards identified are 3.OA.1, 3.OA.5, and 3.OA.7. 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, 3.OA.1, “First Quarter: Represent multiplication as equal groups with concrete objects and drawings. Second Quarter: Represent multiplication as equal groups with arrays. Third Quarter: Interpret products of whole numbers, e.g., interpret 5 x 7 as the total number of objects in 5 groups of 7 objects each. For example, describe a context in which a total number of objects can be expressed as 5 x 7. Fourth Quarter: Ongoing practice and application.” Mastery is expected in the Third 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 1, Math Tools, Time, and Multiplication, Organizer, Coherence, includes an overview of how the content in 3rd grade builds from previous grades and extends to future grades. “In Grade 2, children partitioned shapes into equal shares and described the whole as two-halves, three-thirds, or four-fourths. In Grade 4, children will extend the use of equal-sharing strategies to help develop an understanding of fraction equivalence. In Grade 5, children will interpret fraction and mixed-number quotients of whole numbers and will solve number stories that lead to quotients in the form of fractions or mixed numbers.” 

• Unit 4, Measurement and Geometry, Organizer, Coherence, includes an overview of how the content in 3rd grade builds from previous grades and extends to future grades. “In Grade 2, children focused on specific attributes such as the number of sides, angles, and faces. They explored parallel sides. In Grade 4, children will begin more formal geometry work with angles.”

• Unit 9, Multidigit Operations, Organizer, Coherence includes an overview of how the content in 3rd grade builds from previous grades and extends to future grades. “In Grade 2, children used Commutative and Associative properties of operations to add and subtract. In Grade 4, children will solve larger whole-number problems using strategies based on place value and the properties of operations.”

The materials reviewed for Everyday Mathematics 4 Grade 3 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 3 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 Third 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 3, Operations, Unit 3 Organizer, Unit 3 Materials, teachers need, “pattern blocks; 25 centimeter cubes; number cards 1-9 (4 of each), slate; 1-foot square cardboard templates; colored paper; scissors; straightedge (optional); masking tape (optional); collection of objects (optional) in lesson 7.” 

• Lesson 5-5, Multiplication Facts Strategies, Doubling, Part 1, Overview, Materials, “Math Masters, p. TA19; centimeter cubes (50 per partnership); rectangles; Class Data Pad; slate; Math Journal 2, pp. 164-165; Minute Math; Math Journal 2, p. 167; Math Masters, p. 167.” Math Message, “Use centimeter cubes and grid paper to show your thinking.”

• Unit 7, Fractions, Unit 7 Organizer, Unit 7 Materials, teachers need, “Quick Look Cards 164, 165, 174; fraction circles; Class Fraction Number-Line Poster; fraction strips; straightedge; Fact Strategy Logs (optional); slate; fraction cards; comparison-symbol cards; paper (optional); scissors (optional); The Area and Perimeter Game Action Deck, Deck B. in lesson 10.” 

• Lesson 7-10, Justifying Fraction Comparisons, Math Message, “Use your fraction circles to solve this problem.” Focus: Modeling Fraction Comparisons, “Have children use their fraction strips and the Fraction Number-Line Post on journal page 229 (or the Class Fraction Number-Line Poster) to show that $\frac{1}{6}>\frac{1}{8}$ with other tools.”

The materials reviewed for Everyday Mathematics 4 Grade 3 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 6, More Operations, Lesson 4, include:

• Readiness, “To prepare children for playing Beat the Calculator, have them use calculators to solve missing-factor problems. As needed, remind children how to enter numbers and clear calculators. Pose problems verbally or display them in a table. Explain that they should use multiplication to find the answers. For example: Start with 3. Change to 6. How? Start with 5. Change to 30. How? Allow children to experiment with their calculators to find the missing factors, including using guess-and-check. If needed, rephrase and display problems as 3 times what equals 6? $3\times$ ___ $= 6$? Have children explain their thinking to the group.”

• Enrichment, “To apply children’s understanding of multiplication and division, have them complete two-rule Frames-and-Arrows puzzles on Math Masters, page 196. Note the rules will not always alternate ABAB. Have children color-code their arrows with crayons to distinguish one rule from another. When children finish, have them discuss their solution strategies and explain any useful patterns they identified while solving.”

• Extra Practice, “To provide additional practice with multiplication and division facts, have children generate sets of multiplication facts and compare strategies used to solve them. Some children may benefit from a small strip of tape to secure the fact wheel to their slate.”

• English Language Learner, Beginning ELL, “Use gestures to scaffold the terms Caller, Calculator, and Brain from Beat the Calculator. Point to your mouth and a multiplication fact and say: I am the Caller. I will call out a fact. I will not say the answer. Point to a child’s head and say: You are the Brain. You will solve the problem in your head. Point to the calculator and demonstrate entering the multiplication fact as you point to another child and say: You are the Calculator. You will solve the multiplication fact on the calculator.”

The materials reviewed for Everyday Mathematics 4 Grade 3 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 1, Math Tools, Time, and Multiplication, Challenge, Problem 2, “Don and Molly played Number-Grid Difference. The object of the game is to have the lower sum of 5 scores. Don picked 3 and 5 and made the number 35. Molly picked 8 and 5. What number should Molly make? Explain your answer.”

• Lesson 6-3, Taking Inventory of Know Fact Strategies, Enrichment, “To extend children’s application of multiplication strategies, have them choose strategies and develop rules to multiply 11 by single-digit numbers on Math Masters, page 193.”

• Lesson 7-6, Fractions on a Number Line, Part 2, Enrichment, “To apply children’s understanding of fractions greater than 1, have them locate fractions between whole numbers on a number line. Ask children to defend the placement of their fractions and make changes as needed.”

The materials reviewed for Everyday Mathematics 4 Grade 3 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 3-5, Counting-Up Subtraction, Differentiation Options, English Language Learner Beginning ELL, “Display a number line vertically, with the smaller numbers at the bottom. Demonstrate counting up as you move your hand up along the number line. Orally and with gestures, direct children to count up on a number line. For example, say, Count up from 25, gesturing to 25, then 26, 27, 28, and so on as children count. Once children can count up as you gesture, point to a number and have them count up without gestures. Provide for oral practice by asking children short-response questions, such as: ``How will you count?” 

• Lesson 6-3, Taking Inventory of Known Fact Strategies, Differentiating Lesson Activities, Analyzing Multiplication Facts Strategies, “Scaffold to help partnerships think together about the facts to identify efficient and appropriate strategies for solving them, as well as to plan for using multiple representations to communicate their thinking. Post a menu of questions and response starters. For example: Partner A: Does this fact remind you of other facts we already know? Partner B: It reminds me of ___ Do you agree? Do you think we should try the ___ strategy? Partner A: I think/don’t think that strategy would be appropriate because ___Perhaps we could ___ Partner B: Do you think it would be efficient to use the ___ strategy? Partner A: That strategy would/would not help us get the answer quickly.”

• Lesson 8-5, Playing Factor Bingo, English Language Learner Beginning ELL, “Scaffold the terms factor and product to prepare children to play Factor Bingo. Display a multiplication fact. Circle and label the factors, and then underline and label the product. Draw a square around the multiplication symbol and label it groups of, times, and multiplied by. Point to the labels as you explain the directions for the game.”

• 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 3 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 2-8, (Day 1): Picturing Division, Focus: Solving the Open Response Problem, materials reference use of counters and drawings. “Make slates, markers, and counters available so children can act out the problem, but remind them to record drawings and words that describe their thinking on paper.” 

• Lesson 5-2, Representing Fractions, Focus: Math Message, materials reference use of fraction circles. “The pink fraction circle piece is the whole. Show 1-third of the pink piece. Explain to your partner how you know it shows 1-third.”

• Lesson 7-4, Fraction Stirps, Focus: Math Message, materials reference use of fraction strips. “Cut out five fraction strips. Each strip is one whole. Fold one strip in half. What fraction names each part of the strip?”