The instructional materials reviewed for Everyday Mathematics 4 Grade 5 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 instructional materials include problems that develop conceptual understanding throughout the grade level, especially where called for in the standards. Examples include: 

• Lesson 2-7, U.S. Traditional Multiplication, Part 3, Focus: Introducing U.S. Traditional Multiplication with 2-Digit Factors, Math Journal 1, Problem 5, students use the U.S. traditional multiplication to multiply 2-digit numbers by 2-digit numbers and estimation to determine whether their products make sense. “Complete the area model. Explain how it relates to your work for Problem 3. Area model.” Students create an area model for $87\times46$ by breaking 46 into 40 and 6. Students develop a conceptual understanding of 5.NBT.5, “Fluently multiply multi-digit whole numbers using the standard algorithm.”

• Lesson 4-11, Addition and Subtraction of Decimals with Hundredths Grids, Focus: Subtracting Decimals with Grids, Math Journal 1, students discuss strategies for using grids to solve decimal subtraction problems. “Problem 4, “ $0.74-0.36=?$ Problem 5, Choose one of the problems above. Clearly explain how you solved it.” Students develop a conceptual understanding of 5.NBT.7, “Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.”

• Lesson 5-4, Subtraction of Fractions and Mixed Numbers, Practice: Math Journal 2, Problem 1, students compare decimals and shade grids to represent the decimals. “Shade the first grid to represent one-tenth. Shade the second grid to represent ninety-nine thousandths. Write <, >, or = to make a true number sentence. 0.1 ___ 0.099.” Students develop a conceptual understanding of 5.NBT.3, “Read, write, and compare decimals to thousandths.” 

• Lesson 5-9, Understanding an Algorithm for Fraction Multiplication, Focus: Math Message, Math Journal 2, Problem 8, students find how many total parts and how many shaded parts in an area model while using patterns to derive a fraction multiplication algorithm. “Choose one of the above problems. Draw an area model for the problem. Explain how it shows that your answer is correct.” Students can choose from these problems: $\frac{1}{2}\star\frac{3}{6}$, $\frac{2}{3}\star\frac{1}{4}$, $\frac{3}{5}\star\frac{1}{6}$, $\frac{3}{4}\star\frac{3}{8}$, $\frac{2}{5}\star\frac{4}{10}$, or $\frac{7}{4}\star\frac{2}{12}$.” Students develop conceptual understanding of 5.NF.4 “Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.”

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

• Lesson 3-9, Introduction to Adding and Subtracting Fractions and Mixed Numbers, Home Link, Problem 2, students draw pictures and use number lines to solve mixed-number addition and subtraction number stories. “Ethel had 4 feet of ribbon. She used $1\frac{1}{2}$ feet for a craft project. How many feet of ribbon does she have left?” Students independently demonstrate conceptual understanding of 5.NF.2, “Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators .”

• Lesson 5-11, Explaining the Equivalent Fraction Rule, Home Link, Problem 3, students use the multiplication rule to find equivalent fractions. “Addison wanted to find a fraction equivalent to $\frac{3}{8}$ with 16 in the denominator. He thought: “, so I need to multiply $\frac{3}{8}$ by 2.” He got an answer of $\frac{3}{16}$. a. Is $\frac{3}{16}$ equivalent to $\frac{3}{8}$? How do you know? b. What mistake did Addison make?” Students independently demonstrate a conceptual understanding of 5.FN.4a, “Interpret the product as a part of a partition of q into b equal parts.”

• Lesson 7-7, Playing Property Pandemonium, Practice: Math Boxes, Math Journal 2, Problems 4 and 5, “Problem 4, Which expressions have a value equal to 6? Check all that apply. $6\star\frac{2}{2}$, $6\star\frac{8}{7}$, $\frac{3}{2}\star\frac{6}{1}$, $6\star\frac{9}{10}$, $6\star1$. Problem 5, Explain how you solved Problem 4 without multiplying.” Students independently demonstrate conceptual understanding of 5.NF.5, “Interpret multiplication as scaling.”

The instructional materials reviewed for Everyday Mathematics 4 Grade 5 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 2-8, U.S. Traditional Multiplication, Part 4, Focus: Extending U.S. Traditional Multiplication to Larger Numbers, students learn how to solve 2-digit numbers multiplied by 2-digit numbers. “Display the problem $417\star99$. Have students work independently or in partnerships to solve it in two ways: using partial-products multiplication and using US traditional multiplication ” Students develop procedural skill and fluency of 5.NBT.5, “Fluently multiply multi-digit whole numbers using the standard algorithm.”

• Lesson 3-13, Fraction-Of Problems, Part 1, Practice: Student Reference Book, students practice multiplying fractions and whole numbers by playing the game Fraction Of, “Players take turns. On your turn, draw 1 card from each deck. Place the cards on your gameboard to create a fraction-of problem. The fraction card shows what fraction of the whole you must find. The whole card offers 3 possible choices. Choose a whole that will result in a fraction-of problem with a whole-number answer.” Students develop procedural skills and fluency of 5.NF.4, “Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.”

• Lesson 4-7, Playing Hidden Treasure, Practice: Math Journal 1, Problem 6, students estimate and solve problems using U.S. traditional multiplication, “.” Students develop procedural skills and fluency of 5.NBT.5, “Fluently multiply multi-digit whole numbers using the standard algorithm.” 

• Lesson 6-9, Multiplication of Decimals, Focus: Using Estimation as a Strategy for Decimal Multiplication, students multiply decimals as if they were whole numbers and use estimation to place decimal points in the products. “Display the following problems: $1.2\star0.8=?$; $1.2\star8=?$; $12\star8=?$ What do you notice about the factors in all three problems? Do you think this pattern is true for all multiplication problems with factors that have the same digits in the same order?” Students develop procedural skills and fluency of 5.NBT.7, “Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used,” and 5.NBT.5, “Fluently multiply multi-digit whole numbers using the standard algorithm.”

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 3-1, Connecting Fractions and Division, Part 1, Home Link, Problem 4, students make an estimate and solve problems using the standard algorithm of multiplication, “” and Problem 5, “.” Students independently demonstrate procedural skill and fluency of 5.NBT.5, “Fluently multiply multi-digit whole numbers using the standard algorithm.”

• Lesson 4-7, Playing Hidden Treasure, Practice: Math Journal 1, Problem 1, students solve problems using the standard algorithm for multiplication, “” and Problem 4, “Another student estimated and began solving problems 4-6 using U.S. traditional multiplication. Finish solving the problems.” Students independently demonstrate procedural skill and fluency of 5.NBT.5, “Fluently multiply multi-digit whole numbers using the standard algorithm.”

• Lesson 5-5, Connecting Fraction-Of Problems to Multiplication, Practice; Math Boxes, Math Journal 2, Problem 6, students use the standard algorithm of multiplication to solve problems involving multi-digit numbers. “Estimate and solve. $912\times87=?$” Students independently demonstrate procedural skill and fluency of 5.NBT.5, “Fluently multiply multi-digit whole numbers using the standard algorithm.”

• Multiplication Top It, online game, students multiply 2-digit numbers by 2 or 3-digit numbers and compare the products. “Directions: Players multiply the numbers shown on gems, then compare the products. The player with the greater product takes all of the gems. Players earn points for correctly multiplying the numbers on their gems, comparing the products, and having the greater product.” Students independently demonstrate procedural skill and fluency of 5.NBT.5, “Fluently multiply multi-digit whole numbers using the standard algorithm.”

The materials reviewed for Everyday Mathematics 4 Grade 6 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics.

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

• Lesson 3-5, Game Strategies, Focus: Solving the Open Response Problem, Problem 2, students create a rule for making the largest possible fraction. “Write a rule you can use to make the largest possible fraction with any 3 number cards. Explain why your rule works.” This activity provides the opportunity for students to apply their understanding of 5.NF.3, “Interpret a fraction as division of the numerator by the denominator.”

• Lesson 5-5, Connecting Fraction of Problems to Multiplication, Home-Link, Problem 4, students match number models to fraction-division situations. “Erica’s garden has an area of 24 square feet. She will use $\frac{3}{4}$ of the space for vegetables and $\frac{1}{4}$ of the space for flowers. How much space will she use for vegetables?” This activity provides the opportunity for students to apply their understanding of 5.NF.6, “Solve real world problems involving multiplication of fractions and mixed numbers.”

• Lesson 7-10, Identifying and Visualizing Patterns, Focus: Visualizing Patterns and Relationships, Math Journal 2, Problem 3, students form ordered pairs using corresponding terms of two sequences and graph them in a real-world problem. “a. Use the rules to fill in the column. In (x), Rule: - 1, 5, ___, ___, ___, ___. Out (y), Rule: - 3, 15, ___, ___, ___, ___. b. What rule relates each in number to its corresponding out number? c. Write the numbers from the table as ordered pairs. Then graph the ordered pairs. Draw a line to connect the points.” This activity provides the opportunity for students to apply their understanding of 5.G.2, “Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation.”

• Lesson 8-4, Focus: A Treasure Hunt Day 1, Solving the Open Response Problem, Problem 1, students solve real-world problems as they find volume for rectangular prisms. “A fifth-grade class read an adventure story. In the story, an explorer named Miriam traveled to South America in search of a lost treasure. Traps had been set to guard the treasure. Miriam fell through a trap door into a 9-foot-high rectangular room that measured 4 feet wide by 6 feet long. Suddenly, the room began to fill with water! It stopped when the water was 3 feet deep. Miriam sighed with relief, but her relief didn’t last long. The two 4-foot wide walls of the room began to move, making the room smaller and causing the water level to rise. Every 10 minutes, the walls were 1 foot closer together. To the right is a picture of what the room looked like after 10 minutes, when the 4-foot wide walls had moved 1 foot closer together. What is the approximate height of the water? Show how you found your answer.” Problem 2, “The only way to get out of the room is through the trap door in the ceiling. About how much time will pass before the water lifts Miriam to the trap door? Show your work. Explain how you used representations to help you solve the problem.” This activity provides the opportunity for students to apply their understanding of 5.MD.3, “Recognize volume as an attribute of solid figures and understand concepts of volume measure.”

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 2b, “to be used after Lesson 2-13”, Problem 2, students write division number stories. “Write a number story that can be modeled with the expression $197\div12$ that has a different solution than the number story you wrote for Problem 1. Solve your number story. Explain why your solution is different.” This activity provides the opportunity for students to independently demonstrate understanding of 5.NBT.6, “Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division.”

• Independent Problem Solving 4a, “to be used after Lesson 4-9”, Problem 1, students compare batting averages using place value understanding with decimals to thousandths. “A batting average in softball is calculated by dividing a player’s number of hits by the player’s number of at-bats. A batting average is typically reported as a decimal rounded to the nearest thousandths place. Caroline has 3 hits and 7 at-bats. Vivien has 42 hits and 117 at-bats. Find each player’s batting average. You may use a calculator. Caroline’s batting average: ___. Vivien’s batting average: ___. Make an argument for why Caroline is a better batter than Vivien. Make an argument for why Vivien is a better batter than Caroline.” This activity provides the opportunity for students to independently demonstrate understanding of 5.NBT.3, “Read, write, and compare decimals to thousandths”, and 5.NBT.4, “Use place value understanding to round decimals to any place.”

• Independent Problem Solving 5b, “to be used after Lesson 5-14”, Problem 1, students use division of fractions by whole numbers to solve problems. “The librarian, Mr. Gates, received a shipment of new books. He put half the books on the shelves in the library. He put $\frac{2}{3}$ of the other half on a book display in the hallway. The remaining books were split equally among the 6 fifth-grade teachers. What fraction of the entire shipment of books did each fifth-grade teacher receive? Show your work and explain your thinking.” This activity provides the opportunity for students to independently demonstrate understanding of 5.NF.7, “Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.”

• Independent Problem Solving 7b, “to be used after Lesson 7-12”, Problem 1, students use multiplication strategies to solve fraction problems. “The Smith School student council is collaborating with a local artist to create a tile mosaic. The space they have for the mosaic is $6\frac{1}{2}$feet long and $4\frac{1}{2}$feet in height. The student council has to select the tiles they are going to use. They have narrowed it down to 2 choices described in the table below. The student council would prefer to choose the tile that costs less so that they have money left over for other projects. Which tile should the student council choose? Explain your answer.” This activity provides the opportunity for students to independently demonstrate understanding of 5.NF.6, “Solve real world problems involving multiplication of fractions and mixed numbers.”

The materials reviewed for Everyday Mathematics 4 Grade 5 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 instructional materials attend to conceptual understanding, procedural skill, and fluency, or application include:

• Lesson 2-6, Application: Unit Conversions, Home-Link, Problem 6, students practice multiplying multi-digit whole numbers. “” Students develop procedural skills and fluency of 5.NBT.5, “Fluently multiply multi-digit whole numbers using the standard algorithm.”

• Lesson 3-1, Connecting Fractions and Division, Focus: Creating Other Models for Fair Share Stories, Math Journal 1, Problem 3, “A school received a shipment of 4 boxes of paper. The school wants to split the paper equally among its 3 printers. How much paper should go to each printer?” Students extend their conceptual understanding of 5.NF.3, “Interpret a fraction as division of the numerator by the denominator (). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers.” 

• Lesson 4-6, Introduction to the Coordinate System, Home-Link, Problem 4, students use addition to solve number stories involving decimals. “At the 2012 Summer Olympics in London, Usain Bolt won the men’s 100-meter race with a time of 9.63 seconds and the men’s 200-meter race with a time of 19.32 seconds. How long did it take the sprinter to run the two races combined?” Students engage in the application of 5.NBT.7, “Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.”

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

• Lesson 2-3, Applying Powers of 10, Focus: Estimating with Powers of 10, Math Journal 1, Problem 1, students estimate with powers of 10 to solve multiplication problems while checking the reasonableness of products. “A hardware store sells ladders that extend up to 12 feet. The store’s advertising says: Largest inventory in the country? If you put all our ladders end to end, you could climb to the top of the Empire State Building! The company has 295 ladders in stock. The Empire State Building is 1,453 feet tall. Is it true the ladders would reach the top of the building? Explain how you solved the problem.” Students engage with conceptual understanding and application of 5.NBT.2, “Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.”

• Lesson 4-4, Comparing and Ordering Decimals, Interpreting Real-World Remainders, Math Journal 1, Problem 2, students solve division number stories and interpret the remainders, “Elisabeth is 58 inches tall. What is her height in feet? Number model: Quotient: Remainder: Answer: Elisabeth is ___ feet tall. Circle what you did with the remainder. Ignored it, Reported it as a fraction, Rounded the quotient up; Why?” Students engage with procedural skill and application of 5.NF.3, “Interpret a fraction as division of the numerator by the denominator (). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers.”

• Lesson 5-7, Fractions of Fractions, Focus: Finding Fractions of Fractions, students find fractions of whole numbers and fractions of fractions as they make sense of and solve problems, “Larry has $\frac{1}{2}$ of a fruit bar. He wants to give $\frac{1}{2}$ of what he has to his brother. What part of a whole fruit bar will Larry give to his brother? Tell students they will use a sheet of paper to represent Larry’s fruit bar. Have them fold it in half vertically and then unfold it. Demonstrate to students how to orient their sheet so that the fold line is vertical, and ask them to shade in $\frac{1}{2}$. Ask: If the paper is an entire fruit bar, what does the shaded part of the paper model represent? What part of the $\frac{1}{2}$ fruit bar are we trying to find? Have students fold their sheets in half in the opposite direction, unfold them, and orient the sheets so that the new fold is horizontal...” Teachers finish the activity by having students shade in the $\frac{1}{2}$ of a $\frac{1}{2}$ to demonstrate how much of the whole Larry is giving his brother.” Students engage with procedural skill and conceptual understanding of 5.NF.4a, “Interpret the product $(a/b) × q$ as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations $a × q ÷ b$.”, 5.NF.4b, “Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas.”, and 5.NF.6, “Solve real-world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.”

The materials reviewed for Everyday Mathematics 4 Grade 5 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 2-1, Understanding Place Value, Home-Link, Problem 2, students analyze and make sense of place value problems to solve number riddles. “I have 5 digits. My 9 is worth $9+10,000$. My 2 is worth 2 thousand. One of my 7s is worth 70. The other is worth 10 times as much. My other digit is a 6. What number am I?”

• Lesson 3-6, Fraction Estimation with Number Sense, Home-Link, Problem 2, students utilize fraction number sense to determine if their answers make sense. “Renee calculated $\frac{3}{6}+\frac{2}{4}$ and said the answer was $\frac{5}{10}$. Josie solved the same problem and said the answer was 1. Whose answer is more reasonable? Explain how you know.”

• Lesson 6-3, Application: Converting Measurement in the Metric System, Focus: Converting Metric Units, Math Journal 2, Problem 4, students make sense of information in different problems to make metric unit conversions. “There are 43 milligrams of caffeine in a bottle of ice tea. How many grams of caffeine is that? Answer the questions below to solve. a. What units do you need to convert? b. How are those units related? c. Write your answers to Parts a and b in the “What’s My Rule?” table. Fill in the rule. d. How many grams is 43 milligrams?” Teacher’s Lesson Guide, “Draw students’ attention to Problem 4. Point out how this problem required them to convert a measurement to solve a real-world problem. Ask: How did you decide which units you needed to convert?”

• Independent Problem Solving 8a, “to be used after Lesson 8-7”, Problem 1, students analyze and make sense of problems as they use estimation strategies. “Plastic pollution is considered a pressing environmental issue. Some scientists estimate that plastic waste makes up 80% of the pollution in the oceans. One source of plastic waste is bottled water. One estimate is that in the United States alone, 1,500 bottles of water are consumed each second! Another estimate is that only about 2 out of every 10 water bottles are recycled. The rest are thrown away. Use this information to estimate how many bottles of water are thrown away in the United States every month. Assume that 1 month has 30 days. Show your work and explain your thinking. You may use a calculator to help you.” 

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 1-5, Introduction to Volume, Practice: Math Boxes, Math Journal 1, Problem 3, students consider units involved in a problem and attend to the meaning of quantities as they find the area of a rectangle. “Jo’s closet is 6 ft wide and $1\frac{1}{2}$ ft deep. Find the area of the closet floor.” Problem 5, “Explain how you found the area of Jo's closet floor in Problem 3.”

• Lesson 4-1, Decimal Place Value, Focus: Extending Place-Value Patterns to the Thousandths Place, students make sense of representations to model decimal numbers. “Students consider a picture of a large square (representing 1) and then 10 strips or 100 small squares that compose the large square, using language like 10 times and $\frac{1}{10}$ of to describe the relative sizes of the squares and strips. Abstracting from these ideas, they represent decimals on a place-value chart and with numerals. What might $\frac{1}{10}$ of the small square look like?”

• Lesson 5-8, Area Models for Fraction Multiplication, Focus: Math Message, students represent situations symbolically as they solve fraction-of problems by using folding paper strategies. “Fold and shade a sheet of paper to find $\frac{2}{3}$ of $\frac{2}{3}$.” Focus, Making Sense of Area Models, Teacher’s Lesson Guide, “How should we label the other tick marks? Why? How does this area model show the product of $\frac{2}{3}$ and $\frac{2}{3}$?”

• Lesson 7-1, Multiplication of Mixed Numbers, Part 1, Focus: Solving Mixed-Number Multiplication Problems, Math Journal 2, Problem 2, students understand the relationships between problem scenarios and mathematical representations as they multiply fractions and mixed numbers. “Use the rectangle to make an area model. Label the sides. Find and list partial products. Label the partial products in the area model. Add the partial products to find your answer. you may need to rename fractions with a common denominator. $2\frac{3}{5}\star\frac{1}{4}=?$”

The materials reviewed for Everyday Mathematics 4 Grade 5 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 6-8, Estimating Decimal Products and Quotients, Focus, Math Message, students justify their strategies and thinking as they make conjectures about the relative size of a decimal product based on the factors. “You know that $2.4\star1=2.4$. Will $2.4\star1.8$ be greater than or less than 2.4? How do you know? Share your conjecture and argument with a partner.”

• Independent Problem Solving 3a, “to be used after Lesson 3-7”, Problem 2, students construct mathematical arguments as they divide fractions. “Keiko has a goal of running 10 miles this month. She plans to run on Mondays, Wednesdays, and Fridays. On Mondays she will run $\frac{3}{4}$ mile, on Wednesdays she will run $\frac{1}{2}$ mile, and on Fridays she will run $\frac{2}{3}$ mile. Keiko thinks she will meet her goal of running 10 miles in a month on this training schedule. Do you agree with Keiko? Use words and a drawing to explain why you agree or disagree.”

• Independent Problem Solving 8a, “to be used after Lesson 8-7”, Problem 2, students construct viable arguments as they use place value strategies to multiply. “There are approximately 330 million people in the United States. Problem 1 told you that it is estimated that 1,500 bottles of water are consumed every second in the United States. Does that estimate seem reasonable? Show your work and explain your thinking. You may use a calculator to help you.”

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 1-3, Area and Volume, Focus: Solving the Open Response Problem, Problem 1, students critique the reasoning of others when they use the area to compare. “Allyson and Justin are working together to sew a quilt. Justin wrote down the length and width of the quilt and started to sketch a plan for the design. He showed Allyson his sketch and told her they will use 54 square feet of fabric. Allyson disagrees and says they will only use $13\frac{1}{2}$ square feet of fabric. Why might Justin think they will use 54 square feet of fabric? Do you agree or disagree with Justin's answer? Why?” Problem 2, “Why might Allyson think they will use $13\frac{1}{2}$ square feet of fabric? Do you agree or disagree with Allyson’s answer? Why?” 

• Lesson 5-9, Operations with Fractions, Focus: Multiplying Fractions, Math Journal 2, Problem 11, students critique the reasoning of others as they multiply fractions. “Ben tried to solve Problem 9 and got the answer $\frac{4}{7}$. He said, “That can’t be right because $\frac{1}{3}$ is less than $\frac{4}{7}$.” Do you agree with Ben? Explain.”

• Independent Problem Solving 4b, “to be used after Lesson 4-14”, Problem 2, students critique the reasoning of others as they perform operations with decimals. “Jamella is helping her brother mix green paint to repaint the car he is restoring. They must mix 39.5 pounds of white paint with 0.593 pounds of yellow paint and 0.593 pounds of blue paint. Jamella is trying to figure out how much more white paint she needs than yellow and blue paint together. Her work is below. What was Jamella’s mistake? How would you explain it to Jamella so that she understands how to solve this kind of problem in the future? Use drawings, tools, or pictures in your explanation.”

The materials reviewed for Everyday Mathematics 4 Grade 5 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 5-7, Focus, Finding Fractions of Fractions, students model fractions of fractions by folding paper and using shading to illustrate understanding. Student Math Journal, Problem 2, “Fold paper to help you solve this number story. Carolyn had $\frac{1}{3}$ liter of water. She drank $\frac{3}{4}$ of the water. What part of a liter did she drink? Fold and shade paper to show $\frac{1}{3}$. Then fold and double-shade it to show $\frac{1}{4}$ of $\frac{1}{3}$. Add double-shading to your paper to show $\frac{3}{4}$ of $\frac{1}{3}$. What part of a liter of water did Carolyn drink?”

• Independent Problem Solving 1a, “to be used after Lesson 1-4”, students use the math they know to solve problems and everyday situations as they find areas of rectangles with fractional sides. Independent Problem Solving Masters, Problem 2, “A mosaic is a piece of art created by covering a surface with small pieces of colorful material. Kha-Minh has a box of 36 square tiles with a side length of $\frac{1}{3}$ inch that she would like to use to make a mosaic. She wants the mosaic to be a rectangle and she wants to use all the tiles. Draw and label two different rectangles that Kha-Minh could make. Are the areas of each rectangle the same? Why or why not?” 

• Independent Problem Solving 4a, “to be used after Lesson 4-9”, students describe and explain what they do with a model and how it relates to the problem situation. Independent Problem Solving Masters, Problem 2, “Angelo’s mom pays him $5 per hour for doing work in the yard. His mom makes him save half of what he earns in a bank account and lets him spend the other half however he wants. Angelo wants to buy himself a pair of sneakers that cost $75. How many hours will Angelo have to work to earn enough money to buy the sneakers? Use the grid to help you model and solve the problem. Angelo will have to work ___ hours to earn enough money to buy the sneakers. Explain how you used the grid to model the problem.”

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 5-6, Multiplication of Fraction and Whole Numbers, Focus: Discussing Strategies for Multiplying Fractions by Whole Numbers, Math Journal 2, Problem 5, students choose and use appropriate tools as they multiply fractions and whole numbers. “Solve problems 2-5 using any strategy. Show your work. In a classroom, there are 14 cubbies along a wall. Each cubby is $\frac{5}{6}$ foot wide. How long is the line of cubbies?”

• Independent Problem Solving 2a, “to be used after Lesson 2-8”, Problem 2, students recognize both the insight to be gained and limitations of tools and strategies as they convert measurements. “Jamie has a new fitness tracker. Her tracker gives her a goal of taking 10,000 steps each day. Jamie went for a 4-mile run on Saturday. She estimates that when she runs her stride length (the distance from the toe of one foot to the toe of the other foot) is 4 feet. Did Jamie meet her goal of 10,000 steps by running 4 miles? Choose and use an appropriate tool to help you solve the problem. Explain your answer and describe the tool you used.”

• Independent Problem Solving 3b, “to be used after Lesson 3-12”, Problem 1, students recognize both the insight to be gained and limitations of tools and strategies as they solve problems with fractions. “Write a number story to match the following expression. Then solve your number story. Choose and use an appropriate tool to solve your number story. Describe how you used the tool. $(3\div4)+2\frac{1}{8}$.”

The materials reviewed for Everyday Mathematics 4 Grade 5 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 3-2, Connecting Fractions and Division, Part 2, Practice: Math Boxes, Math Journal, Problem 1, students make use of the structure as they explore patterns in multiplying with powers of 10. “Solve a. $4\times10=$ ___. b. $4\times10^2=$ ___. c. $6\times10^2=$ ___. d. $6\times1,000$.” Problem 5, “Describe a pattern you noticed in Problem 1.” 

• Lesson 8-12, Pendulums, Part 2, Practice: Using the Quadrilateral Hierarchy, Math Journal 2, Problem 3, students look for and explain the structure within mathematical representations as they answer questions about quadrilateral hierarchy. “a. Classify the shape at the right on the hierarchy. List all of the categories you passed through while you were classifying. b. Explain how you knew when to stop classifying the quadrilaterals. How did you know that the shape couldn't move down into more subcategories?” 

• Independent Problem Solving 2a, “to be used after Lesson 2-8”, Problem 1, students look for patterns or structures to make generalizations and solve problems as they use place value knowledge of base-10. “DeShaun’s mom is a computer programmer. She explained that computers use a base-2 place value system. She showed him the following base-2 place-value chart and numbers written in base 2 and base 10. a. How is the base-2 place-value chart similar to a base-10 place-value chart? How is it different? b. How would you write the number 10 in base 2? Explain your thinking.”

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 1-9, Two Formulas for Volume, Focus: Finding a Formula for Volume, students use repeated reasoning to generalize the formula and . “Ask students to multiply the area of the base by the height of Prism E and verify that the product is the same as what they found for the volume of Prism E in the Math Message. $12cm^2\times3cm=36cm^3$. Ask: What else do we need to know to find the volume? How could we find the number of layers in Prism E without using cubes? Why does this work? Summarize by asking: Why does multiplying the area of the base by height of a prism give the volume? Display the formula and the formula  . Ask: What is the relationship between these two formulas?”

• Lesson 4-10, Folder Art, Solving the Open Response Problem, Math Journal 1, Problem 1, students notice repeated calculations to understand algorithms and look for patterns when they create and apply a rule for enlarging an image on a coordinate grid. “Jake is designing a picture for the cover of his reading folder. He draws a picture of a book on a coordinate grid. After drawing the picture, Jake decides he wants to put a picture of a larger book on his folder. Write a new rule that Jake can use to make the picture of the book larger.”

• Lesson 8-2, Applying the Rectangle Method for Area, Practice: Using a Table, a Rule, and a Graph, Math Journal 2, Problem 2, students evaluate the reasonableness of their answers and thinking as they find a rule that relates to a table. “Organic apples are on sale for $1.50 per pound. Anita started making the table in Problem 1 to show how much different amounts of apples will cost. She thinks of $1.50 as 1.5 dollars. She says, “Each time I add 1 pound of apples, I add another $1.50 to the total cost.” She writes the rule “+ 1” in the Pounds of Apples column and the rule “+ 1.5” in the Total Cost in Dollars column. a. Look at the table in Problem 1. What rule relates the Pounds of Apples column to the Total Cost in Dollars column? b. Why does the rule you found in Part a make sense?”

The materials reviewed for Everyday Mathematics 4 Grade 5 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 2, Whole Number Place Value and Operations, Organizer, Coherence, provides an overview of content and expectations for the unit. “In Grade 4, students used partial-products multiplication and lattice multiplication to solve multi-digit multiplication problems. Through Grade 5, students will use U.S. traditional multiplication to solve multiplication problems in mathematical and rich, real-world contexts. In Grade 6, students will use U.S.traditional multiplication to solve multi-digit decimal multiplication problems.”

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 2-5, U.S. Traditional Multiplication, Part 2, Focus: Assessment Check-In, teacher guidance supports students in solving multiplication problems. “Expect most students to be able to solve $423\star3$, which does not involve writing any digits above the line, using U.S. traditional multiplication. Some may also be able to solve $2,681\star5$, which does involve writing digits above the line and remembering to add them.”

• Lesson 4-3, Representing Decimals in Expanded Form, Focus: Introducing Expanded Form for Decimals, Math Message Follow-Up, teacher guidance connects students' prior knowledge to new concepts. “Ask: What did you notice as you added different colors of shading to your grid? Display a shaded grid for the Math Message problem. Ask: What number do the first 3 columns of shading represent? What number does the 1 square in the second color represent? What number do the 2 tiny rectangles shaded in the third color represent? Point out that the shading shows how 0.3, 0.01, and 0.002 combine to make 0.312. Ask: What operation is used to show combining or putting together? Display the number sentence $0.312=0.3+0.01+0.002$. Remind students that this way of writing a number is called expanded form. Highlight how expanded form shows the number broken apart by place value and shows the sum of the value of each digit.”

• Lesson 7-3, Multiplication of Mixed Numbers, Part 2, Common Misconception, teacher guidance addresses common misconceptions as students use tiling to differentiate square foot. “Students may confuse $\frac{1}{2}$ of square foot, as shown in Figure 1, with a $\frac{1}{2}$ foot square (a square that is $\frac{1}{2}$ foot by $\frac{1}{2}$ foot), as shown in Figure 2. When speaking, be sure to differentiate between fractions of a square area and squares with fractional side lengths.”

The materials reviewed for Everyday Mathematics 4 Grade 5 meet expectations for containing adult-level explanations and examples of the more complex grade-level concepts and concepts beyond the current grade so that teachers can improve their own knowledge of the subject.

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

• Lesson 1-9, Two Formulas for Volume, Professional Development, explains finding the volume of rectangular prisms. “Counting the cubes that fit into the bottom of a rectangular prism and finding the area of the base of the prism result in the same numerical value, but they are two different measurements. The number of cubes that pack the first layer of a prism is the volume of the first layer, which is a 3-dimensional measurement, while the area of the base is a 2-dimensional measurement. They have the same numerical value because, to find the volume, the area is multiplied by 1 to account for the height of the layer. Multiplying by 1 does not change the numerical value, but it does change the unit of measurement from square units to cubic units, thereby making it a different measurement.”

• Lesson 2-4, U.S. Traditional Multiplication, Part 1, Professional Development, explains the traditional multiplication algorithm. “Everyday Mathematics calls the algorithm introduced in this lesson U.S. traditional multiplication because it is not the standard algorithm in other parts of the world. The Grade 5 standards require students to fluently multiply multi-digit whole numbers using U.S. traditional multiplication, so it is the focus of Fifth Grade Everyday Mathematics lessons. Students should learn this method, but if they prefer a different multiplication strategy, they should be allowed to use it to solve multiplication problems.”

• Unit 3, Fraction Concepts, Addition, and Subtraction, Unit 3 Organizer, 5.NF.5, provides support with explanations and examples of the more complex grade/course-level concepts. “Links to the Past: In Grade 4, students used visual fraction models and other strategies to solve number stories involving multiplication of a fraction by a whole number.”

• Lesson 4-7, Playing Hidden Treasure, Professional Development, supports teachers with concepts for work beyond the grade. “There are two common ways to look at distances in the coordinate plane. One type of distance is practical distance. On a coordinate grid, the practical distance between two points is along horizontal and vertical grid lines. Practical distance is sometimes called taxicab distance because it measures the distance along a route a taxicab might take. The other type of distance is straight line distance. On a coordinate plane, the straight-line distance between two points is the length of the line segment that connects the points. In Grade 5 students explore practical distance only. They will explore straight-line distance in later grades.”

• Lesson 5-13, Fraction Division, Part 1, Professional Development, supports teachers with concepts for work beyond the grade. “Lessons 5-13 and 5-14 build conceptual understanding of fraction division. According to the standards (5.NF.7), fraction division problems in Grade 5 are limited to the division of unit fractions by whole numbers and vice versa. Students are expected to use visual models and informal reasoning to solve them. They are not expected to use a generalized method of fraction division. Students will be introduced to an algorithm for fraction division in Grade 6. To build a conceptual understanding of division, students are asked to write multiple number models. The initial number model with a variable is intended to help students identify problems as division situations. The number models for the summary division sentence and multiplication check are designed to emphasize the relationship between division and multiplication as described in 5.NF.7a.”

• Unit 6, Investigations in Measurement: Decimal Multiplication and Division, Unit 6 Organizer, 5.MD.1, supports teachers with concepts for work beyond the grade. “Links to the Future: In Grade 6, students will use ratio reasoning to convert measurement units.”

The materials reviewed for Everyday Mathematics 4 Grade 5 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:

• 5th 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. 

• 5th Grade Math, Unit 1, Area and Volume, 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 8-4, Extending Line Symmetry, Core Standards identified are 5.NBT.2, 5.MD.3, 5.MD.5, and 5.MD.5b. 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, 5.NBT.4, “First Quarter: No expectations for mastery at this point. Second Quarter: Use grids, number lines, or a rounding shortcut to round decimals to the nearest tenth or hundredth in cases when rounding only affects one digit. Third Quarter: Use place value understanding to round decimals to any place. 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 2, Fraction Concepts, Addition, and Subtraction, Organizer, Coherence, includes an overview of how the content in 5th grade builds from previous grades and extends to future grades. “In Grade 4, students worked with place-value concepts in whole numbers through 1,000,000. In Grade 6, students will extend their understanding of place value by applying their reasoning to make sense of decimal computation.” 

• Unit 5, Operations with Fractions, Organizer, Coherence, includes an overview of how the content in 5th grade builds from previous grades and extends to future grades. “In Grade 4, students explored and explained a multiplication rule for producing equivalent fractions. In Grade 6, students will consider the sizes of dividends and divisors to help them make sense of the size of quotients when they divide fractions.”

• Unit 6, Investigations in Measurement: Decimal Multiplication and Division, Organizer, Coherence includes an overview of how the content in 5th grade builds from previous grades and extends to future grades. “In Grade 4, students made line plots to display fractional measurement data and answered questions about that data. In Grade 6, students will collect and display data using dot plots, histograms, and box plots.”

The materials reviewed for Everyday Mathematics 4 Grade 5 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 5 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 Fifth Grade Everyday Mathematics.” Each unit includes a Materials Overview section outlining supplies needed for each lesson within the unit. Additionally, specific lessons include notes about supplies needed to support instructional activities, found in the overview of the lesson under Materials. Examples include:

• Lesson 3-2, Connecting Fractions and Division, Part 2, Overview, Materials, “slate; fraction circles; Math Journal 1, pp. 74-75; Student Reference Book p. 318; per partnership: Math Masters p. G11; two 6-sided dice; Math Journal 1, p. 76; Math Masters, p. 82.” Math Message, “Use your fraction circle pieces to help you.”

• Unit 5, Operations with Fractions, Unit 5 Organizer, Unit 5 Materials, teachers need, “fraction circles; per partnership: cards 0-10 (4 of each), 4 counters; slate; per partnership: calculator (optional); coin (optional) in lesson 1.” 

• Unit 7, Multiplication of Mixed Numbers; Geometry; Graphs, Unit 7 Organizer, Unit 7 Materials, teachers need, “fraction circles (optional); per partnership: number cards 1-8 (4 of each); Fraction Number Lines Poster; per group: Math Journal 2, Activity Sheet 19 (Spoon Scramble cards); scissors; 3 spoons in lesson 2.” 

• Lesson 7-2, Multiplication of Mixed Numbers, Part 2, Math Message, “You may use fraction circle pieces or the Fraction Number Lines Poster to help you.”

The materials reviewed for Everyday Mathematics 4 Grade 5 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 3, Fraction Concepts, Addition and Subtraction, Lesson 7, include:

• Readiness, “To prepare for using benchmarks to estimate, students compare fractions to $\frac{1}{2}$ and locate them on a number line. Have students represent fractions between 0 and 1 with fraction circles. Ask them whether each fraction is less than, equal to, or greater than $\frac{1}{2}$ and have them explain their thinking. They record the fractions in the appropriate box on Math Masters, page 97, and then complete the problems at the bottom of the page. Discuss students’ responses.”

• Enrichment, “To further develop fraction number sense and explore using benchmarks, students play Fraction Top-It (Estimation Version). In this version, they apply their knowledge of benchmarks to estimate sums and compare them.”

• Extra Practice, “For more practice using benchmarks to estimate sums and differences, students complete Math Masters, page 98. They make sense of number stories and match the stories to estimates shown on a number line.”

• English Language Learner, Beginning ELL, “Make think-aloud statements using the term scaffold student’s understanding of the term benchmark as a point of reference, or a standard according to which things are judged. For example, I will use the fraction $\frac{1}{2}$ to help me think about the size of $\frac{5}{8}$. I will use $\frac{1}{2}$ as a benchmark to help me think about the size of $\frac{5}{8}$. Students may benefit from seeing how benchmarks can be useful in the same way helper facts are useful for thinking about nearby facts.”

The materials reviewed for Everyday Mathematics 4 Grade 5 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 5, Operations with Fractions, Challenge, Problem 4, “What is $\frac{1}{2}$ of $\frac{2}{3}$ of $\frac{3}{4}$ of 1? Explain how you found your answer.”

• Lesson 6-5, Working With Data in Line Plots, Enrichment, “To extend their work using line plots to solve problems, students create line plots showing the scores of two competitive divers. They calculate the divers’ final scores in two ways: first by using all seven judges’ scores and then by following competitive diving rules, where the two highest and two lowest scores are thrown out. Students compare results and consider why using the scoring rule makes sense.”

• Lesson 7-12, Rules, Tables, and Graphs, Part 2, Enrichment, “To extend their understanding of constructing graphs from data, students represent race results with multiple graphs. Students conduct two different types of races along a 5-meter course. They graph the results for each participant, then compare and discuss the resulting graphs.”

The materials reviewed for Everyday Mathematics 4 Grade 5 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 1-8, Measuring Volume By Iterating Layers, Differentiating Lesson Activities, Using Layers to Solve Cube-Stacking Problems, “Scaffold to help students justify their volume calculations by providing questions-and-response prompts. For example: Can you explain why you ___? If I ___ then we need to ___ because ___ What does that mean? Let me show you what I mean. To paraphrase what you just said you ___.”

• Lesson 3-9, Introduction to Adding and Subtracting Fractions and Mixed Numbers,  English Language Learner Beginning ELL, “Build on students’ understanding of the word remainder as what remains so that they make a connection between the terms remain and remainder. Display a number of objects that do not divide evenly and say: We are going to share these ____ between the 3 of us: 1 for you, 1 for you, and 1 for me…. Point to the remaining objects at the end, saying: These are left over. This is what remains. These are the remainder. Partners do the same using another set of objects, repeating the think-aloud. Ask students to point to the remainder after each round.” 

• Lesson 7-9, Collecting and Using Fractional Data, Differentiation Options, English Language Learner Beginning ELL, “In this lesson students work with the word span, which begins with the consonant cluster sp. English language learners may find it difficult to pronounce this sound in the initial position if it does not occur in that position in their home language. Students may add the vowel sound /e/ to the English pronunciation of span to make it easier to pronounce. Point out the correct pronunciation and articulate it carefully. List other words that begin with the sp sound.” 

• 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 5 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 1-7, Measuring Volume by Counting Cubes, Focus: Math Message, materials reference use of unit cubes. “Cut out and assemble Rectangular Prisms A, B, and C. Take 25 cubes. Estimate how many cubes will fit into each prism. Record your estimates in the second column of the table on Journal page 18.” 

• Lesson 3-1, Connecting Fractions and Division, Part 1, Focus: Modeling with Fraction Circle Pieces, materials reference use of fraction circles. “Make the point that fraction circle pieces are being used to model the situation.” 

• Lesson 4-14, Addition and Subtraction of Money, Focus: Introducing Spend and Save, materials reference use of coins, bills, and counters. “Then distribute one Spend and Save Record Sheet to each student and 1 coin and 1 counter to each partnership. Make play bills and coins available to students who might need them to play the game.”