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 which includes Planning for Rich Math Instruction. This component indicates where conceptual understanding is emphasized within each lesson of the Unit. Lessons include Focus, “Introduction of New Content”, designed to help teachers build their students’ conceptual understanding. The instructional materials include problems that develop conceptual understanding throughout the grade level, especially where called for in the standards (5.NF.B, 5.NBT.A, 5.NBT.B). Examples include: 

• In Student Math Journal, Lesson 2-7, U.S. Traditional Multiplication, Part 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 x 46 by breaking 46 into 40 and 6. (5.NBT.5)
• In Teacher’s Lesson Guide, Lesson 3-9, Introduction to Adding and Subtracting Fractions and Mixed Numbers, Math Message, students explore strategies and tools for adding and subtracting fractions and mixed numbers to develop conceptual understanding of addition and subtraction of fractions and mixed numbers. “Deena and her brother are making two kinds of bread. Both recipes call for $$\frac{2}{3}$$ cup of flour. How much flour do they need in all? You may use fraction circles or the Fraction Number Lines Poster to help you.” In the Teacher’s Lesson Guide, sample strategies are provided for teachers to discuss with students. For instance, “Use fraction circle pieces with the red circles as the whole. Use 2 orange pieces to show $$\frac{2}{3}$$ and 2 more orange pieces to show another $$\frac{2}{3}$$. Put 3 orange third pieces together to make a whole. The total is 1 whole and 1 third, or 1 $$\frac{1}{3}$$.” (5.NF.1)
• In Teacher’s Lesson Guide, Lesson 4-1, Decimal Place Value, Focus, Extending Place-Value Patterns to The Thousandths Place, students extend place-value patterns to decimals, and practice reading and writing decimals to thousandths. Students determine the value of a 1-by-10 rectangular strip and 10-by-10 square when a small square is defined as 1. Next, students discuss times 10 and $$\frac{1}{10}$$ of place value relationships and extend the patterns through the thousandths place. (5.NBT.1) 
• In Student Math Journal, Lesson 4-2, Representing Decimals through Thousandths, Problem 4, students develop conceptual understanding of representing decimals to the thousandths place using three different ways. “Use words, fractions, equivalent decimals, or other representations to write at least three names for each decimal in the name-collection box. Then shade the grid to show the decimal 0.8.” (5.NBT.A)
• In Student Math Journal, Lesson 4-11, Addition and Subtraction of Decimals with Hundredths Grids, students use grids to represent and solve decimal addition and subtraction problems. For example: “Problem 2, 0.18 + 0.35 =? Problem 5, Choose one of the problems above. Clearly explain how you solved it.” (5.NBT.7)
• In Student Math Journal, Lesson 5-4, Subtraction of Fractions and Mixed Numbers, Practice, students compare decimals and shade grids to represent the decimals. For example, Problem 1, “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.” (5.NBT.3a,b) 
• In Teacher’s Lesson Guide, Lesson 5-7, Fractions of Fractions, Focus, Finding Fractions of Fractions, students find a fraction of a fraction. “Remind students that when they solve fraction-of problems, such as $$\frac{2}{3}$$ of 6, they start by thinking about what $$\frac{1}{3}$$ of 6 would be. Ask them to think similarly about this problem to find $$\frac{2}{3}$$ of $$\frac{1}{2}$$, they should first think about what $$\frac{1}{3}$$ of $$\frac{1}{2}$$ would look like. Ask: How could we use shading to show $$\frac{1}{3}$$ of $$\frac{1}{2}$$ of our paper?” (5.NF.4a,4b,.6)
• In Student Math Journal, Lesson 5-9, Understanding an Algorithm for Fraction Multiplication, 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: “$$\frac{1}{2}$$ * $$\frac{3}{6}$$, $$\frac{2}{3}$$ * $$\frac{1}{4}$$, $$\frac{3}{5}$$ * $$\frac{1}{6}$$, $$\frac{3}{4}$$ * $$\frac{3}{8}$$, $$\frac{2}{5}$$ * $$\frac{4}{10}$$, or $$\frac{7}{4}$$ * $$\frac{2}{12}$$.” (5.NF.4)

The materials provide opportunities for students to independently demonstrate conceptual understanding throughout the grade. These include problems from Math Boxes, Home Link, and Practice. Examples include:

• In Math Masters, Lesson 3-9, Introduction to Adding and Subtracting Fractions and Mixed Numbers, Home Link 3-9, students draw pictures and use number lines to solve mixed-number addition and subtraction number stories. Problem 2, “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?” (5.NF.2)
• In Student Reference Book, Lesson 5-1, Using Equivalent Fractions to Find Common Denominators, Practice, students play the game, Decimal Top-It: Subtraction, to practice subtracting decimals. The directions state: “1. Shuffle the deck and place it number-side down on the table. 2. Each player turns over 6 cards and, using the counter as a decimal point, forms 2 numbers with digits in the ones, tenths, and hundredths places. Players may put their cards in any order. 3. Each player finds the sum of his or her numbers. Players then compare their sums. The player with the larger sum takes all the cards. Players can check their answers with a calculator. 4. The game ends when there are not enough cards left for each player to have another turn. The player with the most cards wins.” (5.NBT.A,B)
• In Math Masters, Lesson 5-11, Explaining the Equivalent Fraction Rule, Home Link 5-11, students use the multiplication rule to find equivalent fractions. Problem 3, “Addison wanted to find a fraction equivalent to $$\frac{3}{8}$$ with 16 in the denominator. He thought: “8 * 2 = 16, 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?” (5.NF.4a,5b)
• In Student Math Journal, Lesson 6-5, Working with Data in Line Plots, Math Boxes, students shade a rectangle to represent a multiplication problem. Problem 1, “Shade a rectangle to represent $$\frac{2}{3}$$ * $$\frac{5}{6}$$ = ?” (5.NF.4)
• In Student Math Journal, Lesson 7-7, Playing Property Pandemonium, Math Boxes, Problems 4 and 5, “4. Which expressions have a value equal to 6? Check all that apply. 6 * $$\frac{2}{2}$$, 6 * $$\frac{8}{7}$$, $$\frac{3}{2}$$ * $$\frac{6}{1}$$, 6 * $$\frac{9}{10}$$, 6 * 1. 5. Explain how you solved Problem 4 without multiplying.” (5.NF.5a,5b)

The instructional materials reviewed for Everyday Mathematics 4 Grade 5 meet expectations for attending to those standards that set an expectation of procedural skill and fluency.

The instructional materials develop procedural skill and fluency throughout the grade level. Each Unit begins with Planning for Rich Math Instruction where procedural skills and development activities are identified throughout the unit. Each lesson includes Warm-Up problem(s) called Mental Math Fluency. These provide students with a variety of leveled problems to practice procedural skills. The Practice portion of each lesson provides students with a variety of spiral review problems to practice procedural skills from earlier lessons. Additional procedural skill and fluency practice is found in the Math Journal, Home Links, Math Boxes, and various games. Examples include:

• In Teacher’s Lesson Guide, Lesson 2-7, U.S. Traditional Multiplication, Part 3, Focus, students discuss how to solve 2-digit numbers multiplied by 2-digit numbers, “Suppose you already figured out that 54 * 8 = 432. How could you use that information to help you solve 54 * 18?” This activity provides an opportunity for students to develop fluency of 5.NBT.5, “Fluently multiply multi-digit whole numbers using the standard algorithm.”
• In Teacher’s Lesson Guide, Lesson 4-1, Decimal Place Value, Practice, students practice multiplying fractions and whole numbers by playing the game Fraction Of. In the Student Reference Book, “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.” This activity provides an opportunity for students to develop fluency of 5.NF.4, “Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.”
• In Teacher’s Lesson Guide, Lesson 4-7, Playing Hidden Treasure, Practice, students estimate and solve problems using U.S. traditional multiplication. The Student Math Journal includes 6 multi-digit multiplication problems where students first estimate the product and then solve using U.S. traditional multiplication. For example, Problem 6, “511*219.” (5.NBT.5) 
• In Teacher’s Lesson Guide, Lesson 6-9, Multiplication of Decimals, Focus, 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 * 0.8 = ?; 1.2 * 8 = ?; 12 * 8 = ? 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?” This activity provides an opportunity for students to develop procedural skill 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.”
• In Teacher’s Lesson Guide, Lesson 6-10, Fundraising, Focus, students determine and discuss strategies for choosing the correct product without actually calculating the exact answer. In the Student Math Journal, “For each set of problems, circle the number sentence with the correct product. Do not calculate the exact answer. Explain how you knew which product was correct without finding the exact answer.” In the Teacher’s Lesson Guide, “As students share strategies, consider listing them on the Class Data Pad or chart paper. Display the list for reference when students solve the open response problem.” This activity provides an opportunity for students to develop procedural skill 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.”

The instructional materials provide opportunities to independently demonstrate procedural skill and fluency throughout the grade level as identified in 5.NBT.5, “Fluently multiply multi-digit whole numbers using the standard algorithm.” Examples include:

• In Teacher’s Lesson Guide, Lesson 2-9, One Million Taps, Focus, students estimate the amount of time it takes to address 10 and 100 envelopes based on the amount of time it takes to address 1 envelope. Student Math Journal, Problem 1, “Zoey is mailing invitations for a fifth-grade party. It takes her about 30 seconds to address 1 envelope. About how many seconds would it take Zoey to address 10 envelopes? Show your work. " (5.NBT.2,5)
• In Math Masters, Lesson 3-1, Connecting Fractions and Division, Part 1, Home Link, students make an estimate and solve problems using the standard algorithm of multiplication. Problem 4, “2,598 x 3”; Problem 5, “417 x 63.” (5.NBT.5)
• In Student Math Journal, Lesson 4-7, Playing Hidden Treasure, students solve problems using the standard algorithm for multiplication such as Problem 1, “25 x11” and Problem 4, “Another student estimated and began solving problems 4-6 using U.S. traditional multiplication. Finish solving the problems.” (5.NBT.5)
• In Math Masters, Lesson 4-11, Addition and Subtraction of Decimals with Hundredths Grids, Home Link, students use the standard algorithm of multiplication to solve problems involving multi-digit numbers. Problem 5, “Make an estimate. Then solve using U.S. traditional multiplication. 412 * 176.” (5.NBT.5) 
• Students have the opportunity to independently demonstrate procedural skill and fluency for 5.NBT.5 in the online game, Multiplication Top It. In this 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, correctly comparing the products, and having the greater product.”

The instructional materials reviewed for Everyday Mathematics 4 Grade 5 meet expectations that the three aspects of rigor are not always treated together and are not always treated separately. All three aspects of rigor are present in the instructional materials. Student practice includes all three aspects of rigor, though there are fewer questions for conceptual understanding.

There are instances where conceptual understanding, procedural skill and fluency, and application are addressed independently throughout the instructional materials. Examples include:

• In Math Masters, Lesson 2-7, U.S. Traditional Multiplication, Part 3, students demonstrate procedural skill and fluency as they use the standard algorithm to multiply multi-digit whole numbers. Problem 4. “82 * 11.” (5.NBT.5)
• In Student Math Journal, Lesson 3-1, Connecting Fractions and Division, Part 1, students use conceptual understanding as they draw pictures to solve division stories with non-circular wholes. 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?” (5.NF.3)
• In Math Masters, Lesson 4-12, Decimal Addition Algorithms, students engage in application as they use addition to solve number stories involving decimals. Problem 4, “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?” (5.NBT.7) 

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

• In Student Math Journal, Lesson 2-3, Applying Powers of 10, students engage with conceptual understanding and application as they estimate with powers of 10 to solve multiplication problems while checking the reasonableness of products. Problem 1, “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.” (5.NBT.2)
• In Student Math Journal, Lesson 4-4, Comparing and Ordering Decimals, students engage with procedural skills and application as they solve division number stories and interpret the remainders. Problem 2, “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?” (5.NF.3)
• In Teacher’s Lesson Guide, Lesson 5-7, Fractions of Fractions, Focus, Finding Fractions of Fractions, students use procedural skill and conceptual understanding to solve application problems as they 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. (5.NF.4a,4b,6)