5th Grade - Gateway 2

The materials reviewed for i-Ready Classroom Mathematics, Grade 5 meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings.

Within the materials, Program Implementation, Program Overview, i-Ready has three different types of lessons to address the unique approaches of the standards and to support a balance of conceptual understanding, application, and procedural fluency. “Understand Lessons focus on developing conceptual understanding and help students connect new concepts to familiar ones as they learn new skills and strategies. Strategy Lessons focus on helping students persevere in solving problems, discuss solution strategies, and compare multiple representations through the Try- Discuss-Connect routine.” The Math in Action Lessons “feature open-ended problems with many points of entry and more than one possible solution.” Lessons are designed to support students to explore and develop conceptual understanding of grade-level mathematics. Additional Practice and Interactive Games are also provided so that students can continue to practice the skills taught during lessons if needed.

Students develop conceptual understanding with teacher guidance and support. Examples include:

• Unit 2, Lessons 6, 7, 8, and 9, students develop conceptual understanding 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 6, Understand Decimal Place Value, Session 1, Explore, Model It, Problem 2, “Write the missing numbers in the equation below these place-value models for decimals to show how hundredths, tenths, and ones are related.” A picture is shown of a place-value chart with Ones, Tenths, and Hundredths. Each column has a hundred grid for students to shade. Each column also has an equation to complete: “$$1.0=$$___ \times0.1, 0.1=___$$\times0.01$$, 0.01.” Teacher Edition, Model It, “Tell students they are going to use what they know about the relationships between places in whole numbers to think about relationships between places in decimals less than 1.” Session 2, Model It: Decimal Grids, Problem 2, “Complete four different equations that show relationships between the numbers in models A, B, and C above. ___$$\times10=$$___, ___$$\times10=$$___, ___$$\div10=$$___, ___$$\div10=$$___.” Session 3, Refine, Apply It, Problem 3, “Compare, How does the value of the digit 8 in 1.8 compare to the value of the digit 8 in 480? Explain.” Lesson 7, Understand Powers of 10, Session 2, Develop, Connect It, Problem 5, “Look at the first decimal point pattern diagram and the place-value charts. How do the position and the value of the digit 5 change when multiplying and dividing by 10?” Session 3, Refine, Apply It, Problem 2, “Kosumi says that the product 30\times10^4 has exactly four zeros. Is he correct? Explain.” Teacher Edition, Apply It, Inspect, “Look for understanding that both factors can contribute zeros to the product. Prompt discussion with questions such as: How many zeros does 10^4 contribute to the product? [four] Why does the product 30\times10^4 have more than four zeros? [because 30 already has one zero] What is the product 30\times10^4? [300,000].” Lesson 8, Read and Write Decimals, Session 1, Connect It, Problem 2, Look Ahead, “On the previous page, you used words to say the standard form of the decimal 0.32. You can also use expanded form to break apart a decimal by place value. You can use either decimals or fractions to write the expanded form of a decimal. a. Complete each missing number to show two ways to write 0.32 in expanded form using decimals. 0.32=____$$+$$____, 0.32=____$$\times0.1+$$____$$\times0.01$$; b. Complete each missing number to show two ways to write 0.32 in expanded form using fractions. 0.32=\frac{}{10}+\frac{}{100}; ____$$\times\frac{1}{10}+$$___$$\times\frac{1}{100}$$.” Session 2, Develop, “Rico buys dried chiles to make adobo sauce. The dried chiles weigh 0.604 pounds. How does Rico say the weight of the chiles aloud? Explain your thinking.” Model It, “You can use place-value understanding to write the expanded form of 0.604. You can then write the number as a fraction. With decimals: 0.604=0.6+0.004, ___$$=(6\times\frac{1}{10})+(4\times0.001)$$ with fractions: 0.604=(6\times\frac{1}{10})+(4\times\frac{1}{1000}), =\frac{6}{10}+\frac{4}{1,000}, =\frac{600}{1,000}+\frac{4}{1,000}), =\frac{604}{1,000}.” Session 3, Develop, “Kyle’s sisters, Bridget and Sylvia, run in a race. Silvia finishes the race one and sixteen thousandths behind the winner. Bridget finishes the race two and thirty-five hundredths behind Silvia. Use a model to support your answer.” Model It, “Model the measurements with mixed numbers and expanded form.” Model It, “Use a place-value chart to write the measurements.” Teacher Edition, Facilitate Whole Class Discussion, “Call on students to share selected strategies. Encourage students who have not presented a strategy to reword or rephrase what a classmate has said. Guide students to Compare and Connect the representations. Remind students that one way to agree and build on ideas is to give reasons that explain why the idea makes sense. Ask Where does your model show the whole seconds? the fractional part of a second?” Session 4, Refine, Apply It, Problem 2, “Carlos measures the length of a sideline on a football field. It is one hundred nine thousandths of a kilometer. Write this length as a decimal and in expanded form. Show your work.” Lesson 9, Compare and Round Decimals, Session 1, Explore, Connect It, Problem 2, “You can compare decimals to help you round decimals to a given place. Rounding decimals is similar to rounding whole numbers.” Problem 2a, “A number line can be a useful tool for rounding. Place and label the numbers 1.3 and 1.8 on the number line below.” Problem 2b, “What is 1.3 rounded to the nearest whole number? Explain how you know.” Problem 2c, “What is 1.8 rounded to the nearest whole number? Explain how you know.” Teacher Edition, Look Ahead, “Point out that strategies used to round whole numbers can be used to round decimals. Students should be able to place 1.3 and 1.8 on the number line and explain that 1.3 is closer to and rounds to the nearest whole number 1 while 1.8 is closer to and rounds to the nearest whole number 2. Ask What decimal comparisons do you use when rounding the numbers 1.3 and 1.8 to the nearest whole number?”

• Unit 2, Lesson 12, Add Fractions, students develop conceptual understanding of 5.NF.1 (Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators.) Session 1, Explore, Teacher Edition, Connect It, Look Back, “Look for understanding that an equivalent amount in fourths had to be shown for \frac{1}{2} in order to find the fraction of an hour Amy practices in all.” Look Ahead, “Point out that when the fractions in a sum have unlike denominators and one denominator is not a multiple of the other denominator, each addend must be replaced with an equivalent fraction. Students should be able to explain the terms common denominator, multiple, and equivalent fractions and use this language to discuss the process of writing two fractions with denominators that show same-size parts of the whole. Ask Look at the first pair of fraction models in problem 2b. How do you know what to write as the numerator when you rewrite \frac{1}{2} as an equivalent fraction with a denominator of 6? Listen For You need to multiply the denominator of \frac{1}{2} by 3 to get a denominator of 6, so you also need multiply the numerator by 3: \frac{1}{2}=\frac{1\times3}{2\times3}=\frac{3}{6}. Reflect Look for understanding that to be added, fractions must have same-sized parts of a whole. Students responses should include the common multiple of 2 and 3 to be used as the common denominator, as well as the equivalent fractions for \frac{1}{2} and \frac{2}{3}. Common Misconception If students correctly name a multiple to use as the common denominator in each equivalent fraction but neglect to make a corresponding change in the numerators, then provide students with fraction tiles. Have students find the numerator of \frac{1}{6}s that equal \frac{1}{2} and the number of \frac{1}{6}s that equal \frac{2}{3} and write the corresponding fractions. Discuss what students notice about the numerators and denominators.” Session 4, Refine, Apply It, Problem 8, Math Journal, “Adrian says that \frac{5}{7}+\frac{3}{2} is \frac{31}{14}. Kwame says the sum is \frac{8}{9}. Who is correct? Explain your answer.” 

• Unit 3, Lesson 21, students develop conceptual understanding of 5.NF.5a (Interpret multiplication as scaling (resizing), by: a. Comparing the size of a product to the size of one factor on the basis of the size of the other factor…). Lesson 21, Understanding Multiplication as Scaling, Session 1, Explore, Model It, Problem 1, “Changing the size of a quantity by multiplication is called scaling. Stretching and shrinking are two different ways to scale a quantity. This bar has a length of 6 units. Use the bars at the bottom of the page to complete parts a and b.” A picture is shown of two bar models, the first partitioned into 12 equal parts, and the second partitioned into three equal parts. Problem 1a, “Circle the bar that shows the length of 6 units being doubled, or stretched. Underline the bar that shows the length of 6 units being halved, or shrunk.” Problem 1b, “Write a multiplication equation for each bar. Circle the factor that describes how the length of 6 units has been stretched or shrunk.” Teacher Edition, Model It, “Tell students they are going to use what they know about multiplication to explore changing the size of a quantity by multiplication, which is called scaling. Have students turn and talk to help them think through the task, and then have students complete the problem.” Session 2, Develop, Model It: Area Models, Problem 4, “Use an area model to show scaling. Shade the area model to show \frac{1}{3}\times\frac{3}{4}.” A picture is shown of a 3 by 4 grid. “Is \frac{1}{3}\times\frac{3}{4} less than, greater than, or equal to \frac{3}{4}?” Teacher Edition, Model It, “As students complete the problems, have them identify that they are being asked to use an area model to show the same product as in problem 1 so that they can compare the number line and area models.”

Students have opportunities to independently demonstrate conceptual understanding. Examples include:

• Unit 1, Lessons 2 and 3, students independently engage with 5.MD.3 (Recognize volume as an attribute of solid figures and understand concepts of volume measurement) as they use concrete and semi-concrete representations to create rectangular prisms filled with cubes. Lesson 2, Find Volume Using Unit Cubes, Session 2, Additional Practice, Practice Finding Volume Using Unit Cubes, Problem 3, “What is the volume of the rectangular prism at the right? Show your work.” A picture is shown of a rectangular prism labeled with the dimensions of 2 by 4 by 2 ft. Lesson 3, Find Volume Using Formulas, Session 3, Develop, Apply It, Problem 8, “What is the volume of the solid figure below? Show your work.” A picture is shown of a figure with labeled side lengths.

• Unit 2, Lesson 9, Compare and Round Decimals, Session 2, Develop, Apply It, Problem 7, students independently engage with 5.NBT.3b (Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons) as they use place value strategies to compare decimals. “The long jump is a sport where athletes try to jump as far as possible. A long jumper with low vision practices their long jump. The length of their first jump is 5.46 meters. The length of their second jump is 5.62 meters. Which of their jumps is longer? Show your work.” 

• Unit 3, Lesson 18 Fractions as Division, Session 1, Explore, Additional Practice, Problem 2, students independently engage with 5.NF.3 (Interpret a fraction as division of the numerator by the denominator (\frac{a}{b}=a\div b), as they use their understanding of fractions are division problems. Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers). “Write the fraction \frac{3}{4} as a decimal expression. How could you use multiplication to check your answer?”

The materials reviewed for i-Ready Classroom Mathematics, Grade 5 meet expectations for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency.

Within the materials, Program Implementation, Program Overview, i-Ready has three different types of lessons to address the unique approaches of the standards and to support a balance of conceptual understanding, application, and procedural fluency. The materials include problems and questions, interactive practice, and math center activities that help students develop procedural skills and fluency, as well as opportunities to independently demonstrate procedural skills and fluency throughout the grade. Additional Practice and Interactive Games are also provided so that students can continue to practice the skills taught during lessons if needed.

Students develop procedural skills and fluency with teacher guidance and support. There are also interactive tutorials embedded in classroom resources to help develop procedural skills and fluency. Examples include:

• Unit 1, Lesson 4, Multiply Multi-Digit Numbers, students build procedural skills of 5.NBT.5 (Fluently multiply multi-digit whole numbers using the standard algorithm.) Session 1, Connect It, Problem 2 Look Ahead, “When you multiply with multi-digit numbers, you need to pay special attention to place value. To help, you can use patterns in the number of zeros in the factors and the number of zeros in the product.” Problem 2a, “Look at the partial products used to find the product of 512 and 24. Compare the number of zeros in each partial product to the number of zeros in its factors. Look at the zeros. Describe any patterns you see.” Problem 2b, “Explain why the partial product 4\times500 has three zeros, not just two zeros.” Session 2, Teacher Edition, Model It, “If no student presented these models, have students analyze key features and then point out the ways each model represents: the product 128\times35, factors decomposed as hundreds, tens, and ones. Ask Where does each model break apart factors? Listen For The length and width of the area model are labeled 100+200+8 and 30+5. The equations in the second Model It use 30+5. The products by the arrows show 128 and 35 broken apart. For the area model, prompt students to identify how the model represents the problem. Why can you use a rectangle to represent 128\times35? Look at the equation for the first row. What does 3,840 represent, or tell you about the situation? For the second Model It, prompt students to connect the term distributive property with breaking apart numbers to multiply. Look at the second equation. How does this equation show a factor being distributed? How are the ways of recording partial products in this model and the area model similar? different?” Session 3, Develop, Model It, “You can use the standard algorithm. An algorithm is a set of routine steps used to solve problems. With the standard algorithm for multiplication, you multiply by place value, regrouping and adding as you go. You record the sum of the partial products for each place value in a single row. Step 1: Multiply by ones. Regroup as needed. Record the partial product in one row. Step 2: Multiply by tens. Regroup as needed. Record the partial product in a second row. Now you can add the partial products to find the product.” The problem, 1,429\times42 is shown under each step. Teacher Edition, For the second Model It, “Prompt students to compare the standard algorithm model with the partial products model. Guide them to understand the regrouping numbers written above each step. Why are there more numbers in the partial products model than in the standard algorithm model?” Connect It, “Now you will use the problem from the previous page to help you understand how to connect partial products to the standard algorithm for multiplication.” Problem 2, “Look at the second Model It. How do the partial products 18, 40, 800, and 2,000 relate to the first step of the standard algorithm model?” Problem 3, “Look at the second Model It. How do the partial products 360, 800, 16,000, and 40,000 relate to the second step of the standard algorithm model?” Teacher Edition, Monitor and Confirm Understanding, “Check for understanding that: the partial products are the result of multiplying the value of each digit of 1,429 from right to left, first by 2 ones and second by 4 tens.”

• Unit 1, Lesson 4, Interactive Tutorials, contains one 17-minute tutorial to help students develop procedural skills and fluency with multiplying multi-digit numbers. The video focuses on multiplying whole numbers. 5.NBT.5 (Fluently multiply multi-digit whole numbers using the standard algorithm.)

• Unit 2, Lesson 12, Session 2, Develop, Hands-On Activity, Model equivalent sums, students to develop procedural skills and fluency as they add and subtract fractions with unlike denominators, 5.NF.1 (Add and subtract fractions with unlike denominators.) “If students are unsure about equivalent expressions, then use this activity to connect equivalent sums. Have students write \frac{1}{2}+\frac{1}{3} at the top of a sheet of paper. Right below this expression have them model each addend using fraction tiles. Together list 6 multiples of 2 and 3. [2, 4, 6, 8, 10, 12; 3, 6, 9, 12, 15, 18] Ask: What are two common multiples of 2 and 3? [6, 12] Keeping the half and third tiles in place, have students find the number of \frac{1}{6} tiles it takes to cover each. Below the tiles have them write the equivalent sum now shown. [\frac{3}{6}+\frac{2}{6}] Discuss how the part of the total represented by each addend and the total itself did not change.” 

• Unit 3, Lesson 16, Session 3, Develop, Hands-On Activity, students develop procedural skills and fluency as they perform operations with decimals, 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.) “If students are unsure about why the product of two decimals in the tenths is a decimal in the hundredths, then use this activity to help them make a model showing why. Tell students for this activity, flats represent ones, rods represent tenths,  and units represent hundredths. Have students model 4\times2 with flats [8 flats] and write the equation 4\times2=8. Have students model multiplying 4\times2 by one tenth by trading their 8 flats for 8 rods and writing the equation 4\times2\times0.1=0.8 point out that 0.8 is in tenths. Have students model multiplying 4\times2\times0.1 by one more tenth by trading their 8 rods for 8 units and writing the equation 4\times2\times0.1\times0.1=0.08. Discuss how multiplying by 0.1 two times gives a product in the hundredths. Have students rewrite 4\times2\times0.1\times0.1 as 0.4\times0.2, emphasizing that multiplying two decimals in the tenths gives a product in the hundredths.” 

The instructional materials provide opportunities for students to independently demonstrate procedural skill and fluency throughout the grade level. Examples include:

• Unit 1, Lesson 4, Center Activity, Equivalent Multiplication Expressions, students work with a partner to develop procedural skill and fluency with multiplying multi-digit numbers. 5.NBT.5 (Fluently multiply multi-digit whole numbers using the standard algorithm.)

• Unit 1, Lesson 4, Interactive Practice has one 15-minute interactive practice session to help students multiply two-and three-digit numbers by two-digit numbers. 5.NBT.5 (Fluently multiply multi-digit whole numbers using the standard algorithm.)

• Unit 1, Lesson 4, Multiply Multi-Digit Numbers, Sessions 2 and 3, students independently demonstrate procedural skill and fluency of 5.NBT.5 (Fluently multiply multi-digit whole numbers using the standard algorithm.). Session 2, Additional Practice, Fluency & Skills Practice, “In this activity students estimate the product of two multi-digit numbers and then find the exact values of some of the expressions.” Problem 2, “$$247\times15$$”. Problem 15, “$$285\times27$$”. Session 3, Develop, Apply It, Problem 9, “What is the product of 257\times34? Use the standard algorithm. Show your work.” Additional Practice, Teacher Edition, Fluency & Skills Practice, “In this activity students solve multiplication problems using the standard algorithm.” Fluency and Skills Practice, “The answers are mixed up at the bottom of the page. Cross out the answers as you complete the problems.” Problem 2, “$$3,104\times18$$”. Problem 5, “$$1,236\times55$$.” Problem 12, “$$306\times62$$”. All problems are written vertically.

• Unit 2, Lesson 17, Divide Decimals, Session 2, Develop, Apply It, Problem 8, students demonstrate procedural skills to divide decimal numbers 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). “What is 0.99\div11?  Show your work.” Problem 9, “What is 51.2\div4?” Answer choices: 1.28, 12.08, 12.8, and 120.8.”

• Unit 3, Lesson 15, Multiply Decimals, Session 2, Develop, Additional Practice, Practice Multiplying a Decimal by a Whole Number, students use partial products to multiply decimals by whole numbers 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).  Problem 2, “Complete the steps to find the product. Use decimal grids to help, if needed.  0.35\times3.”  The problem is written vertically and contains blanks to support the development of partial products.  Problem 3, “Look at problem 2. Why is no partial product shown for the zero in the ones place?”

The materials reviewed for i-Ready Classroom Mathematics, Grade 5 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics.

Within the materials, Program Implementation, Program Overview, i-Ready has three different types of lessons to address the unique approaches of the standards and to support a balance of conceptual understanding, application, and procedural fluency. “Understand Lessons focus on developing conceptual understanding and help students connect new concepts to familiar ones as they learn new skills and strategies. Strategy Lessons focus on helping students persevere in solving problems, discuss solution strategies, and compare multiple representations through the Try- Discuss-Connect routine." The Math in Action Lessons "feature open-ended problems with many points of entry and more than one possible solution." Lessons are designed to support students as they apply grade-level mathematics. Additional Practice and Interactive Games are also provided so that students can continue to practice the skills taught during lessons if needed.

There are multiple routine and non-routine application problems throughout the grade level, including opportunities for students to work with support of the teacher and independently. While single and multi-step application problems are included across various portions of lessons, independent application opportunities are most often found within Additional Practice, Refine, and Math in Action lessons.

Examples of routine applications of the mathematics include:

• Unit 1, Lesson 3, Finding Volume Using Formulas, Session 2, Develop, Apply, Problem 7, students use formulas to solve volume problems as they independently demonstrate application of 5.MD.5b (Apply the formulas V=l\times w\times h and V=b\times h for rectangular prisms to find volumes of right rectangular prisms with whole-number edge lengths in the context of solving real world and mathematical problems). “Winnie keeps her special items in a wooden box in the shape of a rectangular prism. The box has a length of 12 inches, a width of 4 inches, and a height of 3 inches. What is the volume of the box? Show your work.”

• Unit 2, Lesson 14, Add and Subtract in Word Problems, Session 1, Additional Practice,  Problem 3, students apply strategies for adding fractions to solve word problems to independently demonstrate application of 5.NF.2 (Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers). “Mao and his mom have a 1-gallon jug of water. They drink \frac{1}{8} gallon of water before lunch and \frac{2}{3} gallon of water during the rest of the day. How much water do they drink all day?” 

• Unit 5, Lesson 32, Represent Problems in the Coordinate Plane, Session 2, Connect It, Teacher Edition Page 665, with teacher support and guidance, students determine the location of points on the coordinate plane through routine problems. 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). “Remind students that one thing that is alike about all the representations is the locations of the buildings in Nolan’s town. Explain that on this page, students will examine how to find distances between some of these points. Monitor and Confirm Understanding, Check for understanding that: (2,2) is the ordered pair for the school, the park and the library are on the same horizontal or vertical movement between the points. Facilitate Whole Class Discussion, Tell students that they will use the distances they found in problems 2 and 3 to find the total distance required in problem 4. Ask How would you describe the path Nolan took from the school to the park? to the library? Listen For Starting at the school, Nolan walks 3 units to the right and 5 units up to get to the park. Starting at the park, Nolan walks 4 units to the left to get to the library. Ask How could you find the total distance that Nolan walks? Listen For The total distance Nolan walks is the sum of the horizontal and vertical distances he walks following the grid lines from the school to the park and then from the park to the library. Ask Could Nolan take a different path from the school to the park that is also 8 units long? Explain. Listen For, Yes, he could walk the horizontal and/or vertical distances in parts, such as 2 units up, 2 units right, 3 units up, and 1 unit right. 5. Look for the idea that if two points are on the same grid line, the distance between them is the number of units of horizontal or vertical movement between the two points. 6. Reflect Have all students focus on the strategies used to solve this problem. If time allows, have students share their preferences with a partner.” 

Examples of non-routine applications of the mathematics include:

• Unit 2, Lesson 14, Add and Subtract in Word Problems, Session 2, Develop, Apply It, Problem 8, students apply strategies for estimating fractions to solve word problems, 5.NF.2 (Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers). “Quinn likes to run at least 5 miles each day. She plans a new course from home to the park is 1\frac{1}{3} miles, from the park to the library is 2\frac{2}{5} miles and from the park to home is \frac{2}[3} mile. Will Quinn run at least 5 miles on this new course? Use only estimation to decide. Then explain if you are confident in your estimate or if you need to find an actual sum. Show your work.” 

• Unit 3, Math in Action, Use Fractions and Decimals, Session 2, Math in Action, Persevere on Your Own, students independently apply division strategies to solve fraction word problems. 5.NF.7c (Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions). “A local nursery hears about the shrub planting project that G.O. and his neighbors are planning. The nursery gives them 50 pounds of compost to use. G.O. reads about using compost on a website. When you plant a shrub, it can help to mix the soil with some compost. You can use a scoop of compost for each shrub. An average scoop of compost is between \frac{1}{4} pound and \frac{1}{2} pound. About how many shrubs can G.O. plant with the compost that the nursery gave him?”

• Unit 5, Math in Action, Work With Coordinates and Patterns, Session 1, Math in Action, Try Another Approach, with teacher support and guidance, students represent a real-world problem in the first quadrant of a coordinate plane. 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). “Nikia is working on a new video game, Shape Shake-Up. She uses a coordinate plane to represent the screen. It helps her decide where to place graphics. Read one of Nikia’s ideas. Game Idea A shape that looks like a ‘C’ traps players. The shape’s perimeter is 14 to 16 units. Its area is 6 to 8 square units. The shape is located more than 2 units above the x-axis and more than 2 units to the right of the y-axis. Draw a shape in the coordinate plane that words with Nikia’s game idea and explain why it works. Label each vertex with an ordered pair.” Teacher Edition, Facilitate Whole Class Discussion, “Read the questions aloud. Prompt students to recognize that they are being asked to draw a shape on the coordinate plane that works with Nikia’s game idea. Ask What are some locations on the plane where you cannot place the shape? Ask What are some locations on the grid where it is okay to place the top of the shape? How do you know?”

The materials reviewed for i-Ready Classroom Mathematics, 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.

Within the materials, Program Implementation, Program Overview, i-Ready has three different types of lessons to address the unique approaches of the standards and to support a balance of conceptual understanding, application, and procedural fluency. The materials include problems and questions, interactive practice, and math center activities that help students develop skills. 

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

• Unit 1, Lesson 5, Divide Multi-Digit Numbers, Session 4, Additional Practice, Practice Using Area Models and Partial Quotients to Divide, Problem 4, students practice procedural fluency of 5.NBT.B (Perform operations with multi-digit whole numbers and with decimals to the hundredths). “A box in the shape of a rectangular prism has a volume of 504 cubic inches. The width of the box is 7 inches, and the height of the box is 6 inches. Use the partial quotient method shown in the example to find the length of the box. Show your work.”

• Unit 2, Lesson 12, Add Fractions, Session 4, Refine, Problem 2, students demonstrate application of 5.NF.2 (Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominator. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers) as they add mixed numbers with unlike denominators. “Chen is in a bike club at his school. Last week, the students in the club rode their bikes 2\frac{2}{3} miles. This week, they rode 1\frac{5}{6} miles. How many miles did the students ride in both weeks combined? Show your work.”

• Unit 3, Lesson 19, Understand Multiplication by a Fraction, Session 1, Explore, Model It, Problem 1, 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). “Shade and label three \frac{1}{2} inch sections of the ruler. Then complete the sentence and the multiplication equation that represents the total length you shaded. 3 sections of \frac{1}{2} inch are ____ inches.  4\times\frac{1}{2}=.”

Multiple aspects of rigor are engaged simultaneously throughout the materials in order to develop students’ mathematical understanding of grade-level topics. Examples include:

• Unit 1, Math in Action, Session 2, Persevere On Your Own, Problem Compost Model,  students develop procedural skill and fluency, conceptual understanding, and application as they solve problems involving relating volume to multiplication and to addition. 5.MD.C (Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition.) and 5.NBT.5 (Fluently multiply multi-digit whole numbers using the standard algorithm.) “One way to recycle plant parts and food scraps is to make compost. Compost is a type of natural fertilizer. Sweet T is making a model for the fair that shows how to make compost. Each layer of the model is shaped like a rectangular prism. Read Sweet T’s notes. Compost Model Notes: Gather grass clippings, brown leaves, and food scraps; First layer is grass clippings; Second layer is brown leaves; Third layer is food scraps; Use more than 3 layers. You choose the height, or thickness, of each layer; Repeat layers as many times as you want to fill the container. The picture above shows the container  Sweet T uses to make the model. How many layers should Sweet T use? How thick should each layer be? Solve It Help Sweet T make a plan; Tell which item is in each layer; Give the length, width, and volume of each layer; Find the total volume of the completed model.”

• Unit 2, Lesson 14, Add and Subtract in Word Problems, Session 4, Refine, Problems 5 and 6, students engage all three aspects of rigor as they solve word problems that involve estimation and reasonableness of answers. 5.NBT.7 (Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value…). Problem 5, “Mrs. Washington is at the grocery store. For a family New Year’s Day dinner, she buys black-eyed peas that cost $4.79, pork that costs $33.54, and rice that costs $12.67. About how much money does Mrs. Washington spend at the grocery store? Will the actual cost be more or less than your estimate? Explain. Show your work.” Problem 6, “A certain liquid boils at 175.62\degree F. The liquid is currently at 68.8\degree F. Enrique says that the temperature needs to rise by about 125\degree F to boil. Part A: Without finding the actual difference, explain why Enrique’s estimate is or is not reasonable. Part B: Find the actual amount the temperature must rise for the liquid to boil. Show your work.”

• Unit 3, Lesson 22, Multiply Fractions in Word Problems, Session 2, Additional Practice, Practice Multiplying Fractions in Word Problems, Problem 3, students use their conceptual understanding of fractions and apply to solve real-world problems. 5.NF.6 (Solve real world problems involving multiplication of fractions and mixed numbers). “In a skyscraper, \frac{4}{5}of the floors are offices. The rest are apartments. A phone company rents \frac{3}{4}of the office floors. What fraction of the total number of floors does the phone company rent? Draw a picture to find the fractions. Then write the answer.”

The materials reviewed for i-Ready Classroom Mathematics, 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. 

Within Program Implementation, Standards for Mathematical Practice in Every Lesson, “The Standards for Mathematical Practice (SMPs) are embedded within the instructional design of i-Ready Classroom Mathematics.” Embedded SMPs Within Lessons, “In addition to SMPs 1, 2, 3, 4, 5, and 6, which are integrated into the instructional framework, the Teacher’s Guide includes additional opportunities for students to develop the habits of mind described by the Standards of Mathematical Practice. The table of contents indicates all of the embedded Standards for Mathematical Practice for each lesson (both integrated SMPs and the specific SMPs highlighted within the lesson). The Lesson Overview includes the Standards for Mathematical Practice addressed in each lesson. In the Student Worktext, the Learning Target also highlights the SMPs that are included in the lesson.” SMPS Integrated in Try-Discuss-Connect Instructional Framework, “i-Ready Classroom Mathematics infuses SMPs 1, 2, 3, 4, 5, and 6 into every lesson through the Try-Discuss-Connect instructional framework (found in the Explore and Develop sessions of Strategy lessons , with a modified framework used in understand lessons).” Try It, “Try It begins with a language routine such as Three Reads, which guides students to make sense of the problem (SMP 1): For the first read, students begin to make sense of the problem (SMP 1) as the teacher reads the problem aloud. For the third read, students read the problem in unison or in pairs.  Students reason quantitatively and abstractly (SMP 2) by identifying the important information and quantities, understanding what the quantities mean in context, and discussing relationships among quantities.” Discuss It, “All students reason abstractly and quantitatively (SMP 2) as they find similarities, differences, and connections among the strategies they have discussed and relate them to the problem they are solving.” Connect It, “As students think through the questions and problems, they connect the quantitative, concrete/representational approaches to a more abstract understanding (SMP 2).”

MP1 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students make sense of problems and persevere in solving them as they work with support of the teacher and independently throughout the modules. Examples include:

• Unit 1, Math In Action, Solve Multiplication and Division Problems, Session 1, Teacher Edition, Try Another Approach, Plan It, Facilitate Whole Class Discussion, students make sense of the problem and use a variety of strategies that make sense to solve the problem. “Read the questions aloud. Prompt students to recognize that they are being asked to determine the number of worms Beau needs and the total cost of the worms. Ask What operation can you use to solve this problem? Listen for Beau’s solution uses multiplication, addition and subtraction. You could use division instead of multiplication or use the other operations in different ways. Ask How will the solution be different if you use a lesser or greater amount of scraps? Listen For The more scraps you have, the more worms you need. The fewer scraps you have, the fewer worms you need. Ask Do you need to find the number of worms needed for each day of the week? Why or Why not Listen For No; the worms that eat scraps on Monday will be there to eat scraps the other days as well.” 

• Unit 2, Math In Action, Use Decimals and Fractions, Session 1, Discuss Models and Strategies, Farm Animals, students discuss models and strategies to make sense of the problem. “The farm where Alex works has agreed to bring some animals to the pet fair. Guests can pay to feed and play with the animals. Alex has to decide which animals to bring and how much food they will need. Farm Animal Notes: Include 2 or 3 different kinds of animals, Include at least 6 but no more than 9 animals, Have enough food to feed each animal a day’s worth of food. Alex reads the farmer’s notes to find out how much an average animal eats in a day. How much food should Alex bring to the pet fair?”

• Unit 3, Lesson 24, Divide Unit Fractions in Word Problems, Session 3, Develop, Try It, students use questioning to make sense of a fraction division problem. “Jeffrey has 2 pounds of birdseed. He uses \frac{1}{4} pound of birdseed to make one ornament. How many same-size ornaments can Jeffrey make?” Teacher Edition, Try It, Make Sense of the Problem, “Before students work on Try It, use Co-Craft Questions to help them make sense of the problem. Read the problem situation aloud and then have students write questions that might be answerable by doing math. Students can compare questions with a partner and make revisions if they choose. Have students share questions with the class. After a brief discussion, display the actual problem question.”

MP2 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students reason abstractly and quantitatively as they work with support of the teacher and independently throughout the modules. Examples include:

• Unit 3, Lesson 18, Fractions as Division, Session 2, Develop, Try It, students create a model in order to understand the relationships between problem scenarios and mathematical representations. “Jade, Miguel, and Hugo are decorating hallways at their school for HIspanic Heritage Month. They have 5 hallways to decorate, and they share the work equally. How many hallways does each student decorate?” Teacher Edition, Discuss It, Support Partner Discussion, “Encourage students to name the model they used as they discuss their solutions. Support as needed with questions such as: How would you describe your model? What was it about this problem that made you think of using that model?”

• Unit 4, Lesson 26, Solve Word Problems Involving Conversions, Session 2, Develop, Try It, Discuss It, students consider units involved in a problem and attend to the meaning of the quantities. “Ask your partner: How did you get started? Tell your Partner: I started by…” Teacher Edition, Discuss It, Support Partner Discussion, “Encourage students to share what did not work for them as well as what did as they talk to each other. Support as needed with questions such as: What was the first step you took to solve the problem? How did you use the relationship between minutes and hours to solve the problem?”

• Unit 5, Lesson 32, Represent Problems in the Coordinate Plane, Teacher Edition, Session 1, Explore, Discuss It, Facilitate Whole Class Discussion, students understand relationships between a problem scenario and a mathematical representation on the coordinate plane. “Call on students to share selected strategies. Ask students to use precise language, such as vertex, x-coordinate and y-coordinate, in their explanations. Guide students to Compare and Connect the representations. Call on several students to rephrase important ideas so that everyone hears them more than once and in more than one way. Ask: How do (student name)’s and (student name)’s models and explanations show the location of point G and why the location is correct? Listen For: The four sides of a square are all the same length, so point G must be located 5 units from point S and point B at the point with coordinates (8,5). SM, MB, BG, and GS are all 5 units long.”

The materials reviewed for i-Ready Classroom Mathematics, 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.

Within Program Implementation, Standards for Mathematical Practice in Every Lesson, “The Standards for Mathematical Practice (SMPs) are embedded within the instructional design of i-Ready Classroom Mathematics.” Embedded SMPs Within Lessons, “In addition to SMPs 1, 2, 3, 4, 5, and 6, which are integrated into the instructional framework, the Teacher’s Guide includes additional opportunities for students to develop the habits of mind described by the Standards of Mathematical Practice. The table of contents indicates all of the embedded Standards for Mathematical Practice for each lesson (both integrated SMPs and the specific SMPs highlighted within the lesson). The Lesson Overview includes the Standards for Mathematical Practice addressed in each lesson. In the Student Worktext, the Learning Target also highlights the SMPs that are included in the lesson.” SMPS Integrated in Try-Discuss-Connect Instructional Framework, “i-Ready Classroom Mathematics infuses SMPs 1, 2, 3, 4, 5, and 6 into every lesson through the Try-Discuss-Connect instructional framework (found in the Explore and Develop sessions of Strategy lessons , with a modified framework used in understand lessons).” Discuss It, “Discuss It begins as student pairs explain and justify their strategies and solutions to each other.  Partners listen to and respectfully critique each other’s reasoning (SMP 3). As students/pairs share their different approaches, the teacher facilitates the discussion by prompting students to listen carefully and asking them to repeat or rephrase the explanations to emphasize key ideas (SMP 3).” 

Students construct viable arguments and critique the reasoning of others in connection to grade- level content as they work with support of the teacher and independently throughout the units. Examples include:

• Unit 1, Lesson 3, Find Volume Using Formulas, Session 1, Explore, Try It, explain their own thinking and critique the reasoning of others when solving a volume problem. “Tiana uses 1-inch cubes to build a model for a gift bag she is making for Kwanzaa. Her model is a rectangular prism. What is the volume of Tiana’s model?” Discuss It, “Ask your partner: Do you agree with me? Why or why not? Tell your partner: I agree with you about…because…”

• Unit 2, Lesson 9, Compare and Round Decimals, Session 3, Additional Practice, Practice Rounding Decimals, Problem 2, students critique the reasoning of others and justify their answer when rounding decimals on a number line. “Indira and Tyrell are rounding $16.50 to the nearest dollar. On a number line, they see that $16.50 is exactly halfway between $16 and $17. Indira says to round to the greater amount, $17. Tyrell says that because $16.50 is in the middle, you can round to either $16 or $17. Who is correct? Explain.”

• Unit 4, Lesson 26, Solve Word Problems Involving Conversions, Session 3, Develop, Discuss It, students critique the reasoning when solving problems involving measurement conversions. “Ask your partner: Do you agree with me? Why or why not? Tell your partner: I agree with you about…because…” Teacher Edition, Support Partner Discussion, “Encourage students to use the Discuss It question and sentence starter on the Student Worktext page as part of their discussion. Support as needed with questions such as: What conversion did you decide to make? Why? How does your work show more than one step to solve the problem?”

• Unit 5, Lesson 30, Evaluate, Write, and Interpret Expressions, Session 3, Additional Practice, Practice Writing and Interpreting Expressions, students explain their thinking when evaluating expressions. Problem 2, “Suppose you wrote a numerical expression for the phrase 20 minus the product of 5 and 2. To evaluate the expression should you subtract or multiply first? Explain.”

• Unit 5, Lesson 31, Understand the Coordinate Plane, Session 1, Explore, Model It, students explain their thinking about how to label a coordinate plane, and comparing and contrasting a coordinate plane and a number line. “What does a point in the coordinate plane represent?” Discuss It, “Compare how you label the coordinate plane with how your partner labeled the coordinate plane. Are they the same? Are they different? I think a coordinate plane is like a number line because…I think a coordinate plane is different from a number line because…”

The materials reviewed for i-Ready Classroom Mathematics, 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.

Within Program Implementation, Standards for Mathematical Practice in Every Lesson, “The Standards for Mathematical Practice (SMPs) are embedded within the instructional design of i-Ready Classroom Mathematics.” Embedded SMPs Within Lessons, “In addition to SMPs 1, 2, 3, 4, 5, and 6, which are integrated into the instructional framework, the Teacher’s Guide includes additional opportunities for students to develop the habits of mind described by the Standards of Mathematical Practice. The table of contents indicates all of the embedded Standards for Mathematical Practice for each lesson (both integrated SMPs and the specific SMPs highlighted within the lesson). The Lesson Overview includes the Standards for Mathematical Practice addressed in each lesson. In the Student Worktext, the Learning Target also highlights the SMPs that are included in the lesson.” SMPs Integrated in Try-Discuss-Connect Instructional Framework, “i-Ready Classroom Mathematics infuses SMPs 1, 2, 3, 4, 5, and 6 into every lesson through the Try-Discuss-Connect instructional framework (found in the Explore and Develop sessions of Strategy lessons, with a modified framework used in understand lessons).” Try It, “Try it continues as students work individually to model important quantities and relationships (SMP 4) and begin to solve the problem (SMP 1). They may model the situation concretely, visually, or using other representations, and they make strategic decisions about which tools or manipulatives may be appropriate (SMP 5).” Connect It, “For each problem students determine which strategies they feel are appropriate, and they model and solve (SMP 4) using pictures, diagrams, or mathematical representations. Students can also choose from a variety of mathematical tools and manipulatives (SMP 5) to support their reasoning.”

MP4 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students model with mathematics as they work with support of the teacher and independently throughout the modules. Examples include:

• Unit 1, Lesson 2, Finding Volume Using Unit Cubes, Session 2, Develop, Teacher Edition, Discuss It, Select and Sequence Student Strategies, with teacher guidance, students model the situation with an appropriate representation and discuss different strategies for finding volume. “One possible order for whole class discussion: models of the full prism built with unit cubes; models or drawings showing one horizontal layer and one column of 3 cubes; models or drawings that use vertical ‘layer’; equations that show repeated addition; equations that show multiplication.”

• Unit 3, Lesson 23, Understand Division with Unit Fractions, Session 1, Explore Division with Unit Fractions, students draw appropriate area model representations to divide fractions. Model It, Problem 1, states, “Mrs. Gomez shares \frac{1}{4} pound of fish equally among 3 cats. How much fish does each cat get?” 1a states, “Draw on the area model to solve the problem.” An area model is pictured for students to shade. Teacher Wrap, Support Partner Discussion, “Encourage students to refer to their area model and equations as they discuss dividing \frac{1}{4} by 3. Remind students they can ask partners to say something in a different way when some words and phrases are not clear. Look for models and equations showing: 12 equal parts in the whole, a quotient or product of \frac{1}{12}.”

• Unit 4, Lesson 26, Solve Word Problems Involving Conversions, Session 3, Develop, Apply It, Problem 6, students model with mathematics as they create equations to solve word problems involving measurement conversions. “The Socotra dragon tree grows only on the island of Socotra in the Indian Ocean. One dragon tree is 6.7 meters tall. A second dragon tree is 730 centimeters tall. Which tree is taller? How much taller? Show your work.”

MP5 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students choose tools strategically as they work with support of the teacher and independently throughout the modules. Examples include:

• Unit 1, Lesson 2, Find Volume Using Unit Cubes, Session 2, Develop, Apply It, Problem 8, students choose appropriate tools and strategies as they utilize unit cubes, 1-cm grid paper, or isometric dot paper to solve problems involving volume. “Gavin uses 1-inch cubes of fruit to make the fruit cube shown. What is the volume of the fruit cube? Show your work.”

• Unit 2, Lesson 13, Subtract Fractions, Session 2, Develop, Apply It, Problem 8, students choose an appropriate strategy to subtract fractions with unlike denominators. “What is \frac{7}{8}-\frac{1}{6}? Show your work.” Teacher Edition, Apply it, “For all problems, encourage students to draw some kind of model to support their thinking. Allow some leeway in precision; drawing same-size parts in a whole can be difficult. \frac{17}{24} or any equivalent fraction; See possible work on the Student Worktext page. Students may also show \frac{21}{24} on a number line and show 4 jumps of \frac{1}{4} each to the left. They may also show a fraction model representing \frac{21}{24} with \frac{4}{24} crossed out.”

• Unit 3, Lesson 21, Understand Multiplication as Scaling, Session 2, Develop, Connect It, Problem 6, students choose appropriate tools and strategies to solve a fraction problem. “Choose any model you like to show how the product \frac{3}{2}\times\frac{4}{3} compares to \frac{4}{3}. Then complete the comparison.”

The materials reviewed for i-Ready Classroom Mathematics, Grade 5 meet expectations for supporting the intentional development of MP6: Attend to precision; and attend to the specialized language of mathematics, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

Within Program Implementation, Standards for Mathematical Practice in Every Lesson, “The Standards for Mathematical Practice (SMPs) are embedded within the instructional design of i-Ready Classroom Mathematics.” Embedded SMPs Within Lessons, “In addition to SMPs 1, 2, 3, 4, 5, and 6, which are integrated into the instructional framework, the Teacher’s Guide includes additional opportunities for students to develop the habits of mind described by the Standards of Mathematical Practice. The table of contents indicates all of the embedded Standards for Mathematical Practice for each lesson (both integrated SMPs and the specific SMPs highlighted within the lesson). The Lesson Overview includes the Standards for Mathematical Practice addressed in each lesson. In the Student Worktext, the Learning Target also highlights the SMPs that are included in the lesson.” SMPs Integrated in Try-Discuss-Connect Instructional Framework, “i-Ready Classroom Mathematics infuses SMPs 1, 2, 3, 4, 5, and 6 into every lesson through the Try-Discuss-Connect instructional framework (found in the Explore and Develop sessions of Strategy lessons , with a modified framework used in understand lessons).” Try It, “Multiple students share a word or phrase that describes the context of the problem as the teacher guides them to consider precision (SMP 6) of the mathematical language and communication.” Discuss It, “The teacher guides students to greater precision (SMP 6) in their mathematics, language, and vocabulary.”

Students attend to precision, in connection to grade-level content, as they work with teacher support and independently throughout the units. Examples include:

• Unit 1, Lesson 2, Find Volume Using Unit Cubes, Session 2, Develop, students work towards precision as they solve problems involving volume and labeling solutions correctly. Try It, “Some of the beads Kimaya uses when braiding hair are shaped like 1-centimeter cubes. These cube beads fill a box shaped like the rectangular prism shown below when stacked without gaps or overlap. What is the volume of the prism?” A picture is shown of a cube with labeled sides of 3 cm, 2 cm, and 4 cm. A 1 cm cube is shown next to it. Teacher Edition, Support Partner Discussion, “Encourage students to use the term cubic centimeters as they discuss their solutions.” Differentiation, Deepen Understanding, Unit of Volume, “Introduce students to exponent notation, such as cm^3 or in^3, used in cubic units. Ask The volume of each unit cube in the model is 1 cubic centimeter. You may see this volume written as 1cm^3. What do you think the small, raised 3 means? Listen For The unit of volume has three dimensions, each of which is measured in centimeters. Tell students that the small, raised 3 is called an exponent. They will learn more about exponents in a later lesson. Ask How do you think you would use an exponent to write a volume of 35 cubic centimeters? to write the volume of 20 cubic feet? Invite volunteers to write the abbreviations on the board. [$$35cm^3$$, $$20ft^3$$] Ask How do exponents show whether a measurement is for length, area, or volume? Listen For The exponent shows the number of dimensions being measured. Length is 1 dimension, area is 2 dimensions, and volume is 3 dimensions.”

• Unit 3, Lesson 18, Fractions as Division, Session 1, Try It, students attend to precision when they explore the idea that dividing to find equal shares, the size of each share is sometimes a fraction. “Mrs. Meier shares 4 fluid ounces of red paint equally among 5 art students. How many fluid ounces of red paint does each student get?” Teacher Edition, Support Partner Discussion, “After students work on Try It, have them respond to Discuss It with a partner. Listen for understanding that: 4 fluid ounces is the amount to be shared equally; 5 students get equal shares; The amount of each share is the unknown.”

• Unit 5, Lesson 32, Represent Problems in the Coordinate Plane, Session 1, Additional Practice, Practice Representing Problems in the Coordinate Plane, Problem 3, students attend to precision as they plot and label shapes in the coordinate plane. “Solve the problem. Show your work. Points P, G, and R are three vertices of a rectangle. Plot and label the fourth vertex, A, of the rectangle. What are the x- and y-coordinates of A? How do you know?” A picture is shown of a coordinate grid with x- and y-axes labeled 1 through 10.

Students attend to the specialized language of mathematics, in connection to grade-level content, as they work with teacher support and independently throughout the units. Examples include:

• Unit 2, Lesson 9, Compare and Round Decimals, Session 1, Explore, Problem 1, students work towards using specialized language and symbols of mathematics. “Which weighs more, the shrimp or the tofu? Use the greater than symbol (>) or the less than symbol (<) to write an inequality statement that compares the weights of the shrimp and tofu for the pad Thai.”

• Unit 4, Lesson 28, Understand Categories of Two-Dimensional Figures, Session 2, Additional Practice, Practice with Categories of Two-Dimensional Figures, Problem 4, students attend to the specialized language of mathematics when categorizing two-dimensional shapes. “Name one attribute that isosceles and equilateral triangles always share. Name one attribute that they only sometimes share.”

• Unit 5, Lesson 30, Evaluate, Write, and Interpret Expressions, Session 2, Try It, students use specific mathematical language to develop strategies for using the order of operations to evaluate expressions that contain grouping symbols. “A school group visits the Tuskegee Airmen National Historic Site. Of the 32 people in the group, 8 are teachers and the rest are students. Each student gets a $6 souvenir poster. The expression $$6\times(32-8)$$ represents the cost, in dollars, to buy the posters. What is the total cost of the posters?” Discuss It, Support Partner Discussion, “Encourage students to use the terms expression, and order of operations as they discuss their solutions.”

The materials reviewed for i-Ready Classroom Mathematics, 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.

Within Program Implementation, Standards for Mathematical Practice in Every Lesson, “The Standards for Mathematical Practice (SMPs) are embedded within the instructional design of i-Ready Classroom Mathematics.” Embedded SMPs Within Lessons, “In addition to SMPs 1, 2, 3, 4, 5, and 6, which are integrated into the instructional framework, the Teacher’s Guide includes additional opportunities for students to develop the habits of mind described by the Standards of Mathematical Practice. The table of contents indicates all of the embedded Standards for Mathematical Practice for each lesson (both integrated SMPs and the specific SMPs highlighted within the lesson). The Lesson Overview includes the Standards for Mathematical Practice addressed in each lesson. In the Student Worktext, the Learning Target also highlights the SMPs that are included in the lesson.” Structure and Reasoning, “As students make connections between multiple strategies, they make use of structure (SMP7) as they find patterns and use relationships to solve particular problems. Students may also use repeated reasoning (SMP 8) as they construct and explore strategies. SMPs 7 and 8 may be particularly emphasized in selected problems throughout the lesson. As students look for patterns and generalize about strategies, they always consider the reasonableness of their work.”  

MP7 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students look for and make use of structure as they work with support of the teacher and independently throughout the modules. Examples include:

• Unit 1, Lesson 3, Find Volume Using Formulas, Session 1, Explore, Teacher Edition, Connect It, 2 Look Ahead, students create, describe, explain a general formula, process, method, algorithm, or model. “Write the formulas V=b\times h and V=l\times w\times h on the board. Ask a volunteer to explain what a formula is and to give another example of a formula. Students may mention area and perimeter formulas. Ask another volunteer to explain the the new term base and the meaning of the letter b in the volume formula V=b\times h. Students should recognize how the volume formula captures the pattern of the volume calculations they have been doing during the previous two lessons as they worked with unit cube models. They will explore these connections further in the next session.” 

• Unit 3, Lesson 15, Multiply a Decimal by a Whole Number, Session 2, Differentiation, Deepen Understanding, students look for and make use of structure with teacher guidance when they develop strategies for multiplying a decimal by a whole number. “When discussing the partial products model, prompt students to consider how the model makes use of the decimal expanded form and properties of operations. Write the expanded form for 2.75 on the board. Then use the distributive property to multiply each part by 3. You may ask volunteers to write the steps. 3\times2.75=3\times(2+0.7+0.5);  ___ =(3\times2)+(3\times0.7)+(3\times0.05);  ___ =6+2.1+0.15. Ask How do the factor pairs in the second row relate to the partial products model on the Student Worktext page? Listen For Each pair of factors in parenthesis is equivalent to one of the pairs of factors next to the partial products. 3\times2 = 3 ones\times2 ones, 3\times0.7 = 3 ones \times7 tenths, and 3\times0.05 = 3 ones\times5 hundredths. Have students confirm that the sum 6+2.1+0.15 is 825 hundredths as shown on the student worktext page. [$$8.25=825$$ hundredths.]"

• Unit 4, Lesson 29, Classify Two-Dimensional Figures, Session 2, Teacher Edition, Facilitate Whole Class Discussion students look at and decompose “complicated ”into “simpler” things. “Call on students to share selected strategies. Ask students to use precise language, such as category, subcategory, and property, in their explanations. Guide students to Compare and Connect the representations. Reword any unclear statements, or ask a student to do so, so that others understand. Confirm with the speaker that the rewording is accurate. Ask How does your model show the relationships among the shapes? Is there more than one correct way to place the shapes in the Venn diagram? Listen for Students should recognize Venn diagrams that show correct category/subcategory relationships of the shapes. Representations of properties may be in lists or tables, and Venn diagrams may include letters or shapes as well as the category labels for each region. Rhombuses and rectangles may be on either side of the region of overlap at the center of the diagram.” Teacher Edition, Model It, “Ask How are the shapes shown in the problem represented in each Model It? Listen For In the first Model It, the shape names are shown in a table with properties of each shape identified as Xs. The second Model It shows the shapes sorted into a Venn diagram. For the table model, prompt students to consider a table as an aid to identify properties. How does the table show the properties a shape has? the properties a shape does not have? How might using this type of table help you make a Venn diagram? For the Venn diagram model, prompt students to relate the Venn diagram to the table. How does the placement of a shape in the Venn diagram relate to the number of Xs shown for that shape in the table? Rectangles and Rhombuses each have four Xs: three for the same properties and one for different properties. What does this mean about the ovals for Rectangles and Rhombuses in the Venn diagram?”

MP8 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students look for and express regularity in repeated reasoning as they work with support of the teacher and independently throughout the modules. Examples include:

• Unit 1, Lesson 4, Multiply Multi-Digit Numbers, Session 2, Develop, Model It, students create, describe, explain a general formula, process, method, algorithm, model, etc. “Model It Use an area model to show partial products. To find the product 128\times35, sketch a rectangle with dimensions 128 by 35. 128 is 100+20+8 35 is 30+5 First row: 3,000+600+240=3,840 second row: 500+100+40=640. Model It Use the distributive property to find partial products and add them 128\times35=128\times(30+5); 128\times(30+5)=(128\times30)+(128\times5).”

• Unit 2, Lesson 13, Subtract Fractions, Session 1, Explore, Try It, students look for and express regularity in repeated reasoning when subtracting fractions. “In the previous lesson, you learned about adding fractions. Now you will learn about subtracting fractions. Use what you know to try to solve the problem below. Ines makes a wooden tongue drum with her brother. The top of the drum is \frac{3}{4}-inch thick with an “H” cut out to make two ‘tongues’ for different sounds. One tongue is \frac{1}{8} inch thicker than the top and the other tongue is \frac{1}{8} inch thinner. What are the thicknesses of the two tongues?”

• Unit 3, Lesson 24, Divide Unit Fractions in Word Problems, Session 3, Develop, Teacher Edition, Differentiation, Deepen Understanding, students look for and express regularity in repeated reasoning when discussing equivalent fractions. “When discussing the equation method shown in the second Model It, prompt students to consider where else they have used renaming as a strategy. Ask In your own words, how do you describe the strategy shown for dividing a whole number by a unit fraction? Listen For You rename the whole number, 2, as the fraction \frac{8}{4} so that the dividend and the divisor describe the same-size parts of a whole. You can then think of dividing 8 fourths into equal groups of 1 fourth.”