4th Grade - Gateway 2

The materials reviewed for i-Ready Classroom Mathematics, Grade 4 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 1, Lessons 1, 2, and 3, students develop conceptual understanding of 4.NBT.2 (Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. Compare two multi-digit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons). Lesson 1, Understand Place Value, Session 1, Explore, Model It, Problem 4, “Use expanded form and word form to understand greater numbers.” Problem 4a, “To say or write the word form of a number, you read each group of three digits followed by the period name. You do not say the period name for the ones period. The word form for the number you wrote in problem 3 is four hundred sixty-seven____, ____.” Problem 4b, “Expanded form is a way to write a number to show the value of each digit. Complete the expanded form below for the number in the place-value chart above. ___00,000+___0,000+___,000+___00+___0+___.” Session 2, Develop, Model It: Expanded Form, Problem 4, “Complete to show different ways you can expand and show 25,049. 25,049=___ ten thousands + ___ thousands + ___ hundreds + ___ tens + ___ ones; 25,049 = ___ thousands + ___ ones; 25,049 = ___ ones.” Teacher Edition, Discuss It, Support Partner Discussion, “Encourage partners to connect the expanded forms by recognizing that each representation shows 25,049 but with a different number of ten thousands, thousands, hundreds, tens, and/or ones. Support as needed with questions such as: How are the ways to represent 25,049 the same? How are the ways to represent 25,049 different?” Session 3, Refine, Apply It, Problem 2, “Suppose you only have hundreds, tens, and ones blocks. What are two different ways you could represent the number 1,718?” Lesson 2, Compare Whole Numbers, Session 1, Connect It, Problem 2, “You can use place value to compare numbers. Start with the greatest place-value positions. Sometimes numbers you compare have the same number of digits. Sometimes they have different numbers of digits.” Problem 2a, “Circle the box with the greater number of staples. 1,250; 1,500.” Problem 2b, “What place value helps you tell which box has more staples? Explain.” Problem 2c, “Circle the greater price. 1,402  958.” Problem 2d, “What place value helps you tell which price is greater? Explain.” Session 2, Develop, Connect It, Problem 1, “Write the numbers 23,643 and 23,987 so that they line up by place. Explain how to line them up.” Teacher Edition, Monitor and Confirm Understanding, “Check for understanding that: the numbers are written one above the other and the digits in each place line up with each other.” Session 3, Refine, Apply It, Problem 8, “A city saves 14,128 gallons of water by repairing leaks. The city also saves 14,210 gallons of water by reducing watering time. Which activity saves more water? Use >, <, or = to write a comparison. Show your work.” Lesson 3, Round Whole Numbers, Session 1, Explore, Problem 2, “You round to estimate and to make numbers easier to work with when you do not need an exact answer.” Problem 2a, “Mark and label 36,219 on the number line below.” A picture is shown of a number line from 36,000 to 37,000 with tics marked in between the numbers. Problem 2b, “Between which two thousands is the number 36,219? Write both numbers of thousands as numerals. ___ and ___.” Teacher Edition, Look Ahead, “Point out that sometimes a number is less than halfway between two thousands, sometimes it is exactly between two thousands, and sometimes it is more than halfway between two thousands.” Session 3, Refine, Apply It, Problem 3, “A website streams 264,398 movies to its customers one year. To the nearest ten, how many movies does the website stream?” Answer choices: 264,300, 264,390, 264,400, 265,000. “Efia chose c as the correct answer. How did she get that answer?” 

• Unit 3, Lessons 11 and 12, students develop conceptual understanding of 4.NBT.5 (Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.). Lesson 11, Multiply by One-Digit Numbers, Session 3, Develop, pg. 242, “People who take part in the Holi festival throw colored powders in the air and on each other. At one event, each person who attends gets 3 packs of colored powders. A total of 1,125 people attend this event. How many packs of colored powders are given out?” Picture It, “You can use an area model to help understand the problem.” A picture is shown of an area model to represent 1,125\times3. Model It, “You can also multiply the numbers using partial products.” A picture is shown of the steps to take when using partial products to multiply 1,125\times3. Teacher Edition, Picture It and Model It, “If no student presented these models have students analyze key features, and them point out the ways each model represents: the number of thousands, hundreds, tens, and ones in 1,125; multiplying the value of each place in 1,125 by 3; the partial products. Ask How are the models the same? Different? Ask Why does Picture It show an area model instead of base-ten blocks to represent this problem?” Lesson 12, Multiply by Two-Digit Numbers, Session 2, Develop, “Folding chairs are set up in a school auditorium for a play. There are 16 rows of chairs. Each row has 28 chairs. How many chairs are set up for the play?” Students are shown pictures of an area model and partial products to solve the problem. Connect It, Problem 4, “Would the problem change if 20+8 on the top of the area model were changed to 10+10+8? Explain.” Problem 5, “How could you estimate to check the reasonableness of your answer to 28\times16 by multiplying with easier numbers?” Teacher Edition, Facilitate Whole Class Discussion, “Be sure students understand that problem 4 is asking them about breaking apart the factor 28 in a different way and that problem 5 is asking how to check that the answer to a two-digit multiplication problem is reasonable. Ask Why might you be likely to break apart 28 into 10+10+8 instead of into 20+8? How does this affect the partial products you get and the product?” 

• Unit 4, Lesson 25, Fractions as Tenths and Hundredths, Sessions 1 and 2, students develop conceptual understanding of 4.NF.5 (Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100.). Session 1, Explore, Try It, “Calvin and his parents are visiting New Orleans. They walk seven tenths of a mile to Congo Square. Write an equivalent fraction to show how far Calvin and his parents walk in hundredths of a mile.” Teacher Guidance, Facilitate Whole Class Discussion, “Call on students to share selected strategies. Prompt students to use academic language and math vocabulary so that their ideas are clear. Guide students to Compare and Connect the representations. ASK How do [student name]’s and [student name]’s models how the distance Calvin and his parents Walk in tenths? in hundredths? Listen For models show 7 parts out of 10 or the fraction \frac{7}{10}; they show 70 parts out of 100 or the fraction \frac{70}{100}.” Session 2, Develop, Connect It, Teacher Guidance, “Monitor and Confirm Understanding 1 - 4 Check for understanding that: the denominators of 10 and 100 are different, multiplying both the numerator and denominator of \frac{4}{10} by the same number, 10, results in an equivalent fraction with a denominator of 100, adding the numerators of \frac{40}{100} and \frac{50}{100} results in the sum, the sum is \frac{90}{100} of a dollar.” 

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

• Unit 1, Lessons 2-3, students use place value understanding to compare numbers and independently engage with 4.NBT.2 (Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. Compare two multi-digit numbers 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 whole numbers. Lesson 2, Compare Whole Numbers, Session 3, Refine, Apply It, Problem 9, Math Journal, “Choose 2 six-digit numbers. Use symbols and words to write comparison statements. Explain how you know the comparisons are correct.” Lesson 3, Round Whole Numbers, Session 2, Develop, Apply It, Problem 8, “Mr. Gomez’s company collects 32,376 water bottles to recycle. Mr. Gomez rounds this amount to the nearest ten thousand. What number does Mr. Gomez round the number of water bottles to? Show your work.” Problem 9, “Gavin and his dad volunteers at a book drive for International Book Giving Day on February 14. The book drive collected 468,500 books. To the nearest thousand, how many books does the book drive collect? Show your work.” 

• Unit 3, Lesson 14, Divide Three-Digit Numbers, Session 4, Refine, Apply it, Problem 9, Math Journal, independently engage with 4.NBT.6 (Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models) as they use their understanding of multiplication to estimate division quotients. “Look at the expression 228\div6. What two multiples of 10 is the quotient between? Explain how you know.” 

• Unit 4, Lessons 17 and 18, students independently engage with 4.NF.2 (Compare two fractions with different numerators and different denominators. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions.) as they use concrete and semi-concrete representations to create equivalent fractions with like denominators to compare fractions. Lesson 17, Understand Equivalent Fractions, Session 1, Additional Practice, Prepare for Equivalent Fractions, Problem 3, “Shade each model to represent the fraction shown.” Pictures of three bar models are shown, one partitioned into halves, one partitioned into fourths, and one partitioned into eighths. The corresponding labels on the models are \frac{1}{2}, \frac{2}{4}, and \frac{4}{8}. Problem 3a, “Is the area you shaded in each model the same?” Problem 3b, “How do you know that \frac{1}{2}, \frac{2}{4}, and \frac{4}{8} are equivalent fractions?” Problem 3c, “Compare the models. How many times as many equal parts and shaded parts does each model have than the model above it?” Lesson 18, Comparing Fractions, Session 3, Additional Practice, Practice Using a Benchmark to Compare Fractions, Problem 2, “Compare \frac{5}{6} and \frac{1}{3} using the benchmark fraction \frac{1}{2}.” Problem 2a, “Label \frac{5}{6} and \frac{1}{3} on the number line below.” A number line is shown from 0 through 1, with tics marked at each sixth. The one-half tic is labeled as well. Problem 2b, “Which fraction is greater than \frac{1}{2}?” Problem 2c, “Which fraction is less than \frac{1}{2}?” Problem 2d, “Write <, >, or = to show the comparison. Explain how you found your answer. \frac{5}{6}\bigcirc\frac{1}{3}.”

The materials reviewed for i-Ready Classroom Mathematics, Grade 4 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, Interactive Tutorials, contains one 17-minute tutorial to help students develop procedural skills and fluency with adding multi-digit numbers. (4.NBT.4)

• Unit 1, Lessons 4 and 5, students build procedural skills and fluency of 4.NBT.4 (Fluently add and subtract multi-digit whole numbers using the standard algorithm.) with teacher support and guidance. Lesson 4, Add Whole Numbers, Session 1, Explore, Connect It, Problem 2 Look Ahead, “There are many ways to add numbers. For example, you can use drawings or base-ten blocks. You can also break apart numbers to add, add numbers by place value, or use an algorithm. An algorithm is a set of steps used to solve a problem. When you line up numbers by place value, you use an algorithm. Suppose you want to add greater numbers such as 35,705 and 23,241.” Problem 2a, “Without adding, circle the strategy that might be best for adding. Draw a Picture; Line Up by Place Value.” Problem 2, “Explain your choice.” Session 2, Develop, “Lupe’s mom and dad own a tamale restaurant. Around Christmas, the restaurant sells 5,657 pork tamales and 3,804 beef tamales. How many tamales does the restaurant sell around Christmas? Use any strategy to add.” Model It, “You can use place value to add.” Model It, “You can record the sums by showing regrouping above the problem. You regroup when the sum of the digits in a place is 10 or greater.” Session 3, Develop, p.60, “Find the sum of 57,541 and 23,098. Use the standard algorithm for addition. Then estimate to check whether your answer is reasonable, or makes sense.” Model It, “You can use the addition algorithm to add. Line up the numbers. Add from right to left.” The problem is shown vertically. “Add the ones. Add the tens. Regroup if you need to. Write the regrouped 1 hundred above. Then add hundreds, thousands, and ten thousands.” Teacher Edition, Model It, “For the addition algorithm, prompt students to identify when regrouping is needed. How do you know whether or not to regroup ones? Which place has been regrouped so far? Why? How does the model show the regrouping?” Session 4, Refine, Apply It, Problem 2, “Find the sum of the three numbers below. 13,728; 15,419; 12,399. Show your work.” Teacher Edition, “Students could solve the problem by using the standard addition algorithm. They may add the numbers in any order and the sum will be the same. Students should regroup 26 ones as 2 tens and 6 ones, 14 tens as 1 hundred and 4 tens, 15 hundreds as 1 thousand and 5 hundreds, and 11 thousands as 1 ten thousand and 1 thousand.” Lesson 5, Subtract Whole Numbers, Session 3, Develop, “During the first weekend of the cherry blossom festival at Ueno Park in Tokyo, Japan, there are 41,923 visitors. The next weekend there are 68,408 visitors. How many more visitors are there during the second weekend than during the first? Use the standard algorithm to subtract. Then use addition to check your answer.” Model It, “You can use the standard algorithm for subtraction to subtract. Line up the numbers. Subtract from right to left. Subtract the ones. Regroup if you need to. There are not enough tens to subtract. Write the regrouped 4 hundreds as 3 hundreds and 10 tens above the problem. Now subtract the tens. You will finish solving the problem on the next page.” Teacher Edition, Model It, “For the subtraction algorithm, prompt students to identify where regrouping is needed. Why are no tens regrouped as ones? Look at the zero in the tens place of the number you are subtracting from. How does the zero help you tell whether or not regrouping is needed? What do the blank boxes in the difference represent?”

• Unit 3, Lesson 14, Divide Three-Digit Numbers, Session 2, Develop, Teacher Edition, Model It, students develop procedural skills using area models and arrays to divide. 4.NBT.6 (Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division, illustrate and explain the calculation by using equations, rectangular arrays, and/or area models). “If no student presented these models, have students analyze key features and then point out the ways each model represents: the dividend of 136, the divisor of 4, the quotient. Ask How are the models alike and different? Listen For Both models show the problem broken into four parts with the number 10 above the first 3 parts and 4 above the last part. The array shows 136 as the total number of squares in the rows and columns while the area model shows 136 as the total area of a rectangle. For an array, prompt students to identify how the array represents the division problem. Why is 10 the first number multiplied by 4? How do the subtraction problems relate to the array? Why does the model have four parts? For an area model, prompt students to identify how the model represents the division problem. Look at the labels on the first rectangle in the area model. What is known, and what is unknown? How does the area model show the division problem broken into smaller parts? What do the numbers above the area model represent?”

• Unit 4, Lesson 20, Add and Subtract Fractions, Session 3, Develop, students develop procedural skills and fluency as they add and subtract fractions. 4.NF.4c (Solve word problems involving multiplication of a fraction by a whole number.) Try It, “Soo has a box with \frac{5}{6} of a liter of broth to make pho. He uses \frac{4}{6} of a liter. What fraction of a liter of broth is left in the box.” Picture It, “You can use a picture to help understand the problem. The picture shows the whole liter divided into 6 equal parts. Five shaded parts show how much broth is in the box. Soo uses 4 sixths of a liter, so take away 4 shaded parts. The 1 shaded part that is left shows the fraction of a liter that is left.” Model It, “ You can also use a number line to help understand the problem. The number line at the right is divided into sixths, with a point at \frac{5}{6}. Start at \frac{5}{6} and count back 4 sixths to subtract \frac{4}{6}. Teacher Guide, Picture It & Model It, “If no student presented these models, have students analyze key features and then point out the ways each model represents: the whole; the number of equal parts; the number of parts Soo uses. Ask What number tells the whole in the picture? in the number line? Is it the same or different? For a sketch of the box of broth, prompt students to identify how the box is labeled to represent the problem. Is there any way that this picture is more or less helpful than the one drawn by [student name]? How is it helpful that the box shows 1 liter divided into sixths? Why are some parts of the box shaded and some clear? For a number line model, prompt students to identify the greatest number on the number line and the number of divisions. How is the number line divided? Why is the point \frac{5}{6} marked?” 

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

• Unit 1, Lessons 4 and 5, Learning Games, Hungry Fish, Pizza, and Match help students develop procedural skills and fluency with adding and subtracting multi-digit numbers. (4.NBT.4)

• Unit 1, Lessons 4 and 5, students independently demonstrate procedural skill and fluency of 4.NBT.4 (Fluently add and subtract multi-digit whole numbers using the standard algorithm.). Lesson 4, Add Whole Numbers, Session 2, Develop, Fluency & Skills Practice, “In this activity students use different strategies, such as place-value and the addition algorithm, to add two four-digit numbers with or without regrouping.” Problem 3, “$$4,121+6,215$$.” Problem 5, “$$2,999+6,871$$.” Session 3, Develop, Additional Practice, Practice Using the Standard Algorithm to Add Greater Numbers, Problem 2, “Each year, the Aloha Festivals in Hawaii include a flower parade. One float in the parade has 12,818 flowers. The performers on the float wear 1,342 flowers. What is the total number of flowers on the float and its performers?” Answer choices: 13,150; 14,160; 23,150; 24,160. Session 4, Refine, Apply It, Problem 1, “On Saturday, the subway has 246,440 riders. On Sunday, the subway has 175,756 riders. What is the total number of subway riders on the two days? Show your work.” Lesson 5, Subtract Whole Numbers, Session 2, Additional Practice, Teacher’s Edition, Fluency & Skills Practice, “In this activity students practice estimating and finding differences of two five-digit numbers using the standard algorithm.” Fluency & Skills Practice, “Estimate. Circle all the problems with differences between 30,000 and 60,000. Then find the differences of only the circled problems.” Problems are written vertically. Problem 2, “$$62,554-31,618$$.”  Problem 9, “$$90,434-51,533$$.”

• Unit 3, Lesson 11, Multiply by One-Digit Numbers, Session 3, Additional Practice, Practice Multiplying a Four-Digit Number by a One-Digit Number, Problem 3, students independently demonstrate procedural skills of multiplying using expanded form of 4.NBT.5 (Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.) “Use the expanded form of 3,569 to find 4\times3,569.”

• Unit 3, Lesson 16, Finding Perimeter and Area, Session 3, Additional Practice, Practice Finding Area,  Problem 1, students independently demonstrate procedural skills of multiplying to find the area of a room 4.MD.3 (Apply the area and perimeter formulas for rectangles in real world and mathematical problems). “Vinh puts flooring in a rectangular room. The picture shows the length and width of the room. How many square feet of flooring does Vinh use? A= ___ \times ___; A= ___. Vinh uses ___ square feet of flooring.” A picture is shown of a rectangle with labeled side lengths of 30 ft and 25 ft.

The materials reviewed for i-Ready Classroom Mathematics, Grade 4 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 2, Lesson 6, Session 2, Additional Practice, Practice Multiplication as Comparison,  Problem 7, students independently apply multiplication or division strategies to solve multiplicative comparison problems. 4.OA.2 (Multiply or divide to solve word problems involving multiplicative comparison…) “A pet caretaker walks dogs 9 times a day. They walk dogs 5 days a week from Monday to Friday. Draw and label a bar model to show the total number of times the caretaker walks dogs each week.”

• Unit 3, Lesson 13, Use Multiplication to Convert Measurements, Session 2, Develop, Apply It, Problem 7, students use multiplication and division to solve word problems involving conversions to independently demonstrate application of 4.OA.3 (Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.) “Chen buys 14 ounces of kiwis and 2 pounds of peaches. How many more ounces do the peaches weigh than the kiwis? Show your work. (1 pound = 16 ounces).”

• Unit 4, Lesson 21, Add and Subtract Mixed Numbers, Session 2, Develop, students have discussions to make sense of the mixed numbers as they apply 4.NF.3c (Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction) with teacher support and guidance. Try It, “Markers come in boxes of 8. For an art project, one group of students uses 1\frac{5}{8} boxes of markers, and another group uses 1\frac{6}{8} boxes. How many boxes of markers do the two groups use altogether?” Teacher Edition, Make Sense of the Problem, “Before students work on Try It, use Three Reads to help them make sense of the problem. For the third read, have students read chorally. As students discuss the important quantities, listen for understanding that each group of students in the problem uses 1 full box of makers and part of another box.”

Examples of non-routine applications of the mathematics include:

• Unit 2, Lesson 7, Session 2, Develop, Apply It, Problems 8-9, allows students to apply multiplication and division strategies to solve word problems, 4.OA.2 (Multiply or divide to solve word problems involving multiplicative comparison…). Problem 8, “A food truck sells shawarma on plates and in wraps. During lunch, the food truck sells 10 shawarma plates. It sells 3 times as many shawarma wraps as shawarma plates. Write an equation with an unknown to find the number of shawarma wraps the food truck sells. Then solve the equation. Show your work.” Problem 9, “Elu eats 6 times as many raisins as Finn. Finn eats 9 raisins. Write an equation with an unknown to find the number to find the number of raisins Elu eats. Then solve the equation. Show your work.”

• Unit 3, Lesson 13, Use Multiplication to Convert Measurements, Additional Practice, Practice Converting Unit of Liquid Volume, Problem 6, students independently by comparing units of measurement to see which holds more, to demonstrate application of 4.MD.1 (Know relative size of measurement units within one system of units including km, m, cm; kg, g,; lb, oz; l, ml; hr, min, sec. Within a single system of measurement, express measurements in a larger unit in terms of smaller unit…). “A small bottle contains 2 cups of soy sauce. Do 5 small bottles of soy sauce have a greater amount of soy sauce than a 1-quart bottle of soy sauce? Explain. (1 quart = 4 cups).”

• Unit 4, Lesson 28, Problems About Time and Money, Session 4, Refine, Apply It, Problem 9, students independently demonstrate application of addition and subtraction strategies to solve word problems about money/decimals. 4.MD.2 (Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale). “Jamie has three $5.00 bills. He buys a wristband for $1.75 and a basketball for $12.50. How much money does Jamie have left? Explain how to find the answer.”

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

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 3, Lesson 12, Multiply by Two-Digit Numbers, Session 2, Additional Practice, Problem 6,  students demonstrate conceptual understanding of 4.NBT.5 (Multiply a whole number of up to four digits by a one-digit whole number and multiply two two-digit numbers) as they show more than one way to solve a multi-step multiplication problem. “Ms. Lonetree teaches 6 computer classes a day at school. Each class is 52 minutes long. She teaches 5 days a week. How much time does she spend teaching each week? Show two different ways to solve this problem. Show your work.”

• Unit 4, Lesson 29, Problems About Length, Liquid Volume, Mass, and Weight, Session 1, Additional Practice, Prepare for Problems About Length, Liquid Volume, Mass, and Weight, Problem 3, students develop application of 4.MD.2 (Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale) as they solve word problems involving converting measurement units. “Nadia and her dad have 3 yards of rope. They use 20 yards of the rope to make a clothesline. They use 1 yard 2 feet of the rope to make a chew toy for their dog. How many feet of rope do Nadia and her dad have left?”

• Unit 5, Lesson 31, Angles, Sessions 2 and 3, students develop procedural skill as they measure and draw angles. 4.MD.6, (Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure.) Session 2, Develop, p. 678, Model It, “You can use a protractor to measure the angle. First, line up either 0\degree mark on the protractor exactly with one ray of the angle.” A picture of a protractor with an angle and the words vertex and ray labeled is provided. “Next, line up the center point of the protractor exactly with the vertex of the angle. Recall that the vertex is the point where two rays meet to form an angle. Then look at the other ray to read the number of degrees.” Session 2, Develop, Apply It, Problem 7, “What is the measure, in degrees, of the angle shown?” A 235\degree angle is pictured on a protractor. Session 3, Develop, Apply It, Problem 8, “Angle D measures 80\degree. One ray of angle D is shown. Draw another ray to make angle D.” Problem 9, “Draw a 75\degree angle.”

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 Yearly Blog Visits, students develop procedural skill and fluency, conceptual understanding, and application as they solve problems that involve understanding multi-digit whole numbers when multiplying up to four-digits by a one-digit number 4.NBT.5 (Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.) “Max’s blog site now shows the monthly visitors through June. He asks you to write a report about the number of visitors he had during this time. He also wants you to estimate numbers for the whole year. Max’s Video Game Blog Visitors January visitors 30,000+2,000+50+1; February visitors 28,486; March visitors thirty thousand, eighteen; April visitors 50,000+9,000+600+30+2; May visitors 62,187; June visitors sixty-three thousand, nine hundred two. How many visitors should Max expect to get on his blog in one year? Solve It Write a report for Max about visitors to his blog site. Use rounding and estimation to help you write a report. Include: the approximate number of visitors each month and a 6-month total; a prediction of the total number of visitors there will be for the whole year; an explanation of how you made the one-year prediction. Reflect Use Mathematical Practices After you complete the task, choose one of these questions to discuss with a partner. Look for Structure What number patterns helped you make a prediction? Make an Argument Why is your prediction a reasonable estimation?” 

• Unit 2, Lesson 9, Explore Number and Shape Patterns, Session 4, Refine, Apply It, students refine procedural skill and fluency and conceptual understanding as they solve problems involving number and shape patterns. 4.OA.5 (Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself.)  Problem 7, “Draw a shape pattern that follows the rule that shapes go back and forth between four sides and five sides. Show your work.” Problem 8, “Write a number pattern that follows the rule subtract 6 and also has all odd numbers. Show your work.” Problem 10, Math Journal, “Heidi says that a number pattern with the rule add 2 always has even numbers. Is Heidi correct. Explain.”

• Unit 4, Lesson 23, Understand Fraction Multiplication, Session 3, Refine, Apply It, Problem 3, students develop conceptual understanding and application as they use multiplication to solve a word problem involving fractions. 4.NF.4 (Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.). “Olivia makes quinoa vegetable soup. She needs \frac{3}{2}cups of quinoa. She has a \frac{1}{2}-cup measuring cup. How many times does she fill the measuring cup with quinoa? Make a drawing and write a multiplication equation to model the situation.”

The materials reviewed for i-Ready Classroom Mathematics, Grade 4 meet expectations for supporting the intentional development of MP1: Make sense of problems and persevere in solving them; and MP2: Reason abstractly and quantitatively, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

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, Lesson 5, Subtract Whole Numbers, Session 2, Develop, Try It, students utilize a variety of strategies to solve that make sense to solve a word problem. “Athletes are training for a para athletic competition. There are 2,153 athletes training for the high jump event. There are 4,002 athletes training for the cycling event. How many more athletes are training for cycling than for high jump? Use any strategy to subtract.” Teacher Edition, Make Sense of the Problem, “Before students work on Try It, use Three Reads to help them make sense of the problem. Each time, ask the class to consider one of the following questions: What are you trying to find out? What questions are you trying to answer? What are the important numbers and relationships in the problem?”

• Unit 2, Lesson 9, Number and Shape Patterns, Session 3, Develop Shape Patterns, Try It, students actively engage in solving a word problem by working to understand the information in the problem. “Liv is knitting a Fana sweater. The design of a Fana sweater comes from Norway. Fana sweaters have shape patterns. The shape patterns that Liv makes goes back and forth between a triangle and a square. Show the pattern that Liv makes.” Student Worktext, Discuss It, p. 181, “Ask your partner: Can you explain that again? Tell your partner: At first, I thought…” Teacher Edition, Discuss It, Support Partner Discussion, “Encourage students to use the Discuss It questions and sentence starters on the Student Worktext page as part of their discussion. Support as needed with questions such as: How did you begin to think about this problem? Can you explain how your answer is different from or the same as your partner’s answer?”

• Unit 4, Lesson 22, Add and Subtract Fractions in Line Plots, Session 1, Explore, Try It, students actively engage in solving a word problem by working to understand the information in the problem. “Jana’s family has a worm compost bin. Jana measures the length of some worms and records the data in a line plot. What is the difference between the lengths of the shortest and the longest worm?” A picture is shown of a line plot from 1 to 4, with tics at every fourth. Student Worktext, Discuss It, p. 463, “Ask your partner: Can you explain that again? Tell your partner: I knew…so I…” Teacher Wrap, Discuss It, Support Partner Discussion, “After students work on Try It, have them respond to Discuss It with a partner. Listen for understanding of: subtracting whole numbers, subtracting fractions, subtracting mixed numbers by decomposing them, subtracting the whole numbers and the fractions, and combining the results.”

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 2, Lesson 6, Understand Multiplication as a Comparison, Session 1, Explore, Model It, students work to understand the relationships between problem scenarios and mathematical representations. “You can think about multiplication as joining equal groups. a. Draw 3 groups of 5 stars b. Write a multiplication equation to find the total numbers of stars.” Teacher Edition, Model It, “Read the question at the top of the student worktext page. Remind students that one way to think about multiplication is joining equal groups. Tell students that they are going to use what they already know about multiplication to think about multiplication in a new way - as a comparison.”

• Unit 3, Lesson 12, Multiply by Two-Digit Numbers, Session 2, Develop, Discuss It, students discuss what the numbers or symbols in an expression/equation represent. “Folding chairs are set up in a school auditorium for a play. There are 16 rows of chairs. Each row has 28 chairs. How many folding chairs are set up for the play?” Teacher Edition, Discuss It, Support Partner Discussion, “Encourage students to use the terms partial product and hundreds, tens, and ones as they discuss their solutions. Support as needed with questions such as: How did you break apart the number 16 and 28? How did you find the total represents the product?”

• Unit 4, Lesson 19, Understand Fraction Addition and Subtraction, Session 2, Develop, Model It: Area Models, students consider the fractions involved in a problem and work to understand the relationship between the fractions and visual model. Problem 3, “Show \frac{1}{8}+\frac{2}{8}.” A picture is shown of a circle partitioned into 8 equal parts. Teacher Edition, Model It, “As students complete the problems, have them identify that they are being asked to use area models to show fraction addition and subtraction. Clarify as needed that students should shade, color, or cross out parts in each model.”

The materials reviewed for i-Ready Classroom Mathematics, Grade 4 meet expectations for supporting the intentional development of MP3: Construct viable arguments and critique the reasoning of others, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

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 2, Lesson 6, Understand Multiplication as Comparison, Session 3, Refine, Problem 3, students critique the reasoning of another students' mathematical thinking and understanding of multiplication. “Elan planted 4 seeds. His stepmom said she planted 2 times as many seeds.  Elon figured out that his stepmom planted 6 seeds. What did Elon do wrong?”

• Unit 3, Lesson 13, Use Multiplication to Convert Measurements, Session 1, Explore, Discuss It, students justify their own mathematical thinking about converting measurements, then critique the reasoning of others. “Ask your partner: Do you agree with me? Why or Why not?  Tell your partner: I agree with you about…because…”

• Unit 4, Lesson 19, Understand Fraction Addition and Subtraction, Session 3, Refine, Apply It, Problem 5, students justify their own reasoning and understanding about adding and comparing fractions. “Look at the expression \frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}. Is this sum greater than, less than, or equal to \frac{5}{6}? Explain how you know.”

• Unit 5, Math In Action, Classify Shapes and Angles, Session 1, Bella’s Solution, Teacher Edition, Differentiation, Extend, Deepen Understanding, Understanding Ways to Organize Shapes, students explain their own geometric reasoning when looking at a problem to solve using a variety of tools, such as a checklist and asking questions. “As you look at the checklist, encourage students to describe what they see in the problem and tell how it relates to each checklist item. Encourage students to ask each other questions and explain their thinking. Ask Why does Bella’s table have 3 rows? Listen For The problem states she must have at least one category about the shapes’ sides and at least 2 categories about the shapes’ angles. So, the solution must have at least 3 categories. Each of the rows in Bella’s table shows a different category. Ask How can you tell that the solution works? Listen For Each shape from the problem is in the table. Each shape is included at least once. Ask Why are some shapes shown more than once in the table? Listen for Those categories overlap. The shapes can fit one or more categories.” 

• Unit 6, Lesson 32, Add and Subtract with Angles, Session 4, Refine, Apply It, Problem 9, students justify their reasoning to a geometry problem. “Cristobal wants to make a wooden table top in the shape of a circle. He has three pieces of wood with the angle measures shown. Can Cristobal use the three pieces of wood to make the table top so that there are no gaps or overlaps between the pieces? Explain.”

The materials reviewed for i-Ready Classroom Mathematics, Grade 4 meet expectations for supporting the intentional development of MP4: Model with mathematics; and MP5: Use appropriate tools strategically, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

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 2, Lesson 8, Multiples and Factors, Session 1, Additional Practice, Prepare for Multiples and Factors, Problem 3, students model with mathematics as they use appropriate strategies and equations to solve word problems. “Solve the problem. Show your work. A park has several rows of trees. Each row has 5 trees. How many trees could be in the park?” 

• Unit 3, Lesson 15, Divide Four-Digit Numbers, Session 1, Connect It, Problem 2, students model with mathematics as they use an area model to divide numbers. “You can divide four-digit numbers in many ways.” Problem 2a, “Complete the area model to show 3,200\div5.” A picture is shown of an area model, with blank labels for students to fill in. “The quotient of 3,200\div5 is ___.” Problem 2b, “Another way to find 3,200\div5 is by using partial quotients. Complete the division that shows using partial quotients.”

• Unit 4, Lesson 20, Add and Subtract Fractions, Session 2, Develop, p. 418, students model with mathematics as they use pictures and visual models to solve problems with fraction operations with teacher guidance. “Francisca and Nahele are painting a fence green. Francisca starts at one end and paints \frac{3}{10} of the fence. Nahele starts at the other end and paints \frac{4}{10} of the fence. What fraction of the fence do they paint altogether?” Picture It, “You can use a picture to help understand the problem. Think what the fence might look like if it has 10 equal-size parts. Each part is \frac{1}{10} of the whole. They paint 3 tenths and 4 tenths of the fence.” Teacher Edition, Facilitate Whole Class Discussion, “Guide students to Compare and Connect the representations. Reword any unclear statements, or ask a student to do so. Ask Where does your model show the total number of equal parts in the fence? the part Francisca paints? the part Nahele paints? the total number of tenths the two friends paint?”

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 2, Lesson 7, Multiplication and Division in Word Problems, Session 1, Connect it, Problem 1, students choose appropriate tools and/or strategies as they solve multiplicative comparison problems. “Look Back Explain how you could find how many violins the band has.” Teacher Edition, Hands-On, Use counters to represent multiplicative comparison situations, “If students are unsure about the concept of multiplication as a comparison, then use this activity to have them model multiplicative comparison situations. Distribute counters or grid paper. Tell students to put 6 counters in a row or draw an array of 6. Have students add another row of 6 more to their array. Explain to students that this shows 2 times as many as 6. Have each pair find the number in all. [12] Then ask volunteers to tell how they and their partner found the number in all. Discuss how students could use either addition or multiplication to solve. Repeat the activity for finding 3 times as many as 6 and 4 times as many as 6.”

• Unit 3, Lesson 14, Divide Three-Digit Numbers, Session 1, Explore, Connect It, Problem 2, students choose appropriate strategies as they explore the different ways to solve division problems. “You can solve division problems in many ways. You can use place value, rectangular arrays, area models, equations, and the relationship between multiplication and division. An area model shows both multiplication (4\times50=200) and division (200\div4=50). You can also use area models to break apart a problem into smaller parts. Fill in the missing labels on two other area models for $$(200\div4)$$.” Two area models are shown. One is divided into two parts and the other is divided into four parts.

• Unit 5, Lesson 24, Multiply Fractions by Whole Numbers, Session 1, Additional Practice, Prepare for Multiplying Fractions by Whole Numbers, Problem 3, students choose appropriate tools/strategies to multiply fractions. “Solve the problem. Show your work. Kyleigh cares for a flock of turkeys on her family farm. She feeds the turkeys \frac{3}{8}of a whole bag of grain in 1 week. What fraction of the bag of grain do the turkeys eat in 2 weeks?” Teacher Edition, “Assign problem 3 to provide another look at solving a problem by multiplying a fraction by a whole number…Students may want to use fraction bars, use fraction tiles or circles, or draw models with pencil and paper.”

The materials reviewed for i-Ready Classroom Mathematics, Grade 4 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 2, Lesson 10, Model and Solve Multiple-Step Problems, Session 2, Develop, Teacher Edition, Differentiation, Bar Model, students work towards precision and accuracy with bar models. “When discussing the bar model, prompt students to think about the length of the boxes in each bar and the amounts they represent. Ask Why is the box labeled $6 slightly larger than the one box labeled $4? Why is the box labeled $9 slightly longer than the two boxes labeled $4? Listen For 6 is greater than 4; 9 is greater than$$2\times4$$, or 8. On the board, draw the bar model with each of the boxes the same length. Ask In your opinion, why might an imprecise bar model lend you to an incorrect conclusion? Listen For You might think Garrett has $12 left because the bar representing the unknown amount is the length of three boxes labeled $4. Generalize Is drawing a bar model with boxes of relative size important when representing any number? Have students explain their reasoning. Listen for understanding that boxes need to accurately represent the problem and solution.” 

• Unit 3, Lesson 14, Divide Three-Digit Numbers, Session 3, Apply It, Problem 7, students attend to precision when they explore the idea that remainders can impact the answer in division word problems. “A store orders 315 hats. The hats are shipped in boxes of 8. How many boxes are needed to ship all the hats? First, find which two multiples of 10 the quotient is between. Then find the quotient using an area model. Show your work.”

• Unit 4, Lesson 18, Compare Fractions, Sessions 1 and 4, students attend to precision when they compare two fractions with different numerators and different denominators. Session 1, Explore, Try It, “Ayana and Lisa have tahini bars that are the same size. Ayana eats \frac{2}{4} of her tahini bar. Lisa eats \frac{2}{5} of her tahini bar. Who eats more of her tahini bar?” Teacher Edition, Discuss It, Support Partner Discussion, “After students work on Try It, have them respond to Discuss It with a partner. Listen for understanding of: both wholes as the same size; 4 as the number of equal parts in one whole; 5 as the number of equal parts in the other whole; 2 as the number of parts considered in each whole.” Session 4, Refine, Apply It, Problem 1, “Ethan and Skyler work on the same set of homework problems. Ethan finishes \frac{5}{6} of the problems, and Skylar finishes \frac{2}{3} of the problems. Who finishes more of their homework problems? Show your work.”

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 8, Session 1, Explore, Connect It, Problem 2, students learn the specialized language of mathematics as they work with multiples and factors. “You can extend your thinking about multiplication by looking at factors of a number, factor pairs, multiples, prime numbers, and composite numbers.” Problem 2a, “A factor pair is two numbers that are multiplied to give a product. Since 1\times20=20, a factor pair of 20 is ___ and ___.” Problem 2b, “Fill in the multiplication equations to show the other factor pairs of 20. ___$$\times$$ ___ =20 Factor pair: ___ and ___ and ___ \times ___ =20 Factor pair: ___ and ___.” Problem 2c, “What are the six different factors of 20?” Problem 2d, “A multiple is the product of a given number and any other whole number. When you multiply numbers, the product is a multiple of each factor. So, 20 is a multiple of each of its factors. The number 20 is a multiple of ___, ___, ___, ___, ___, and ___.” Problem 2e, “A number with more than one factor pair is called a composite number. A prime number has only one factor pair, the number itself and 1. The number 1 is neither prime nor composite. The factors of 11 are 1 and 11. Is 11 a prime or composite number?”

• Unit 4, Lesson 25, Fractions as Tenths and Hundredths, Session 1, Additional Practice, Prepare for Fractions as Tenths and Hundredths, Problem 1, students attend to the specialized language of mathematics as they work with fractions. “Think about what you know about fractions. Fill in each box. Use words, numbers, and pictures. Show as many ideas as you can. Chart with: Word: numerator, denominator, tenths, hundreds, In My Own Words, Example.” 

• Unit 5, Lesson 33, Classify Two-DImensional Figures, Session 2, Develop, Connect It, Problem 3, students attend to the specialized language of mathematics to sort shapes. “Explain how to sort shapes based on parallel and perpendicular sides.”

The materials reviewed for i-Ready Classroom Mathematics, Grade 4 meet expectations for supporting the intentional development of MP7: Look for and make use of structure; and MP8: Look for and express regularity in repeated reasoning, for students, in connection to grade-level content standards, as expected by the mathematical practice standards.

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 2, Lesson 10, Model and Solve Multiple-Step Problems, Session 1, Explore, Connect It, Problem 2, students analyze a problem and look for more than one approach. “You can model and solve problems in different ways. You can use a bar model to model the problem. Complete the bar model to show the number of students who play drums in all. You can also model the problem with an equation. In an equation, you can use a letter, such as n in the bar model above to  represent the unknown. The unknown in this problem is the total number of students who play the drums. Complete one way to write an equation (___\times___) + (___\times___)=n. How many students play the drums in all?” 

• Unit 3, Lesson 13, Use Multiplication to Convert Measurements, Session 1, Explore, Try It,  students look at and decompose “complicated” into “simpler” things when converting measurements. “Emerson hears the announcer on a TV show say, ‘We will return in 4 minutes.’ It takes Emerson 300 seconds to wash the dishes. Does he have enough time to wash the dishes before the TV show returns? Change the number of minutes until the show returns to a number of seconds to find out.” A “Units of Time” conversion chart is pictured. 

• Unit 5, Lesson 33, Classify Two-Dimensional Figures, Session 4, Develop, with teacher guidance, students look for and make use of structure as they develop strategies to classify triangles. Try It, “A website sells 7 kinds of triangular flags based on sides and angles.” Teacher Edition, Deepen Understanding, Tables, “When discussing the two tables at the bottom of the Student Worktext page, prompt students to consider how the tables serve as a tool to help them classify triangles. Ask What information is shown in the first table? In the second table? Listen For The first table shows triangle names based on the number of sides of equal length. The second table shows triangle names based on the kinds of angles. Read the names of the triangles in the first table aloud so students become familiar with them. Tell students that triangles can be described with two names, one from each table: for example, an acute scalene triangle. Ask According to the table, what types of sides and angles does an acute scalene triangle have? Listen For All 3 angles are acute and all 3 sides are different lengths. Ask a volunteer to draw this type of triangle on the board and write the name beneath the triangle. Repeat with other types of triangles as time permits.”

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, Add Whole Numbers, Session 2, with teacher guidance, students look for and express regularity in repeated reasoning as they develop strategies for adding numbers with more than three digits. Try It, “Lupe’s mom and dad own a tamale restaurant. Around Christmas, the restaurant sells 5,657 pork tamales and 3,804 beef tamales. How many tamales does the restaurant sell around Christmas? Use any strategy to add.” Teacher Edition, Deepen Understanding, Add Whole Numbers, “When discussing both models, prompt students to identify patterns in adding multi-digit numbers as they describe methods and shortcuts used to add. Ask How is the strategy used in the second Model It similar to the strategy used in the first Model It? How are they different? Listen for Both Strategies use place-value reasoning to add. In the first Model It, each partial sum is shown separately. The addition algorithm in the second Model It uses a shortcut method to show each partial sum. Students may say, ‘In the first step, putting a 1 in the ones place of the sum and a small 1 above the 5 in the tens place is similar to saying that 7+4=11, or 1 ten and 1 one. The small 1 reminds you that you need to include another ten when adding the tens.’ Generalize How can you be sure that using both models will result in the same answer? Have students explain their reasoning. Listen For understanding that the notation is just record keeping, so if you add ones to ones and tens to tens and so forth, the total will be the same no matter how you record it.”

• Unit 2, Lesson 9, Number and Shape Patterns, Session 1, Try It, students look for and express regularity in repeated reasoning when they explore the idea that a rule that describes a number pattern can be used to extend the pattern. “What are the next two numbers in the pattern below? 5, 10, 15, 20, 25, ___, ____.” Teacher Edition, Discuss It, Support Partner Discussion, “After students work on Try It, have them respond to Discuss It with a partner. Listen for understanding that: the number pattern has a rule; each number in the pattern follows the same rule; the rule is applied to each number to get the next number in the pattern.” 

• Unit 3, Lesson 15, Divide Four-Digit Numbers, Session 1, Explore, Try It, students look for and notice repeated calculations to understand algorithms and make generalizations or create shortcuts. “Previously, you learned about dividing three-digit numbers by one-digit numbers. Use what you know to try to solve the problem below. What is 1,400\div4?”