4th Grade - Gateway 2

The materials reviewed for Eureka Math² Grade 4 meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings.

Materials provide opportunities for students to develop conceptual understanding throughout the grade level.These opportunities are most often found within the Launch and Learn portions of lessons. Examples include:

• Module 1, Topic B, Lesson 9: Compare numbers within 1,000,000 by using >, =, and <, Learn, Compare Values of Digits, students develop conceptual understanding by reasoning about place value units and the value of the digits in two numbers to compare numbers within 1,000,000. Students also compare numbers in different forms and order more than two numbers. Teachers, “Write 16,300 and 1,650. Invite students to think–pair–share about which number is greater. How might the place value chart be helpful in comparing these numbers? Write 16,300 and 1,650 on the place value chart as students do the same. Write sentence frames for students to complete. Have students complete the sentences. As they say each sentence, record the comparison by using the greater than or less than symbol. Draw students’ attention to the symbol and the phrase it represents. Direct students to write two statements for the numbers 16,300 and 1,650 by using comparison symbols under the place value chart.” (4.NBT.2)

• Module 2, Topic B, Lesson 5: Multiply by using place value strategies and the distributive property, Launch, students develop conceptual understanding as they determine which representation does not belong. “Introduce the Which One Doesn’t Belong? routine. Display the pictures of the number bond, tape diagram, and equations. Give students 2 minutes to find a category in which three of the pictures belong, but a fourth picture does not. When time is up, invite students to explain their chosen categories and to justify why one picture does not fit. Highlight responses that emphasize reasoning about decomposing a factor to multiply. Ask questions that invite students to use precise language, make connections, and ask questions of their own. Consider asking the following questions to guide the discussion. Which one doesn’t belong? Why do you think 31 is decomposed in all 4 pictures? What do you notice about how 31 is decomposed in pictures A, C, and D? Invite students to turn and talk about why they think most of the pictures show 31 decomposed into tens and ones.” (4.NBT.5) 

• Module 4, Topic A, Lesson 1: Decompose whole numbers into a sum of unit fractions, Learn, Decompose into a Sum of Unit Fractions, students develop conceptual understanding of fractions by relating fraction strips and number bonds to equations that decompose 1 and 2 into a sum of unit fractions. The teacher, “Distribute two paper strips to each student. Direct students to fold one of the strips into 2 equal parts. Then lay the strip on a whiteboard and use a bracket to label the strip as 1. The whole paper strip represents 1. How can we name each part? How do you know? Each part is \frac{1}{2}. There is 1 rectangle partitioned into 2 equal parts. One of those parts is 1 half. Invite students to label each part in unit form and fraction form. How many halves are equivalent to, or the same amount as, 1? Begin a number bond with a total of 1 as students do the same. How can we use a number bond to show how we decomposed 1? We decomposed 1 into \frac{1}{2} and \frac{1}{2}. Complete the number as students do the same. Write an equation to show the decomposition of 1 into halves as students do the same.” (4.NF.3)  

Materials provide opportunities for students to independently demonstrate conceptual understanding throughout the grade level. The Problem Set, within Learn, consistently includes these opportunities. When appropriate, teachers may use other portions of lessons for independent demonstration of conceptual understanding. Examples include:

• Module 2, Topic C, Lesson 16: Divide by using the break apart and distribute strategy, Land Exit Ticket, students demonstrate conceptual understanding by dividing, explaining or showing the strategy used. “Find. 172\div4.Show or explain your strategy” (4.NBT.6) 

• Module 4, Topic D, Lesson 18: Estimate sums and differences of fractions by using benchmarks, Learn, Problem 1, Estimate a Sum, students demonstrate conceptual understanding by using estimation and fraction addition to solve real-world problems. “Use the Read-Draw-Write process to solve the problem, Carla makes food for a bake sale. Each recipe lists vanilla as an ingredient. The cake recipe uses \frac{23}{8} teaspoons. The pie recipe uses \frac{9}{8} teaspoons. The cookie recipe uses \frac{3}{8} teaspoons. The cupcake recipe uses \frac{11}{8} teaspoons. About how many teaspoons of vanilla does Carla use?” (4.NF.3) 

• Module 5, Topic A, Lesson 3: Represent tenths as a place value unit, Land, Exit Ticket, students demonstrate conceptual understanding as they use area models and place value disks to represent tenths, and they write equations to show the equivalence of numbers in fraction form and decimal form. Problem 1, “Write the decimal fraction as a decimal number. 6 tenths, \frac{6}{10}=___.” An area model is provided. Problem 2, “Write the decimal number as a decimal fraction. 4 tenths, 0.4 = ___.” Four place value disks are provided. (4.NF.6)

The materials reviewed for Eureka Math² Grade 4 meet expectations for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency. 

The materials develop procedural skill and fluency throughout the grade level, within various portions of lessons, including Fluency, Launch, and Learn. Examples include: 

• Module 1, Topic D, Lesson 16: Add by using the standard algorithm, Fluency, Whiteboard Exchange: Estimate Sums, students develop procedural skill and fluency as they estimate a sum within 1,000 to prepare for using estimation to assess the reasonableness of an answer. The teacher displays “469+228=m,” and asks: “How could you round each addend to help you estimate the sum? Whisper your idea to your partner. Provide time for students to share with their partners. Write an equation that shows an estimated sum and how you rounded both addends.” (4.NBT.4)

• Module 1, Topic D, Lesson 18: Subtract by using the standard algorithm, decomposing larger units once, Fluency, Beep Counting by Ten Thousands, students develop procedural skill and fluency as they complete a pattern to build fluency with finding one ten thousand more and less than a given number from topic C. “Invite students to participate in Beep Counting. Listen carefully as I count on and count back by ten thousands. I will replace one of the numbers with the word beep. Raise your hand when you know the beep number. Ready? Display the sequence 47,000, 57,000, beep. 47,000, 57,000, beep Wait until most students raise their hands, and then signal for students to respond. Display the answer.” (4.NBT.4)

• Module 1, Topic D, Lesson 19: Subtract by using the standard algorithm, decomposing larger units up to 3 times, Launch, students develop procedural skill and fluency as they subtract three-digit numbers using the standard algorithm. The teacher leads a discussion about renaming in order to subtract using the standard algorithm. “Write 612-437 horizontally. We can use what we know about renaming more than once with smaller numbers to help us rename more than once with larger numbers.” (4.NBT.4)

Materials provide opportunities for students to independently demonstrate procedural skill and fluency throughout the grade level. The Problem Set, within Learn, consistently includes these opportunities. When appropriate, teachers may use other portions of lessons for independent demonstration of procedural skill and fluency. Examples include:

• Module 1, Topic D, Lesson 16: Add by using the standard algorithm, Learn, Problem Set, Problems 1-3, students independently demonstrate procedural skill and fluency as they add numbers with up to six digits, “Add by using the standard algorithm. Problem 1, 5,212+367; Problem 2, 5,212+1,367; Problem 3, 5,252+1,367.” (4.NBT.4)

• Module 1, Topic D, Lesson 19: Subtract by using the standard algorithm, decomposing larger units up to 3 times, Problem Set 1 - 3, students independently demonstrate procedural skill and fluency as they subtract using the standard algorithm. Problem 1, “3,570-2,490.” Problem 2, “3,570-2,590.” Problem 3, “96,873-48,900.” (4.NBT.4)

• Module 5, Topic D, Lesson 12: Apply fraction equivalence to add tenths and hundredths, Fluency, Whiteboard Exchange: Multiply Whole Numbers, students independently demonstrate procedural skill and fluency as they multiply two-digit numbers to build multiplication fluency with multi-digit numbers. “Display 20\times34=. Multiply. Show your strategy. Give students time to work. When most students are ready, signal for students to show their whiteboards. Provide immediate and specific feedback. If students need to revise, briefly return to validate their corrections. Display the total. Repeat the process with the following sequence: 62\times40=___, 12\times14=___, 32\times29=___” (4.NBT.5)

The materials reviewed for Eureka Math² Grade 4 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics. 

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 Problem Sets or the Lesson Debrief, Learn and Land sections respectively.

Examples of routine applications of the math include:

• Module 3, Topic F, Lesson 22: Represent, estimate, and solve division word problems, Learn, Estimate Quotients, Classwork, Problem 1, students solve routine word problems using a letter to represent the unknown with teacher support. “A grocery store aisle has 378 cans of food. The cans are arranged equally among 6 shelves. How many cans of food are on each shelf?” Sample student response: “378\div6=c.”(4.OA.3)

• Module 4, Topic E, Lesson 28: Represent and solve word problems with mixed numbers by using drawings and equations, Learn, Solve a Word Problem, Classwork, Problem 1, students solve routine problems by selecting representations and strategies to solve word problems with mixed numbers. “Use the Read–Draw–Write process to solve the problem. There were 6\frac{3}{8} pizzas on a table when lunch started. At the end of lunch, there were 2\frac{6}{8} pizzas left. How many pizzas were eaten during lunch? Guide students to reason about the problem by asking questions such as the following: What information does the problem give us? What does the question ask? What can you draw to represent the problem?” (4.NF.3d)

• Module 5, Topic D, Lesson 14: Solve word problems with tenths and hundredths, Land, Exit Ticket, students independently solve routine word problems with tenths and hundredths. “Use the Read–Draw–Write process to solve the problem. Write the solution statement by using a decimal number. Miss Diaz’s class drinks 2.9 liters of juice. Mrs. Smith’s class drinks 3.15 liters of juice. How many liters of juice do both classes drink altogether?” (4.MD.2)

Examples of non-routine applications of the math include:

• Module 1, Topic C, Lesson 15: Apply estimation to real-world situations by using rounding, Land, Exit Ticket, students independently solve non-routine problems by applying estimation to real-world situations using rounding. “Mr. Lopez plans to buy snacks for his students. He has 24 students in his first class, 18 students in his second class, and 23 students in his third class. Estimate how many snacks Mr. Lopez should buy. Explain how you estimated and why.” (4.OA.3)

• Module 4, Topic F, Lesson 33: Solve word problems involving multiplication of a fraction by a whole number, Practice Set, Problem 4, students independently solve non-routine problems as they analyze different methods for multiplying a fraction by a whole number. “Mr. Davis puts \frac{7}{10} kilograms of soil into each of his flowerpots. He fills 1 dozen flowerpots. He uses a measuring container that holds \frac{1}{10} kilogram of soil. How many times does Mr. Davis fill the measuring container for all the flowerpots? (Hint: 1 dozen flowerpots = 12 flowerpots).” (4.NF.4c)

• Module 6, Topic A, Lesson 6: Relate geometric figures to a real world context, students use geometric figures to create a floor plan, Learn, Create a Floor Plan, Classwork, Problems 1 and 2, students solve non-routine problems with a partner by examining the requirements of the project, drawing a floor plan, using their knowledge of geometric figures, and discussing their floor plan features. Problem 1, “Use dot paper to create a floor plan of a home. Follow these guidelines: ___ Include only one level. Do not include stairs. ___ Use straight line segments. ___ Include hallways. ___ Include at least one example of each figure listed in the table. ___ Label each room.” Problem 2, “Use the floor plan to complete the table.” The table provides floor plan criteria including: line segments, parallel line segments, perpendicular line segments, right angles, acute angles, and obtuse angles. (4.G.1)

The materials reviewed for Eureka Math² 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. 

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:

• Module 1, Topic A, Lesson 2: Solve multiplicative comparison problems with unknowns in various positions, Learn, Model Multiplicative Comparison Problems, “students attend to conceptual understanding as they represent various multiplicative comparison situations with sticky notes, tape diagrams, and equations. Direct students to use sticky notes, a tape diagram, and an equation to represent the problem: 18 is 3 times as many as___. How does your tape diagram show what is unknown? How does your tape diagram show how many times the unit is repeated? What equation did you write to represent the problem? Why? How can you use a multiplication equation with an unknown factor to think about this problem? What is the value of the unknown? How do you know? What is the value of the unknown? How do you know?” (4.OA.1 and 4.OA.2)

• Module 2, Topic C, Lesson 13: Divide three-digit numbers by one-digit numbers by using an area model, Fluency, Choral Response: 10 Times as Much, students attend to procedural skill and fluency as they solve for a product. “Display 1 on the place value chart. How many ones are on the place value chart? Say the answer in unit form. What is 10\times1 one? Say the answer in unit form. On my signal, say the equation in standard form. Repeat the process with the following sequence: 10\times1 ten = ___ hundred, 10\times3 tens = ___ hundreds, 10\times1 hundred = ___ thousand, 10\times5 hundreds = ___ thousands, 10\times1 thousand = ___ ten thousand, 10\times7 thousands = ___ ten thousands.” (4.NBT.1)

• Module 4, Topic A, Lesson 6: Rename mixed numbers as fractions greater than 1. Launch, Classwork, Problem 1, “students attend to application as they rename a whole number as a fraction and discuss using similar strategies to rename a mixed number. Ray needs 3 cups of flour to make bread. He only has a \frac{1}{2} cup measuring scoop. What did you draw to represent the problem? How can he use the \frac{1}{2} cup scoop to measure 3 cups of flour? Present the equation: 3=\frac{6}{2}.  Does renaming the whole number as a fraction help you think about the problem? Why? What is different about the number of cups of flour Ray needs to make cookies?” (4.NF.3a and 4.NF.3b)

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

• Module 2, Topic A, Lesson 2: Divide two- and three-digit multiples of 10 by one-digit numbers, Fluency, Counting on the Number Line by 2 Tens, “students engage with conceptual understanding and procedural skills and fluency as they build place value understanding counting by a unit of 2 tens in unit and standard from from 100 to 200 to operate with multi-digit numbers. Let’s count forward and back by twos from 10 to 20. Ready? Display the number line. Use the number line to count forward and back by 2 tens in unit form from 10 tens to 20 tens. The first number you say is 10 tens. Ready? Display each number, one at a time on the number line, as students count. Now count forward and back by 2 tens again. This time say the numbers in standard form. The first number you say is 100. Ready? Display each number, one at a time on the number line, as students count.” (4.NBT.6) 

• Module 3, Topic B, Lesson 9: Solve multiplication word problems, Learn, Multiplicative Comparison: Share, Compare, and Connect, Equal-Groups Problem, students engage with procedural skills and fluency and application as they multiply to solve real-world problems. “There are 48 pencils in each box. Mr. Lopez buys 7 boxes of pencils. How many pencils does he buy in all?” (4.NBT.5)

• Module 4, Topic B, Lesson 11: Represent equivalent fractions by using tape diagrams, number lines, and multiplication or division, Learn, Decompose and Compose Fractional Units, students engage with conceptual understanding and procedural skills and fluency as they solve problems involving equivalent fractions using tape diagrams and writing equations to represent them. “Label \frac{6}{8}on the number line and direct students to do the same. Invite students to turn and talk about how the tape diagram and number line represent the multiplication. Write \frac{6}{10}=\frac{6}{10}+\frac{2}{2}=\frac{3}{5} and direct students to do the same.” (4.NF.1)

The materials reviewed for Eureka Math² 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.

Students have opportunities to engage with MP1 and MP2 across the year and they are identified for teachers within margin notes called, “Promoting the Standards for Mathematical Practice.” According to the Grade 3-5 Implementation Guide, “Promoting the Standards for Mathematical Practice highlights places in the lesson where students are engaging in or building experience with the Standards for Mathematical Practice (MPs). Although most lessons offer opportunities for students to engage with more than one Standard for Mathematical Practice, this guidance identifies a focus MP within each lesson. The notes also provide lesson-specific information, ideas, and questions that you can use to deepen students’ engagement with the focus MP. Often, the suggested questions for a particular MP repeat. This intentional repetition supports students in understanding the MPs in different contexts.”

Materials provide intentional development of MP1 to meet its full intent in connection to grade-level content. Students make sense of problems and persevere in solving them as they work with the support of the teacher and independently throughout the units. Examples include:

• Module 2, Topic C, Lesson 16: Divide by using the break apart and distributive strategy, Learn, Number of Groups Unknown, Classwork, students make sense of problems and persevere in solving them as they use the Read-Draw-Write process to solve a word problem. “A factory made 304 chair legs. How many chairs can the factory make with 4 legs on each chair? Ask the following questions to promote MP1: What is your plan to find how many chairs can be made? Is your strategy working? Is there something else you could try? Does your answer make sense? Why?”

• Module 3, Topic B, Lesson 4: Apply place value strategies to divide hundreds, tens, and ones, Learn, Draw an Area Model to Divide, Classwork, “When students draw an area model and write an equation to divide, they are making sense of problems and persevering in solving them (MP1). Complete the equation and direct students to do the same. 639\div3=200+10+3=213. Invite students to talk about the similarities between the area model and the equation. Write 498\div2=___. Draw a rectangle and direct students to do the same. ‘What is the width of the rectangle?’ (2) Students label the width. ‘What is the largest unit in the total? How many hundreds are in 498? How many hundreds compose the length of the rectangle? How do you know?’” The teacher is instructed to “Draw a line on the area model to indicate the partial area. Label the length as 200. Direct students to do the same. Point to the area model and ask the following question. What is the area of this part of the rectangle? How do you know? Label the area as 400 and direct students to do the same. How much of the area is left to represent? How many tens compose the length of the rectangle? How do you know? What is the area of this part of the rectangle? How do you know? Label the area as 80 and direct students to do the same. How much of the area is left to represent? Repeat the process with 18 ones. Relate the area of the rectangle to the width and the length of the rectangle by writing an equation. Then relate the length of the rectangle to the quotient. Invite students to add the areas of the small rectangles to verify that the area of the rectangle is equal to 498.”

• Module 6, Topic C, Lesson 16: Find unknown angle measures around a point, Learn, Find Multiple Unknown Angle Measures, Problem Set and Debrief, “When students find multiple unknown angle measures around a point, they make sense of problems and persevere in solving them (MP1).” Problem Set, Problem 5, “Write and solve equations to find the unknown angle measures. \overline{EF} and \overline{GH} intersect at N. a. The measure of \angle ENG is ___. b. The measure of \angle GNF is ___.” During the Debrief, the teacher facilitates a class discussion about using addition and subtraction to find unknown angle measures. “What are some ways we can find the measure of an angle without using a protractor? How can knowing the angle measure of an adjacent angle help you find unknown angle measures?”

Materials provide intentional development of MP2 to meet its full intent in connection to grade-level content. Students reason abstractly and quantitatively as they work with the support of the teacher and independently throughout the units. Examples include:

• Module 2, Topic C, Lesson 11: Divide by using familiar strategies, Learn, Choose a Strategy to Divide, Classwork, “Students reason abstractly and quantitatively (MP2) as they choose a strategy and divide a two-digit number by a one-digit number to solve a word problem. Find the unknown. Show or explain your strategy. 1. 6 people equally share 78 dollars. How much money does each person get? Ask the following questions to promote MP2: How does the context of 6 people equally sharing 78 dollars help you decide which model and strategy to use? Does your solution make sense mathematically?” 

• Module 4, Topic F, Lesson 33: Solve word problems involving multiplication of a fraction by a whole number, Learn, Multiplicative Comparison Word Problem, Classwork, Problem 2, “Students reason abstractly and quantitatively (MP2) as they use the Read–Draw–Write process to solve a word problem, decontextualize to multiply a fraction by a whole number, and decide to express the product as a fraction or as a mixed number. Use the Read-Draw-Write process to solve the problem. 2. A kitten weighs \frac{4}{5} kilograms. A puppy is 6 times as heavy as the kitten. How many kilograms does the puppy weigh?” The teacher is prompted to ask the following questions to promote MP2: “What does the given information in the problem tell you about how to draw a tape diagram to represent the situation? How does your tape diagram represent how the weights of the kitten and puppy are related? Does your answer make sense mathematically?”

• Module 5, Topic B, Lesson 8: Represent decimal numbers in expanded form, Practice Set, Problems 5 - 10, as students “use the relationship between one-dollar bills, dimes, and pennies to understand expanded form of decimal numbers, they are reasoning abstractly and quantitatively (MP2). Write each number in expanded form. Use decimal form or fraction form. 5. 3 ones 2 tenths, 6. 5 ones 4 tenths 9 hundredths. Write the value of each digit. Use decimal form. 9. 3.84, 10. 7.09”

The materials reviewed for Eureka Math² 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. 

Students have opportunities to engage with MP3 across the year and it is explicitly identified for teachers within materials in margin notes called, “Promoting the Standards for Mathematical Practice.” According to the Grade 3-5 Implementation Guide, “Promoting the Standards for Mathematical Practice highlights places in the lesson where students are engaging in or building experience with the Standards for Mathematical Practice (MPs). Although most lessons offer opportunities for students to engage with more than one Standard for Mathematical Practice, this guidance identifies a focus MP within each lesson. The notes also provide lesson-specific information, ideas, and questions that you can use to deepen students’ engagement with the focus MP. Often, the suggested questions for a particular MP repeat. This intentional repetition supports students in understanding the MPs in different contexts.”

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:

• Module 1, Topic C, Lesson 15: Apply estimation to real-world situations by using rounding, Learn, Classwork, Estimate by Rounding in a Different Way, “Students construct viable arguments and critique the reasoning of others (MP3) when they explore rounding to a place value unit other than the nearest unit. 934,242 people visited a museum last year. Will rounding last year’s number of visitors to the nearest ten thousand give a useful estimate for the number of visitors next year? Why?” The teacher is prompted to ask the following questions to promote MP3: “Why does rounding 934,242 to the nearest thousand or nearest ten thousand not give a good estimate for the number of visitors next year? How would you change rounding 934,242 to the nearest unit to better estimate the number of visitors next year?”

• Module 2, Topic E, Lesson 22: Use division and the associative property of multiplication to find factors, Launch, “When students work with a partner to find all the factors of a given large number and then share their thinking and reasoning, they are constructing viable arguments (MP3). Find as many factors of 96 as you can.” The teacher is prompted to ask the following questions to promote MP3: “Have you found all the factors of 96? How do you know for sure? Why does your strategy work? Convince your partner. What questions can you ask your partner to make sure you understand their strategy?”

• Module 3, Topic F, Lesson 24: Solve multi-step word problems and assess the reasonableness of solutions, Learn, Solve a Word Problem and Examine Solution Paths, Classwork, Problem 2, as “students solve multi-step word problems and share their solution path and listen to and analyze their peers’ solution path for solving a word problem, they construct viable arguments and critique the reasoning of others (MP3). A restaurant uses 161 gallons of milk each week. A school uses 483 gallons of milk each week. How many more gallons of milk does the school use than the restaurant in 4 weeks?” The teacher is prompted to ask the following questions to promote MP3: “Why does your method work? Convince the other pair in your group. What questions can you ask the other pair to make sure you understand their method?”

• Module 5, Topic B, Lesson 6: Represent hundredths as a place value unit, Learn, Hundredths as a Fractional Unit and a Place Value Unit, students construct viable arguments and critique the reasoning of others as they, “decide if fractional units are always, sometimes, or never place value units and discuss their reasoning with a partner. Use the Always Sometimes Never routine to engage students in constructing meaning and discussing their ideas.” Students are given time to think about this statement, then discuss with a partner. The teacher is prompted to ask the following questions to promote MP3: “Can you think of a situation where a fractional unit is not a place value unit? When do you think fractional units are place value units? Why?”

The materials reviewed for Eureka Math² 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. 

Students have opportunities to engage with MP4 and MP5 across the year and they are explicitly identified for teachers within materials in margin notes called, “Promoting the Standards for Mathematical Practice.” According to the Grade 3-5 Implementation Guide, “Promoting the Standards for Mathematical Practice highlights places in the lesson where students are engaging in or building experience with the Standards for Mathematical Practice (MPs). Although most lessons offer opportunities for students to engage with more than one Standard for Mathematical Practice, this guidance identifies a focus MP within each lesson. The notes also provide lesson-specific information, ideas, and questions that you can use to deepen students’ engagement with the focus MP. Often, the suggested questions for a particular MP repeat. This intentional repetition supports students in understanding the MPs in different contexts.”

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:

• Module 1, Topic D, Lesson 21: Solve two-step word problems by using addition and subtraction, Learn, Two-Step Comparison Word Problem, Classwork, “When students draw tape diagrams to represent information about the miles Mia has driven, they are modeling with mathematics (MP4).” Teachers, “Display the word problem and chorally read it with the class. Mia is a bus driver. She drives 196,000 miles in two years. She drives 100,723 miles in year 1. How many fewer miles does Mia drive in year 2? Invite students to use the Read–Draw–Write process to solve the problem. Direct students to estimate how many fewer miles Mia drives in year 2 before they determine the actual answer. Circulate and observe student work. Use the following questions to advance student thinking: What can you draw to represent the number of miles she drives in year 1? Year 2? Which tape should be longer? How do you know? How can you represent the total number of miles she drives? Where is the unknown represented in the tape diagram? What letter can you use to represent the unknown? What operations will you use to find the solution? Why? What solution statement can you write? Select two or three students to share in the next segment. Purposely choose work that allows for rich discussion about using efficient subtraction strategies and using the tape diagram in different ways to find a solution.”

• Module 4, Topic A, Lesson 4: Represent fractions by using various fraction models, Learn,  How Many Ways, “Students model with mathematics (MP4) as they represent fractional distances by using different models, (e.g., a number line or an area model), and describe how the model represents the fractions.” Teachers, “Present the situation: Gabe’s home is \frac{4}{10} miles from his school. To get to school, Gabe runs \frac{7}{10} miles and walks \frac{2}{10} miles. Invite students to work with a partner to draw as many different models as they can to represent the total distance of  \frac{4}{10} miles. Select two or three pairs to share their work. Purposefully choose work that allows for rich discussion about the various models used to represent \frac{4}{10}. The student work samples shown demonstrate drawing a tape diagram, a number line, a circular fraction model, and an area model.” Teachers are prompted to ask the following questions to promote MP4: “How can you write the total distance Gabe walked and ran mathematically? What key ideas about how far Gabe ran and walked do you need to include in your model? How does your model show \frac{4}{10}=\frac{2}{10}+\frac{2}{10}?”  

• Module 5, Topic D, Lesson 14: Solve word problems with tenths and hundredths, Learn, Solve a Comparison Word Problem, Classwork, “When students use the Read–Draw–Write process to solve problems with decimal numbers, they are modeling with mathematics (MP4). Use the Read–Draw–Write process to solve the problem. Write the solution statement by using a decimal number. A wall in Mrs. Smith’s living room is 0.78 meters longer than a wall in her bedroom. The wall in Mrs. Smith’s bedroom is 4.32 meters long. How long is the wall in her living room?” Teachers are prompted to ask the following questions to promote MP4: “What key ideas in the problem do you need to make sure are in your diagram? How can you improve your diagram to better represent the context? How can you simplify the problem to help estimate the answer to the problem?”

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:

• Module 2, Topic B, Lesson 9: Solve multiplication word problems, Learn, Exit Ticket, “When students draw to represent a multiplicative comparison situation and then select a method to multiply, they use appropriate tools strategically (MP5).” Students choose between different strategies to solve word problems, for example: Multiply Tens and Ones, Distributive Property, and Compensation. “Use the Read-Draw-Write process to solve the problem. A puppy weighs 18 pounds. Jayla says her dog weighs 4 times as much as the puppy. What is the weight of Jayla’s dog?”

• Module 4, Topic C, Lesson 14: Compare fractions with related denominators, Learn, Justify a Comparison, “Students use appropriate tools strategically (MP5) when they select a method and use either a number line, an equation, or another model or set of words to justify a comparison between two fractions with related units.” Teachers, “Present the fractions: \frac{2}{6}___ \frac{3}{12}. Give students 2 minutes to use pictures, words, or numbers to justify a comparison of the fractions. Circulate as students work. Identify two students that showed their thinking in different ways. Purposefully choose work that allows for rich discussion about connections between methods. Facilitate a class discussion. Invite students to share their thinking with the whole group. Ask questions such as the following that invite students to make connections and encourage them to ask questions of their own: How does your method help you compare the fractions? How is your method similar to or different from your partner’s method? Which fraction did you rename? Why? How did related units help you find a common denominator? Repeat the process with \frac{5}{9}___\frac{2}{3}.”

• Module 5, Topic C, Lesson 10: Use pictorial representations to compare decimal numbers. Launch, “When students choose to represent a decimal number with an area model, a tape diagram, a number line, or place value disks and explain the reason for their selection, they are using appropriate tools strategically (MP5).” Teachers, “Introduce the Take a Stand routine to the class. Draw students’ attention to the signs hanging in the classroom that say Area Model, Tape Diagram, Number Line, and Place Value Disks. Present the decimal number 0.5. Invite students to stand beside the sign that best describes the way they would prefer to represent the number pictorially. When all students are standing near a sign, allow 1 minute for groups to discuss the reasons why they chose that sign.” Teachers are prompted to ask the following questions to promote MP5: “Why did you choose to use the pictorial representation you did? Which pictorial representation would be the most helpful to you to compare numbers in decimal form? Why?”

The materials reviewed for Eureka Math² 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. 

Students have opportunities to engage with MP6 across the year and it is explicitly identified for teachers within materials in margin notes called, “Promoting the Standards for Mathematical Practice''. According to the Grade 3-5 Implementation Guide, “Promoting the Standards for Mathematical Practice highlights places in the lesson where students are engaging in or building experience with the Standards for Mathematical Practice (MPs). Although most lessons offer opportunities for students to engage with more than one Standard for Mathematical Practice, this guidance identifies a focus MP within each lesson. The notes also provide lesson-specific information, ideas, and questions that you can use to deepen students’ engagement with the focus MP. Often, the suggested questions for a particular MP repeat. This intentional repetition supports students in understanding the MPs in different contexts.”

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

• Module 1, Topic C, Lesson 13: Round to the nearest ten thousand and hundred thousand, Learn, Round Six-Digit Numbers to the Nearest Ten Thousand, “When students round a six-digit number to the nearest ten thousand they are attending to precision (MP6).” Teachers “Invite students to work with a partner to draw a number line to plot 634,921 and round it to the nearest ten thousand. Circulate as students work and provide support as needed by asking questions such as the following: How many ten thousands are in 634,921? What is 1 more ten thousand than 63 ten thousands? What is halfway between 63 ten thousands and 64 ten thousands? How do we write 63 ten thousands and 64 ten thousands in standard form? What is halfway between 630,000 and 640,000? How do you know? Where do we plot 634,921? Which ten thousand is 634,921 closer to? What statement do we write to show 634,921 rounded to the nearest ten thousand? What is 634,200 rounded to the nearest ten thousand? How do you know? Invite students to turn and talk about how they find which 2 ten thousands the number they are rounding is between.” Teachers are prompted to ask the following questions to promote MP6: “When rounding 634,921 to the nearest ten thousand, what steps do you need to be extra careful with? Where might you make an error when rounding 634,921 to the nearest ten thousand?” 

• Module 2, Topic C, Lesson 14: Divide two-digit numbers by one-digit numbers by using place value strategies, Land, Exit Ticket, “When students represent division on a place value chart and record the division in equations by using the break apart and distribute strategy, they are attending to precision (MP6). Draw on the place value chart and complete the equations to find 84\div4. 84\div4=( ___tens +___ ones )\div4; = (___ tens\div4) + (___ ones\div4); = __ + ___; = ___.” A two column table labeled tens and ones is provided.

• Module 4, Topic A, Lesson 3: Decompose fractions into a sum of fractions, Learn, Decompose a Fraction into Non-Unit Fractions, “Students attend to precision (MP6) as they decompose a fraction into a sum of unit fractions and then group the unit fractions to show a different decomposition.” Teachers, “Distribute one paper strip to each student. Guide students to fold the strip into eighths. What equation can we write to show \frac{7}{8} as a sum of unit fractions? Write the equation. Invite students to decompose \frac{7}{8} into two parts and shade the two parts with two different colors. Circulate as students work and identify three students who shade the parts in different ways. Gather the class and invite one of the selected students to share their work. Guide a discussion to relate decomposing \frac{7}{8} into unit fractions to decomposing \frac{7}{8} into non-unit fractions. How many eighths are represented by the first color? Invite students to turn and talk about how to group the unit fractions in the equation to show \frac{2}{8}. Place parentheses in the equation to group the first two unit fractions. How many eighths are represented by the second color? Place parentheses in the equation to group the unit fractions to show \frac{5}{8}. How could we record this decomposition as a sum of non-unit fractions? Write the equation. Invite students to turn and talk about how the shaded strip relates to the sum of fractions. Repeat the process with a few more student work samples to show the decomposition of \frac{7}{8} in different ways. Give partners 1 minute to record an equation on the back of their paper strip to represent the way they decomposed \frac{7}{8} into a sum of fractions. Invite students to turn and talk about how decomposing a fraction into unit fractions and non-unit fractions is similar and different.” Teachers are prompted to ask the following questions to promote MP6: “How does your tape diagram represent a decomposition of \frac{7}{8} into a sum of two fractions? What details are important to think about when using a tape diagram to represent decomposing \frac{7}{8} into two parts? Where might you make an error when writing an equation to show the decomposition represented in your tape diagram?”

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

• Module 4, Topic B, Lesson 10: Generate equivalent fractions with larger units, Learn, Demonstrate Equivalence, Classwork, Problem 5, “When students draw an area model, partition it, and use division to demonstrate fraction equivalence, they are attending to precision (MP6). 5. Draw an area model and use division to demonstrate the equivalence. \frac{6}{10}=\frac{3}{5}. Give pairs 2 minutes to demonstrate the equivalence with an area model and division. What number did you divide the numerator and denominator in \frac{6}{10} by? Why? 2 is a factor of both 6 and 10. How can that help you find a fraction equivalent to \frac{6}{10}?” Teachers are prompted to ask the following questions to promote MP6: “What details are important to think about as you decide how to partition the area model to demonstrate that \frac{6}{10} and \frac{3}{5} are equivalent? When using division to express this equivalence, what steps do you need to be extra careful with?”

• Module 6, Topic A, Lesson 3: Draw right, acute, obtuse, and straight angles, Learn, Draw and Name an Angle, Classwork, Problem 1, “When students draw different-size angles and name them, they are attending to precision (MP6).” Teachers, “Direct students to problem 1 in their books and prompt them to use their straightedges to draw an angle using \vec{LM}. Draw an angle with the given ray. Label the angle. 1. Use \vec{LK} to draw \angle{KLM}. As students work, circulate and ensure that students are using a straightedge to construct the angle and are labeling their drawings (e.g., labeling point K on \vec{LK}, drawing an arc at the vertex to indicate the angle, or drawing a small square to indicate a right angle if a right angle is drawn). After students finish drawing, invite them to turn and talk about how their angles are similar to and different from their partners’ angles. Prompt students to use the terms right, acute, obtuse, and straight angle to describe their angles as they share.” Teachers are prompted to ask the following questions to promote MP6: “What details should be considered when drawing and naming an angle? When naming an angle, what steps need to be precise? Why?”

• Module 6, Topic A, Lesson 4: Identify, define, and draw perpendicular lines, Learn, Perpendicular Sides in Polygons, “When students identify or draw and use proper notation for perpendicular lines, rays, and line segments, they are attending to precision (MP6).” Teachers, “Invite students to remove Identify Sides in Polygons from their books and insert it into their whiteboards. Direct students to polygon 1. Ask students to trace \overline{AD} and \overline{AB} in polygon 1, and then use their right-angle tool to check or a right angle formed by the two sides. What can we say about \overline{AD} and \overline{AB}? How do you know? Direct students to mark the right angle with a small square. Write: \overline{AD}\perp\overline{AB}. Invite students to write \overline{AD}\perp\overline{AB} next to the polygon. Have partners work together to find all pairs of perpendicular sides in polygon 1 and record their findings with symbols. Invite students to share each pair of perpendicular sides. Direct students to work with a partner to identify and record as many pairs of perpendicular sides as they can in polygon 2. Which sides are perpendicular in polygon 2? Are any other sides perpendicular? How do you know? Repeat the process with polygon 3. Invite students to turn and talk about how to find and name two perpendicular sides in a polygon.” There is a polygon (rectangle) labeled ABCD shown. Teachers are prompted to ask the following questions to promote MP6: “How can we write \overline{BC} is perpendicular to \overline{CD} using the new symbol we learned? When drawing perpendicular lines, what steps do you need to be careful with? Why?”

The materials reviewed for Eureka Math² 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 the grade-level content standards, as expected by the mathematical practice standards. 

Students have opportunities to engage with MP7 and MP8 across the year and they are explicitly identified for teachers within materials in margin notes called, “Promoting the Standards for Mathematical Practice.” According to the Grade 3-5 Implementation Guide, “Promoting the Standards for Mathematical Practice highlights places in the lesson where students are engaging in or building experience with the Standards for Mathematical Practice (MPs). Although most lessons offer opportunities for students to engage with more than one Standard for Mathematical Practice, this guidance identifies a focus MP within each lesson. The notes also provide lesson-specific information, ideas, and questions that you can use to deepen students’ engagement with the focus MP. Often, the suggested questions for a particular MP repeat. This intentional repetition supports students in understanding the MPs in different contexts.”

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:

• Module 2, Topic A, Lesson 1: Multiply multiples of 10 by one-digit numbers by using the associative property of multiplication, Learn, Multiply by Using Unit Form, “When students represent 6\times30 on a place value chart and rewrite the expression in unit form to help find the product, they are looking for and making use of structure (MP7).” Teachers, “direct students to work with a partner to draw a place value chart to represent 6\times30, write the expression in unit form, and find the product. What fact did you use to help you find the product?  Say the product of 6 and 30 in unit form. What is 18 tens in standard form? Invite students to turn and talk about how unit form and the facts they know can help them multiply with multiples of 10.” Teachers are prompted to ask the following questions to promote MP7: “How can you use what the unit form and the place value representation of 6\times30 have in common to help you find the product? How can you use multiplication facts you know to help you find  6\times30?”

• Module 4, Topic A, Lesson 1: Decompose whole numbers into a sum of unit fractions, Learn, Tape Diagram and Number Line, “As students decompose a whole number into unit fractions by using a tape diagram and a number line and write an equation showing the decomposition as a sum of unit fractions, they are looking for and making use of structure (MP7).” Teachers, “Display the picture of the tape diagram of 1. The tape represents 1.” An image of a tape Diagram of 1 is shown. “How is 1 decomposed? Invite pairs to write an equation to show how 1 is decomposed into unit fractions. What equation did  you write? Display the picture of the tape diagram of 2.” An image of a tape diagram of 2 is shown. “What does the tape diagram represent? Where is 1 represented in the tape diagram? How do you know? Invite students to think–pair–share about how 2 is decomposed. Ask pairs to write an equation with parentheses to show how 2 is decomposed. How did you use parentheses in your equation? Invite students to remove Number Line 0 to 2 from their books and insert it into their whiteboards. Where do you see a length of 1 on the number line? How could we use the number line to help us draw our own tape diagram with the same length of 1? Draw a tape diagram above the number line from 0 to 1 as students do the same.” An image of a number line is shown. “How can we partition the tape diagram and number line into fourths? Guide students in partitioning the tape diagram into fourths.What fraction does each part of the tape diagram represent? What fraction does each part of the tape diagram represent? Where do you see 1 fourth on the number line?Trace the length from 0 to \frac{1}{4} and label the first tick mark. Direct students to do the same. Invite students to write an equation below the tape diagram and number line to show how 1 is decomposed. Write the equation.” Image of a number line with the points 0, 1, 2 labeled is shown. “What can we do to our drawing to show 2 decomposed into fourths?How will the equation change? Give students 1 minute to extend their drawings and rewrite the equation to show 2 as a sum of fourths. Then show the completed tape diagram, number line, and equation.” Image of a number line with the points 0, 1, 2 labeled is shown. “Invite students to turn and talk about how the number line and tape diagram represent the equation.” Teachers are prompted to ask the following questions to promote MP7: “How is the decomposition of 1 by using a tape diagram and by using a number line related? How does this help you write an equation showing 1 as a sum of unit fractions? How is decomposing 2 into thirds similar to decomposing 1 into thirds?”

• Module 6, Topic A, Lesson 2: Identify right, acute, obtuse, and straight angles, Learn, Use a Right-Angle Tool, Classwork, Problem 1, “Students look for and make use of structure (MP7) as they use their right-angle tool to determine whether an angle is right, acute, or obtuse.” Teachers, “Direct students to problem 1 in their books. Read the problem chorally with the class. Use your right-angle tool to classify the angle as a right angle, an acute angle, or an obtuse angle. Trace the rays of the angle with your finger, beginning at P, moving to the vertex, R, and then extending to S. We can call this \angle{PRS}. Write \angle{PRS}. Invite students to think–pair–share about how else the angle could be named. Write \angle{SRP}. When we name an angle using the names of three points, we must list the points in order. We cannot call this \angle{RSP} or \angle{RPS}. There is also a third way to name the angle. You can sometimes name an angle by its vertex, or the endpoint of the two rays. What is the vertex of this angle? Write \angle{R}. Support student understanding of the term vertex by inviting them to label the vertex of \angle{R}. Write vertex and draw a line pointing to the vertex. Invite students to do the same. Does \angle{R} look like a right angle? Let’s use our right-angle tool to check. Use the right-angle tool to show students how to determine whether \angle{R} is the same size as, smaller than, or larger than a right angle. Think aloud and use precise language as you use the right-angle tool. I can line up the bottom of the right-angle tool with \vec{RS}. Then I slide the right-angle tool so that its corner is at the vertex R. Prompt students to use their right-angle tools to do the same. Is \angle{R} the same size as, smaller than, or larger than a right angle? How do you know? Point to the square at the vertex of the angle. We know from learning about squares and rectangles that this square at the vertex identifies this angle as a right angle. Direct students to problems 2 and 3. Point to the arc of each angle. This is an arc. An arc is a symbol that we use to identify which angle we are looking at. Write arc and draw a line pointing to the arc in problem 2. Invite students to do the same in their books.” Teachers are prompted to ask the following questions to promote MP7: “How are acute and obtuse angles related? How can that help you identify them? How can you use what all angles have in common to help identify right, acute, and obtuse angles?”

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: 

• Module 2, Topic D, Lesson 17: Express measurements of length in terms of smaller units, Learn, Conversion Strategies, “Students look for and express regularity in repeated reasoning (MP8) as they determine relationships between customary units.” Teachers, “Write 6 yards = ____ feet. Invite students to work with a partner to use a number line and a conversion table to help them complete the equation. One partner can draw and complete the number line and the other partner can draw and complete a conversion table. How many feet are in 6 yards? What multiplication equation represents how to convert 6 yards to feet? Record the multiplication equation. Point to each factor and the product as you ask the following questions. What does 6 represent? What does 3 represent? What does 18 represent? Display the picture of the ribbon. How has your thinking changed about the lengths labeled on the spool of ribbon? Invite students to turn and talk about different strategies they can use to convert customary length measurements.” Teachers are prompted to ask the following questions to promote MP8: “What is similar about your reasoning when converting feet to inches and when converting yards to feet? What patterns do you notice when converting feet to inches or yards to feet? How can that help you convert more efficiently?” 

• Module 5, Topic A, Lesson 2: Decompose 1 one and express tenths in fraction form and decimal form, Learn, Tenths on a Number Line, Classwork, Problem 2, “When students repeatedly label tenths in fraction form and decimal form on the number line to make sense of decimal form, they are looking for and expressing regularity in repeated reasoning (MP8).” Teachers, “Use the meter stick to draw tick marks and partition the number line into tenths. Direct students to problem 2. Invite them to label the tenths along the bottom of the number line in fraction form as you do the same. 2. Label the number line. Label \frac{1}{10}  as 0.1 above the number line. Invite students to do the same. This is another way to write 1 tenth. We can read this number as zero point one or as 1 tenth. Repeat the process with 0.2 and 0.3. Point to the next tick mark. How do you think we will label this tick mark? Why? Invite students to label 0.4 through 0.9 as you do the same. Point to the fractions written below the number line. What is the same about all the fractions? A fraction with a denominator of 10 is an example of a decimal fraction. Decimal fractions can be written by using a decimal point. Point to the numbers above the number line. A number that is written with a decimal point is written in decimal form. A number written in decimal form is called a decimal number. We can write fractions with the unit tenths in decimal form. We can write a number in decimal form or fraction form. Both are different ways to record the same number. Invite students to turn and talk about how writing tenths in fraction form and in decimal form is similar and different.” Teachers are prompted to ask the following questions to promote MP8: “What patterns did you notice when you labeled tenths in fraction form and decimal form? What is similar about how a number is written in fraction form and decimal form?”

• Module 6, Topic B, Lesson 8: Use a circular protractor to recognize a 1\degree angle as a turn through ​​\frac{1}{360} of a circle, Learn, Benchmark Angles, Classwork, Problem 1, “When students make and then measure benchmark angles they are looking for and expressing regularity in repeated reasoning (MP8). 1. Use your angle-maker tool and protractor to make and measure each benchmark angle. Then complete the table.” The teacher then directs students to work with a partner and use their angle-maker tools and protractors to create the remaining benchmark angles and complete the table. As students work, circulate and check for understanding. After students finish working, direct them to look at the Angle Measure column. Do you notice a pattern?” Teachers are prompted to ask the following questions to promote MP8: “When you make benchmark angles, does anything repeat? How can that help you? What patterns do you notice as you make and measure benchmark angles? How can that help you find the angle measures of these angles?”