5th Grade - Gateway 2

The materials reviewed for Reveal Math Grade 5 meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings.

The materials develop conceptual understanding throughout the grade level, with teacher guidance, through discussion questions and conceptual problems with low computational difficulty. Examples include:

• In Lesson 4-6, Represent Subtraction of Tenths and Hundredths, Explore and Develop, Develop the Math, the teacher is directed to “Make a false claim for students to critique. Write 0.04 - 0.01 = 0.03. Point to the equation and say This equation is correct. Yes or No? Ask students to correct the statement. Revisit this routine throughout the lesson to provide reinforcement.” This opportunity allows students to engage with their teacher in the conceptual development of 5.NBT.7, add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies...

• In Lesson 12-5, Solve Problems Involving Measurement Data on Line Plots, Guided Exploration, “What are the steps you would perform to solve this problem? Can you understand other students’ plans? How are their plans similar to yours? How are they different?” The teacher facilitates mathematical discourse and deepens conceptual understanding of 5.MD.2, make a line plot to display a data set of measurements in fractions of a unit. 

• In Lesson 13-4, Classify Triangles by Properties, Bring It Together, “How do you know if a triangle can be classified as a scalene, isosceles, or equilateral? What is similar about categories and subcategories in a hierarchy? What is different?” The teacher facilitates mathematical discourse and deepens conceptual understanding of 5.G.4, classify two- dimensional figures in a hierarchy based on properties.

The materials provide opportunities for students to independently demonstrate conceptual understanding through concrete, semi-concrete, verbal, and written representations. Examples include:

• In Lesson 2-1, Understand Volume, Activity Based Exploration, “demonstrate how to form rectangular prisms using the nets. Have students determine how many of each unit can fit inside the rectangular prism.” Students build conceptual understanding by using unit cubes, marbles, beans or other measurement units, 5.MD.3, recognize volume as an attribute of solid figures and understand concepts of volume measurement.

• In Lesson 2-4, Determine the Volume of Composite Figures, On My Own, Exercise 6, students “draw line(s) to show how you decomposed the figure. What is the volume of the figure?” This helps students build conceptual understanding of 5.MD.4, measure volumes by counting cubes, using cubic cm, cubic in, cubic ft, and improvised units.

• In Lesson 6-4, Represent Multiplication of Decimals, Differentiate, Differentiation Resource Book, Item 6, “Use an area model to solve. 6.2 x 2.1 = ___.” Students use concrete models or drawings to multiply decimals,  5.NBT.7, add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies...

• In Lesson 7-4, Represent Division of 2-Digit Divisors, On My Own, Exercise 1, “What is the quotient? Use an area model to solve.”Students use area models to independently demonstrate 5.NBT.6, find whole number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value.

The materials reviewed for Reveal Math Grade 5 meet expectations that the materials develop procedural skills and fluency throughout the grade level. The materials provide opportunities for students to independently demonstrate procedural skills and fluency throughout the grade level. 

The materials develop procedural skill and fluency throughout the grade with teacher guidance, within standards and clusters that specifically relate to procedural skills and fluency, and build fluency from conceptual understanding. Examples include:

• Fluency Practice exercises are provided at the end of each unit. Each Fluency Practice includes Fluency Strategy, Fluency Flash, Fluency Check, and Fluency Talk. “Fluency practice helps students develop procedural fluency, that is, the ‘ability to apply procedures accurately, efficiently, and flexibly.’ Because there is no expectation of speed, students should not be timed when completing the practice activity.” Fluency Practice exercises in Grade 5 progress toward 5.NBT.5, fluently multiply multi-digit whole numbers using the standard algorithm.

• In Lesson 3-1, Generalize Place Value, Number Routine: Where Does it Go?, “Students determine the location of a decimal on two number lines with different marked endpoints. Remind students that this is an estimation activity and exact locations are not needed.” Students build fluency of decimals, 5.NBT.3, read, write, and compare decimals to thousandths.

• In Lesson 5-4, Use Area Models to Multiply Multi-Digit Factors, Explore & Develop, Develop the Math, Activity-Based Exploration, students explore area models to determine different ways to decompose them to form partial products. “Ask students to write a multiplication problem using one 3-digit factor and a one 1-digit factor and draw an area model to represent the product. Have students record as many ways as possible to decompose the area model. Invite students to share ways they decomposed the area model, focus attention on similar methods of decomposing, such as decomposing by place value. ‘Do you think these methods of decomposing will work for multiplying two multi-digit numbers?’” This exploration provides an opportunity for students to develop procedural skill and fluency of 5.NBT.5, fluently multiply multi-digit whole numbers using the standard algorithm. 

• In Lesson 5-7, Multiply Multi-Digit Factors Fluently, Differentiate, Reinforce Understanding,  “Work with students in pairs using a spinner that contains 2-digit numbers. One student spins the spinner to obtain a factor. The other student rolls a number cube twice to produce a 2-digit factor. Help students multiply the two factors and then check their work using estimation. Have students repeat the process with new numbers.” The teacher works with students to build procedural skill and fluency of 5.NBT.5, fluently multiply multi-digit whole numbers using the standard algorithm.

• In Unit 7, Divide Whole Numbers, Fluency Check, Fluency Talk, “Explain how you can use properties of operations to find the product of a number and a multiple of 10.” This discussion helps students build fluency of multiplication, 5.NBT.5, fluently multiply multi-digit whole numbers using the standard algorithm.

The materials provide opportunities for students to independently demonstrate procedural skill and fluency. Examples include:

• In Lesson 2-3, Use Formulas to Determine Volume, Exit Ticket, Exercise 1, “Use a formula to find the volume of the rectangular prism,” Students independently demonstrate procedural  fluency of 5.MD.5, relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume.

• In Lesson 5-4, Use Area Models to Multiply Multi-Digit Factors, Differentiate, Building Proficiency, Student Practice Book, Item 3, students use area models and partial products to solve. “18 x 221 = _____.” Students independently demonstrate procedural skill and fluency of 5.NBT.5, fluently multiply multi-digit whole numbers using the standard algorithm.

• In Unit 14, Algebraic Thinking, Fluency Practice, Fluency Strategy, “You can choose a strategy to multiply. You can use an area model, partial products, or an algorithm.” Students practice different strategies to independently demonstrate procedural skill and fluency of 5.NBT.5, fluently multiply multi-digit whole numbers using the standard algorithm.

The materials reviewed for Reveal Math Grade 5 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics. Additionally, the materials provide students with the opportunity to independently demonstrate multiple routine and non-routine applications of the mathematics throughout the grade level. 

The materials provide specific opportunities within each unit for students to engage with both routine and non-routine application problems. In the Digital Teacher Center, Program Overview: Learning & Support Resources, Implementation Guide, Focus, Coherence, Rigor, Application, “Students encounter real-world problems throughout each lesson. The On My Own exercises include rich, application-based question types, such as ‘Find the Error’ and ‘Extend Thinking.’ Daily differentiation provides opportunities for application through the Application Station Cards, STEM Adventures, and WebSketch Explorations. The unit performance task found in the Student Edition offers another opportunity for students to solve non-routine application problems.” 

The materials develop application throughout the grade as students solve routine problems in a variety of contexts, and model the contexts mathematically within standards and clusters that specifically relate to application, both dependently and independently. Examples include:

• In Lesson 2-3, Use Formulas to Determine Volume, Practice & Reflect, On My Own, Problem 8, “A freezer, shaped like a rectangular prism, is 6 feet long, 2 feet wide, and 3 feet tall. What is the volume of this freezer.”This exercise allows students to develop and apply mathematics of 5.MD.5b, relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume.

• In Lesson 8-1, Division Patterns with Decimals and Powers of 10, On My Own, Exercise 10, “Danny walks 567.3 miles in 100 days. Michelle walks a 567.3 miles by walking 0.1 miles each day. Who walked for more days? Who walked farther each day? Explain.” This exercise allows students to develop and apply mathematics of 5.NBT.2, explain patterns in the placement of the decimal point when the decimal is multiplied or divided by a power of 10.

• In Lesson 11-2, Solve Problems Involving Division, Additional Practice, Exercise 3, “A 10-kilometer race is divided into 3 equal sections. How long is each section of the race?” This problem allows students to apply mathematics  of 5.NF.3, interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers.

The materials develop application throughout the grade as students solve non-routine problems in a variety of contexts, and model the contexts mathematically within standards and clusters that specifically relate to application, both dependently and independently. Examples include:

• In Lesson 3-5, Use Place Value to Round Decimals, Extend Thinking, Item 3, “Quentin drives 632.074 miles from Sacramento, California to Las Vegas, Nevada one day and then drives 632.32 miles from Las Vegas to Santa Fe, New Mexico the Next Day. If the distance Quentin traveled on the first day was rounded to 632.32, what is a possible distance he could have traveled on that day?” This is a non-routine problem because students can answer with any number less than 632.325 and equal to or greater than 632.315. This exercise allows students to develop and apply mathematics of 5.NBT.4, use place value understanding to round decimals to any place.

• In Unit 10, Multiply Fractions, Application Station, Connection Card, Fraction of a Fraction, “Sometimes, a recipe makes much more food than you need. Find a recipe that uses healthy ingredients and makes a lot of food. Be sure to choose a recipe that lists the amount of each ingredient as a fraction. Next, create three different stories for why you are only making \frac{1}{5}, \frac{2}{3} , and \frac{7}{8} of the recipe. Draw area models to represent \frac{1}{5}, \frac{2}{3} , and \frac{7}{8} of the amount of each ingredient needed. 1. Explain how you determined the side lengths of each area model? 2. Explain how you know whether you are making less, the same amount, or more food than the original recipe? 3. Compare the different amounts of each ingredient after determining \frac{1}{5}, \frac{2}{3} , and \frac{7}{8} of each. This exercise allows students to develop and apply mathematics of 5.NF.6, solve real world problems involving multiplication of fractions and mixed numbers. 

• In Unit 14, Algebraic Thinking, Application Station, Real World Card, Earning an Income, “People are paid money they earn for performing a job, either as an hourly wage or a salary. An hourly wage means getting paid the same amount for every hour that you work. But a salary is a fixed amount of money divided up over the year and usually paid every week or two. Choose 5 jobs that interest you. Research and record the starting income and whether it is an hourly wage or a yearly salary. Determine the amount of money earned for each 40−hour week. Discuss your strategies for how you will determine this with your group. This material may be reproduced for licensed classroom use only and may not be further reproduced or distributed. Then, create 5 tables, one for each job, so that the first column is the number of weeks and the second column is the amount of money earned each week. Describe the relationship between the terms in each column as you extend the table. Write the corresponding terms as ordered pairs, plot the ordered pairs on a coordinate plane, and connect the points. Finalize each by describing the relationship between the corresponding terms in the table. 1. What are some other ways, besides income, that employers attract employees to work for them? 2. What are some reasons for choosing one job over another? 3. What are some reasons for choosing a job that might pay less?” This exercise allows students to develop and apply mathematics of 5.NBT.7, apply and extend previous understandings of division to divide unit fractions by whole numbers by unit fractions.

The materials reviewed for Reveal Math Grade 5 meet expectations in that the three aspects of rigor are not always treated together, and are not always treated separately. There is a balance of the three aspects of rigor within the grade. Additionally, multiple aspects of rigor are engaged simultaneously to develop students’ mathematical understanding of a single topic/unit of study throughout each grade level. 

All three aspects of rigor (conceptual understanding, procedural skill & fluency, and application) are present independently throughout the grade level. Examples include:

• In Lesson 2-1, Understand Volume, On My Own, Work Together, students develop conceptual understanding of volume as the amount of space taken up by a solid object. “One student used marbles to pack a rectangular prism. Another student used unit cubes. What do you notice about these strategies?”

• In Lesson 4-1, Estimate Sums and Differences of Decimals, On My Own, Problems 1-8, students build fluency with place-value concepts and learn procedures for estimating sums and differences of decimals. For example, Problem 1, “9.86 + 4.30.” Problem 3, “3.92 + 6.14.” Problem 5, “8.32 - 5.9.”

• In Lesson 9-9, Solve Problems Involving Fractions and Mixed Numbers, On My Own, Problem 3, students add and subtract mixed numbers involving unlike denominators to solve real world problems. “Alyana buys 4\frac{3}{10} pounds of potatoes. She uses 2\frac{3}{4} pounds in a recipe. How many pounds does she have left?

The materials provide a balance of the three aspects of rigor as multiple aspects of rigor are engaged simultaneously to develop students’ mathematical understanding of a single topic/unit of study throughout the grade level. Examples include:

• In Lesson 2-3, Use Formulas to Determine Volume, On My Own, Problem 9, students use their conceptual understanding of volume to develop the formula to calculate volume of rectangular prisms, and apply the formula to solve real-world problems. “An Olympic swimming pool is 2 meters deep. What is the volume of the swimming pool?” A pool with dimensions of 50m by 25m is shown.

• In Lesson 6-2, Estimate Products of Decimals, On My Own, Problems 7-12, students extend their conceptual understanding of estimation to build procedural skill and fluency of estimating products of decimals. Problem 7, “Estimate each product by finding a range. Show your work. 4.93 x 7.88.” Problem 10, “Estimate each product by finding a range. Show your work. 4.1 x 13.5.” Problem 12, “Estimate each product by finding a range. Show your work. 16.12 x 3.55.”

• In Lesson 12-3, Solve Multi-Step Problems Involving Measurement Units, On My Own, Problem 4, students build their procedural skill and fluency with multiplication involving whole numbers and fractions to solve real-world problems involving measurement conversions. “A track at the school is 400 meters long. Jackson walks around the track  3\frac{1}{2} times. How many kilometers did Jackson walk?

The materials reviewed for Reveal Math Grade 5 meet expectations for supporting the intentional development of MP1: Make sense of problems and persevere in solving them; and MP2: Reason abstractly and quantitatively, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

Each Unit Overview, Math Practices and Processes section, identifies one mathematical practice that is prevalent in the unit, and gives an overview of its use within the unit. In the Standards section of each lesson, mathematical practices for the lesson are also identified; however, in both sections, the mathematical practice is labeled as MPP Reason abstractly and quantitatively, rather than MP1 or MP 2. Within each of the lesson components, mathematical practices are not labeled or identified, leaving where they are specifically addressed up for interpretation and possible misidentification.

The materials provide intentional development of MP1: Make sense of problems and persevere in solving them, in connection to grade-level content. Examples include: 

• In Lesson 6-6, Explain Strategies to Multiply Decimals, Differentiate, Extend Thinking, Differentiation Resource Book, Exercise 4, “Numbers 1-6 are solutions to multiplication problems. 4.  24.30 _____ and _____. Match each problem A-L to its solution above in 1 to 6.” Students engage with MP1 as they make connections between equations, pictorial representations, and word problems.

• In Lesson 9-8, Add and Subtract mixed Numbers with Regrouping, On My Own, Reflect, “When is regrouping necessary when adding and subtracting mixed numbers?” Students engage in MP1 as they reflect on problem solving strategies.

• In Unit 11, Divide Fractions, Unit Review, Exercise 12, What equation does this model most likely represent? A. 5 \div 3 = n, B. 3 \div \frac{1}{5} =  n, C. 5 \div \frac{1}{3} = n, D. 3 \div 5 = n”  Students engage with MP1 as they make sense of a model and match it to the corresponding equation. 

The materials provide intentional development of MP2: Reason abstractly and quantitatively, in connection to grade-level content. Examples include:

• In Teacher’s Guide, Lesson 3-4, Compare Decimals, Guided Exploration, “Students extend their understanding of comparing whole numbers using place value to decimal numbers. What different ways can you write the comparison statement? How are they the same? How are they different? Explain your reasoning. Think About it: Are there other models or tools you could use to compare decimal numbers? How could writing them in expanded form help?” Students engage with MP2 as they explain/discuss what the numbers or symbols in an expression/equation represent. 

• In Lesson 7-1, Division Patterns with Multi-Digit Numbers, Differentiate, Build Proficiency, Student Practice Book, Exercise 11, “There are 32,000 quarters in rolls of 40, how many rolls of quarters are there?” Students engage with MP2 as they consider units involved and attend to the meaning of quantities.

In Lesson 11-5, Represent Division of Unit Fractions by Non-Zero Whole Numbers, Explore & Develop, Learn, Work Together, “Peter has \frac{1}{4} gallon of water. He equally shares the water among his 2 dogs. How much water will each dog get?” Students engage with MP2 as they consider units involved and attend to the meaning of quantities.

The materials reviewed for Reveal Math Grade 5 meet expectations for supporting the intentional development of MP3: Construct viable arguments and critique the reasoning of others, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

Each Unit Overview, Math Practices and Processes section, identifies one mathematical practice that is prevalent in the unit, and gives an overview of its use within the unit. In the Standards section of each lesson, mathematical practices for the lesson are also identified; however, in both of these sections, the mathematical practice is labeled MPP: Construct viable arguments and critique the reasoning of others, rather than MP3 Construct viable arguments and critique the reasoning of others. Additionally, the math practices are not identified within the lesson sections, therefore leaving where they are specifically addressed up for interpretation and possible misidentification.

Examples of intentional development of students constructing viable arguments in connection to grade-level content, including guidance for teachers to engage students in MP3 include:

• In Unit 2 , Volume of Rectangular Prisms, Math Probe, Exercise 1, students construct viable arguments as they determine which expression(s) can be selected to determine the volume. “Which expression(s) can be used to determine the volume of the rectangular prism shown. Select all that apply. Do not actually find the volume of the prism. Explain your choice(s).”

• In Unit 3, Unit Review, Performance Task, Part B, students construct viable arguments as they compare two decimals to thousandths. “Jupiter has 67 confirmed moons. Each moon orbits at different speeds. One moon takes 259.22 Earth days to orbit Jupiter and another one takes 259.653 Earth days. Use >, <, or = to compare the orbit speeds. Explain your answer.”

• In Lesson 13-5, Properties of Quadrilaterals, Practice & Reflect, On My Own, Exercise 13, students construct viable arguments as they classify two-dimensional figures in a hierarchy based on properties. “How are all quadrilaterals the same? How are they different?”

Examples of intentional development of students critiquing the reasoning of others in connection to grade-level content, including guidance for teachers to engage students in MP3 include:

• In Lesson 3-3, Read and Write Decimals, Differentiate, Additional Practice, Exercise 6, students critique the reasoning of others. “Colby says that \frac{27}{100} written in word form is twenty-seven thousandths. Do you agree? Explain?”

• In Lesson 5.2, Patterns When Multiplying a Whole Number by Powers of 10, Differentiate, Student Practice Book, Exercise 13, students critique the reasoning of others as they explain patterns in the number of zeros of the product when multiplying a number by powers of 10. “Herschel thinks that 30 x 1,000 = 30,000. How would you respond to Herschel?”

• In Lesson 11-1, Relate Fractions to Division, Own My Own, Exercise 12, students critique the reasoning of others. “Spencer divides 6 pounds of food from the food drive into 3 boxes. He says each box has \frac{3}{6} pounds of food. Is he right? How do you know?”

The materials reviewed for Reveal Math Grade 5 meet expectations for supporting the intentional development of MP4: Model with mathematics; and MP5: Choose tools strategically, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

Each Unit Overview, Math Practices and Processes section, identifies one mathematical practice that is prevalent in the unit, and gives an overview of its use within the unit. In the Standards section of each lesson, mathematical practices for the lesson are also identified; however, in both sections, the mathematical practice is MPP Model with mathematics, rather than MP4. Additionally, the math practices are not identified within the lesson sections, therefore leaving where they are specifically addressed up for interpretation and possible misidentification. 

Examples of intentional development of students modeling with mathematics in connection to grade-level content, including guidance for teachers to engage students in MP4 include:

• In Lesson 2-2, Use Unit Cubes to Determine Volume, Reinforce Understanding, Small Group, “Give each student 30 unit cubes. Have each student create a rectangular solid using some or all of the cubes to find the volume of the figure. Have students switch figures with another student and find the volume of the figure. Then have students switch again so that each student finds the volume of all three figures. If necessary, remind students that they can count the number of cubes used to find the volume.” Students engage with MP4 as they check to see whether an answer makes sense and change the model when necessary as they measure volumes by counting unit cubes. 

• In Lesson 2-3, Use Formulas to Determine Volume, Differentiate, Build Proficiency, Student Practice Book, Exercise 7, “A window air conditioner can cool a space of up to 50 cubic meters. The floor of a room has an area of 16 square meters, and the height of the walls is 3 meters. Will the air conditioner be able to cool the room? Explain.” Students engage in MP4 as they use the math they know to solve problems and everyday situations.

• In Lesson 8-3, Represent Division of Decimals by a Whole Number, Guided Exploration, Math is...Modeling, students answer, “How do decimal grids help you understand dividing decimals by a whole number?” Students reflect on using decimal grids as a strategy to help them divide decimals by whole numbers.

Examples of intentional development of students using appropriate tools strategically in connection to grade-level content, including guidance for teachers to engage students in MP5 include:

• In Lesson 4-1, Estimate Sums and Differences of Decimals, Practice & Reflect, On My Own, Exercise 8,  “What is a reasonable estimate for the sum or difference? Explain the strategy you used? 5.42 - 1.7=   .” Students engage in MP5 as they choose an appropriate strategy to make a reasonable estimate.

• In Unit 9, Add and Subtract Fractions, Unit Resources, Application Station, Real World Card, Create and Solve, “Create a multi-step problem that adds and subtracts mixed numbers to solve. Then use a digital tool to present the problem to your group and find the solution together. 1. What digital tool did you use to present your problem? 2. What did you like about this digital tool? 3. Was there something you could not do using this digital tool? 4. How might you use this digital tool again?” Students engage in MP5 to choose tools (and create a digital tool) to solve multi-step word problems.

• In Lesson 10-3, Multiply Mixed Numbers, Assess, Exit Ticket, Exercise 3, “Jacob chooses a pumpkin that weighs 6\frac{3}{5} kilograms. Kaleigh chooses a pumpkin that weighs 1\frac{3}{4} times as much as Jacob’s pumpkin. How many kilograms does Kaleigh’s pumpkin weigh?” Students engage in MP5 as they choose tools and strategies to solve fraction word problems.

The materials reviewed for Reveal Math Grade 5 meet expectations that there is 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.  

Each Unit Overview, Math Practices and Processes section, identifies one mathematical practice that is prevalent in the unit, and gives an overview of its use within the unit. In the Standards section of each lesson, mathematical practices for the lesson are also identified; however, the mathematical practice is labeled MPP Attend to precision, rather than MP6: Attend to precision. Additionally, the math practices are not identified within the lesson sections, therefore leaving where they are specifically addressed up for interpretation and possible misidentification.

The instructional materials address MP6 in the following components:

• In the Digital Teacher Center, Program Overview: Learning & Support Resources, Implementation Guide, Language of Math, Unit-level Features, “The Language of Math feature highlights math terms that students will use during the unit. New terms are highlighted in yellow. Terms that have a math meaning different from everyday means are also explained.” Math Language Development, “This feature targets one of four language skills - reading, writing, listening, speaking - and offers suggestions for helping students build proficiency with these skills in the math classroom.” Lesson Level Features, “The Language of Math feature promotes the development of key vocabulary terms that support how we talk about and think about math in the context of the lesson content.” Each Unit Review also includes a vocabulary review component which references specific lessons within the unit.

Examples of intentional development of MP6: attend to precision, in connection to the grade-level content standards, as expected by the mathematical practice standards, including guidance for teachers to engage students in MP6 include:

• In Lesson 5-7, Multiply Multi-Digit Factors Fluently, Reinforce Understanding, Independent Work, Exercise 8, “Find the product of each equation using an algorithm. 1,786 x 62.” Students attend to precision by multiplying a four-digit whole number by a two-digit whole number with accuracy.

• In Lesson 6-1, Patterns When Multiplying Decimals by Powers of 10, On My Own, Problem 6, “Juan walks 4.7 x 10³ meters from his house to the museum. Mary walks 9.3 x 10² meters from her house to the museum. Who walks farther, Juan or Mary? How do you know?” Students attend to precision in calculations with exponents.

• In Unit 8, Divide Decimals, Performance Task Exercise 11, “What is the quotient? 9.72 \div3” Students attend to precision as they divide decimals.

Examples of where the instructional materials attend to the specialized language of mathematics, including guidance for teachers to engage students in MP6 include:

• In Unit 3, Teacher Edition, Math Probe, Take Action, “Build place-value ideas by using language that reinforces place value. For example, rather than reading 3.45 as three point four five, students should read it as three and forty-five hundredths.” Students attend to the specialized language of math.

• In Lesson 7-1, Division Patterns with Multi-Digit Numbers, Own My Own, Reflect, “How does using place-value patterns and basic facts help you divide whole numbers by multiples of 10?” Students attend to the specialized language of math including place-value, division, and multiples.

• In Unit 9, Add and Subtract Fractions, Unit Review, Vocabulary Review, Exercise 2, “In order to find the sum of fractions with unlike denominators, you can rewrite the fraction using ______ so that the denominators are alike.” Students attend to the specialized language of math by completing a vocabulary review to check their understanding of fractions.

The materials reviewed for Reveal Math Grade 5 meet expectations for supporting the intentional development of MP7: Look for and make use of structure; and MP8: Look for and express regularity in repeated reasoning, for students, in connection to grade-level content standards, as expected by the mathematical practice standards.

Each Unit Overview, Math Practices and Processes section, identifies one mathematical practice that is prevalent in the unit, and gives an overview of its use within the unit. In the Standards section of each lesson, mathematical practices for the lesson are also identified; however, the mathematical practice is labeled MPP Look for and make use of structure, rather than MP7: Look for and make use of structure; and MP8: Look for and express regularity in repeated reasoning. Additionally, the math practices are not identified within the lesson sections, therefore leaving where they are specifically addressed up for interpretation and possible misidentification. 

Examples of intentional development of students looking for and making use of structure, to meet its full intent in connection to grade-level content, including guidance for teachers to engage students in MP7 include:

• In Lesson 6-1, Patterns When Multiplying Decimals by Powers of 10, Reinforce Understanding, Exercise 5, “Use patterns to help you find the value of each expression. 1.3 x 10² = ____, 1.3 x 10³ = ____, 1.3 x 10⁴ = ____.” Students engage in MP7 as they make connections between multiplying whole numbers by powers of ten to multiplying decimals by powers of ten.

• In Lesson 9-8, Launch, Notice and Wonder, “What do you notice? What do you wonder? Pose Purposeful Questions: The questions that follow may be asked in any order. They are meant to help advance students’ exploration of using regrouping to add and subtract mixed numbers and are based on possible comments and questions that students may make during the share out. What is missing from each part? How can you make wholes? Let’s think about when we need to, and how we can, use regrouping to subtract mixed numbers.”  Students engage in MP7 as they look for and explain the structure within mathematical representations. 

• In Lesson 13-5, Properties of Quadrilaterals, Guided Exploration, Math is...Structure, students answer “How can you compare the attributes of quadrilaterals and triangles?” Students engage with MP7 as they determine the properties that define categories.

Examples of intentional development of students looking for and expressing regularity in repeated reasoning, including guidance for teachers to engage students in MP 8 include:

• In Lesson 5-6, Relate Partial Products to an Algorithm, On My Own, Reflect, “How are partial products and an algorithm for multiplication related?” Students engage with MP8 as they make generalizations about multiplication strategies.

• In Lesson 11-4, Divide whole Numbers by Unit Fractions, On My Own, Exercise 15, Extend Your Thinking, “When a whole number is divided by a fraction that is less than 1, will the quotient always be greater than the whole number? Explain why or why not.” Students engage with MP8 as they evaluate the reasonableness of the answers and thinking.

• In Lesson 14-5 Relate Numerical Patterns, Guided Exploration, Math is...Structure, students answer “How are the terms in Pattern A related to their corresponding terms in Pattern B?” Students engage with MP8 as they make inferences about inverse relationships.