5th Grade - Gateway 2

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

Materials develop conceptual understanding throughout the grade level. According to the Teacher’s Edition’s Program Overview, “conceptual understanding and problem solving are crucial aspects of the curriculum.” In the Topic Overview, Math Background: Rigor, “Conceptual Understanding Background information is provided so you can help students make sense of the fundamental concepts in the topic and understand why procedures work.” Each Topic Overview includes a description of key conceptual understandings developed throughout the topic. The 3-Act Math Task Overview indicates the conceptual understandings that students will use to complete the task. At the lesson level, Lesson Overview, Rigor, the materials indicate the Conceptual Understanding students will develop during the lesson.

Materials provide opportunities for students to develop conceptual understanding throughout the grade level. The Visual Learning Bridge and Guided Practice consistently provide these opportunities. Examples include: 

• Topic 3, Lesson 3-1, Lesson Overview, Conceptual Understanding states, “Students extend their understanding of place values and powers.” In Solve & Share, students apply their knowledge of place value to find products of whole numbers and powers of 10 using patterns and mental math. The materials state, “At Izzy’s Party Store, party invitations come in packages of 8.  How many invitations are in 10 packages? 100 packages? 1,000 packages? Solve this problem any way you choose.” The image of a girl states, “You can use appropriate tools. Place-value blocks are useful for picturing problems that involve powers of 10.” The materials show place-value blocks: one single cube, one stack of 10, one square of 100, and one cube of 1000. Students develop conceptual understanding as they use patterns in the number of zeros of a product to multiply a number by a power of 10. (5.NBT.2)

• Topic 8, Lesson 8-8, Lesson Overview, Conceptual Understanding states, “Students use number sense to decide whether multiplying a number n by a scale factor b results in a product that is greater than or less than n.” In Guided Practice, Do You Understand?, Problem 2, students reason about how multiplying a given number by another number changes its value. “2. Which of the following are less than 8?” Choices are 8 \times \frac{9}{10}, 8 \times \frac{7}{6}, and \frac{3}{5} \times 8. Students develop conceptual understanding as they compare the size of a product to the size of a fractional factor. (5.NF.5a and 5.NF.5b)

• Topic 11, Lesson 11-2, Lesson Overview, Conceptual Understanding states, “Students use unit cubes to develop a formula for the volume of a rectangular prism.” In the Visual Learning Bridge, students consider the essential question, “How can you use a formula to find the volume of a rectangular prism?” The materials show three frames: A) shows a 6 unit \times 4 unit \times 3 unit rectangular prism with grid lines. “Remember that volume is the number of cubic units (units³) needed to pack a solid figure without gaps or overlaps. Find the volume of the rectangular prism if each cubic unit represents 1 cubic foot.” An image of a girl states, “You can find the volume of a rectangular prism by counting cubes or using a formula. A formula is a rule that uses symbols to relate two or more quantities.” B) shows a transparent 6 ft \times 4 ft \times 3 ft rectangular prism and a step-by-step volume calculation. “If the dimensions of a rectangular prism are given as length l, width w, and height h, then use this formula to find the volume : V = l \times w \times h. The volume of the rectangular prism is 72 cubic feet or 72 ft³.” C) shows a rectangular prism whose base is shaded and a step-by-step volume calculation. “Another formula for the volume of a rectangular prism is V = b \times h, where b is the area of the base.” Classroom Conversation asks students the following questions: “A) What three measurements can you use to calculate the volume of a rectangular prism? What is a cubic foot? B) What formula can you use to calculate the volume of the prism? C) What is another formula that you can use to calculate the volume of a prism? Why is the volume given in cubic feet?” Students develop conceptual understanding as they measure volume by counting unit cubes and applying the volume formulas. (5.MD.4 and 5.MD.5b)

Materials provide opportunities for students to independently demonstrate conceptual understanding throughout the grade level. The Practice problems consistently provide these opportunities. Examples include:

• Topic 5, Lesson 5-4, Lesson Overview, Conceptual Understanding states, “Students begin to develop conceptual understanding of algorithms for dividing by 2-digit numbers.” In Independent Practice, Problem 7, students use partial quotients to divide. “7. 631 \div 14. Add the partial quotients: + + = _ with ___ left over.” Students independently demonstrate conceptual understanding by finding whole-number quotients of whole numbers dividends with two-digit divisors. (5.NBT.6)

• Topic 10, Lesson 10-1, Lesson Overview, Conceptual Understanding states, “Students learn to read a line plot.” In Independent Practice, Problems 6–8, students use a line plot to answer questions. “6. How many orders for cheese does the line plot show? 7. Which amount of cheese was ordered most often? 8. How many more orders for cheese were for \frac{3}{4} pound or less than for 1 pound or more?” The materials show a line plot entitled “Orders for Cheese.” The number line, Amount (in pounds), shows two Xs above \frac{1}{4}, four Xs above \frac{1}{2}, three Xs above \frac{3}{4}, two Xs above 1, and one X above 1\frac{1}{2}. An image of a girl states, “Data in a line plot can be shown with dots or Xs.” Students independently demonstrate conceptual understanding by using operations on fractions to solve problems involving information presented in a line plot. (5.MD.2)

• Topic 15, Lesson 15-1, Lesson Overview, Conceptual Understanding states, “Students extend their understanding of numerical patterns.” In Independent Practice, Problem 7, students use a rule to answer a word problem. “7. If Tim and Jill continue saving in this way, how much will each have saved after 10 weeks? Explain how you decided.” The materials show a table of weekly “Piggy Bank Savings” for Tim and Jill. Students independently demonstrate conceptual understanding by generating two numerical patterns given a rule. (5.OA.3)

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

The materials develop procedural skills and fluency throughout the grade level within various portions of lessons. The Teacher’s Edition Program Overview indicates, “Students perform better on procedural skills when the procedures make sense to them. So procedural skills are developed with conceptual understanding through careful learning progressions. … A wealth of resources is provided to ensure all students achieve success on the fluency expectations of Grades K-5.” Various portions of lessons that allow students to develop procedural skills include Solve & Share, Visual Learning Bridge, Convince Me!, Guided Practice, and 3-ACT MATH; in addition, the materials include Fluency Practice Activities. Examples include: 

• Topic 3, Lesson 3-4, Lesson Overview, Procedural Skill states,  “Students extend their proficiency with multiplication as they multiply two 2-digit numbers.” In Solve & Share, students develop procedural skills and fluency as they solve a problem by multiplying two 2-digit numbers. Their work shows prior and emerging understandings.” The materials pose the problem, “Ms. Silva has 12 weeks to train for a race. Over the course of one week, she plans to run 15 miles. If she continues this training, how many miles will Ms. Silva run before the race?” A girl suggests, “You can use partial products to help make sense of and solve the problem.” (5.NBT.5)

• Topic 7, Lesson 7-5, Lesson Overview, Procedural Skill states, “Students add and subtract fractions with unlike denominators to solve problems.” In Guided Practice, Do You Know How?, Problem 6, students develop procedural skills and fluency as they rewrite fractions with a common denominator to add and subtract them. “Find the sum or difference. \frac{7}{8}+(\frac{4}{8}-\frac{2}{4})”  (5.NF.1)

• Topic 11, Lesson 11-3, Lesson Overview, Procedural Skill states, “Students find the volume of solid figures made up of nonoverlapping rectangular prisms.” In Guided Practice, Problem 3, students develop procedural skills and fluency when they find the volume of each solid figure. The materials show a solid figure that consists of two rectangular prisms; side lengths are labeled. A sample student response is "15 \times 7 \times 2 = 210; 5 \times 7 \times 6 = 210; 210 + 210 = 420 in³.” (5.MD.5c)

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

• Topic 4, Lesson 4-4, Lesson Overview, Procedural Skill states, “Students’ proficiency builds as they use their knowledge of place value to accurately place the decimal point in the product.” In Independent Practice, Problem 20, students independently demonstrate procedural skills and fluency by using what they know about whole-number multiplication and place value to multiply decimals to the hundredths. The materials prompt, “Find each product. 20. 2. 54 \times 12.” (5.NBT.7)

• Topic 9, Lesson 9-4, Lesson Overview, Procedural Skill states, “Students continue to learn procedures for using models of drawings to divide a whole number by a unit fraction.” In Problem Solving, Problem 15, students independently demonstrate procedural skill and fluency by applying previous understandings of division to divide a whole number by a unit fraction.The materials present the context and prompt, “Dan has 4 cartons of juice.  He pours \frac{1}{8} carton for each person on a camping trip. How many people can he serve? Draw a picture to help you answer the question.” The materials provide images of 4 cartons that students may partition into eighths. (5.NF.7b)

• Topic 15, Lesson 15-2, Lesson Overview, Procedural Skill states, “Students use tables and given rules to learn procedures for extending two patterns and to find relationships between the two patterns.” In Problem Solving, Problem 10, students independently demonstrate procedural skill and fluency as they generate two numerical patterns using two given rules. “Higher Order Thinking At their family’s pizzeria, Dan makes 8 pizzas in the first hour they are open and 6 pizzas each hour after that.  Susan makes 12 pizzas in the first hour and 6 pizzas each hour after that.  If the pizzeria is open for 6 hours, how many pizzas will they make in all? Complete the table using the rule “add 6” to help you.” The materials show a table entitled “Number of Pizzas Made”; the columns are labeled Hour, Dan, and Susan. Students fill in the Dan and Susan columns for 1–6 hours. (5.OA.3)

The materials for enVision Mathematics Grade 5 meet expectations that the materials are designed so that teachers and students spend sufficient time working with engaging applications of the mathematics.

Engaging applications—which include single and multi-step, routine and non-routine applications of the mathematics—appear throughout the grade level and allow for students to work with teacher support and independently. In each Topic Overview, Math Background: Rigor provides descriptions of the concepts and skills that students will apply to real-world situations. Each Topic is introduced with a STEM Project, whose theme is revisited in activities and practice problems in the lessons. Within each lesson, Application is previewed in the Lesson Overview. Practice & Problem Solving sections provide students with opportunities to apply new learning and prior knowledge.

Examples of routine applications of the math include:

• In Topic 2, Lesson 2-6, Solve & Share, students add and subtract decimals to hundredths. “At a baseball game, Sheena bought a sandwich for $6.95 and two pretzels for $2.75 each. She paid with a $20 bill. How much change did she receive? Solve this problem any way you choose. Use bar diagrams to help.” The materials show two blank bar diagram templates. (5.NBT.7)

• In Topic 7, Lesson 7-3, Independent Practice, Problem 5, students add fractions with unlike denominators by making equivalent fractions. “Find each sum. Use fractions strips to help. 5. \frac{1}{6}+\frac{1}{3}+\frac{1}{6}=” The materials include the image of a boy saying, “Remember that you can use multiples to find a common denominator.” (5.NF.1)

• In Topic 14, Lesson 14-2, Assessment Practice, Problem 25, students independently solve a routine word problem involving representing a real-world problem by graphing points in the first quadrant of the coordinate plane. “Talia draws a map of her neighborhood on a coordinate grid. Her map shows the school at S(1,6), her house at H(4,3), and the library at L(7,2). Graph and label each location on the grid at the right.” The materials provide an 8-unit by 8-unit first quadrant graph of the coordinate plane. (5.G.2)

Examples of non-routine applications of the math include:

• In Topic 9, Lesson 9-7, Guided Practice, Problem 3, students solve real-world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions. “Tamara needs tiles to make a border for her bathroom wall. The border will be 9 feet long and \frac{1}{3}foot wide. Each tile measures \frac{1}{3} foot by \frac{1}{3} foot. Each box of tiles contains 6 tiles. How many boxes of tiles does Tamara need? Write two equations that can be used to solve the problem.” (5.NF.7c)

• In Topic 11, 3-Act Math Task, Fill ‘er Up, students apply their understanding of volume as they engage with the main question, “How many bags of ice do you need to fill the cooler?” Before watching a video in Act 1, the materials ask students to think about this statement, “Ice is frozen below 32°F (or 0°C), but most picnics and cookouts happen when it’s warm out. The insulated walls of a cooler help keep ice from melting, which keeps my juice nice and cold.” Teachers ask students what they know about thermoses and coolers and ask them to share stories about picnics, coolers, and thermoses. During Act 1 students watch a video of someone pouring a bag of ice into a partially filled cooler. The students do not know exactly how many bags of ice will fill the cooler but make a prediction about how many bags of ice fit in the cooler. During Act 2 the materials provide students with images of a single ice cube next to a can of water, a bag of ice next to a cooler, a cooler with one bag of ice poured in, and a uniform stack of eleven ice cubes that marks the height of the cooler. Students consider how they can use this new information and apply their understanding of volume of cylinders, cubes, and rectangular prisms to determine how many bags of ice are needed to fill the cooler. During Act 3, the materials reveal the answer. Students discuss why some predictions were closer to the answer in the video than others and come to accept a model as useful even if it is not perfect. In the Sequel, students answer the question, “What fraction of the cooler’s volume is drinks and what fraction is ice?” (5.MD.5a)

• In Topic 15, Lesson 15-4, Problem Solving, Performance Task, Problem 6-8, students describe how they could use different methods to solve a problem, write two rules, and graph the information from the rules. “Jordan is running in a track-a-thon to raise money for charity. Who will make a larger donation, Aunt Meg or Grandma Diane? Explain. 6. Make Sense and Persevere How can you use tables and a graph to solve the problem? 7. Use Appropriate Tools For each pledge, write a rule and complete the table. 8. Use Appropriate Tools On the grid, graph the ordered pairs in each table.” The materials include a chart “Track-a-Thon Pledges” indicating that Aunt Meg will pledge “$8 plus $2 per lap” and Grandma Diane will pledge “$15 + $1 per lap.” (5.OA.3)

​The materials for enVision Mathematics Grade 5 meet expectations that the three aspects of rigor are not always treated together and are not always treated separately. 

Each Topic Overview contains Math Background: Rigor, where the components of Rigor are addressed. Every lesson within a topic contains opportunities for students to build conceptual understanding, procedural skills and fluency, and/or application. During Solve and Share and Guided Practice, students explore alternative solution pathways to master procedural fluency and develop conceptual understanding. During Independent Practice, students apply the content in real-world applications, use procedural skills and/or conceptual understanding to solve problems with multiple solutions, and explain/compare their solutions.

The three aspects of rigor are present independently throughout the grade. For example:

• Topic 2, Lesson 2-3, Problem Solving, Problem 13, students attend to conceptual understanding as they use place-value blocks to show how to subtract decimals to hundredths. “Write an expression that is represented by the model below.” The model shows place value blocks worth 1.45 with 0.31 outlined and removed. Students write the expression 1.45 - 0.31. (5.NBT.7)

• Topic 6, Lesson 6-6, Independent Practice, Problems 4-6, students attend to application as they solve real-world problems involving multiplication of fractions and decimal equivalents.  “Reasoning Sue made chicken soup by combining the entire can of soup shown with a full can of water. How many 10-fluid ounce bowls can she fill with the soup?  How much soup will be left over? 4 . Explain what each of the quantities in the problem means. 5. Describe one way to solve the problem. 6. What is the solution to the problem? Explain.” The materials show a can of soup indicating 18.6 fl oz.  (5.NF.7)

• Topic 14, Lesson 14-3, Independent Practice, Problem 5, students attend to procedural skills and fluency as they represent real-world problems by graphing points in the first quadrant of the coordinate plane and interpreting coordinate values of points in the context of the situation. “Find the missing coordinates and tell what the point represents.” The materials show a graph of Yosemite Wildlife Sightings, with a point in red that students will have to find the coordinates for, the horizontal axis represents deer and the vertical axis represents elk. (5.G.2)

Multiple aspects of Rigor are engaged simultaneously to develop students’ mathematical understanding of a single topic/unit of study throughout the materials. For example:

• Topic 1, Lesson 1-5, Problem Solving, Problem 12, students attend to conceptual understanding and procedural skills and fluency as they compare decimals to the thousandths to satisfy a given criteria. “Number Sense Carlos wrote three numbers between 0.33 and 0.34. What numbers could Carlos have written?” (5.NBT.3b)

• Topic 9, Lesson 9-6, Independent Practice, Problem 11, students apply to conceptual understanding and application as they extend their understanding of division with whole numbers and unit fractions by completing a picture that represents the solution of a problem. “Keiko divided 5 cups of milk into \frac{1}{4}-cup portions. How many \frac{1}{4}-cup portions did Keiko have? Complete the picture to show your solution.” The materials show an array of 5 squares. (5.NF.7a)

• Topic 16, Lesson 16-3, Independent Practice, Problem 11, students attend to application, conceptual understanding and procedural skills and fluency as they apply understandings of attributes of a two-dimensional figure to determine if statements are true or false. “Write whether each statement is true or false. If false, explain why. 11. What properties does the shape have? Why is it not a parallelogram?” The materials show a kite. (5.G.B)

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

Students have opportunities to engage with the Math Practices across the year, and they are explicitly identified for teachers within the Program Overview within the Topic Contents at the Lesson-level. The Math Practices and Problem Solving Handbook introduces each of the Math Practices with specific emphasis on making connections among representations to develop meaning and corresponding Thinking Habits. The Teacher’s Edition provides support for developing, connecting, and assessing each math practice. Topic Planners include the Math Practices at the lesson level; relevant practices are specified in Lesson Overviews. 

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

• Topic 1, Lesson 1-6, Problem Solving, Problem 24, students make sense of problems and persevere in solving them as they figure out the fractional amount of ham sandwiches. “Make Sense and Persevere Robert slices a large loaf of bread to make 12 sandwiches. He makes 3 turkey sandwiches and 5 veggie sandwiches. The rest are ham sandwiches. What fraction of the sandwiches Robert makes are ham?” 

• Topic 7, Lesson 7-3, Solve & Share and Look Back!, students make sense of problems and persevere in solving them as they add fractions with unlike denominators by replacing given fractions with equivalent fractions having like denominators. Solve & Share, “Over the weekend, Eleni ate \frac{1}{4} box of cereal, and Freddie ate \frac{3}{8} of the same box. What portion of the box of cereal did they eat in all? Solve this problem any way you choose.” The materials show a box of cereal; vertical measurements \frac{1}{4} and \frac{3}{8} are shown on the side of the box. An image of a girl states, “You can use fraction strips to represent adding fractions. Show your work!”  Look Back!, “Make Sense and Persevere What steps did you take to solve this problem?” 

• Topic 14, Lesson 14-4, Problem Solving, Performance Task, Problem 7, students make sense of problems and persevere in solving them as they use a graph to represent a real-world mathematical problem by graphing points in the first quadrant of the coordinate plane and interpret coordinate values of points in the context of the situation. “Rozo Robot A toy company is testing Rozo Robot. Rozo is 18 inches tall and weighs 2 pounds. The employees of the company marked a grid on the floor and set Rozo at (2, 5). They programmed Rozo to walk 3 yards east and 4 yards north each minute. What will Rozo’s location be after 7 minutes? Make Sense and Persevere Do you need all of the information given in the problem to solve the problem? Describe any information that is not needed.” The materials show a blank first quadrant of the coordinate plane. 

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

• Topic 3, Lesson 3-7, Problem Solving, Problem 24, students reason abstractly and quantitatively as they explain which trips would cost more between two different choices. “Use the table. Which would cost more: 15 trips to Boston or 11 trips to New York? Explain.” The materials show the data table, “Airfare Prices,” which lists four destinations and their corresponding ticket costs. 

• Topic 8, Lesson 8-4, Solve & Share and Look Back!, students reason abstractly and quantitatively as they explain if their answer should be less than or greater than 1. Solve & Share, “The art teacher gave each student half of a sheet of paper. Then she asked the students to color one fourth of their pieces of paper. What part of the original sheet did the students color? Solve this problem any way you choose.” A girl states, “You can draw a picture to represent the problem.” Look Back!, “Reasoning Should your answer be less than or greater than 1? How do you know?” 

• Topic 12, Lesson 12-1, Problem Solving, Problem 21, students reason abstractly and quantitatively as they consider why to use one conversion over another. “Reasoning The dimensions of the nation’s smallest post office are 8 feet 4 inches by 7 feet 3 inches. Why would you use the measurement 8 feet 4 inches instead of 7 feet 16 inches?”

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

Students have opportunities to engage with the Math Practices across the year, and they are explicitly identified for teachers within the Program Overview within the Topic Contents at the Lesson-level. The Math Practices and Problem Solving Handbook introduces each of the Math Practices with specific emphasis on making connections among representations to develop meaning and corresponding Thinking Habits. Topic Planners include the Math Practices at the lesson level; relevant practices are specified in Lesson Overviews.

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

• Topic 2, Lesson 2-2, Convince Me!, students construct viable arguments and critique the reasoning of others as they use estimation involving addition to explain and justify their thinking. “Critique Reasoning Tomàs said, ‘We did great in Week 4! We collected just about twice as many pounds as in Week 1!’ Use estimation to decide if he is right. Explain your thinking.” The materials show a data chart with the pounds of dog food collected each week for 5 weeks.

• Topic 7, Lesson 7-2, Problem Solving, Problem 12, students construct viable arguments and critique the reasoning of others as they explain errors in others’ work and provide the correct answer. “Critique Reasoning Explain any mistakes in the renaming of the fractions below.  Show the correct renaming. \frac{3}{4}=\frac{9}{12} \frac{2}{3}=\frac{6}{12}” 

• Topic 11, Lesson 11-2, Solve & Share and Look Back!, students construct viable arguments and critique the reasoning of others as they find the volume of rectangular prisms by using a formula. Solve & Share, “Kevin needs a new aquarium for his fish. The pet store has a fish tank in the shape of a rectangular prism that measures 5 feet long by 2 feet wide by 4 feet high. Kevin needs a fish tank that has a volume of at least 35 cubic feet. Will this fish tank be big enough? Solve this problem any way you choose.” The materials show a girl looking inside an aquarium saying, “Read the problem carefully to make sure that you understand what you are trying to find. Show your work!” Look Back!, “Critique Reasoning Malcolm says the volume of the aquarium would change if its dimensions were 2 feet long, 4 feet wide, and 5 feet high. Do you agree? Explain.” 

• Topic 16, Lesson 16-4, Problem Solving, Performance Task, Problems 8 and 10, students construct viable arguments and critique the reasoning of others as they solve problems about geometric figures. “Flag Making Mr. Herrera’s class is studying quadrilaterals. The class worked in groups, and each group made a ‘quadrilateral flag.’ The materials show four flags that each contain three or four quadrilaterals. “8. Construct Arguments Which flags show parallelograms? Construct a math argument to justify your answer…10. Critique Reasoning Marcia’s group made the red flag.  Bev’s group made the orange flag. Both girls say their flag shows all rectangles. Critique the reasoning of both girls and explain who is correct."

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

Students have opportunities to engage with MP4 and MP5 across the year, and they are explicitly identified for teachers within the Program Overview within the Topic Contents at the Lesson-level.  The Math Practices and Problem Solving Handbook introduces each of the Math Practices with specific emphasis on making connections among representations to develop meaning and corresponding Thinking Habits. The Teacher’s Edition provides support for developing, connecting, and assessing each math practice. Topic Planners include the Math Practices at the lesson level; relevant practices are specified in Lesson Overviews. 

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 the support of the teacher and independently throughout the Topics. Examples include:

• Topic 2, Lesson 2-6, Independent Practice, Problems 3-5, students model with mathematics as they use bar diagrams to solve multi-step problems involving addition and subtraction of decimals. “Model with Math Luz Maria has $15. She buys a ticket to a movie and a smoothie. How much money does she have left? 3. What do you need to find before you can solve the problem? 4. Draw two bar diagrams to solve the problem. 5. What is the solution to the problem? Show the equations you used to solve the problem.” The materials show a sign with prices: Ticket $9.50, Popcorn $4.50, Smoothie $2.85.

• Topic 7, Lesson 7-12, Solve & Share and Look Back!, students model with mathematics as they use different representations to solve multi-step real-world problems involving adding and subtracting mixed numbers. Solve & Share, “Annie found three seashells at the beach. How much shorter is the Scotch Bonnet seashell than the combined lengths of the two Alphabet Cone seashells? Solve this problem any way you choose. Use a diagram to help.” The materials show that the Scotch Bonnet is 2\frac{1}{8} inches in length and the Alphabet Cone is 1\frac{3}{4} inches. Look Back!, “Model with Mathematics What is another way to represent this problem?” 

• Topic 10, Lesson 10-4, Problem Solving, Performance Task, Problem 10, students model with mathematics as they use line plots to solve multi-step word problems and to determine if their answer makes sense. “Television Commercials Ms. Fazio is the manager of a television station. She prepared a line plot to show the lengths of the commercials aired during a recent broadcast. She concluded that the longest commercials were 3 times as long as the shortest ones because 3 \times \frac{1}{2} = 1\frac{1}{2}.” The materials show a line plot, “TV Commercials,” that indicates time durations of \frac{1}{2} a minute, 1 minute, and 1\frac{1}{2} minutes. “10. Model with Math Did Ms. Fazio use the correct operation to support her conclusion? Explain.” 

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

• Topic 1, Lesson 1-1, Solve & Share, students use appropriate tools strategically as they attend to explaining patterns in the number of zeros of the product when multiplying a number by powers of 10 as they use appropriate tools to represent their thinking. “A store sells AA batteries in packages of 10 batteries. They also sell boxes of 10 packages, cases of 10 boxes, and cartons of 10 cases. How many AA batteries are in one case? One carton? 10 cartons? Solve these problems any way you choose.” The materials show place-value blocks: a thousand cube, a hundred square, and a ten stack. A girl states, “You can use appropriate tools, such as place-value blocks, to help solve the problem. However you choose to solve it, show your work!”

• Topic 9, Lesson 9-4, Solve & Share, students use appropriate tools strategically as they use area models, number lines, and drawings to solve real-world word problems involving division of a whole number by a unit fraction. “One ball of dough can be stretched into a circle to bake a pizza. After the pizza is cooked, it is cut into 8 equal slices. How many slices of pizza can you make with 3 balls of dough? Solve this problem any way you choose.” The materials show a circle that is partitioned into eighths. A girl states, “You can use appropriate tools to help find the answer. Show your work!”

• Topic 16, Lesson 16-3, Visual Learning Bridge, students use appropriate tools strategically as they show how two-dimensional figures are related to each other using a Venn Diagram.  (A) “This Venn diagram shows how special quadrilaterals are related to each other. How can you use the Venn diagram to describe other ways to classify a square? What does the diagram show about how a trapezoid relates to other special quadrilaterals?” Teacher guidance prompts, “Reason Abstractly Why is Quadrilaterals written at the top of the rectangle in the Venn diagram? What properties does a square have? Why is trapezoids shown in a separate circle?” The materials show a Venn Diagram that relates parallelograms, rectangles, squares, rhombuses, and trapezoids. A boy states, “A Venn diagram uses overlapping circles to show relationships between items.”

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

Students have opportunities to engage with MP6 across the year, and they are explicitly identified for teachers within the Program Overview within the Topic Contents at the Lesson-level. The Math Practices and Problem Solving Handbook introduces each of the Math Practices with specific emphasis on making connections among representations to develop meaning and corresponding Thinking Habits. Topic Planners include the Math Practices at the lesson level; relevant practices are specified in Lesson Overviews.

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

• Topic 1, Lesson 1-5, Solve & Share and Look Back!, students attend to precision when they order and compare numbers by recognizing that the value of a digit depends upon its place value. Solve & Share, “The lengths of three ants were measured in a laboratory. The lengths were 0.521 centimeter, 0.498 centimeter, and 0.550 centimeter. Which ant was the longest? Which ant was the shortest?” The materials show a cascade of place value from ones to thousandths and three ants of different sizes. Look Back!, “Be Precise What are the lengths of the ants in order from least to greatest?”

• Topic 6, Lesson 6-6, Problem Solving, Performance Task, Problem 10, students attend to precision when they divide a decimal by a two-digit whole number to solve a multi-step problem. “Cooking Competition Lucas’s cooking class is having a cooking competition.  There are 6 teams. Each student brought supplies that will be shared equally among the teams. The table shows the supplies Lucas brought. If the supplies are shared equally among the teams, how much of each supply will each team get? 10. Be Precise What is the solution to the problem? Explain.” The materials show a data table of cooking supplies and their corresponding unit prices. 

• Topic 12, Lesson 12-8, Visual Learning Bridge and Convince Me!, students use unit conversion to solve real world problems. Visual Learning Bridge (A), “A city pool is in the shape of a rectangle with the dimension shown. What is the perimeter of the pool?” The materials show a pool, with dimensions 60 feet and 25 yards, and the image of a boy who states, “You can convert one of the measures so that you are adding like units.” (B), “What do you know? The dimensions of the pool: l = 25 yards, w = 60 feet What are you asked to find? The perimeter of the pool.” The boy states, “You can use feet for perimeter.”  Teacher guidance: “Attend to Precision Why do we decide to use only one unit for the perimeter?” Convince Me!, “Be Precise If the width of the is increased by 3 feet, what would be the new perimeter of the pool? Explain.” Teacher guidance: “Attend to Precision Students can use number sense to find the new perimeter. If the width of the pool is increased by 3 feet on both sides, that means there are an additional 6 feet added to the total perimeter.”

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

• Topic 2, Review What You Know/Vocabulary Cards and Activity, Topic 2 Vocabulary, students use specialized language to reinforce understanding of terms such as Associative Property of Addition and compatible numbers. Teacher guidance: “Have students use Teaching Tool 27 (Vocabulary: Word Map) to display information about each vocabulary word. For example, have students complete the Word Map by writing one of the terms in the middle, showing examples of the term on the left, related words on the right, and writing what they know about the term at the bottom.”

• Topic 8, Review What You Know/Vocabulary Cards and Activity, Vocabulary, Problem 1, students use specialized language when they apply their understanding of terms such as equivalent fractions and mixed number to complete sentences. “Choose the best term from the box. Write it on the blank. 1. To estimate the sum of two or more fractions, replace the addends with ______.” Students complete the sentence with the appropriate term from a given word bank.

• Topic 16, Lesson 16-2, Problem Solving, Problem 15, students use specialized language when they identify triangles based upon the measure of its angles and the lengths of its sides. “Be Precise Suppose you cut a square into two identical triangles. What types of triangles will you make?”

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

Students have opportunities to engage with the Math Practices across the year, and they are explicitly identified for teachers within the Program Overview within the Topic Contents at the Lesson Level. The Math Practices and Problem Solving Handbook introduces each of the Math Practices with specific emphasis on making connections among representations to develop meaning and corresponding Thinking Habits. The Teacher’s Edition provides support for developing, connecting, and assessing each math practice. Topic Planners include the Math Practices at the lesson level; relevant practices are specified in Lesson Overviews.

MP7 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 the support of the teacher and independently throughout the Topics. Examples include:

• Topic 4, Lesson 4-1, Convince Me!, students look for and make use of structure when they multiply decimals by powers of 10 using the structure of the number system. “Use Structure Complete the chart. What patterns can you use to place the decimal point?” The materials show a 4 by 4 chart with row headings 1.275, 26.014, and 0.4 and column headings \times 10^1, \times 10^2, \times 10^3.

• Topic 6, Lesson 6-1, Solve & Share, students look for and make use of structure when they use their understanding of the structure of the decimal place-value system for multiplication by ten and apply that structure to dividing by ten. “An object is 279.4 centimeters wide. If you divide the object into 10 equal parts, how wide will each part be? Solve this problem any way you choose.” The materials show an image of a boy who states, “How can you use structure and the relationship between multiplication and division to help you?”

• Topic 13, Lesson 13-3, Problem Solving, Problem 13, students look for and make use of structure when they determine who has a greater amount by comparing the structure of numerical expressions without evaluating them. “Use Structure Peter bought 4 \times (2\frac{1}{4} + \frac{1}{2} + 2\frac{7}{8}) yards of ribbon. Marilyn bought 4 \times (2\frac{1}{4} + \frac{1}{2} + 3) yards of ribbon. Without doing any calculations, determine who bought more ribbon. Explain.”

MP8 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 the support of the teacher and independently throughout the Topics. Examples include:

• Topic 4, Lesson 4-2, Convince Me!, students look for and express regularity in repeated reasoning when they use rounding and compatible numbers to estimate decimal products and evaluate whether the estimates are overestimates or underestimates. “Reasoning About how much money would 18 pounds of cheese cost if the price is $3.95 per pound? Use two different ways to estimate the product. Are your estimates overestimates or underestimates? Explain.”

• Topic 10, Lesson 10-2, Solve & Share and Look Back!, students look for and express regularity in repeated reasoning as they look to generalize methods for making line plots and using them to evaluate data. Solve & Share, “A fifth-grade class recorded the height of each student. How could you organize the data? If all the students in the class lay down in a long line, how far would it reach? Make a line plot to solve this problem.” The materials show a set of data that represents the “Heights of Students in Grade 5 (to the nearest \frac{1}{2} inch)”; heights range from 50 inches to 60 inches. A girl states, “Organizing data makes it easier to understand and analyze.” Look Back!, “Generalize How does organizing the data help you see the height that occurs most often? Explain.”

• Topic 15, Lesson 15-2, Convince Me!, students look for and express regularity in repeated reasoning as they extend a pattern beyond the given table and explain if the relationship will always hold true. “Generalize Do you think the relationship between the corresponding terms in the table Jack created will always be true? Explain.” The data table provided shows the number of Weeks (up to 5 weeks) and for each week the “Total Miles Run”, and “Total Miles Biked”.