5th Grade - Gateway 2

The instructional materials for enVision Mathematics Common Core Grade 5 meet expectations that the materials develop conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings.

Each lesson is structured to include background information for the teacher and problems and questions that develop conceptual understanding. Examples include, but are not limited to:

• Conceptual understanding for each topic is outlined in the Teacher Edition’s section Math Background: Rigor. For example, the Topic 8 Overview explains multiplying fractions. The Conceptual Understanding states, “Various models are used to develop understanding of multiplying two fractions. One very useful model is an area model. Using an area model also helps develop understanding of finding the area of a rectangle with fractional side lengths.” 
• The Teacher Edition contains a Rigor section for each lesson explaining how conceptual understanding is developed in the lesson. In Lesson 2-2=3, the Rigor section states, “Students use place-value blocks to show how decimals can be added or subtracted. They use tens and ones blocks to show part of a whole and they combine or remove blocks, regrouping as needed”  
• Each lesson begins with a Visual Learning Bridge activity that provides the opportunity for a classroom conversation to build conceptual understanding for students.  For example, Lesson 1-2 states, “Students extend their understanding that each place has a value equal to 10 times the value of the place to its right, and develop the understanding that each place has a value equal to $$\frac{1}{10}$$ of the value of the place to its left.”  Teachers are prompted to ask the following questions, “In 1,440,000, which digit has the greatest place value? Why does the 1 have a greater value than either 4?”
• Each lesson is introduced with a video: Visual Learning Animation Plus, to promote conceptual understanding. For example, the Lesson 4-5 video states, “How can you model decimal multiplication?”  The scenario begins with the following: “A farmer has a square field that is 1 mile wide by 1 mile long. Her irrigation system can water the northern 0.5 mile of her field. If her tomatoes are planted in a strip 0.3 mile wide, what is the area of her watered tomatoes?”   
• Each lesson contains a Convince Me! section that provides opportunities for conceptual understanding. For example, in Lesson 7-10, students use estimation to determine the reasonableness of answers: “Estimate $$8\frac{1}{3}-3\frac{3}{4}$$. Tell how you got your estimate. Susi subtracted and found the actual difference to be $$5\frac{7}{12}$$. Is her answer reasonable? Explain.”

The instructional materials for enVision Mathematics Common Core Grade 5 meet expectations that they attend to those standards that set an expectation of procedural skill and fluency.

The instructional materials develop procedural skill and fluency throughout the grade-level. Problem sets provide opportunities to practice procedural fluency. Regular opportunities for students to attend to Standard 5.NBT.5: Fluently multiply multi-digit whole numbers using the standard algorithm. Examples include, but are not limited to:

• Each topic contains a Math Background: Rigor page with a section entitled “Procedural Skill and Fluency.” In Topic 3, this section states, “Topic 3 carefully develops understanding and fluency with using the standard multiplication algorithm for whole numbers. Students start by multiplying 1-digit numbers by 2-, 3-, and 4-digit numbers in Lesson 3-3. This is extended to multiplying two 2-digit numbers in Lesson 3-4 and then to multiplying two 2-digit numbers in Lessons 3-5 and 3-6. Finally, students practice the multiplication algorithm for all of these in Lesson 3-7. The procedure involves multiplying one place at a time.”
• Fluency Practice Activities are found at the end of Topics 3 through 16 to support multi-digit multiplication. In Topic 3, the Fluency Activity states, “Solve each problem. Then follow multiples of 10 to shade a path from START to FINISH. You can only move up, down, right, or left.”  For example, “70 x 89 = __."
• The Performance Task for Topic 3 provides students the opportunity to demonstrate fluency of using the standard algorithm when multiplying. Students are given information on how much elephants eat daily and asked, “How much does Maxie eat in 2 weeks? Complete the computation below." Students calculate 308 x 14.
• Students can practice fluency skills when accessing the Game Center Online at PearsonRealize.com.

he instructional materials for enVision Mathematics Common Core Grade 5 meet expectations that the materials are designed so that teachers and students spend sufficient time working with engaging applications of grade-level mathematics. Engaging applications include single and multi-step problems, routine and non-routine, presented in a context in which the mathematics is applied.

Materials provide opportunities for students to solve a variety of problem types requiring the application of mathematics in context. Additionally, the materials support teachers by explaining how the students will apply concepts they have learned within each topic in the Math Background: Rigor section of the Topic Overview. 

Students are provided opportunities to work with routine problems presented in context that require application of mathematics.  Examples include, but are not limited to:

• In Lesson 3-8, the Visual Learning Bridge states, “In 1980, a painting sold for $1,575.  In 2015, the same painting sold for 5 times as much. What was the price of the painting in 2015?”
• In Lesson 9-7, the Solve & Share section states, “Organizers of an architectural tour need to set up information tables every $$\frac{1}{8}$$ mile along the 6-mile tour, beginning $$\frac{1}{8}$$ mile from the start of the tour.  Each table needs 2 signs. How many signs do the organizers need?”

Students are provided opportunities to work with non-routine problems presented in context that require application of mathematics.  Examples include, but are not limited to:

• In Lesson 9-5, Visual Learning Bridge, students engage in non-routine applications of dividing a unit fraction by a whole number: “Half of a pan of cornbread is left over.  Ann, Beth, and Chuck are sharing the leftovers equally. What fraction of the original cornbread does each person get?” (5.NF.7c)
• In Lesson 9-7, Question 3 states, “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 Lesson 11, the Performance Task states, “Hiroto works in a sporting goods store. #1  Hiroto stacks identical boxes of golf balls to form a rectangular prism. Each box is a cube. Part A:  How many boxes are in the Golf Ball Display? Part B: Explain how the number of boxes you found in Part A is the same as what you would find by using the formula V=l x w x h.” (5.MD.C) 

Students are provided opportunities to independently demonstrate the use of mathematics flexibly in a variety of contexts. Examples include, but are not limited to:

• In Lesson 3-8, Question 5 states, “A hardware store ordered 13 packs of nails from a supplier.  Each pack contains 155 nails. How many nails did the store order?” 
• In Lesson 5-5, Question 17 states, “The Port Lavaca fishing pier is 3,200 feet long.  There is one person fishing for each ten feet of length. Write and solve an equation to find how many people are fishing from the pier."
• In Lesson 6-6, the Visual Learning Bridge states, “Ms. Watson is mixing 34.6 fluid ounces of red paint and 18.2 fluid ounces of yellow paint to make orange paint. How many 12-fluid ounce jars can she fill? Use reasoning to decide.”
• In Topic 9, the Performance Task states, “Guadalupe likes to bake.  The recipe shown is for blueberry scones with a lemon glaze. Last week, Guadalupe only wanted to make a small batch of scones.  So, she used only half of each amount in the recipe. How much baking powder did Guadalupe use? Write and solve a division equation to answer the question.” 

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

All three aspects of rigor are present independently throughout the program materials. Examples include, but are not limited to:

• Conceptual Understanding is needed to solve in Lesson 5-5. Students use conceptual understanding of place value and area models to divide 3-digit dividends by 2-digit divisors. Question 1 is solved by making a model, “If the orchard has 200 seedlings and 12 are planted in each row, how many rows will be filled? Draw place value blocks to show your answer.”   
• Fluency is practiced in Lesson 3-4, Questions 7-14. Students use the standard algorithm to multiply 2-digit numbers. In Question 7, students solve, “53 x 17." 
• Students apply mathematics to solve problems in context.  In Lesson 8-9, Question 3, students solve, “Isabel is buying framing to go around the perimeter of one of her paintings.  Each inch of framing costs $0.40. What is the total cost of the framing for the painting?” The question displays a picture with a length of $$10\frac{1}{4}$$ inches and a width of $$6\frac{1}{4}$$ inches.

Multiple aspects of rigor are engaged simultaneously to develop students’ mathematical understanding of a single topic/unit of study throughout the materials. Examples include, but are not limited to:

• In Lesson 3-3, students use conceptual understanding of regrouping when using the standard algorithm to multiply 2, 3, and 4-digit numbers by 1-digit. To solve Question 13, students find the product and use estimation to determine reasonableness of their estimate for “2204 x 6."
• In Lesson 4-4, students use conceptual understanding of place value to fluently multiply decimals.  Question 22 states, “The airline that Vince is using has a baggage weight limit of 41 pounds. He has two green bags, each weighing 18.4 pounds, and one blue bag weighing 3.7 pounds. Are his bags within the weight limit? Explain.”
• In Lesson 12-5, students use their understanding of metric units to convert capacity in context. Question 24 states, “Carla makes 6 liters of punch. She pours the punch into 800 mL bottles. How many bottles can she fill?”
• In Lesson 16-2, students use their understanding of ordered pairs to solve problems in context. Question 2 states, “Suppose you have a graph of speed that shows a lion can run four times as fast as a squirrel. Name an ordered pair that shows this relationship. What does this ordered pair represent?”

The instructional materials reviewed for enVision Mathematics Common Core Grade 5 meet expectations that the Standards for Mathematical Practice are identified and used to enrich mathematics content within and throughout the grade-level.

All eight Standards for Mathematical Practice (MPs) are clearly identified throughout the materials. Math Practices are identified in the Topic Planner by lesson. In addition, Math Practices and Effective Teaching Practices (ETP) are identified for each topic, within each lesson, and for specific problems. For example, in Topic 5:

• The Topic Planner states, “MP.1, and MP.5 are addressed in Lesson 3.”
• The Math Practices and ETP addresses MP6: “Attend to precision. Students attend to precision when calculating answers to division problems. (e.g., p. 212, Item 9).”
• In Lesson 5-5, Mathematical Practice states, “MP.5 Use Appropriate Tools Strategically: Students use tools, such as money, place-value blocks, and area models to divide by 2-digit divisors. Also MP.2, MP.4.”

The MPs are used to enrich the mathematical content and are not treated separately. A Math Practices and Problem Solving Handbook is available online at PearsonRealize.com.  This resource provides a page on each math practice for students and teachers to use throughout the year. Math Practice Animations are also available for each practice to enhance student understanding. For example:

• MP1: In Lesson 7-6, Question 24, students make sense of problems and persevere in solving them. For example: “Cal has $12.50 to spend. He wants to ride the roller coaster twice and the Ferris wheel once. Does Cal have enough money? Explain. What are 3 possible combinations of rides Cal can take using money he has?”
• MP2: In Lesson 2-2, Question 21, students reason abstractly and quantitatively. For example: “The size and shape of the Golden Gate Park are often compared to the size and shape of Central Park, About how many more acres does Golden Gate Park cover than Central Park?”
• MP3: In Lesson 8-2, Question 8, students critique the reasoning of others. For example: “Janice said that when you multiply a fraction less than 1 by a nonzero whole number, the products is always less than the whole number. Do you agree? Explain."
• MP4: In Lesson 5-2, Question 23, students model with math,. For example: “The sign shows the price of baseball caps for different pack sizes. Coach Lewis will buy the medium-size pack of caps. About how much will each cap cost? Write an equation to model the problem.”
• MP5: In Lesson 14-1, Solve & Share, students understand how to use ordered pairs to clearly identify a point on the coordinate grid. For example: “On the first grid, plot a point where two lines intersect. Name the location of the point. Plot and name another point. Work with a partner. Take turns describing the locations of the points on your first grid. Then plot the points your partner describes on your second grid. Compare your first grid with your partner’s second grid to see if they match.” 
• MP6: In Lesson 1-5, Solve & Share, students attend to precision when ordering decimals. For example: “What are the lengths of the ants in order from least to greatest?” 
• MP7: In Lesson 3-6, Question 22, students look for structure in place-value relationships in order to solve multiplication problems. For example: “Trudy wants to multiply 66 x 606. She says that all she has to do is find 6 x 606 and then double that number. Explain why Trudy’s method will not give the correct answer. Then show how to find the correct product.” 
• MP8: In Lesson 15-2, Convince Me!, students use repeated reasoning when they analyze patterns and make generalizations. For example: “Do you think the relationship between the corresponding terms in the table Jack created will always be true?”  Students explain why the pattern extends beyond 5 weeks. 

The instructional materials reviewed for enVision Mathematics Common Core Grade 5 meet expectations that the instructional materials prompt students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics. 

The Solve & Share activities, Visual Learning Bridge problems, Problem Sets, 3-Act Math activities, Problem Solving: Critique Reasoning problems, and Assessment items provide opportunities throughout the year for students to both construct viable arguments and analyze the arguments of others.

Student materials consistently prompt students to construct viable arguments. Examples include, but are not limited to:

• In Lesson 2-2, Question 19 states, “The cost of one DVD is $16.98, and the cost of another DVD is $9.29. Ed estimated the cost of the two DVDs to be about $27. Is his estimate higher or lower than the actual cost? Explain."
• In Lesson 3-5, the Visual Learning Bridge states, “Last month a bakery sold 389 boxes of bagels. How many bagels did the store sell last month? Find 12 x 389.” The next question in the Convince Me! section states, “Is 300 x 10 a good estimate for the number of bagels sold at the bakery? Explain."
• In Lesson 7-2, the Solve & Share states, “Sue wants $$\frac{1}{2}$$ of a rectangular pan of cornbread. Dena wants $$\frac{1}{3}$$ of the same pan of cornbread. How should you cut the cornbread so that each girl gets the size portion she wants? Solve this problem any way you choose.” The next question in the Look Back section states, “Is there more than one way to divide the pan of cornbread into equal-sized parts? Explain how you know."

Student materials consistently prompt students to analyze the arguments of others. Examples include, but are not limited to:

• In Lesson 1-2, Question 16 states, “Paul says that in the number 6,367, one 6 is 10 times as great as the other 6. Is he correct? Explain why or why not."
• In Lesson 7-1, the Visual Learning Bridge states, students are presented with this scenario, “Mr. Fish is welding together two copper pipes to repair a leak. He will use the pipes shown. Is the new pipe closer to $$\frac{1}{2}$$ foot or 1 foot long? Explain.” The next question in the Convince Me! section states, “Nolini says that if the denominator is more than twice the numerators, the fraction can always be replaced with 0. Is she correct? Give an example in your explanation.”
• In Lesson 7-2, Problem Solving, Question 22 states, “Irene wants to list the factors for 88. She writes 2, 4, 8, 11, 22, and 44, and 88. Is Irene correct? Explain.”
• In Lesson 16-4, Question 10 states, “Mr. Herrera’s class is studying quadrilaterals. The class worked in groups, and each group made a “quadrilateral flag. 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 instructional materials reviewed for enVision Mathematics Common Core Grade 5 meet expectations that the instructional materials assist teachers in engaging students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics.

Materials assist teachers in engaging students in both constructing viable arguments and analyzing the arguments of others in a variety of problems and tasks presented to students. Examples include, but are not limited to:

• In Lesson 4-5, Visual Learning Bridge, students solve, “Nita says that the product of an even number and an odd number is always even. Is she correct?”  In the Convince Me!, students solve, “Does multiplying by 8 also always result in an even product? Explain.” Teachers guidance includes: “Students explain that a number multiplied by 8 results in an even number because 8, a multiple of 2, can be divided by 2 with none left over.  Ask students to explain why multiplying by any even number results in an even product.” 
• In Lesson 6-5, Convince Me! states, “Is 3.6 ÷ 1.2 equal to, less than, or greater than 36 ÷ 12? Explain.'' Teacher guidance includes: “Since 3.6 ÷ 1.2 = $$\frac{3.6}{1.2}$$, multiplying the dividend and divisor by the same number to get an equivalent quotient is similar to multiplying the numerator and denominator of a fraction by the same number to get an equivalent fraction.”
• In Lesson 8-6, Convince Me! states, “The fractions on the right ($$\frac{5}{8},\frac{5}{6}$$) refer to the same whole. Kelly said, ‘These are easy to compare. I just think about $$\frac{1}{8}$$ and $$\frac{1}{6}$$.’ Circle the greater fraction. Explain what Kelly was thinking.”  The Teacher’s Edition notes in the margin state, “Students explain a possible reason for Kelly’s thinking to help deepen their understanding of how to compare fractions with unlike denominators. Point out that it is always true that if two fractions have the same numerator (e.g. $$\frac{5}{6}$$ and $$\frac{5}{8}$$) the fraction with the lesser denominator is the greater fraction.” 
• In Lesson 15-1, the Visual Learning Bridge states, “Lindsey has a sage plant that is 3.5 inches tall. She also has a rosemary plant that is 5.2 inches tall. Both plants grow 1.5 inches taller each week. How tall will the plants be after 5 weeks? What is the relationship between the height of the plants?” The next section, Convince Me!  states, “If the patterns continue, how can you tell the rosemary plant will always be taller than the sage plant?” The teacher edition states, “Students explain their reasoning about the relationship between the two patterns in terms of the context. Reiterate how much the two plants grow each week and the difference between the plants’ heights at the beginning."
• In Lesson 16-3, Question 13 states, “Construct Arguments: Draw a quadrilateral with one pair of parallel sides and two right angles. Explain why this figure is a trapezoid.” The teacher edition states, “Emphasize that in order for the figure to be trapezoid, the figure needs to have only one pair of parallel sides. If needed, remind students that a figure with two pairs of parallel sides is a parallelogram." 

The instructional materials reviewed for enVision Mathematics Common Core Grade 5 meet expectations that materials explicitly attend to the specialized language of mathematics.

The materials provide explicit instruction in how to communicate mathematical thinking using words, diagrams, and symbols. Examples include, but are not limited to:

• Each topic contains a Vocabulary Review providing students the opportunity to show their understanding of vocabulary and use vocabulary in writing.
• The Teacher Edition provides teacher prompts to support oral language. For example in Topic 7, the Oral Language prompt states, “Before students complete the page, you might reinforce oral language through a class discussion involving one or more of the following activities. Have students define terms in their own words. Have students say math sentences or ask math questions that use the words." 
• The Game Center at PearsonRealize.com contains an online vocabulary game. 
• A vocabulary column is provided in the Topic Planner that lists words addressed with each lesson in the topic. In Lesson 9-4, the Vocabulary List includes: Unit Fraction. This word is also listed in the Lesson Overview. 
• Online materials contain an “Academic Vocabulary” and an “Academic Vocabulary Teacher’s Guide” section. The guide supports vocabulary instruction by providing information on how teachers can develop word meaning and build word power. The Academic Vocabulary section provides a variety of academic words with definitions and activities to help students learn the words. For instance, when clicking on the word, horizontal, the definition is provided: “Straight across from side to side.” Next, the word is used in context: “Does the figure have a horizontal line of symmetry? Where is it?” Lastly, students are provided a task to help build word power: “Use the word in a sentence."
• A glossary exists in both the Student Edition and the Teacher’s Edition Program Overview. In the glossary, distributive property is defined as: “Multiplying a sum (or difference) by a number is the same as multiplying each number in the sum (or difference) by the number and adding (or subtracting) the products. Example: 3 x (10 + 4) = (3 x 10) + (3 x 4)."
• Visual Learning Bridge activities provide explicit instruction in the use of mathematical language.The words are highlighted in yellow and a definition is provided. 
• A bilingual animated glossary is available online which uses motion and sound to build understanding of math vocabulary.

The materials use precise and accurate terminology and definitions when describing mathematics, and support students in using them. Examples include, but are not limited to:

• In Lesson 2-1, the Visual Learning Bridge states, “Compatible numbers are numbers that are easy to compute mentally." Further explanation is provided with an example, “Add 11.45 + 9.55 first because they are easy to compute mentally. $11.45 + $9.55 = $21. $21 + $3.39 = $24.39." 
• In Lesson 12-3, the Visual Learning Bridge states, “1 ton is equal to 2,000 pounds." The materials display. “1 ton (T) = 2,000 pounds (lb).” Next, students practice with the new term in the Problem Solving portion of the lesson, “What would be the most appropriate unit to measure the combined weight of 4 horses?”
• In Lesson 15-1, Visual Learning Bridge, students learn about corresponding terms. Materials prompt teachers, “Say corresponding terms and have students repeat it. Display these number sequences. 0,3,6,9,12,15. 0,4,8,12,16, 20. Point to pairs of corresponding terms, each time saying corresponding terms as you point."
• In In Lesson 16-1, Visual Learning Bridge, students classify triangles based on the lengths of their sides and their angle measures.  During Independent Practice, students “Classify each triangle by its sides and then by its angles.”