5th Grade - Gateway 2

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

The materials include problems and questions that develop conceptual understanding and provide opportunities for students to independently demonstrate conceptual understanding throughout the grade. Throughout the materials, there are sections that emphasize introducing concepts and developing understanding such as Build Understanding and Spark Your Learning. Students have the opportunity to independently demonstrate their understanding in the Check Understanding and On Your Own problems at the end of each lesson. Evidence includes: 

• Lesson 2.3, Spark Your Learning states “One of Miami’s tallest buildings is 900 Biscayne Bay. It stands 650 feet tall. The building has 63 floors. If each floor is approximately the same height, about how tall is one floor of the 900 Biscayne Bay building?” (5.NBT.6)
• Lesson 3.4, On My Own, states “Tina’s taco truck sold 3 times as many veggie tacos in August than in September. She sold twice as many in September than in October. She sold 927 veggie tacos during the three months. Represent the number of veggie tacos sold with a bar model. Write an equation to show the amount represented by each box of the bar model. Then find the amount. How veggie tacos did Tina sell in September? Explain how you know.” (5.NBT.6)
• Lesson 8.1, Check for Understanding, Question 1 states “At nine o’clock, $$\frac{5}{8}$$ of the 16 cats at the party go home. How many cats go home at nine o’clock? Draw a visual model to find the answer.” (5.NF.4)
• Lesson 8.5, Spark Your Learning states “A contractor buys rectangular floor tiles for a home that he is building. How can you find the area of the tile? Use the square to find the area of the tile. Explain your reasoning.” (5.NF.4) 
• Lessons 10.5, On My Own, Problem 11 states “Mae uses the expression $$5\div\frac{1}{6}$$ to solve a problem. Write a word problem that can be modeled by the expression. Draw a visual representation to show the quotient. Write an equation to model the problem. Interpret the quotient for the situation.” (5.NF.7)
• Lesson 13.2, On My Own provides students the opportunity to reason on what decimal has “$$\frac{1}{10}$$ of the value of 0.08 and 10 times as much as the value of .008? Explain.” (5.NBT.1)
• Lesson 14.4, More Practice and Homework, Question 6, Math on the Spot states “Tania measures the growth of her plant each week. The first week, the plant’s height measured 2.65 decimeters. During the second week, Tania’s plant grew .7 decimeter. How tall was Tania’s plant at the end of the second week? Describe the steps you took to solve the problem.” (5.NBT.7)
• In Lesson 15.2, students use visual models and base ten representations to represent multiplication with decimals and whole numbers. (5.NBT.7)
• In Lesson 15.5, students use an area model to multiply decimals by decimals. (5.NBT.7)
• Lesson 6.1, students utilize visual models such as fraction strips/bar model to add fractions with different denominators. (5.NBT.7)

The instructional materials for HMH Into Math Grade 5 meet expectations that they attend to those standards that set an expectation of procedural skill and fluency. 

The materials include problems and questions that develop procedural skill and fluency and provide opportunities for students to independently demonstrate procedural skill and fluency throughout the grade. Procedural skills and fluencies are intentionally built on conceptual understanding. This is primarily found in two areas of the materials: 

• In the On Your Own section, students work through activities to practice procedural skill and fluency.
• In the More Practice/Homework section, students can find additional fluency practice.  

Students have numerous opportunities to develop and independently demonstrate procedural skill and fluency, especially where called for by Standard 5.NBT.5. Examples include but are not limited to: 

• In Lesson 2.4, students use partial quotients to solve multi-digit division problems. On Your Own, Problem 8, “2,352 ÷ 48”. In the More Practice and Homework, Problem 6 “8,632 ÷ 29.” (5.NBT.6)
• The Module 3 Review provides students practice with division. In Problems 8 and 9, students determine whether an estimated digit is too high or low, adjusting the estimate if needed. Then they divide. Problem 8. “The estimate is 90. 8,645 ÷ 91.” Problem 9, “The estimate is 60. 1,243 ÷ 21.”
• In Lesson 8.3, On Your Own, students write an equation before multiplying. Problem 1, “$$1\frac{1}{4}$$ by $$1\frac{1}{3}$$.” (5.NBT.5)
• Lesson 8.5 presents opportunities for students to practice fluency with multiplication of fractions in both the Check for Understanding and On Your Own section. For example, Problem 4, “Find the product. $$\frac{4}{9}\times\frac{3}{5}$$”. Problem 9 “$$\frac{3}{8}\times\frac{3}{7}$$.” (5.NF.4)
• In Lesson 14.5, On Your Own, students practice subtracting decimals. For example, in Problem 15, students are asked “Find the Difference, 27.64 - 16.98” and in Problem 18, students are asked “Find the unknown number, 1.7 = __ - 4.63.” (5.NBT.7)

The instructional materials for HMH Into Math Grade 5 meet expectations that the materials are designed so that teachers and students spend sufficient time working with engaging applications of mathematics. Engaging applications include single and multi-step problems, routine and non-routine, presented in a context in which the mathematics is applied. 

The instructional materials include multiple opportunities for students to engage in routine and non-routine application of mathematical skills and knowledge of the grade-level. Students also have opportunities to independently demonstrate the use of mathematics flexibly in a variety of contexts. Application contexts are used throughout the curriculum to build conceptual understanding. During Independent Practice and On Your Own, students often engage with problems that include real-world context and present opportunities for application. The More Practice/Homework sections also contain additional application problems. 

• Lesson 3.2, On Your Own, Problem 13 states “Mr. Torres has 212 coins in his collection. He wants to keep all of his coins in a binder. He can store 24 coins on each binder page. He finds 212 ÷ 24 to be 8, r20, so he buys 8 pages. Does Anderson buy the correct number of pages? Explain.” (5.NBT.6)
• Lesson 8.4,On Your Own, Problem 3, Model with Mathematics: Write a story problem that can be modeled with the equation $$\frac{1}{4} \times \frac{8}{12} = \frac{2}{12}$$. Then draw a visual model to represent the problem.” (5.NF.6)
• Lesson 8.7, On your Own, Problem 10, students solve real world problems involving multiplication of fractions and mixed numbers. “Sam is using craft felt to carpet two rooms in a dollhouse. Both rooms are $$\frac{5}{6}$$ ft by $$\frac{7}{8}$$ foot. How many square feet of craft felt does Sam need to carpet both rooms? Explain your reasoning.” (5.NF.6) 
• Lesson 9.2, More Practice/Homework, Problem 1, students solve real world problems involving multiplication of fractions and mixed numbers. “Trail running is an exercise that involves running on trails instead of paved roads to reduce the impact on ankles and knees. Samantha runs on the Lakeside Trail. She runs $$2\frac{1}{2}$$ times around the loop and then walks the remainder of the way. Write an equation to model the distance Samantha runs.” (5.NF.6)
• Lesson 9.1, On Your Own, Problem 5, students are given the dimensions of a square tile used to tile a rectangular-shaped patio and to ask students to choose a length and width for the patio given constraints. Then students find the area of the patio and explain the method used. (5.NF.6 )
• Lesson 9.4, On Your Own, Problem 4, students solve real world problems involving multiplication of fractions and mixed numbers. “The area of a bathroom is 40 square feet. The area of another room is $$2\frac{3}{4}$$ times as great as the area of the bathroom. What is the area of the other bedroom?”  (5.NF.6)
• Lesson 9.1, Problem of the Day, students solve real world problems involving multiplication of fractions and mixed numbers as they determine how much water would go into 4 beakers if they each held $$2\frac{2}{8}$$ liter of water. Students are prompted to draw a model. (5.NF.6)
• Lesson 10.3, On Your Own, Problem 7, students solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions. “Consider the expression $$\frac{1}{5} ÷ 3$$. Write two different word problems that can be represented by this expression. Draw a visual model to represent the problems and then solve. What does the quotient represent in each problem?”  (5.NF.7c) 
• Lesson 11.2, On Your Own, Problem 13, “Maggie has a goal of jogging 100 miles. The distance she runs each day is the same unit fraction.  What are some possible fractions of a mile she can run each day and the number of days will it take her to reach her goal? Explain how you found your answers” (5.NF.7c).
• Lesson 11.4, On Your Own, Problem 13, students solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions. “Kecia cuts a $$\frac{1}{4}$$ pound pepper into 6 equal-sized pieces. How much of one whole pound is each piece? Represent the problem on the number line (number line with range 0 to 1 given, no intervals labeled).” (5.NF.7c)
• Lesson 11.4, On Your Own, Problem 9, students solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions. For the equation $$\frac{1}{4} ÷ 6 = t$$, Lena writes the word problem ‘A string is 6 feet long. Jen wants to cut the string into 1/4-foot pieces. How many pieces will Jen get?” Why does the word problem not make sense for this equation?’” (5.NF.7c)
• Lesson 15.6, On Your Own, Problem 3, students find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, properties of operations, and/or the relationship between multiplication and division. “A giraffe can cover one mile in about 6.4 minutes. An elephant can cover one mile in about 8.2 minutes. About how much longer will it take the elephant to cover 3 miles than the giraffe?” (5.NBT.6 and 5.NBT.7)
• Lesson 17.6, On Your Own, Problem 6, states “Carlos sells coupon booklets for $5.50 apiece. He makes $60.50. Monica sells the same booklets for $4.75 each and makes $57. Who sells more booklets? How many more?” (5.NBT.7)

Each Unit has a Performance Task that involves real world applications of the mathematics from that unit. For example, the Unit 4 performance task is called “Trail Teamwork” and addresses 5.NF.3, 5.MD.1, and 5.MD.2 as students complete the following:

• Determine what fraction of a 10 mile long hiking trail each of 4 people are in charge of cleaning.
• Determine the distance between equidistant signs along a trail (3 signs within $$\frac{1}{2}$$ mile).
• Determine how many trees are planted if every $$\frac{1}{4}$$ mile a tree is planted. (5.NF.7c) 
• Create line plots to show the heights of those trees after a few weeks.

The instructional materials for HMH Into Math Grade 5 meet expectations that the three aspects of rigor are not always treated together and are not always treated separately. In general, two, or all three, of the aspects are interwoven throughout each module. On the Module planning page, the progression is included in a diagram showing the first few lessons focused on understanding and connecting concepts and skills and the last lessons focused on applying and practicing. 

All three aspects of rigor are present independently throughout the program materials. Examples include:

• Lesson 16.1, Check Understanding, Problem 2, students develop conceptual understanding of multiplication with decimals using hundredths grids. “Kali has a banner with a width of 0.3 meter and length of 1.5 meters. What is the area of the banner? Use the decimal model.” (5.NBT.7)
• Lesson 5.5, On Your Own, Problem 16, students use the formula for volume to find the volume of rectangular prisms. Given a cube with the following information: length: 7 cm; width 4 cm, volume 168 cu cm, students “Find the unknown number. Height _ cm.” (5.MD.5b)
• Lesson 3.4, On Your Own, Problem 5,  students solve an application problem. For example, “A local group raised $3,273 during a recent event. The money raised will be shared equally among 3 different charities. How much money will each charity receive? Write an equation to model the situation. Then solve.”  (5.NBT.6)

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

• In Lesson 10.4, Build Understanding, Problem 1, states "The nature preserve has a 3-mile long trail for birdwatchers. The ranger divides the birdwatcher trail into $$\frac{1}{2}$$-mile sections and names each section after a different bird. How many of these sections does the trail have? A. Complete to describe the situation and model it with an expression. 'The trail is ___ miles long and is divided into ____-mile sections. This can be modeled by the expression ___”. 
• In Lesson 11.1, More Practice and Homework, Problem 1, students represent the situation for each problem with a visual model. They write a division equation and a related multiplication equation. “Marcos has 4 gallons of gasoline for his lawn mower. How many lawns can he mow if each lawn uses $$\frac{1}{4}$$ gallon of gasoline?” Students engage in application and conceptual understanding as they complete the problem.  
• In Lesson 11.3, On Your Own, Problem 3  students use application and conceptual understanding to solve “Write and solve a division word problem for the visual model.” The visual model shows five circles partitioned into six pieces each. (5.NF.7c)

The instructional materials reviewed for HMH Into Math 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. 

Student materials consistently prompt students to both construct viable arguments and analyze the arguments of others. Turn and Talk sections often require students to construct viable arguments and analyze the arguments of others. In addition, students are often asked to justify their reasoning in practice problems, especially in problems labeled “Critique Reasoning”.

• Lesson 4.4, On Your Own, Problem 11,  “Deshawn says that he can evaluate the numerical expression 7 + (3 × 8) – 5 without parentheses and get the same answer. Is Deshawn correct? Explain how you know.”
• Lesson 8.6, On Your Own, Problem 12, “Carl and Maeve are asked to think of a fraction and multiply it by 5,267. Carl thinks of $$\frac{5}{6}$$. Maeve thinks of $$\frac{7}{7}$$. They both say their product is less than 5,267. Are they correct? Explain.”
• Lesson 4.2, Critique Reasoning, Problem 4, “James is making banners for his club’s airplanes to pull. Each banner is 5 feet long and is attached by a 10-foot long rope. He models the total length of the banners and rope for six airplanes with the numerical expression 6 × (5 + 10). He says the total length for six planes is five times as great as the total length needed for one plane. Correct his error.” 
• Lesson 8.6, Spark Your Learning, “The painting shown is resized to 3.4 of its original size. How does the height of the resized painting compare to the height of the original painting? Is the height of the resized more than or less than $$\frac{3}{4}$$ foot? Draw a visual model to represent your thinking. Justify your reasoning.” 
• Lesson 15.5, Turn and Talk, “Is your answer reasonable? Explain.”  
• Lesson 11.5, Turn and Talk, “Why should you divide each half of the rectangle into 4 equal groups?” and “Does it matter what visual model you use to find the quotient of a unit fraction divided by a whole number? Why might you choose one model over another? Explain.”

The instructional materials reviewed for HMH Into Math Grade 5 meet expectations for assisting teachers in engaging students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics.

Many of the lesson tasks are designed for students to collaborate, with teacher prompts to promote explaining their reasoning to each other. Independent problems provided throughout the lessons also have teacher guidance to assist teachers in engaging students. Examples include the following:

• The Teacher Edition provides Guided Student Discussion with guiding questions for teachers to create opportunities for students to engage in mathematical discourse. Module 14, “How can you tell that the 4-digit number at the top of the subtraction problem is less than 2,000?” Module 19 “What would have happened if the instruction in step D had the pattern start at 95 instead of 83?”
• Critique, Correct, and Clarify, is a strategy used to assist students in constructing viable arguments. Lesson 1.4, On Your Own, “Have students work out the steps to multiply on their own. Encourage students to describe the error and review explanations with a partner. Students should refine their responses after their discussions with a partner.” Lesson 5.4, On Your Own, Problem 12, “Point out to students that Problem 12 can be solved more than one way. As shown, the volume of the new cube can be multiplied by the number of cubes: $$8 \times (2 \times 3 \times 4)$$. Or the length of each side of the cube can be doubled: $$6 \times 4 \times 8$$. Encourage students to describe different ways of solving the problem with their partners. Students should refine responses after their discussions.”
• In Lesson 2.4, students tell if an estimate is reasonable and explain why. Teacher guidance states, “Problem 3, Construct Arguments shows that students need to determine the reasonableness of a quotient.”
• The Teacher Edition includes Turn and Talk in margin notes to prompt student engagement. Lesson 11.1, Build Understanding, “Have students share their reasoning. For students who are struggling, suggest that they compare the multiplication equation with their visual models.”  The Turn and Talk builds off of earlier discussion questions such as “What numerical information is given in the problem? How can you show the number of pounds of potato salad? How can you show the $$\frac{1}{4}$$-pound servings in your visual model?"

The instructional materials reviewed for HMH Into Math Grade 5 meet expectations for explicitly attending to the specialized language of mathematics. The materials provide explicit instruction on communicating mathematical thinking with words, diagrams, and symbols. The materials use precise and accurate terminology and definitions when describing mathematics and support students in using them. Examples are found throughout the materials. 

• At the beginning of each module, Key Academic Vocabulary is highlighted for the teacher.  These sections include both Prior Learning. Review Vocabulary and Current Development and New Vocabulary. Definitions are given for each vocabulary word. 
• Within the Student pages, new vocabulary is introduced in highlighted sections called Connect to Vocabulary. Lesson 4.1: “You can model a context mathematically using a numerical expression. A numerical expression is a mathematical phrase that uses only numbers and operation signs”. In Lesson 20.3 it states “There are two accepted definitions of a trapezoid. One definition defines a trapezoid as having exactly one pair of parallel sides. The other definition defines a trapezoid as having at least one pair of parallel sides.”
• In the module planning pages, there is a Linguistic Note on the Language Development page that provides teachers with possible misconceptions relating to academic language. Unit 1: “Many English words have multiple meanings that can interfere with comprehension. For example, flat and long are typically used as adjectives; however, they also name the base-ten blocks used for understanding multi-digit place values.”
• In the lesson planning pages, Sharpen Skills in some lessons include Vocabulary Review activities. Lesson 20.1:  “Objective: Students review types of polygons. Have students work in types of polygons. List the following review terms on the board: triangle, decagon, hexagon, octagon, quadrilateral. Ask students to discuss what attributes all of these figures have in common. Then have students identify the specific characteristics of each. Have students form pairs and each student should draw an example of each figure listed. Students should compare their figures.”
• Guide Student Discussion provides prompts related to understanding vocabulary. Module 6: “Listen for students who correctly use vocabulary as part of their discourse. Students should be familiar with the terms fraction, whole, numerator, denominator, and equal parts. Ask students to explain what they mean if they use those terms. How can you tell how many large triangles will fit in the puzzle? How can you use this fact to write a fraction for each large triangle? How can you use these facts to find the fractional area of each small triangle?”
• Vocabulary is highlighted and italicized within each lesson in the materials. 
• There is a vocabulary review at the end of each module. Students do fill-in-the-blank with definitions or examples, create graphic organizers to help make sense of terms, or the teacher is prompted to make an Anchor Chart where students define terms with words and pictures, trying to make connections among concepts.