5th Grade - Gateway 2

The instructional materials for Into Math Florida Grade 5 meet the 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, sections emphasize introducing concepts and developing understanding such as: “Build Understanding” and “Spark Your Learning”. Students have the opportunity to independently demonstrate their understanding with the “Check Understanding” and “On My Own” problems at the end of each lesson. For example: 

• Lesson 1.1, Spark Your Learning, asks students, “On a road trip, Anna and her family stop at a ranch where bales of hay are being weighed. Describe the relationship between the two weights. How can you use the relationship to compare the weight?” (5.NBT.1.1)
• Lesson 2.3, Spark Your Learning, asks students, “One of Florida’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.2.6)
• Lesson 3.4, On My Own, students solve, “Tina’s Treat Truck sold three times as much ice cream in August than in September. She sold twice as much in September than in October. She sold 927 pounds of ice cream during the three months. Represent the amount of ice cream sold with a bar model. Write an equation to show the amount represented by each box of the bar model. How much ice cream did Tina sell in September? Explain how you know.” (5.NBT.2.6)
• Lesson 8.1, Check for Understanding, Question 1, students solve, “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.2.4)
• Lesson 8.5, Spark Your Learning, asks students, “A contractor buys rectangular floor tiles for a home that he is building. How could you find the area of the tile? Draw a visual model to show how you can find the area of the tile. Explain your reasoning.” (5.NF.2.4)
• Lessons 10.5, On My Own Problem 11, “Mae uses the expression 5 ÷ $$\frac{1}{6}$$ to solve a problem. Write a word problem that can be modeled by the expression. Draw a visual representation to show the quotient.” (5.NF.2.7)
• Lesson 13.2, On My Own, students reason on what decimal has “$$\frac{1}{10}$$ of the value of 0.08 and what decimal has 10 times as much as the value of .008? Explain.” (5.NBT.1.1)
• Lesson 14.4, More Practice and Homework, Question 6, Math on the Spot, asks students, “Tania measure 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.2.7)
• Lesson 15.2, students use visual models and base ten representations to represent multiplication with decimals and whole numbers. (5.NBT.2.7)
• Lesson 15.5, students use an area model to multiply decimals by decimals. (5.NBT.2.7)
• Lesson 16.1, students utilize visual models such as fraction strips/bar model to add fractions with different denominators. (5.NBT.2.7)

The instructional materials for Into Math Florida 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 primarily found in two areas of the materials. In “On Your Own,” students work through activities to practice procedural skill and fluency; additional fluency practice is found in “More Practice/Homework.”

• In Lesson 1.6, students develop multiplication fluency. In On My Own, Problems 9 and 10, students refer to a table with costs to find the money earned by selling multiples of items in the table. For example, Problem 9, “If the store sells 35 filing cabinets and 14 tables, how much does it earn?" (5.NBT.2.5)
• In Lesson 2.4, students use partial quotients to solve multi-digit division problems. On My Own, Problem 8, “2,352 ÷ 48.” More Practice/Homework, Problem 6, “8,632 ÷ 29.” (5.NBT.2.6)
• Lesson 8.3, On My Own, students write an equation before multiplying. Problem 3 asks students to use an area model to multiply “1$$\frac{1}{4}$$ by 1$$\frac{1}{3}$$.” (5.NF.2.6)
• Lesson 8.5 presents opportunities for students to practice fluency with multiplication of fractions in both the Check for Understanding and On My Own. For example, Problem 4, “Find the product. $$\frac{4}{9}$$ x $$\frac{3}{5}$$; and Problem 9, $$\frac{3}{8}$$ x $$\frac{3}{7}$$.” (5.NF.2.4)
• Lesson 14.4, On My Own, students practice subtracting decimals. “Problem  15, Find the Difference, 27.64 - 16.98;” “Problem 18, Find the unknown number: ___ - 4.63 = 1.7.” (5.NBT.2.7)

The instructional materials for Into Math Florida Grade 5 meet the expectations that the materials are designed so that teachers and students spend sufficient time working with engaging applications of the 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. During Independent Practice and On My Own, students engage with problems that include real-world context and present opportunities for application. For example: 

• Lesson 3.2, On My Own, Problem 13, “Anderson 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.2.6).
• Lesson 8.4, On My Own, student solve real-world problems involving multiplication of fractions and mixed numbers. For example, “Write a story context that can be modeled with the equation $$\frac{1}{4}$$ x $$\frac{8}{12}$$ = $$\frac{2}{12}$$. Then draw a visual model to represent the problem.” (5.NF.2.6)
• Lesson 8.4. Homework, Problem 1, “Dominick’s doctor tells him to take a one-half dose of medicine. One dose equals $$\frac{2}{3}$$ tablespoon. Draw a visual model to find the amount of medicine Dominick needs. Write and solve an equation to go with your visual model. How many tablespoons is a one-half dose?” (5.NF.2.6)
• Lesson 8.7, On My 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 her dollhouse. Both rooms are $$\frac{5}{6}$$ ft by $$\frac{7}{8}$$ ft. How many square feet of craft felt does she need to carpet both rooms? Explain your reasoning.” (5.NF.2.6)
• Lesson 9.2, More Practice/Homework, Problem 1, student solve real-world problems involving multiplication of fractions and mixed numbers. “Samantha runs on the Lakeside Trail. She runs 2 and one half times around the loop and then walks the remainder of the way. Write and solve an equation to model the distance Samantha runs.” (5.NF.2.6)
• Lesson 9.1, On My Own, Problem 5, students are given mixed number length and width of a step ands asked to find the largest square tile of any size she can use so that the tiles fit exactly, and then explain how the tiles show the area of the step. (5.NF.2.6 )
• Lesson 9.4, On My Own, Problem 4, students solve real-world problems involving multiplication of fractions and mixed numbers. “The area of Milo’s bathroom is 40 square feet. The area of his bedroom is 2$$\frac{3}{4}$$ times as great as the area of his bathroom. What is the area of his bedroom?” (5.NF.2.6)
• Lesson 9.9, Problem of the Day, students solve real-world problems involving multiplication of fractions and mixed numbers. For example, students determine how much water would go into four beakers if they each held $$\frac{2}{8}$$ liter of water. Students are prompted to draw a model. (5.NF.2.6)
• Lesson 10.3, On My 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 problem  and then solve. What does the quotient represent in each problem?” (5.NF.2.7c)
• Lesson 11.2, On My Own, “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.2.7c)
• Lesson 11.4, On My Own, students solve real-world problems involving division of unit fractions by non-zero whole numbers, and division of whole numbers by unit fractions. “Kecia buys $$\frac{1}{4}$$ pounds of peppers. She cuts the peppers into 6 equal sized strips. How much of one whole pound is each strip? Represent the problem on a number line (number line with range 0 to 1 given, no intervals labeled)." (5.NF.7c)
• Lesson 11.5,  On My 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 $$\frac{1}{10}$$ ÷ 4 = b: Write a word problem.” Students draw a visual model for the problem as well. (5.NF.2.7c)
• Lesson 11.6, On My Own, students solve real-world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions. “To make a homemade ‘lava lamp’ you can mix vegetable oil, food coloring, and $$\frac{1}{4}$$ tablet of baking soda. How many lava lamps can you make if you have 5 tablets of baking soda?” (5.NF.2.7c)
• Lesson 15.6, On My Own, student find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. “The average heart rate of a giraffe is 65 beats each minute. The average heart rate of a horse is 44 beats each minute. How many more times does a giraffe’s heart beat in 2.75 minutes than a horse’s heart does?” (5.NBT.2.6,7)
• Lesson 17.6, On My Own, Problem 6, “Carlos sells coupon booklets for $5.50 apiece. He makes $60.50. Monica sells the same books for $4.75 each and makes $57. Who sells more booklets? How many more?” (5.NBT.2.7)

Each Unit has a Performance Task involving real-world applications of the mathematics from the unit. For example, the Unit 4 Performance Task is called Trail Teamwork and has students: determine what fraction of a 10-mile long hiking trail each of four people is in charge of cleaning, determining 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.2.7c), and create line plots to show the heights of those trees after a few weeks.

The instructional materials for Into Math Florida 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. The module planning page includes a progression diagram showing the first few lessons focused on understanding and connecting concepts and skills. The last lessons focus on applying and practicing.

All three aspects of rigor are present independently throughout the program materials. For example: 

• Lesson 3.4, Problem 5, students solve an application problem, “Students at a local elementary school raised $3273 during a charity 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.2.6)
• Lesson 5.5 develops procedures for finding volume of rectangular prisms. Students learn the formula and then use it in 19 different problems. (5.MD.3.5b)
  
• Lesson 16.1 develops conceptual understanding of multiplication with decimals using hundredths grids. In each of the problems, students color grids to represent the products of two decimals. (5.NBT.2.7)

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

• Lesson 10.4, Build Understanding, Problem 1, students use their understanding of fractions to solve a real-world problem. “The nature preserve has a 3-mile long trail for birdwatchers. The ranger divides the birdwatcher trail in $$\frac{1}{2}$$ 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 ___.'”
• Lesson 11.1, More Practice and Homework, students represent the situation for each problem with a visual model. They then write a division equation and a related multiplication equation. Problem 1, “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.  
• Lesson 11.3, On My Own, Problem 3, students use application and conceptual understanding, “Write and solve a word problem that can be represented with the visual model.” The visual model shows five hexagons partitioned into six pieces each. (5.NF.2.7c)

The instructional materials reviewed for Into Math Florida Grade 5 meet the 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 construct viable arguments and analyze the arguments of others. A common strategy in these materials is Turn and Talk with a partner about the related task. Often these Turn and Talks 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 those problems labeled “Critique Reasoning.”

• Lesson 1.2, Practice & Homework Journal, Problem 7, Math on the Spot What’s the Error?, “Sophia said that the expanded form for 605,970 is 6000,000 + 50,000 + 900 + 70. Describe Sophia’s error and give the correct answer.”
• Lesson 4.2, Critique Reasoning, Problem 4, “James is making banners for his 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 x (5 + 10). He says the total length for six planes is five times as much as the total length needed for one plane. Correct his error."
• Lesson 4.4, On My Own, Problem 11, “Deshawn says the he can evaluate the expression 7 + (3 x 8) - 5 without parentheses and get the same answer. Is Deshawn correct? Explain how you know.”
• Lesson 6.6 On My 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 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 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 help you find the quotient of a unit fraction divided by a whole number? Why might you choose one model over another? Explain.”
• Lesson 15.5, Turn and Talk, “Is your answer reasonable? Explain how you know.”
  

The instructional materials reviewed for Into Math Florida Grade 5 meet the 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.

• The Teacher Edition provides Guided Student Discussion with guiding questions for teachers to create opportunities for students to engage in mathematical discourse. For example, in Module 14, “How can you tell that the 4-digit number at the top of the subtraction problem is less than 2,000?” In 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. For example, in Lesson 1.4, On My 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.” In Lesson 5.4, On My 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 x (2 x 3 x 4). Or the length of each side of the cube can be doubled: 4 x 6 x 8. Encourage students to describe different ways of solving the problem with their partners. Students should refine responses after their discussions.”
• Lesson 2.4 asks students to tell if an estimate is reasonable and explain why. Teacher guidance says, “Problem 3 Construct Arguments shows that students need to determine the reasonableness of a quotient.”
• The Teacher Edition includes Turn and Talk in the margin notes to prompt student engagement. For example, in Lesson 11.1, Build Understanding, “Have students share their reasoning. For students who are struggling, suggest that they compare the multiplication expression with their visual models”  The Turn and Talk builds off of earlier discussion questions such as, “What math problem do you need to solve? How can you show the number of pounds of potato salad? How can you show the divisor in your visual model?”.

The instructional materials reviewed for Into Math Florida Grade 5 meet the 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 found throughout the materials include: 

• At the beginning of each module, Key Academic Vocabulary is highlighted for the teacher.  The sections include both Prior Learning - Review Vocabulary, and Current Development - New Vocabulary. Definitions are given for each vocabulary word.
• Within the Student pages, new vocabulary is introduced in highlighted sections called Connect to Vocabulary. For example, in 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, “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, a Linguistic Note on the Language Development page provides teachers with possible misconceptions relating to academic language. For example, in 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. Point out that the articles 'a' and 'the' in front of these words are a strong clue to their meaning.”
• In Sharpen Skills in the lesson planning pages, some lessons include Vocabulary Review activities. For example, in Lesson 20.1, “Objective: Students review ty.” “Materials: markers, poster paper” “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 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 such as in Module 6, “Listen for students who correctly use review vocabulary as part of their discourse. Students should be familiar with the terms fraction, sum, like denominator, and mixed number. Ask students what they mean if they use those terms.” “Why can’t you count the number of shapes in the puzzle to determine the number of equal parts in the whole puzzle?” “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 complete a 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.