4th Grade - Gateway 2

The instructional materials for Into Math Florida Grade 4 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, 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 with the "Check Understanding" and "On My Own" problems at the end of each lesson. For example: 

• Lesson 1.22, Spark Your Learning, “Texas has a land area of two hundred sixty-eight thousand, five hundred ninety-seven square miles. It is the largest state in the southern U.S. How can you write the area of Texas in two different ways using numbers?” (4.NBT.1.2 )
• Lesson 6.4, students use area models and the distributive property to represent division. (4.NBT.2.5)
• Lessons 7.2, Build Understanding, students use base ten blocks to represent division with one-digit divisors. Problem 1, “Fran has 423 scrapbook stickers. She wants to put an equal number of stickers in 3 different scrapbooks. How many stickers can she put in each scrapbook? Find 423 ÷ 3. Use base ten blocks to show the division.” (4.NBT.2.6)
• Lesson 5.4 asks students to use the distributive property and partial products to multiply one-digit by four-digit numbers. (4.NBT.2.5)
• Module 11, Opening Activity, students are provided four different squares partitioned and shaded differently. In a “Turn and Talk” they are asked, “Which square’s shading represents a different amount?” and “How could you change the shading to make it represent the same amount as the others?” (4.NF.1.2)
• Lesson 11.1, Spark Your Learning, students are asked, “Liz and Alvin have the same go-karts in different colors. The fuel tank in Liz’s go-kart is $$\frac{2}{5}$$ full. The fuel tank in Alvin’s go-kart is $$\frac{1}{3}$$ full. Whose go-kart has more fuel? How do you know?” (4.NF.1.2)
• Lesson 11.2, Spark Your Learning, students are asked, “Compare Abbot’s and Rowan’s rope climbs. Who climbed higher? How do you know?" (table provided of $$\frac{5}{8}$$ vs $$\frac{4}{10}$$) - Visual of the same size rope and two students. (4.NF.1.2)
• Lesson 11.3, Check Understanding, students are asked, “Jason makes a $$\frac{5}{6}$$ turn on his skateboard. Kym makes a $$\frac{3}{4}$$ turn, and Sam makes a $$\frac{10}{12}$$ turn. Which two skaters make the same-size turn? Explain.” (4.NF.1.2)
• Lesson 11.5, Question 4, students are asked, “Jerry has two same size circles divided into the same number of equal parts. One circle has $$\frac{3}{4}$$ of the parts shaded, and the other has $$\frac{2}{3}$$ of the parts shaded. His sister says that the least number of pieces each circle could be divided into is 7. Is his sister correct? Explain." (4.NF.1.2)
• Lesson 11.6, Question 5, asks student to reason about “Isaiah hikes $$\frac{11}{12}$$ mile along the Lake View Trail. Cheryl hikes $$\frac{10}{8}$$ mile along the same trail. Who hikes farther? Use a fraction comparison strategy to support your reasoning.” (4.NF.1.2)

The instructional materials for Into Math Florida Grade 4 meet the 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.”

• Module 2 focuses on Addition and Subtraction of Whole Numbers. In Lesson 2.1, On My Own, students “Estimate. Then find the sum,” for problems such as Problem 8, 609,987 + 123,654. In Lesson 2.2, More Practice, students “Estimate. Then find the difference” for problems 3-5. Problem 4 includes, 38,207 - 28,278. (4.NBT.2.4)
• Lesson 5.5, On My Own, Problems 6 - 8, “Estimate. Then write the problem vertically to find the product.” Problem 6, 6 x 523; Problem 7, 9 x 5,181; Problem 8, 8 x 6,719. (4.NBT.2.5)
• Module 7 Review, Divide and Check, Problem 7, 231 ÷ 5; Problem 8, 458 ÷3; Problem 12, 2,551 ÷7. (4.NBT.2.5)
• Lesson 8.1, On My Own, students multiply with tens. “Choose a method. Then find the product.” Problem 5, 29 x 80 = ___; Problem 6, 25 x 30 = ___; Problem 7, 90 x 16 = ___. (4.NBT.2.4)
• Lesson 15.3, On My Own, “Find the Sum.” Problem 7, 3$$\frac{1}{4}$$ + 1$$\frac{2}{4}$$ = ___; Problem 8, $$\frac{5}{6}$$ + $$\frac{5}{6}$$ + $$\frac{5}{6}$$ = ___.” (4.NF.2.3c)
• Additional fluency practice can be found in the More Practice/Homework activities.

The instructional materials for Into Math Florida Grade 4 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 Spark Your Learning, Independent Practice, and On My Own, students engage with problems that include real-world context and present opportunities for application. For example: 

• Lesson 3.4, On My Own, Problem 5, students solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. For example: “Aditi wants to make posters to show animals in the Everglades. She has 4 posters, and each poster can fit 8 pictures.  She wants to show 2 pictures of each type of animal. How many different types of animals can Aditi show on her posters? Write equations to model the problem.” (4.OA.1.3)
• Lesson 3.5, Problem 5, students make a model to solve a problem where a students wants to make a poster to show animals in the Everglades. “She has 4 posters each of which can fit 8 pictures. She wants to show 2 pictures of each animal. How many different animals can be shown on her posters?” (4.OA.1.3)
• Lesson 5.7, Homework, Problem 6, students solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. For example, “The Brown family is driving to Junction City, which is 426 miles away. The family drives 60 miles for each of the first 3 hours. Then they drive 55 miles for each of the next 4 hours. How far are they from Junction City?"  (4.OA.1.3)
• Lesson 14.3. On My Own, Model with Mathematics, students solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators. For example, “Weston walks $$\frac{1}{4}$$ mile to school and $$\frac{1}{4}$$ mile home. How many miles does Weston walk? Use a visual fraction model, write an equation and find the distance, d.” (4.NF.2.3d)
• Lesson 5.6, Check Understanding, students multiply a whole number of up to four digits by a one-digit whole number and multiply two two-digit numbers, using strategies based on place value and the properties of operations. For example, “The Cat in the Hat is a short book with only 236 words. The library has 5 copies of this book. How many words appear in the books?” (4.NBT.2.5,6)
• Lesson 7.2, On My Own, students multiply a whole number of up to four digits by a one-digit whole number and multiply two two-digit numbers, using strategies based on place value and the properties of operations. For example, “Jackie places 552 photos of cats on 4 bulletin boards at the animal shelter. Each board has the same number of photos. How many photos are on each board?” (4.NBT.2.5,6)
• Lesson 7.4, Check for Understanding, students solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. For example, “There are 48 people going on a hike. Each pack has 8 bottles of water. How many packs are needed for each hiker to have 2 bottles? How can you check that your answer is reasonable?” (4.OA.1.3)
• Lesson 8.7, More Practice/Homework, students solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. For example, “A crane operator moves 6 shipping containers that weigh 215 tons each onto a barge. The same crane operator loads 4 more shipping containers that weigh 194 tons each onto the barge. How many tons of shipping containers did the crane operator load onto the barge? Write an equation to model the situation. How can you check if your answer is reasonable?” (4.OA.1.3)
• Lesson 14.5, On My Own, Problem 6, students solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators. For example, “Oliver has a board that is $$\frac{10}{12}$$ foot long. After he cuts some off, he has $$\frac{7}{12}$$ foot left. How much did Oliver cut off? Model the problem with an equation and then answer the Problem." (4.NF.2.3d)  
• Lesson 15.1, Homework 1, students solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators. For example, “Olivia rode her bike for $$\frac{9}{10}$$ hour. She rode her electric scooter for $$\frac{3}{10}$$ hour. How much longer did Olivia ride her bike than her scooter?” (4.NF.2.3d)
• Lesson 16.2, Homework, Problem 1, students solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem. For example, “Adam is restoring old wagon wheels and needs to cut 3 wooden spokes that are each $$\frac{5}{8}$$ yard long. What is the total length of wood that he needs to cut? Write two equations modeling the problem and the solution.” (4.NF.2.4c)
• Lesson 16.3, On My Own Model with Mathematics,  students solve word problems involving multiplication of a fraction by a whole number. For example, “Lana bakes banana bread for a fundraiser. She uses $$\frac{3}{4}$$ cup of bananas in each loaf. She bakes 5 loaves. How many cups of bananas does she use? Describe a fraction model you could draw to represent the problem. Then model it with an equation and solve the problem.” (4.NF.2.4c)

Each Unit has a Performance Task involving real-world applications of the mathematics from the unit. For example, the Unit 3 Performance Task is about “Visiting New York City”. It has students calculate how much it will cost 20 people to go on a tour of Chinatown (4.OA.1.1), for 23 people to go to a show (4.OA.1.1), calculate how much a group comprised of adults and children would save by visiting one attraction vs. another (4.OA.1.2 and 4.OA.1.3), and calculate the number of postcards purchased by 52 tourists (4.NBT.1.2).

The instructional materials for Into Math Florida Grade 4 meet the 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 5.2 builds conceptual understanding of multiplication through the use of area models and the distributive property. “How can you use the Distributive Property to break apart the base-ten blocks and find the product?” (4.NBT.2.5)
• Lesson 8.6 builds procedural fluency in multi-digit multiplication, “Estimate. Then choose a method to find the product.” This section includes six problems where students estimate two digit number multiplication like 43 x 35. (4.NBT.2.5)
• Lesson 12.6 emphasizes application of multi-step problem solving with money. For example, “Four friends earn a total of $5.00 by turning in cans for recycling. If the friends share the amount equally, how much does each get? Give your answer as  a decimal amount.” (4.MD.1.2)
  

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

• Unit 5, Performance Task, Problem 1, students use application to solve problems involving multiplication of fractions by whole numbers. “Enrique lives with his grandmother in an apartment building for senior citizens. He earns extra money by running errands for some of his grandmother’s neighbors. Enrique charges $4 for every $$\frac{1}{4}$$ hour he spends working. He spent $$\frac{2}{4}$$ hour going to the deli for Mr. McGuire, 1$$\frac{1}{2}$$ hours delivering papers for the apartment manager and $$\frac{3}{4}$$ hour picking up Mrs. Shultz’s groceries. Did Enrique ear enough money to buy a $49 video game? Explain your reasoning.”
• Lesson 14.5 engages students in the application of addition and subtraction of fractions. “Ross is helping to make popcorn at the carnival. At the start of his shift, the container of kernels is $$\frac{11}{12}$$ full. During his two-hour shift, they use $$\frac{3}{12}$$ of the container. How full is the container after lunch? You can write an equation without first making a model. Draw a fraction model to solve the problem. How full is the container after lunch?” (4.NF.2.3d)
• Lesson 14.3, Spark Your Learning, students use conceptual understanding to solve application problems. “Caleb enters a frog in a frog-jumping contest. His frog jumps twice. Caleb wants to find the total distance his frog jumps. Explain how can you determine the lengths of each of the frog’s two jumps, then find the total distance the frog jumped. (There is a visual of a frog jumping, a number line split up into 4 equal parts, with 0 and 1 labeled).”
  
• Lesson 16.3, On My Own, Problem 9, “Lana bakes banana bread for a fundraiser. She uses $$\frac{3}{4}$$ cup of bananas in each loaf. She bakes 5 loves. How many cups of bananas does she use? Describe a fraction model you could draw to represent the problem. Then, model it with an equation and solve the problem.” (4.NF.2.4c)

The instructional materials reviewed for Into Math Florida Grade 4 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 12.6, On My Own, students critique other’s reasoning, “Blake has $5.00. He sees some pencil packets that are $1.05. Blake says that he can buy 5 packets. Is he correct? Why or why not?”
• Lesson 7.2, On My Own, students critique others' reasoning, “Mara completes this division. Is her answer correct? Why or why not?”
• In Lesson 11.1, Turn and Talk, “Why is it important that the size of the fuel tanks in the go-karts is the same?” and “How do you know your answer is correct?”
• In Lesson 15.1, Turn and Talk, “One classmate represented this problem with an equation and another used subtraction. Who is correct and how do you know?”
• In Lesson 15.4, Turn and Talk, “A classmate says that you just need to subtract the whole number from the whole number and the fraction from the fraction to solve this problem. How would you respond?”

The instructional materials reviewed for Into Math Florida Grade 4 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 Lesson 21.1, “Why do we need to change hours to minutes?,” “How could you use reasoning to solve this problem without using the table?,” and in Lesson 12.3, “What do you need to know in order to find the time that Inez finishes? How can you find that out?”
• Critique, Correct, and Clarify is a strategy used to assist students in constructing viable arguments. For example, in Lesson 18.2, On My Own, Problem 5, “Point out to students that Problem 5 has an error. 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 19.2, On My Own, Problem 19, asks students to analyze the reasoning of a fictitious student. Teacher guidance says, “Point out that in Problem 19 Jimmy’s reasoning is not complete. Encourage students to describe why his reasoning is incomplete and review explanations with a partner. Students should refine their responses after their discussions with a partner.”
• In Lesson 4.1, Connect Math Ideas, Reasoning, and Language states, “Select students who used various strategies and have them share how they solved the problem with the class. As they explain their thinking, encourage other students to raise their hands and critique the reasoning.”
• In Lesson 1.5, Optimize Output, “Point out to students that the Turn and Talk asks how it is possible for two different estimates to both be correct. Encourage students to describe the thought processes of Anja and Liam and review explanations with a partner. Students should refine their responses after their discussions with a partner.“
• The Teacher Edition includes Turn and Talk in the margin notes to prompt student engagement. For example, in Lesson 15.2, “Select students who used various strategies and have them share how they solved the problem with the class. Encourage students to ask questions of their classmates. Using rectangular arrays is based on prior knowledge and should be shared first. Then have another student share a solution using base ten blocks.”

The instructional materials reviewed for Into Math Florida Grade 4 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 11.2, “A known size or amount that helps you understand a different size of an amount is a benchmark. Common benchmarks are 0, $$\frac{1}{2}$$, and 1.” In Lesson 4.1, “The Identity Property of Multiplication states that the product of any number and 1 is that number.”
• 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 Module 8 it states, “By giving all students regular exposure to language routines in context, you will provide opportunities for students to listen, speak, read, and write about mathematical situations and develop both mathematical language and conceptual understanding at the same time.”
• In Sharpen Skills in the lesson planning pages, some lessons include Vocabulary Review activities. For example, in Lesson 20.1, “Objective: Students review vocabulary used to classify triangles by angles and by sides.” “Materials: markers, poster paper” “Have students work in small groups to create a poster. Have students divide their posters into two sections: Classify Triangles by Sides and Classify Triangles by Angles. In the first section, have them write the terms scalene, isosceles, and equilateral. In the second section, have them write the terms acute, right, and obtuse. Have students draw an example of each type of triangle and write a definition for each term. Have each group share their poster with another group.”
• Guide Student Discussion provides prompts related to understanding vocabulary such as in Module 1, “Listen for student who correctly use review vocabulary as part of their discourse. Students should be familiar with the terms place value, greater than, less than, equal to, and compare. Ask students what they mean if they use those terms.” “Rounding to the nearest 10, what numbers round to 860?” “Rounding to the nearest 100, what numbers round to 900?” “Given the size of the ranges, what is a good strategy for determining how to place numbers in the table?”
• 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.