4th Grade - Gateway 2

The instructional materials for HMH Into Math Grade 4 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 the following:

• Lesson 1.2, Spark Your Learning states “The total area of Texas is two hundred sixty‑eight thousand, five hundred ninety‑six square miles. It is the largest state in the southern U.S. How can you write the total area of Texas in two different ways using numbers?” (4.NBT.2 )
• Lesson 6.4, students use area models and the distributive property to represent division. (4.NBT.5)
• Lessons 7.2, Build Understanding, students use base ten blocks to represent division with one digit divisors. Problem 1 states “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 $$423 \div 3$$.  Use base ten blocks to show the division.” (4.NBT.6)
• Lesson 5.4, students use the distributive property and partial products to multiply 1 digit by 4 digit numbers. (4.NBT.5)
• Module 11, Opening Activity, provides students with four different squares partitioned and shaded differently. In a Turn and Talk they are asked “Which square’s shading represents a different amount? How could you change the shading in that square to make it represent the same amount as the others?” (4.NF.A.2)
• Lesson 11.1, Spark Your Learning states “Liz and Alvin have the same go-karts in different colors. The fuel tank in Liz’s go-kart is $$\frac{3}{5}$$ 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.2)
• Lesson 11.2, Spark Your Learning states “Abbot and Rowan go to the Climb-a-thon. They both climb ropes that are the same length. Who climbs higher than halfway up the rope?” The problem includes a table representing $$\frac{5}{8}$$ and $$\frac{4}{10}$$ as well as visuals of the same size rope and two students. (4.NF.2) 
• Lesson 11.3, Check Understanding states “Jason makes a $$\frac{5}{6}$$ turn on his skateboard. Samantha makes a $$\frac{10}{12}$$ turn. Did they make the same-sized turn? Use the visual models to explain.” (4.NF.2) 
• Lesson 11.5, Question 4 states “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.2) 
• Lesson 11.6, Question 5 states “Isaiah hikes $$\frac{11}{12}$$ mile along the Lake View Trail. Cheryl hikes 3.5 miles along the same trail. Who hikes farther? Use a fraction comparison strategy to support your reasoning.” (4.NF.2)

The instructional materials for HMH Into Math Grade 4 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 4.NBT.5. Examples include but are not limited to: 

• Module 2 focuses on Addition and Subtraction of Whole Numbers. In Lesson 2.1, On Your 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: 38,207 - 28,278.” (4.NBT.4)
• Lesson 5.6, On Your Own, Problems 6 - 8, “Estimate. Then write the problem vertically to find the product.” Problem 6 states “$$6 \times 523$$”; Problem 7 states “$$9 \times 5,181$$”; Problem 8 states “$$8 \times 6,719$$.” (4.NBT.5)
• Module 7, Review,  “Divide and Check, Problem 6: $$231 ÷ 5$$; Problem 7: $$458 ÷3$$; Problem 11 $$2,551 ÷ 7$$.” (4.NBT.5)
• Lesson 8.1, On Your Own, students multiply with tens. Examples include “Choose a method. Then find the product.” Problem 5 “$$80 \times 29 =$$ ___”; Problem 6 “$$35 \times 30 =$$ ___”; and Problem 7 “$$90 \times 16 =$$ ___.” (4.NBT.4)
• Lesson 15.3, On Your Own, Problems 6 -9 states “Find the Sum. Write your answer as a mixed number.” Problem 6 “$$1 \frac{9}{10} + 1 \frac{8}{10} =$$ __”; Problem 7 “$$3\frac{1}{4} + 1\frac{2}{4} =$$ ___”; Problem 8, “$$\frac{5}{6} + \frac{5}{6} + \frac{5}{6} =$$ ___”; and Problem 9, “$$2\frac{5}{8} + 1\frac{7}{8} =$$ ___.” (4.NF.3c)
• Additional fluency practice can be found in the More Practice/Homework activities.

The instructional materials for HMH Into Math, Grade 4 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 Spark Your Learning, and Independent Practice, Check for Understanding, 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 3.5, On Your Own, Problem 4, students solve “Aditi takes 72 photos of animals. That is 9 times as many photos as Shane takes. How many fewer photos does Shane take than Aditi? Write equations to model and solve the problem. Let s = the number of photos Shane takes. Let f = how many fewer photos Shane takes than Aditi”. (4.OA.3)
• Lesson 5.7, More Practice/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. “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 after driving 7 hours?” (4.OA.3)
• Lesson 14.3, On Your Own, Problem 4, “Model with Mathematics, 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.B.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. “A short book only has 236 words. The library has 5 copies of this book. How many words appear in the books?” (4.NBT.5 and 4.NBT.6)
• Lesson 7.2, On Your Own, Problem 3, 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. “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.5 and 4.NBT.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. “There are 48 people going on a hike. Each pack of water has 8 bottles. How many packs are needed for each hiker to have 2 bottles? How can you check that your answer is reasonable?” (4.OA.3)
• Lesson 8.7, More Practice/Homework, Problem 3, 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. “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.3)
• Lesson 14.5, On Your Own, Problem 6, students solve word problems involving addition and subtraction of fractions with 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. Use b for the length of board Oliver cut off.“ (4.NF.3d)
• Lesson 15.1, More Practice/Homework, Problem 1, students solve word problems involving addition and subtraction of fractions with the same whole and having like denominators.“Olivia hikes the Appalachian Trail for $$\frac{9}{10}$$ mile on Saturday. She hikes for $$\frac{3}{10}$$ mile on Sunday. How much farther does Olivia hike on Saturday than on Sunday?” (4.NF.3d)
• Lesson 16.2, More Practice/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. “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 an equation using unit fractions that models the problem and the solution.”  (4.NF.4c)
• Lesson 16.3, On Your Own, Problem 9: Model with Mathematics, students solve word problems involving multiplication of a fraction by a whole number. “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.4c) 

Each Unit has a Performance Task that involves real world applications of the mathematics from that 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) and for 23 people to go to a show (4.OA.1). It continues by asking students to calculate how much a group comprised of adults and children would save by visiting one attraction versus another (4.OA.2 and 4.OA.3) and to calculate the area of a postcard display. (4.MD.3)

The instructional materials for HMH Into Math, Grade 4 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 include a progression 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 5.2, Build Understanding, Problem 1, Part B, 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.5)
• Lesson 12.6, On Your Own, Problem 12, emphasizes the application of multi-step problem solving with money. “Four friends earn a total of $7.00 by turning in cans for recycling. If the friends share the amount equally, how much does each get? Give your answer as a decimal dollar amount.” (4.MD.2)
• Lesson 8.6, On Your Own, Problem 6, builds procedural fluency in multi digit multiplication. “Estimate. Then choose a method to find the product. 43 x 35”. (4.NBT.2.

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

• Lesson 14.5, Step it Out, Problem 2, engages students in the application of addition and subtraction of fractions. “Ross makes popcorn at a carnival. At the start of his shift, the container of kernels weighs $$\frac{11}{12}$$ pound. During his shift, he uses $$\frac{3}{12}$$ pound of the kernels. How many pounds of kernels are left after Ross’s shift?” (4.NF.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).
• Unit 5 Performance Task, Problem 1, students solve application 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 earn enough money to buy a $49 video game? Explain your reasoning.”
• Lesson 16.3, On Your Own, Problem 9 states “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.4c)

The instructional materials reviewed for HMH Into Math Grade 4 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 those problems labeled “Critique Reasoning.” 

• Lesson 12.6, On Your Own, Problem 7, “Blake has $5.00. He sees some pencil packets that are $1.05. Blake says that he can buy 5 packets. Is he correct? Explain.” 
• Lesson 7.2, On Your Own, Problem 11, “Mara completes this division. Is her answer correct? Why or why not?” 
• Lesson 6.1, On Your Own, Problem 4, “Max says 20 objects can be separated into 4 equal groups. Mara says 20 objects can be separated into 5 equal groups. Who is correct? Explain. Draw to show your answer.”
• 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 “The fractions $$\frac{4}{5}$$ and $$\frac{7}{8}$$ each have one piece missing from the whole. How can you use the sizes of the missing pieces to compare the two fractions?”
• Lesson 15.1, Turn and Talk, “One classmate represents this problem with an addition equation and another uses a subtraction equation. Who is correct and how do you know?”
• 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 HMH Into Math Grade 4 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. Lesson 21.1 “Why is it helpful to change hours to minutes? How could you use reasoning to solve this problem without using the table?” 
• Critique, Correct, and Clarify is a strategy used to assist students in constructing viable arguments. Lesson 18.2, On Your 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 Your Own, Problem 19, students 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.”
• Lesson 4.1, Connect Math Ideas, Reasoning and Language states “Select students who used various strategies and tools to share with the class how they solved the problem. Have students discuss why they chose a specific strategy or tool.”
• 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 margin notes to prompt student engagement.  Lesson 3.1, s "Encourage students to discuss how their visual models show 3 groups of 4, which is the same as showing 4 three times. If some students are struggling, pair them with the students who quickly found a different visual model.”

The instructional materials reviewed for HMH Into Math Grade 4 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 11.2 “A known size or amount that helps you understand a different size of an amount is a benchmark. Common benchmarks are 0, $$\frac{1}{2}$$, 1, and $$1\frac{1}{2}$$”. 
• 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. Module 8: “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 the lesson planning pages, Sharpen Skills in some lessons include Vocabulary Review activities. Lesson 20.1 “Objective: Students review vocabulary used to classify triangles by angles and by sides. Materials: markers, poster paper.” The lesson continues “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 students 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. 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? What whole numbers round to 900 when you round them to the nearest hundred? How can you determine which numbers to place in the table?”
• Vocabulary is highlighted and italicized within each lesson in the materials. 
• There is a vocabulary review provided 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.