3rd Grade - Gateway 2

The instructional materials for HMH Into Math Grade 3 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 “Ling is at a fair with her friends Jimmy and Pablo. At a game booth, they each get 4 balls. How many balls do Ling and her friends get?” Show more than one way to find the number of balls. (3.OA.1)
• Lesson 6.2, Spark Your Learning states “Annie wants to arrange 20 photographs in equal groups in a book. She wants to make equal groups. Write questions that can be asked about Annie's photographs. Draw equal groups to show how Annie's photographs can be arranged.” (3.OA.1)
• Lesson 13.1, Build Understanding, students compare pictures of flags to determine which are divided into equal parts and which are not. They are asked to use this knowledge to draw both types of flags and then they name equal parts using words like fourths and eighths. (3.NF.1)
• Lesson 13.3, On My Own, Question 4, students shade four equal parts of a hexagon, then write the fraction in words and numbers. (3.NF.1) 
• Lesson 13.6, Practice and Homework Journal, Question 7 states “The shape represents $$\frac{1}{2}$$ of a whole. To make an amount that is greater than 1, how many shapes will you need? Draw your shapes. Write the mixed number that represents the amount you drew.” (3.NF.1)
• Lesson 15.3, Practice and Homework Journal, Question 8, Math on the Spot, states “James ate $$\frac{4}{8}$$ of his pancake. David ate $$\frac{4}{6}$$ of his pancake. Who ate more of his pancake? James said he knows he ate more because eight is greater than six. Does his answer make sense?” (3.NF.3d)

The instructional materials for HMH Into Math Grade 3 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 3.OA.7. Examples include but are not limited to: 

• Lesson 4.6, Multiply by 9’s, On Your Own, Problem 5: $$9 \times 1 =$$ __. Problem 6: $$4 \times 9 =$$ __. Problem 7: ___ $$=9 \times 7$$. (3.OA.7)
• Lessons 7.3 and 7.4 provide students practice with multiplication and division with 2, 4, 5, 8, and 10.  Different representations are presented for each operation. (3.OA.7)
• In Lesson 7.7 students practice skip counting by 2, 3, and 4. Students circle the numbers that are the same in each set. (3.OA.7)
• Lesson 9.2 asks students to use mental math to find the sum or difference. “Problem 5, 46 + 24 + ____. Problem 6, 639 - 425 = ___.” 
• Lesson 10.2 asks students to estimate and find the sum of two multi-digit numbers. Problem 7, “612 + 75”; Problem 8, “548 + 56”; Problem 9: “324 + 119”. (All problems are presented vertically). (3.OA.7)
• In Lesson 10.4 students find the difference using place value and regrouping. (3.NBT.2)

The instructional materials for HMH Into Math Grade 3 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, Independent Practice, On Your Own, students engage with problems that include real-world context and present opportunities for application. For example:

• Lesson 1.3, On Your Own, STEM problem, students use multiplication and division within 100 to solve word problems. “Solar cells on a solar panel collect sunlight which is changed into electricity. Solar cells are arranged in equal rows on a frame to make a solar panel. Describe one way you could arrange 24 solar cells in an array to make a panel.”  (3.OA.3)
• Lesson 1.4, Spark Your Learning, students use multiplication and division within 100 to solve word problems in situations. “Andy designed the game board shown (picture provided of a flow chart). The game will have 21 squares The squares need to be in equal rows. Show two different game board designs.” (3.OA.3)
• Lesson 3.3, On My Own, Problem 4: “Michael buys 2 packages of hamburger buns. Each package has the number of buns shown. How many hamburger buns are in 2 packages? Show the equal groups. Write a multiplication equation for the problem.” (3.OA.3)
• Lesson 5.4, More Practice/Homework, Problem 9: students solve two-step word problems using the four operations. “Ava's class buys 8 packages of balloons for the class celebration. Each package has 50 balloons. If 21 balloons are left over, how many balloons are used for the party?” (3.OA.D.8)
• Lesson 8.4, On Your Own, Problem 11, students write two equations with letters representing the unknown to solve the problems. “Jamie’s plant grows 3 inches each week for three weeks. During the fourth week, it growth 5 inches. How much does Jamie’s plant grow over the four weeks?” (3.OA.D.8)
• Lesson 10.6, On Your Own, Problem 8: “Write a two-step word problem with an unknown number. Write equations to model the problem. Then solve.” (3.OA.8) 

Each unit has a Performance Task that involves real world applications of the mathematics from that unit. For example, the Unit 3 Performance Task has students follow one child, a baseball card collector, throughout his day. Students estimate the number of baseball cards he has at the end of the day (3.NBT.1), tell the time of his various stops throughout the day (3.MD.1), determine the perimeter of his baseball card display (3.MD.8), and how many coins he has (not counting money) (3.OA.3).

The instructional materials for HMH Into Math Grade 3 meet expectations in 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 pages 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.

• Lesson 16.1 builds conceptual understanding of equivalent fractions. Students draw visual models and use number lines to show fraction equivalence. (3.NF.3b)
• In Modules 4, 7, 9 and 10, students develop procedural skill and fluency as students find products, work with related facts, division, as well as implementing estimation and mental math to support the addition and subtraction for the grade level. (3.OA.7 and 3.NBT.2) 
• Lesson 8.4, On Your Own, Problem 8, students engage in application as they “Write a two-step word problem that can be solved using two equations with different operations.” (3.OA.8)

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

• Lesson 7.6, Check for Understanding, Problem 4, students solve: “The 28 students in Van’s class are on a field trip to a cave. They are divided into groups of 7 for a tour of the cave. How many groups are there?” (3.OA.5).
• The Unit 4 Performance Task attends to conceptual understanding and application. Students create number lines to visually see equal parts and then use it to solve real-world problems. Item 2, “Ava makes a white cake and a strawberry cake that are the same size. She cuts the white cake into fourths and the strawberry cake into eighths. Yoshi puts $$\frac{1}{4}$$ of the white cake on a plate. He wants to put an equal amount of the strawberry cake on the plate. How many pieces of each cake should Yoski put on the plate? Use the number line to show what fraction of each cake Yoshi should put on the plate.” (3.NF.1, 3.NF.2, 3.NF.3, and 3.MD.4.)
• Lesson 13.6, Spark Your Learning, attends to conceptual understanding and application. “Emilio cuts his pizzas into slices. Each slice is a fourth of a whole pizza. Emilio has 9 slices to sell. Show all the different amounts of pizza that Emilio can sell. Name each fraction that you show.” (3.NF.1 and 3.NF.2)

The instructional materials reviewed for HMH Into Math Grade 3 meet expectations that the Standards for Mathematical Practice are identified and used to enrich mathematics content within and throughout the grade-level.

The MPs are identified at the unit, module and lesson level. In addition, the information in the Planning and Pacing Guide also include references to the MPs. For example:

• The Planning and Pacing Guide outlines for teachers where to look for each of the MPs. It states “MP1, MP3, and MP5 are paired with Spark Your Learning tasks. When students connect understanding they have developed with more efficient procedures, MP7 and MP8 are being attended to. This helps students explain and justify their procedures with MP4, MP2, and MP6 are attended to within lessons that ask students to apply procedures in practice.”
• Planning and Pacing Guide pages 17 -19 provide additional detail and clarity about each Mathematical Practice. These pages also include “Questions to Ask” with each MP. These questions provide support to advance students. 
• MPs are clearly identified throughout the materials. For example, MP1 is identified in Lessons 2.5 and Lesson 8.2; MP2 in Lessons 4.6 and Lesson 6.2, MP3 in Lesson 9.6 and  Lesson 13.3; MP4 in Lessons 7.2 and Lesson 7.7; MP5 in Lesson 5.4 and Lesson 7.3; MP6 in Lessons 2.5 and Lesson 10.6; MP7 in Lesson 1.4 and Lesson 4.6; and MP8 in Lesson 7.7 and Lesson 8.1.
• The Planning and Pacing Guide for the teachers has a section identified as Correlations for Mathematical Practices. In this section, the 8 Mathematical Practices are listed in a table with a detailed description (from the common core documents) of the practice as well as “some examples” of where the practice is included in the text series.
• The Module Review includes a labeled question “Use Tools” in the student edition that asks students to choose a tool and explain their choice. The Teacher Edition recommends and provides additional support to have students discuss and share strategies and tools used as part of their review of the Module.

Within the Teacher Edition, in the margin under Homework & Test Prep, there is a section that describes the MPs that can be found within the Homework worksheet for the students. For example: 

• Lesson 4.7, On Your Own, Problem 11, is identified as “Use structure.” 
• Lesson 12.1, On Your Own, Problem 2, states “Critique Reasoning.” 

For the most part, when identified, MPs are used to enrich the mathematical content of the lessons. Examples include:

• Lesson 4.4, Build Understanding, identifies MP8 as students use structure from previous work with the distributive property.
• Lesson 16.1, Build Understanding, identifies MP3 where students construct arguments about equivalent fractions.
• Lesson 10.5, On Your Own, Problem 5, has students engage in MP6 as they explain a strategy used to solve a problem. 

The instructional materials reviewed for HMH Into Math Grade 3 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 those labeled “Critique Reasoning.”

• Lesson 7.1, On Your Own, Problem 3, students critique the reasoning of two students who have different ways to solve the same problem, one using division and the other multiplication. “Eva has 21 squash seeds. She plants groups of 3 squash seeds together in a mound. Eva says she can use division to find the number of mounds she plants. Jenna says that she can use multiplication to find the number of mounds. Who is correct? Explain your thinking.”
• Lesson 10.6, On Your Own, Problem 3, students analyze the work of others. Students are given a box labeled ‘Ed’s Work’ with two equations in it. “Ed has 8 boxes with 7 rocks in each box. Then he finds 9 more rocks. Ed writes these equations to find how many rocks he has now. Is Ed’s work correct? Explain.”
• Lesson 11.4, Turn and Talk, states, “When rectangles have the same area, how do you know which will have the greatest perimeter and the least perimeter?”.
• Lesson 13.2, Turn and Talk, states, “Compare the fractions in Task 1 and 2. How are they alike? How are they different?”
• Lesson 14.1, Turn and Talk: “What fraction of the whole playground does each hamster's play space represent?”
• Lesson 16.2, Spark your Learning, “Thea is a landscaper. According to her design, $$\frac{2}{3}$$ of the garden should contain red roses. She plants red roses in $$\frac{4}{6}$$ of her new garden. DoesThea make a mistake? Show a way to solve the problem.” 
• Lesson 16.2, Build Understanding, “Coach Penny draws a diagram in which a soccer field is divided into 8 equal-sized zones. The 6 zones close to the goal are the scoring zones. Coach Ruiz divided the same field into 4 equal-sized zones with the 3 zones closest to the goal as the scoring zones.” Students write fractions for the sections of Coach Penny's portion of the field that are scoring zones, and Coach Ruiz's portion of the field. They then compare the areas of the two scoring zones and explain similarities and differences. 
• Lesson 5.1, On Your Own, Problem 4, “Pam says that she can write $$8 \times 60$$ as the sum of two products. Is she correct? Explain.”
• 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.”

The instructional materials reviewed for HMH Into Math Grade 3 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. In Lesson 4.7, Step it Out, Sample Guided Discussion, students are asked “How can you tell if a product will be even? How can you represent the number 4 as the sum of two equation addends? So, if you multiply 7 by (2 + 2), what are the two smaller products? What statements can you write about whether the product of two odd numbers is even or odd? How can you prove why your statement makes sense using equal groups?”
• Critique, Correct, and Clarify is a strategy used to assist students in constructing viable arguments. In Lesson 2.2, On Your Own, Problem 7, students analyze a statement made by a fictitious student. Teachers are told to “Point out to students that in Problem 7, Katy’s statement about a gap when measuring the area of a figure may or may not be correct. Encourage students to describe why the statement is or is not correct and to review explanations with a partner. Students should refine their responses after their discussions with a partner.” In Lesson 9.6, On Your Own, Problem 6, students analyze two ways to estimate the difference of 524 - 365 and tell which estimate will be closer to the actual difference. Teachers are told to “Encourage students to describe why they think one estimate is closer to the actual difference and to 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. In Lesson 1.3, it states “Select students who used various strategies and tools and have them share how they solved the problem with the class. Encourage students to share with the class how they solved the problem. Have students discuss why they chose a specific strategy or tool.”
• Lesson 13.6, Connect Math ideas, Reasoning, and Language, it states “Ask students to share their strategies. Prompt discussion by asking ‘How are the approaches similar? How are they approaches different?”

The instructional materials reviewed for HMH Into Math Grade 3 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 lessons, new vocabulary is introduced in highlighted sections called Connect to Vocabulary. Lesson 13.1, “A whole is all of the parts that make up one shape or group. If all of the parts of a whole are the same size, then the whole is divided into equal parts.” Lesson 10.2 “To regroup is to exchange amounts of equal value to rename a number. Examples: 17 ones is 1 ten and 7 ones. 13 tens is 1 hundred and 3 tens.”
• In the lesson planning pages, Sharpen Skills for some lessons include Vocabulary Review activities. Lesson 10.1, “Objective: Students complete graphic organizers for the terms sum and expanded form.” “Materials: Word Descriptions graphic organizer,” “Have students work in small groups to complete a Word Descriptions for the term sum. Have students discuss the definition and state examples and non-examples of a sum. Then have students complete a Word Descriptions for the term expanded form."
• 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 sum, addend, and equal groups.  Ask students what they mean if they use those terms.” “How could you use an array to represent each total?” 
• Student pages include Connect to Vocabulary boxes that define content vocabulary. Lesson 4.1, “The Identity Property of Multiplication states that the product of any number and 1 is that number.”
• 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.