2nd Grade - Gateway 2

The instructional materials reviewed for HMH Into Math Grade 2 meet expectations that the materials develop conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings.

Each module contains two types of lessons specifically designed to engage students with conceptual understanding, Spark Your Learning and Bridging Lessons. The instructional materials present multiple opportunities for students to develop conceptual understanding, and examples include:

• Module 4, Lesson 2, students develop understanding of 3-digit numbers by using tens and ones blocks, connecting cubes, or drawing a picture, to show the number of crayons in boxes. Students answer "How many tens are in 100? Which number is in the hundreds place?" (2.NBT.1)
• Module 18, Lesson 3, students measure to the nearest inch. Students begin the lesson using tiles to measure the length of an object and expand to using an inch ruler. (2.MD.1)
• Module 19, Lesson 1, students estimate lengths using centimeters. Students use a string and connecting cubes at the beginning of the lesson to find everyday objects that are longer than 10 centimeters and to compare measurements. Students move to using unit cubes and paper clips to measure, estimate, and compare length. (2.MD.A)
• Module 20, Lesson 1, students relate inches to a number line. Students use a ruler as a number line to solve a word problem, “What does the ruler remind you of? (A Number line) What can a number line help you with?” (2.MD.6)

The instructional materials present multiple opportunities for students to develop conceptual understanding, and examples include:

• Module 2, Lesson 2, students use equations to represent addition and subtraction situations. Students use cubes, drawings, and equations to show their understanding of addition and subtraction. (2.OA.1)
• Module 20, Lesson 1, students relate inches to a number line. Students use a ruler as a number line to solve a word problem, “What does the ruler remind you of? (A Number line) What can a number line help you with?” (2.MD.6)

The instructional materials reviewed for HMH Into Math Grade 2 meet expectations for attending to those standards that set an expectation of procedural skill and fluency.

Students develop procedural skills and fluencies throughout the grade level. Each module contains Procedural lessons that help students develop the steps in a procedure and determine when the procedure should be used. Module and Lesson components that specifically attend to student’s developing and independently demonstrating procedural skill and fluency include:

• In Module Planning: Teaching for Success, Teacher to Teacher notes give the teacher advice on how to question the student in order to build procedural fluency. For example, in Module 6, Teacher to Teacher suggests teachers ask questions about adding tens to numbers using mental math. (2.NBT.4)
• In Homework and Test Prep, students practice skills and develop fluency through the Spiral Review. For example, Module 16, Lesson 2, Spiral Review, students demonstrate fluency with subtracting two-digit numbers. (2.NBT.5)

Modules 10, 11, 12, 13, and 15 address 2.NBT.5, fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction. For example:

• Module 11, Lesson 1, Sharpen Skills, students build fluency and practice basic math skills to support fluency in adding within 100. In Build Understanding, students knowledge of “making ten” is extended to making multiples of ten. 
• Module 11, Lesson 4, Fluency Builder, students find the missing number in a subtraction equation. Students practice fluency in the More Practice/Homework as they break apart vertical addition problems into units to add.
• Module 12, Lesson 3, students develop fluency adding two-digit numbers using the standard algorithm. During On Your Own and More Practice/Homework, students demonstrate fluency in regrouping to solve problems accurately and efficiently with a numerical model.

Module 1 addresses 2.OA.2, fluently add and subtract within 20 using mental strategies. Students build fluency by using strategies such as: doubles, fact families, properties of operation, and making ten to add or subtract. Specific examples include: 

• Module 1, Lesson 3, On Your Own, students use related facts and make connections between operations using bar models to develop fluency in adding and subtracting within 20.
• Module 1, Lesson 4, More Practice/Homework, students choose a strategy to subtract.

The instructional materials reviewed for HMH Into Math Grade 2 meet expectations for teachers and students spending 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.

Students engage in routine application problems throughout the grade level. In Independent Practice and On Your Own, students apply what they have learned to solve real world problems independently. For example: 

• In Module 13, Lesson 2, On Your Own, Problem 9, students rewrite subtraction problems. “A tennis team practices with 41 tennis balls. Some tennis balls are green and some are yellow. The team practices with 28 yellow tennis balls. How many green tennis balls does the team practice with?” (2.OA.1)
• In Module 15, Lesson 1, On Your Own, Problem 4, students solve addition and subtraction word problems. “Sara has 48 dog treats. She has 32 fewer than Min. How many dog treats does Min have? Write an equation to show the problem. Solve.” (2.OA.1)
• In Module 15, Lesson 2, Check for Understanding, Problem 1, students solve subtraction word problems. “A store sells 38 bags of grapes. There are 27 bags of grapes left at the end of the day. How many bags of grapes are there to start?” (2.OA.1)
• In Module 15, Lesson 3, On Your Own, Problem 10, students solve multi-step addition and subtraction problems. “There are 8 children at the party. 6 more children come to the party. Then 4 children leave. How many children are at the party now?” (2.OA.1)

Examples of non-routine application of the mathematics include:

• In Module 3, Lesson 3, On Your Own, students draw picture graphs to represent data given in context. Students complete a picture graph using data given and answer questions using the picture graph. Students also “Write a new question that you could answer using the graph. Then solve.” (2.MD.10)
• In Module 8, Lesson 3, On Your Own, Problem 2, students solve word problems involving money. For example, “Manny saves 63¢ and Erica saves 30¢. How much money do they save?” (2.MD.8)
• In Module 12, Lesson 5, On Your Own, Problem 7, students add two digit numbers. “There are 54 notebooks in a box. Some notebooks are blue. Some notebooks are red. How many notebooks of each color should be in the box?” Students then create their own problem to equal 54. (2.OA.1)

The instructional materials reviewed for HMH Into Math Grade 2 meet expectations for the three aspects of rigor not always being treated together and not always being treated separately. Overall, two, or all three, of the aspects are interwoven throughout each module.

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

• In Module 1, Lessons 1-7, students develop procedural skill and fluency in adding and subtracting within 20 using mental strategies. (2.OA.2)
• In Module 7, Lesson 1, Build Understanding, students use place value understanding to add coins. Task 2, “Jack puts 4 dimes and 6 pennies in his coin bank today. Show these coins using tens and ones. Then find the total value of the coins. Part A: How can you show 4 dimes and 6 pennies in the chart? Draw to show your work.” (2.NBT.2, 2.MD.8) 
• In Module 12, Lesson 2, Spark Your Learning, students apply their understanding of subtraction. (2.NBT.5) Problem 1, "Steve has 15 baseball cards. He gives 7 baseball cards to Evan. How many baseball cards does Steve have now?"

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

• In Module 12, Lesson 2, More Practice and Homework, students apply their understanding of regrouping in the Attend to Precision problem. “There are 24 fish in a pond. Then 16 fish swim away. How many fish are there now? Do you need to regroup? Explain."
• In Module 17, Lesson 5, students use conceptual understanding to draw a visual model that shows subtraction equations in application problems “On a farm, there are 800 stalks of corn. There are 595 bean plants. How many more stalks of corn than bean plants are on the farm?” Students are asked to draw to show their regrouping (Step It Out, Task 1). (2.NBT.7)

The instructional materials reviewed for HMH Into Math Grade 2 meet expectations for carefully attending to the full meaning of each practice standard (MP). 

The materials attend to the full intent of the MPs. Examples of the instructional materials attending to the full meaning of the MPs include:

• MP1: In Module 1, Lesson 2, Spark Your Learning, “Choose two numbers. How can you show the sum of the numbers.” Guiding questions teachers can ask are, “What tools could you use to represent the addition fact you wrote? How does the addition fact relate to your concrete model or visual model? How are the two addition facts the same? How are they different?” 
• MP2: In Module 14, Lesson 3, students reason abstractly and quantitatively by completing the bar model by deciding where to place each number from the word problem. Students are reminded to use a blank for the unknown number. Teachers ask them how they can add to find the unknown number. The sample guided discussion given states, “What do you need to find? What do you need to find the total? What strategy can you use to add the numbers?” 
• MP4: In Module 15, Lesson 2, Independent Practice, Problem 8, “Write a word problem for this equation. 56 - 37 = 19.”
• MP5: In Module 18, Lesson 3, Spark your Learning, “Emilo’s class measures a leaf. Draw tiles to show the length of the leaf. Then draw the leaf using one of your paper rulers.” Teachers are then instructed to ask “What other tool could you use to solve the problem? Compare using that tool with the one you used in the problem.”
• MP6: In Module 5, Lesson 1, Build Understanding, Task 1, has the teacher read and discuss the problem. Have children show a three-digit number in different ways. The sample guided discussion askis, “Why can 321 be written as 3 hundreds, 2 tens, and 1 one and also as 300+20+1? How does the quick picture help you write the number in expanded form?” 
• MP7: In Module 2, Lesson 5, On Your Own, Problem 2, “Amav puts some watermelon slices in 3 rows. There are 2 slices in each row. Draw the slices. Then write an addition equation to find the total number of slices.” Students use arrays to represent repeated addition as they begin to build underlying concepts of multiplication. 
• MP8: In Module 6, Lesson 1, Step It Out, Task 2, Sample Guided Discussion, “What pattern are you using when you count by fives?" and "What pattern are you using when counting by hundreds?”

The instructional materials reviewed for HMH Into Math Grade 2 meet expectations for prompting students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics. 

Students have opportunities to construct viable arguments through activities such as explaining their thinking or justifying steps, and the materials prompt them to analyze the arguments of others. Examples include: 

• In Module 1, Lesson 4, On Your Own, Problem 5, students “Construct Arguments: There are 12 squirrels sitting in an oak tree. Then 9 squirrels leave the tree. How many squirrels are in the tree now? What strategy did you use to solve the problem? Explain.”
• In Module 7, Lesson 3, On Your Own, Problem 2, students solve “Tasha uses these coins to buy a balloon. What is the total value of the coins? Draw to show your work.” Problem 3: Construct Arguments: “Explain how you solved Problem 2.”
• In Module 12, Lesson 5, On Your Own, Problem 9, students “Construct Arguments: Explain the addition that you did to solve Problem 8. Be sure to tell about the values of the digits in the numbers.”
• In Module 17, Lesson 6, On Your Own, Problem 7, students “Construct Arguments: Lexi and Zach both solved the same subtraction problem, but they got different answers. Check Lexi and Zach’s work. Who got the right answer? Explain how you know." 
• In Module 20, Lesson 3, On Your Own, Problem 2, students “Construct Arguments: Mrs. Morgan has a wood board that is 67 centimeters long. She cuts off a piece of the board. The board is now 49 centimeters long. How many centimeters does Mrs. Morgan cut off? ____ centimeters. Explain how you found your answer.”
• In Module 21, Lesson 1, On Your Own, Problem 3, students “Construct Arguments: Sahil is not sure if he is holding a rectangular prism or a cube. How can we decide which shape he has? Explain.” Students will explain that by looking at the faces of the three-dimensional shape, Sahil can tell it is a rectangular prism because the faces are rectangles.”

The instructional materials reviewed for HMH Into Math Grade 2 meet expectations for assisting teachers in engaging students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics.

The materials provide teachers with Sample Guided Discussions, Turn and Talks, and Leveled Questions to assist teachers in engaging students in discourse. There is also some teacher guidance on how to lead discussions beyond the provided questions. Examples include: 

• In Module 2, Lesson 4, Turn and Talk, teachers ask “How did the number of rows of toy trains help you write the addition equation? Explain.”
• In Module 4, Lesson 4, Turn and Talk, “What if there were 7 tens in the number? How would the number be different? What number would you write? Explain.”
• In Module 7, Lesson 2, Spark Your Learning, “Select children who have used various strategies and tools to share with the class how they solved the problem. Have children discuss why they chose a specific strategy or tool.”
• In Module 11, Lesson 4, Sample Guided Discussion, teachers prompt student discussion by asking, “How can you verify your solution to the problem using a concrete model?”
• In Module 14, Lesson 3, the materials state, “How does completing a bar model help you write an equation to represent and solve a problem?” Teachers are guided to, “Monitor children to note their strategies for writing equations to represent the problem. Ask children to share their strategies and discuss which strategy might be most helpful to solve the problem.” 
• In Module 19, Lesson 4, Build Shared Understanding, teachers, “Select children who have used various strategies and tools to share with the class how they solved the problem. Have children discuss why they chose a specific strategy or tool.”

The instructional materials reviewed for HMH Into Math Grade 2 meet expectations for 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, accurate terminology and definitions when describing mathematics and support students in using them. 

The Planning and Pacing Guide has a section for Language Development that states HMH Into Math is built upon 4 design principles to promote the use and development of language:

• Principal 1: Support Sense-Making;
• Principal 2: Optimize Output to help students describe their mathematical reasoning and understanding;
• Principal 3: Cultivate Conversations to facilitate mathematical conversations among students; and,
• Principal 4: Maximize Linguistic and Cognitive Meta-Awareness to help students evaluate their use of language and see how mathematical ideas, reasoning and language are connected.

Language Routines and new/review vocabulary are summarized on the Language Development page for each module, and this also includes Key Academic Vocabulary for Prior Learning - Review Vocabulary and Current Development - New Vocabulary with definitions. Also in Language Development, Linguistic Notes provide teachers help with possible misconceptions relating to academic language. For example:

• In Module 2, the Linguistic Note states, “Many classroom commands use words that are familiar in isolation, but as phrases may be misunderstood. To help English Language Learners be successful with the lesson, provide additional assistance with phrases such as make pairs, count by twos, and equal groups.”
• In Module 8, the Linguistic Note states, “Mathematics is filled with symbols. Take time to identify the connections between symbols and meaning, such as the dollar sign. To help children succeed with this lesson, relate that meaning to math concepts. When studying money, it is important for children to understand that each coin and bill has a specific value.”
• In Module 14, Lesson 4, Sharpen Skills, “Children will build math vocabulary by using a graphic organizer. Facilitate a discussion with children about what they remember about the word difference. Begin by having them give ideas about real-world situations that involve subtraction”
• In Module 18, Lesson 1, the Connect Math Ideas, Reasoning, and Language states, “Have children use the Interactive Glossary during this conversation to record their understanding. Before beginning the task, have children use their own words to explain how to estimate. Then have partners share their work and discuss how their answers compare and contrast.” 
• In Module 16, Key Academic Vocabulary includes hundreds, ones, regroup, and tens. In Module 20, Review Vocabulary includes centimeter and inches

Student pages include vocabulary boxes defining content vocabulary. Vocabulary is highlighted and italicized within each lesson in the materials. The vocabulary review at the end of each Module requires students to match new vocabulary terms with their meaning and/or examples provided, fill-in-the-blank with definitions or examples, or create a graphic organizer to help make sense of terms. Some lessons include Vocabulary Review. Connect Math Ideas, Reasoning, and Language Compare and Connect encourage students to use vocabulary terms to discuss mathematics with correct terminology. For example:

• In Module 7, Lesson 2, the Connect to Vocabulary box includes Nickel: “A nickel has a value of 5¢.” “A quarter has a value of 25¢.”
• In Module 18, nine new terms are introduced. New terms are used consistently throughout the module. In Lesson 6, for example, the Spark Your Learning states, “Measure a classroom object, like a mirror, in two ways. What do you notice about the measurements?” In Lesson 18.3, Connect Math Ideas, Reasoning, and Language Compare and Connect states, “Remind children they are learning how to measure and estimate length in units. Before beginning the task, have children use their own words to explain the term inch ruler. Then, have partners share their work and discuss their responses..” 
• In Module 21, Lesson 1, Build Understanding, Connect Math Ideas, Reasoning, and Language Compare and Connect states, “Remind children they understand how to solve problems using two- and three-dimensional shapes. Before beginning the task, have children use their own words to describe the face, edge, and vertex of a three-dimensional shape. Have partners share their work and discuss how their answers compare and contrast.” 

Vocabulary cards can be used with vocabulary games. The eGlossary includes vocabulary terms and definitions translated into ten different languages. The Interactive Glossary provides the definition and a visual (diagrams, symbols, etc.) is provided for each vocabulary word.The Interactive Glossary also provides space for students to make graphic organizers or drawings for each new vocabulary term. In the student materials, the instructions state, “As you learn about each new term, add notes, drawings, or sentences in the space next to the definition. Doing so will help you remember what each term means.”