1st Grade - Gateway 2

The instructional materials reviewed for HMH Into Math Grade 1 meet expectations for developing 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 and conceptual understanding, and examples include:

• Module 2, Lesson 2, Count Back, begins with students working in pairs and using connecting cubes, counters, or drawing pictures to show subtraction. During Build Understanding, students use manipulatives and pictures to count back, solve, and write an equation to solve the problem. (1.OA.5)
• Module 2, Lesson 4, students use strategies such as counting on; making ten; decomposing a number leading to a ten; using the relationship between addition and subtraction; and creating equivalent but easier or known sums. In the Teacher’s Edition, "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." (1.OA.6)
• Module 6, Lesson 2, Spark Your Learning, students represent the number of animal cards in two different ways using counters, connecting cubes, or pictures. For example, “Liz has 9 animal cards. They are elephant cards and tiger cards. How many of each animal card could she have?” (1.OA.1)
• Module 9, Lesson 1, students choose a tool to represent teen numbers using tens and ones. Students use a ten frame to answer "How many counters fill a ten frame?" and "How does the quick picture help you know the value of a number?" (1.NBT.2)

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

• Module 7, Lesson 3, students select and draw a number of counters that are more than 3 and fewer than 10. Students then show 3 fewer counters. Students use pictures, counters, and equations to deepen their understanding of unknown problems. (1.OA.1)
• Module 10, Lesson 1, students represent a two digit number using a picture or base ten blocks. They then write the number of tens and ones in the number. (1.NBT.2)
• Module 17, Lesson 2, students compare the length of objects to the length of a string and determine if the string is longer or shorter. Students answer, "How will you use the string and the objects you chose as tools to compare length?" Students also compare the lengths of objects and draw to show which objects are longer or shorter. (1.MD.1)

The instructional materials reviewed for HMH Into Math Grade 1 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 having students think about the problem 12 - 5. The teacher asks questions about what equation they could write and how they would use tally marks to solve the problem. (1.OA.1)
• Module 3, Lesson 1, students use pictures and equations to represent addition in other ways using the commutative property of addition. Check Understanding, Problem 1, “Ron Plants 3 white rose bushes and 9 red rose bushes. How many rose bushes did he plant? ____=____+____ or ____=____+____rose bushes.” (1.OA.3)
• Module 8, Lesson 7, Step It Out , students use information from a chart to solve a problem using tally marks to show how many flowers are in different vases. “How do you know how many tallies to make for Vicki? What is one way to use the tally chart to tell whether Lian or Marcy put more flowers in the vase?” (1.MD.4)

 Unit 1 addresses 1.OA.6, add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Students build fluency through adding 10 and more, making a 10 to add, adding doubles, and using known sums to add. For example:

• Module 1, Lesson 4, On Your Own, Problems 4-9 , students “make ten” to add within 20. Students fluently build 10 and add the remaining value.
• Module 2, Lesson 2, Learn Together and Independent Practice, address addition and subtraction within 20, demonstrating fluency for addition and subtraction within 10 when students use a number line to count back to subtract and solve numerous subtraction problems.
• Module 3, Lesson 7, Step It Out, students find the sum of number cubes to develop fluency within 10 and apply that fluency to find sums within 20. In On My Own, Problem 4, students show three different ways to make 10. and students develop fluency of addition within 10.

The instructional materials reviewed for HMH Into Math Grade 1 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 1, Lesson 7, Independent Practice, students draw and solve, “Julian has 7 marbles. His brother gives him 4 more. How many marbles does Julian have now?” (1.OA.1)
• In Module 2, Lesson 6, Check Understanding, students write an equation and explain the strategy they used to solve a word problem. “Nate has 14 apples. He gives 5 of them to friends. How many apples does he have now?” (1.OA.1)
• In Module 7, Lesson 6, students solve, “There are 13 blueberries. There are 8 more blueberries than cherries. How many cherries are there? Use a strategy to solve the problem. Draw or write to show the strategy. There are ___ cherries.” (1.OA.6) 
• In Module 8, Lesson 2, On Your Own, students use colored marbles to create a graph, interpret the data, and reason about the data. (1.MD.4)
• In Module 12, Lesson 8, students Use Mental Math to Make 10 Less and 10 More. During Independent Practice, students solve, “Susan, Tracy, and Kris each have a rock collection. Susan has 48 rocks in her collection. Tracy has 10 more rocks than Susan. Kris has 10 more rocks than Tracy. How many rocks do Tracy and Kris each have?” (1.NBT.5)

Examples of non-routine application of the mathematics include: 

• In Module 3, Lesson 5, On Your Own, Problem 8, students write their own word problem with three addends, and then solve it. (1.OA.2)
• In Module 5, Lesson 4, students “Write a story problem for the equation. Solve the problem. 17 - 8 = ?” (1.OA.1)
• In Module 6, Lesson 7, Independent Practice, students respond to “Cindy sees 13 butterflies. Some are orange and some are blue. How many butterflies could be orange, and how many could be blue?" Students are given a bar model to help them solve. (1.OA.1)

The instructional materials reviewed for HMH Into Math Grade 1 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, Lesson 3, students develop procedural skill and fluency of addition within 20. (1.OA.6)
• In Module 5, Lesson 1, students develop conceptual understanding as they use objects and drawings to solve add to and take from problems where the results are unknown. (1.OA.1)
• In Module 5, Lesson 4, students solve add to and take from application problems. “Bert has some markers. He gives 8 to Alex. Now he has 5 markers. How many markers does he have to start?” (1.OA.1)

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 4, Lesson 3, students develop conceptual understanding of related facts using two color connecting cubes, and they develop fluency in writing related addition and subtraction equations. (1.OA.6)
• In Module 7, Lesson 7, Step it Out, Problem 5, students use procedural skills and visual models to solve difference unknown word problems. “Sasha has 12 paper clips. Thomas has 7 fewer paper clips than Sasha. How many paper clips does Thomas have?” Students are prompted to use and draw counters and write an equation. (1.OA.1)
• In Module 9, Lesson 1, Build Understanding, Task 1, students build conceptual understanding using a variety of strategies to represent word problems. “There are 6 ladybugs on a bush. 6 more ladybugs join them. How many ladybugs are there now? Part A: How can you make a concrete model to solve the problem? Draw to show what you did. Part B: A quick picture uses sticks and circles to show tens and ones. How can you draw a quick picture to show the number of ladybugs?” (1.NBT.B.2)

The instructional materials reviewed for HMH Into Math Grade 1 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 4, Lesson 2, Build Understanding,Task 1, “What are some ways you can use your concrete model to help you work through this problem? How do you know the facts you wrote in Part B are related facts? ”
• MP2: In Module 3, Lesson 5, On Your Own , Problem 7, “Reason: Find the unknown addend. Complete the equation. 1 + 3 + __ = 5.”
• MP4: In Module 6, Lesson 7, Step It Out, Task 2: “Lia has some animal books. She gives 9 books to Max. Now she has 5 books. How many books did Lia start with?” Students are guided to create a visual model for the problem, write a model they can use to solve the problem and use the models to organize their thinking.
• MP5: In Module 2, Lesson 2, Spark your Learning, “Viktoria makes 7 picture frames. She gives 2 to her friend. How many picture frames does she have now? How can counting help you solve the problem” Teachers are guided to ask, “Which tool could you use to solve the problem? Why is the tool you chose the one that works for you?” 
• MP6: In Module 12, Lesson 5, Step It Out, Task 2, prompts teachers to “Encourage children to explain how their drawings show the problem and explain why it is important to be accurate in their drawings.” “There are 20 paint jars in a store. The store gets 34 more paint jars. How many paint jars does the store have?”
• MP7: In Module 15, Lesson 1, Build Understanding, Turn and Talk, “How is a rectangle and a square the same? How are they different?”
• MP8: In Module 2, Lesson 2, Build Understanding, Task 2, “A kangaroo is on step 6 and takes 2 jumps down. What step is the kangaroo on now?  How can you use the picture to show how to count back? B. How can you write an equation to solve the problem?”

The instructional materials reviewed for HMH Into Math Grade 1 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 8, Lesson 3, Build Understanding, Problem 1C, “Did more friends wear blue or orange shirts? Circle your answer. How do you know?”
• In Module 8, Lesson 5, On Your Own, Problem 3, “Reason. There are 12 children in art class. 5 children paint. The rest make clay pots. Andy made this bar graph to show the data. He made a mistake. Explain the mistake Andy made.”
• In Module 13, Lesson 3, Build Understanding, Turn and Talk, “How can you use tens and ones to solve a subtraction problem? Explain the steps you used to subtract 90 – 40.”
• In Module 15, Lesson 2, On Your Own, Problem 4, students “Construct Arguments: Explain how to draw a triangle.”
• In Module 17, Lesson 3, On Your Own, Problem 7, students “Construct Arguments: How do you know that you colored half of each shape?”
• In Module 17, Lesson 1, On Your Own, students “Construct Arguments: Mercer thinks she circled the longest pencil. What is her mistake?”

The instructional materials reviewed for HMH Into Math Grade 1 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 4, Lesson 3, Learn Together, Turn and Talk states, “Have children name two other related facts and explain their thinking.”
• In Module 11, Lesson 1, Spark Your Learning, “Select children who 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 Lesson 16.2, More Practice/Homework Problems 1 and 2, the materials prompt teachers to ask students, “How do you know the shares are equal? and How do you know the shares are unequal?”

The instructional materials reviewed for HMH Into Math Grade 1 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 5, the Linguistic Note states, “As children solve various problem types using addition or subtraction, they may lack the language to express what is unknown. Work with children to identify Add To and Take From Result Unknown, Change Unknown, and Start Unknown problem types.” Module 6 includes Review Vocabulary: equation.
• Module 7 includes Key Academic Vocabulary, more and fewer.
• In Module 12, the Linguistic Note states, “The language in a math textbook can be challenging for English Language Learners. Many mathematics terms have multiple meanings. Taking time to distinguish between the meanings of these terms will help avoid confusion when asking questions, such as How many tens in 34? Taking time to distinguish between the meanings of these terms will help avoid confusion when asking questions about tens and ones.”
• In Module 17, the Linguistic Note states, “the math topics for measurement and data are rich with opportunities for cooperative grouping and language development. Take time prior to a lesson to highlight key vocabulary. For example, make sure children understand the suffixes -er and -est as used to compare objects by length.”

The Guided Student Discussion often provides prompts related to understanding vocabulary, for example, Lesson 4.3, Sample Guided Discussion, Persevere, “How do you know if facts are related to each other?”

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 2, Lesson 2, Count Back is highlighted in yellow and a visual model of counting back with counters and an equation is represented. 
• In Module 3, Lesson 6, students draw pictures to define equal and unequal. 
• In Module 8, Lesson 3, Connect Math Ideas, Reasoning, and Language Compare and Connect states, “Before beginning this task, have children define the terms tally chart, tally marks, and tallies in their own words. Encourage children to point to examples.”
• In Module 14, Lesson 1, Build Understanding, Connect Math Ideas, Reasoning, and Language Compare and Connect states, “Have children describe the shape of a cube and a rectangular prism. Have partners share their work and discuss how their descriptions 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.”