Kindergarten - Gateway 2

The instructional materials reviewed for HMH Into Math Kindergarten 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 conceptual understanding, examples include:

• Module 2, Lesson 3, Read and Write 4 and 5, Spark Your Learning, the teacher models the number 3 with counters and asks students "What number is one larger than three?" During Learn together, students build understanding with counting, one to one correspondence, identify four and five, and write the corresponding numerals. In independent practice, students show what they know about the numbers four and five and how the two numbers are different from each other. Students count and circle the picture that represents 5 swimmers and draw what they know about 4 and 5. (K.CC.3)
• Module 11, Lesson 3, students solve addition problems within 10 by counting the number of objects in the picture. Spark Your Learning questions develop student understanding by showing different representations and highlighting common errors. (K.OA.1)

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

• Module 11, Lesson 5, students count how many objects are in each group in a picture and write a corresponding addition equation. (K.OA.1)
• Module 12, Lesson 3, Small Group, includes three options to develop students’ conceptual understanding of put together problems within 10. Students use two color counters and crayons to create word problems putting together two groups and writing an equation. (K.OA.1)
• Module 17, Lesson 1, Problem 1, students use 10 cubes and draw some more ones to show the number. (K.OA.3)

The instructional materials reviewed for HMH Into Math Kindergarten 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 include advice for teachers on how to question the student in order to build procedural fluency. For example, in Module 12, Teacher to Teacher suggests having students visualize 6, draw 6, and then talk about a part-part-whole relationship to make the connections about part-whole relationships. (K.OA.5)
• Activating Prior Knowledge is included in lessons. Lesson 5.1, the activity states: "Have children identify the number shown in each five frame." This supports fluency with addition within 5. (K.OA.5)  

Students develop fluency with K.OA.5 in Modules 5 and 6. Specific lesson components, Step it Out and More Practice/Homework, include opportunities for students to engage in procedural skill and fluency. In addition, Reteach opportunities provide additional fluency practice. For example:

• Module 5, Lesson 3, ReTeach, includes independent practice for students to build fluency of addition within 5. Additionally, the More Practice/Homework and Test Prep sections provide students with independent practice of adding within 5.
• Module 5, Lesson 4, Step It Out, On Your Own, and More Practice/Homework, students independently practice take from problems within 5. Test Prep extends independent practice by having students use pictures to demonstrate fluency of take from problems within 5.
• Module 5, Lesson 6, Step It Out, students use pictures to create matching subtraction equations to practice fluency within 5. During On Your Own, students independently practice within 5. The More Practice/Homework provides students with additional practice opportunities to build subtraction skills.
• Module 6, Lesson 2, Practice and Homework Journal, students write subtraction equations to solve, “Molly has five cars. Two of the cars are yellow. The rest are blue. How many cars are blue?” 

In addition, every lesson has an optional Sharpen Skills activity to build procedural skill and fluency. For example, in Module 5, Lesson 3, students use different color connecting cubes to articulate sentences that include addition vocabulary. For example, “I am joining 3 blue and 2 red. The total number is 5. I am adding 2 red cubes to 3 blue cubes. Three plus 2 is equal to 5.” (K.OA.5)

The instructional materials reviewed for HMH Into Math Kindergarten 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 5, Lesson 1, students solve, “How can you act out an addition problem to match the picture? Write the numbers that represent the children in each group.” Guided questions throughout the activity prompt students to apply their understanding of addition to find a total unknown. “How many children are sitting on the rug? How many children come? What happens to the group when one child joins them? How can you find the total number of children there now? How many children join them? What happens to the group of children when one child joins them? What groups are you adding?” (K.OA.2).
• In Module 5, Test Form A, Item 5, students apply mathematics to a real-world context: “Ben and Lisa are playing in the park. Their friends Matt and Kate join them. How many friends are in the park now?” (K.OA.2)
• In Module 12, Lesson 5, Step It Out and On Your Own, students create picture models and equations to solve addition and subtraction word problems, and in More Practice/Homework, students solve word problems within 10 in a variety of contexts. (K.OA.2)

Students engage with non-routine application of the mathematics when they create story problems for pictures and number sentences. For example:

• In Module 5, Lesson 6, students “Tell a subtraction word problem about taking from a group using numbers within 5. Use drawings to solve the problem. Write the equation to model the problem.” (K.OA.2)
• In Module 6, Lesson 1, students represent addition problems within 5 using objects and drawings. During Spark Your Learning, students respond to, “Look at the picture. What addition word problem can you tell about the cat and the dogs? Write how many total pets?” Students create their own story problem based on the picture of 4 pets. (K.OA.2)
• In Module 6, Lesson 7, students “Tell your own addition word problem within 5. Draw to solve the problem. Write the equation to model your story.” (K.OA.2)
• In Module 12, Lesson 1, Spark Your Learning, students look at a picture of dogs. “How can you represent the brown dogs and the white dogs? Complete the equation to model the groups and the total number of dogs.” Students count the small dogs and place a red counter on them while counting aloud. Students count the large dogs and place a yellow counter on them while counting aloud and then are asked, “How can you decide how many dogs there are in all?” (By counting all the counters). Students add the brown (red counters) and white dogs (yellow counters) and write a corresponding equation. Throughout the lesson, students model a story problem with counters, write a corresponding addition sentence using the correct numbers, plus sign, and equal sign, and find the total sum. (K.OA.1)

The instructional materials reviewed for HMH Into Math Kindergarten 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 of rigor are interwoven throughout each module.

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

• Module 1, Lessons 1-3 address developing conceptual understanding of numbers to 5 as students use counters to represent numbers. Students listen to stories and show the number using counters. Lesson 3, Build Understanding, Task 3, “Listen to the story. Draw counters to represent the number of objects in the story. Say each number as you draw the counters. Circle the number you represented. “Shana sees five beach balls.”
• In Module 2, Lesson 4, Step It Out, students develop procedural skill and fluency as they count the flowers in each vase to count and write the number. (K.CC.3)
• In Module 5, Lesson 2, students use pictorial representations to demonstrate story problems to show subtraction within 5. (K.OA.5) 

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 9, Lesson 3, Step It Out, Task 1, students apply what they know about counting to understand how to count on from any given number in a counting sequence. “There are five pumpkins in the crate. Beginning at 5, count forward. Mark an X on each pumpkin as you count. Tell a classmate how many total pumpkins there are.” (K.CC.2)
• In Module 10, Lesson 1, On Your Own, Problem 4, students develop conceptual understanding by using counters, pictures, or connecting cubes to compare two groups of objects within 10. Students practice procedural skill by drawing lines to match the objects in each group. “Use matching to identify which group has a greater number of objects that the other group. Circle the group that has a greater number of balloons. There are two groups of hot air balloons.” (K.CC.6)

The instructional materials reviewed for HMH Into Math Kindergarten 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 11, Lesson 1, students listen to a story, act it out, and then write an equation to match. “There are six butterflies in the meadow. Four more butterflies join them. How many butterflies are in the meadow now?”
• MP2: In Module 12, Lesson 2, students solve “There are ten plates on the picnic table. Six are blue. The rest are red. How many are red?” Teachers are instructed to ask “What kind of equation will you write? What does 10 represent? What number will you write next? What does 6 represent? What number will you write next? What does 4 represent? What signs will you write?”
• MP4: In Module 6, Lesson 4, On Your Own , students “Tell a subtraction word problem about a group being taken apart using numbers within 5. Use drawings to solve the problem. Write the equation to model the problem.”
• MP5: In Module 7, Lesson 3, Spark your Learning, students listen to the story “Zoe is at the carnival with her family. She needs ten tickets to ride the carousel.” Teachers are encouraged 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 9, Lesson 3, Learn Together, “There are five pumpkins in the crate. Beginning at 5, count forward. Mark an X on each pumpkin as you count. Tell a classmate how many total pumpkins there are.” 
• MP7: In Module 7, Lesson 4, Ways to Make Ten, students use two color counters to represent 10, color the ten frame on the page to match their counters, and write in the addition sentence to match.
• MP8: In Module 9, Lesson 2, Step It Out, Task 2, “Point to the yellow box. Count to 100 by tens. Color each number as you count. What number should you start with if you want to count by tens? Start with 10 and count ten more. What number is ten more than 10? Point to the number on the hundred chart. Ten more is 20. The 1 in the number 10 changes to a 2, and the 0 stays the same. Now count ten more. What number is ten more than 20? How has the number changed? What are the other numbers you will say to count to 100 by tens?”

The instructional materials reviewed for HMH Into Math Kindergarten 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 where they explain their thinking or justify, and the materials prompt them to analyze the arguments of others. Examples include:

• In Module 5, Lesson 4, Build Shared Understanding, students explain how they solved a problem and discuss why they chose a specific strategy or tool.
• In Module 9, Lesson 2, On Your Own, students “place rows of numbers as you count by tens to 100. Tell a classmate about counting by tens to 100. How are the numbers you count alike? How are the numbers you count different?” 
• In Module 12, Lesson 5, students draw representations of word problems and write an equation to model the problem. Students “compare their drawings and talk about how they knew how many objects to draw and how they can use their drawings to help them solve the problem.” 
• In Module 19, Lesson 2, On Your Own, Problem 3, “Look at the rugs. Compare the lengths of the two rugs. Use the words longer than or shorter than to describe the lengths. Circle the longer rug. Mark an X on the shorter rug. Explain why the two rugs must be at the same starting point to compare the lengths.” 
• In Module 19, Lesson 3, Learning Mindset, Bounce Back, students discuss that other perspectives are valuable and not everyone sees things in the same way.

The instructional materials reviewed for HMH Into Math Kindergarten 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 1, Build Understanding, Connect Math Ideas, Reasoning, and Language, Compare and Connect, states, “Before beginning the task, have children describe and give examples in their own words of the meaning of the words classify and category. Have students discuss their examples. Prompt discussion by asking, Do you agree or disagree? Why?”
• In Module 7, Lesson 2, Turn and Talk, “Have children explain to partners how they chose what color or colors of counters to use. Did they use all red or all yellow? Did they use one color for each number? Why?”
• In Module 19, Lesson 3, Turn and Talk, “Have children talk about what would happen if the birdbath on the right were placed in a hole instead of on the same surface as the one on the left. What would be different about how children could or could not compare the heights?”

The instructional materials reviewed for HMH Into Math Kindergarten 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 includes Key Academic Vocabulary for Prior Learning, Review Vocabulary and Current Development, and New Vocabulary with definitions. Also in Language Development, Linguistic Notes provide teachers help with possible misconceptions relating to academic language. For example:

• Module 3 identifies and defines review vocabulary, larger, and Current Development, greater than, count, less than, equal to, compare, and match.
• In Module 5, the Linguistic Note states, “Listen for children who mix up the language between addition and subtraction. For example, children should say that when two groups are joined together, the result is the total.” When a group is taken apart, children should say the number that is “left.” Help children use the correct terms throughout each lesson.”
• Module 6 includes Key Academic Vocabulary such as put together and take apart.
• In Module 8, the Linguistic Note states, “Point out the difference between an amount of something and a number of objects. Discuss this idea in terms of ‘how much’ versus ‘how many.’ To help children understand the difference, provide examples from a variety of contexts. For example, ‘how much’ can be used to describe a value of a coin or an amount of liquid in a jar. On the other hand, ‘how many’ could be used to describe the number of coins or the number of ice cubes in a jar.”

The Guided Student Discussion often provides prompts related to understanding vocabulary such as: Module 14, Lesson 4, Task 1, states, “What does a cone look like? Which objects in the picture have curved surfaces? Which of the objects are cones? Why is the garbage can not a cone?”

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 3, Lesson 2, Connect Math Ideas, Reasoning, and Language Compare and Connect states, “Review the vocabulary term greater than. Have children compare the meanings of the terms greater than and less than. Ask questions to help clarify their understanding.”
• In Module 4, Lesson 2, Build Understanding, Connect Math Ideas, Reasoning, and Language Compare and Connect states, “Before beginning the task, have children describe and give examples in their own words of the meaning of the word shape. Have children discuss their examples. Prompt discussion by asking: Do you agree or disagree? Why?”
• In Module 5, Lesson 7, students use visual models and equations to help review the module’s vocabulary for addition and subtraction.
• In Module 16, Lesson 1, students use graphic organizers with the terms, circle, square, rectangle, and triangle. Students draw a picture to illustrate each term and give an example of a real world object of that shape.

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.”