2nd Grade - Gateway 2

The instructional materials for Into Math Florida 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 and independently demonstrate conceptual understanding, and examples include:

• In Lesson 4.2, students develop understanding of three-digit numbers by showing the number of crayons in boxes using ten and one blocks, connecting cubes, or drawing a picture. Students answer, “How many tens are in 100? What number would you write in the hundreds place on a place value chart?”. (2.NBT.1.1)
• In Lesson 14.2, students use equations to represent addition and subtraction situations. Students also use cubes, drawings, and equations to show their understanding of addition and subtraction. (2.OA.1.1)
• In Lesson 18.2, 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.1.1)
• In Lesson 18.1, 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)
• In Lesson 20.1, students relate inches to a number line. Students use a ruler as a number line to solve a word problem and answer, “What does the rule remind you of? (A Number line) What can a number line help you with?” Students use the number line to count up. (2.MD.2.6)

Students are also provided opportunities to build shared understanding via Let’s Talk activities. An example includes:

• Lesson 1.1 uses models to demonstrate finding a solution with the class. Students share their understanding of using doubles to find sums and explain how they solved problems. Students also engage one another by asking questions of their classmates.

The instructional materials for Into Math Florida 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, and 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.1.4)
• In Lesson 2.4, Spark Your Learning, the teacher reads, “Lily wants to put her books in equal rows. Use tools to show how she can do this. Then show how you can add to find the total number of books.” (2.OA.3.4)
• In Lesson 2.5, students practice writing equations using repeated addition to find the total number of objects in arrays. (2.OA.3.4)
• In Homework and Test Prep at the end of each lesson, students practice skills and develop fluency through the Spiral Review at the end of each page. For example, Lesson 16.2, Spiral Review, students practice subtracting two-digit numbers. (2.NBT.2.5)

Modules 10, 11, 12, 13, and 15 address 2.NBT.2.5, 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:

• In Lesson 10.2, students use a number line to add and subtract two-digit numbers.
• In Lesson 11.3, students make an addend of ten to solve addition of two-digit numbers.
• In Lesson 13.7, students use strategies of addition to find the sum of four two-digit numbers.
• In Lesson 14.1, students explain how to write addition equations to solve word problems. 

Unit 1 addresses 2.OA.2.2. The lessons in the unit address developing fluency for addition and subtraction within 20. Students build fluency through use of doubles facts to add, using addition and subtraction strategies or properties, and making ten to add or subtract. Specific examples include: 

• In Lesson 1.1, Sharpen Skills, students build fluency with addition. 
• In Lesson 1.3, On My Own, students use related facts and make connections between operations using bar models to develop fluency in adding and subtracting within 20.
• In Lesson 1.4, More Practice/Homework, students choose a strategy to subtract.

Unit 4 addresses 2.NBT.2.5. The lessons in the unit address developing fluency for addition and subtraction within 100. Students build fluency by decomposing tens into ones to add and subtract and through compensation to add in Modules 11 and 12. Specific examples include:

• In Lesson 11.1, 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. In Lesson 11.2, Build Understanding, students use models and pictures to begin establishing fluency in building “friendly” numbers.
• In Lesson 11.3, Sharpen Skills, students build fluency and practice basic math skills using compensation to build the next ten to solve.
• In Lesson 11.4, Fluency Builder, students use mental math strategies to break apart numbers into units of tens and ones to add efficiently within 100. Students practice fluency in the More Practice/Homework as they break apart vertical addition problems into units to add.
• In Lesson 12.3, students develop fluency adding two-digit numbers using the standard algorithm. During On My Own and More Practice/Homework, students demonstrate fluency in regrouping to solve problems accurately and efficiency with a numerical model.
• In Lesson 12.4, students develop fluency with regrouping and subtraction using the standard algorithm.

In addition, Sharpen Skills are optional activities included with each lesson to build fluency and practice skills. It is optional because this section says, “If time permits.” For example, in Lesson 6.3, students sit in small groups and are given a three-digit number. Students say a number that is ten more or ten less than that number. Play continues in the same way. This activity builds fluency with ten more or ten less with numbers less than 1000 (2.NBT.2.8).

The instructional materials for Into Math Florida 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 My Own, students apply what they have learned to solve real world problems independently. For example: 

• In Lesson 13.4, students rewrite subtraction problems. “Kenneth has 41 tennis balls. At practice, he serves some over the net. Then he has 28 tennis balls left. How many tennis balls did Kenneth serve?” (2.OA.1.1)
• In Lesson 15.1, Independent Practice, students solve addition word problems, “There are some balls in a basket. Now there are 32 balls in the basket. How many balls are in the basket to start?”. (2.OA.1.1)
• In Lesson 15.2, 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.1.1)
• In Lesson 15.3, 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.1.1)

Examples of non-routine application of the mathematics include:

• In Lesson 3.3, Independent Practice, 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.4.10)
• In Lesson 3.5, Step It Out, students solve a multi-step word problem to title, label, and draw a bar graph to interpret the data given. (2.MD.4.10)
• In Lesson 8.3, Independent Practice, students solve word problems involving money, “Manny saves 63¢ and Erica saves 30¢. How much money do they save?”. (2.MD.3.8)
• In Lesson 12.5, 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.1.1)

The instructional materials for Into Math Florida Grade 2 meet expectations for the three aspects of rigor not always being treated together and not always being treated separately. In general, 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 Lessons 1.1 through 1.7, students develop procedural skill and fluency in adding and subtracting within 20 using mental strategies. (2.OA.A.2)
• In Lesson 7.1, Build Understanding, students use place value charts to add coins using strategies based on place value. (2.NBT.1.2, 2.MD.4.8)
• In Lesson 12.2, Spark Your Learning, students use subtraction to solve a problem (2.NBT.2.5). “Steve has 30 baseball cards. He gives 10 to Evan. How many 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 Lesson 12.2, More Practice and Homework, students apply their understanding of regrouping in the Reason problem. “There are 27 fish in a pond. Then 16 fish swim away. How many fish are left? Do you need to regroup? Explain.”
• In Lesson 13.1, students use strategies learned through conceptual understanding to build fluency to add two-digit numbers. (2.NBT.2.5, 2.NBT.2.7)
• In Lesson 17.5, students draw a visual model to show subtraction equations to solve subtraction word problems. (2.NBT.2.7)

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

The materials attend to the full intent of all eight MPs. In the Teacher’s Edition, the Focus and Coherence for each lesson describes how the MPs are addressed with the lesson. The Planning and Pacing Guide includes a description of lesson components that address specific MPs.

• During Spark Your Learning, students encounter a productive perseverance task that engages students with MP.1.1 (Make sense of problems and persevere in solving them), MP.3.1 (Construct viable arguments and critique the reasoning of others), and MP.5.1 (Use appropriate tools strategically).
• Connect Concepts and Skills lessons focus on MP.7.1 (Look for and make use of structure) and MP.8.1 (Look for and express regularity in repeated reasoning) where students connect understanding they have developed with more efficient procedures. These practices help students explain and justify the procedures they use along with MP.4.1 (Model with Mathematics) when students are connecting their understanding to a procedure. 
• Apply and Practice lessons provide opportunities for MP.2.1 (Reason abstractly and quantitatively) as well as provide opportunities for MP.6.1 (Attend to precision) as students apply procedures in practice.

Examples of the instructional materials attending to the full meaning of the MPs include:

• MP.1.1: In Lesson 18.3, Step It Out, Task 3, “Tell students they will use the line plot to answer questions about the measurement data they recorded in the previous activity.” “Can you explain what the problem is asking in another way? What strategy can you use to make sure you count all the lengths?”
• MP.2.1: In Lesson 14.6, students reason abstractly and quantitatively by showing two different ways to add and subtract 2, two-digit numbers. “Mia has a number that is 20 less than the number Tom has. Tom has a number that is 3 tens and 0 ones. What number does Mia have?” 
• MP.4.1: In Lesson 15.2, Independent Practice, Problem 8, “Write a word problem for this equation. 56 - 37 = 19.”
• MP.5.1: In Lesson 2.1, On My Own, Problems 5 and 6 explain, “Allow children to use tools to determine whether the given number is odd or even.” Problem 7 has children demonstrate their understanding of what numbers are even and odd by choosing any odd number of flower stickers and any even number of animal stickers.
• MP.6.1: In Lesson 5.1, Build Understanding, Task 2 Sample Guided Discussion, “How is writing the hundreds, tens, and ones different from the expanded form?”.
• MP.7.1: In Lesson 2.5, On My Own, Problem 2, “Amav and his family went to a watermelon festival. Amav put some watermelon slices into 3 rows. There are 2 slices in each row. How many slices are there?” Students use arrays to represent repeated addition as they begin to build underlying concepts of multiplication. 
• MP.8.1: In Lesson 6.1, Step It Out, Task 2, Sample Guided Discussion, “What pattern are you using when you count by 5’s? What pattern are you using when you count by 10’s? What pattern are you using when you count by 100’s?”

The instructional materials reviewed for Into Math Florida 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 Lesson 1.4, On My 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 can you use to solve the problem? Explain.”
• In Lesson 7.3, On My Own, students solve: “Joe has 1 quarter, 1 dime, and 1 nickel. Does he have enough money to buy a sticker for $.50? Explain.”
• In Lesson 12.5, On My Own , Problem 9, the materials prompt students with, “To find the sum of 15 and 46, Liang says that he will need to regroup. Do you agree with Liang? Explain.”
• In Lesson 14.4, Learning Mindset, students reflect on, “Do I share and accept strategies from others?”.
• In Lesson 17.6, Independent Practice, Problem 8, students “....construct arguments. Lexi and Zach disagree. They 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? Talk with your partner about you could fix the wrong problem.”
• In Lesson 18.6, Learning Mindset, students reflect on, “Did I try to solve Problem 4 in more than one way? Which way worked best?”
• In Lesson 20.3, On My Own, Problem 2, students “....construct arguments. Mrs. Morgan has a cactus plant that is 67 centimeters tall. She cuts off the top piece of the cactus. The cactus is now 49 centimeters tall. How many centimeters does Mrs. Morgan cut off? ____ centimeters. Explain how you found your answer.”

The instructional materials reviewed for Into Math Florida 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 Lesson 2.4, Turn and Talk, teachers are prompted to ask students, “How did the number of rows of toy trains help you write the addition equation? Explain. Discuss the relationship between the addends in the equation array of trains.”
• In Lesson 4.4, Turn and Talk states, “What if there was a 7 in the tens place? How would the number be different? What number would you write? Explain.”
• In Lesson 7.2, Spark, Build Shared Understanding Let’s Talk states, “Have children explain how they used base-10 blocks to show their coins and how they found the value of coins. Have the class use agree signals to show they agree or disagree with the explanation.”
• In Lesson 11.4, Sample Guided Discussion, teachers prompt student discussion by asking, “How can you verify your solution to the problem using your concrete model?”.
• In Lesson 12.5, Spark, Build Shared Understanding Let’s Talk, states, “Challenge children to compare ways of recording addition and select children with clear understanding of writing addition without a place value chart to explain how they solved the problem. Encourage children to ask questions.”
• In Lesson 14.3, 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 finding the sum. Ask children to share their strategies and discuss which strategy might be easier, quicker, etc.” Possible answers are given: “By drawing a bar model, I can see the addends I am adding together to find the total, or sum.” 
• In Lesson 19.4, Build Shared Understanding Let’s Talk, directs teachers to, “Select students who use different strategies to measure the line. Have them explain how they measured using the centimeter ruler and the meter stick. Encourage children to ask questions of their classmates.”

The instructional materials reviewed for Into Math Florida 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. Examples are found throughout the materials.

The Planning and Pacing Guide has a section for Language Development that states Into Math Florida is built upon four 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 how to skip count, count by twos, fives, and tens.”
• 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, Sharpen Skills, “Students will build math vocabulary fluency by using a graphic organizer. Teachers will facilitate a discussion about the term difference and will complete the graphic organizer using words, numbers, and pictures.”
• In Lesson 18.1, the Linguistic Note states, “How is an estimate different than a measurement?” Teachers are prompted to have students use their interactive glossary during this conversation to record their understanding. 
• In Module 16, Key Academic Vocabulary includes: ones, tens, hundreds, addends, sum, difference, place value, and regroup. In Module 20, Review Vocabulary includes: rows, columns, whole, and equal shares, and New Vocabulary includes: halves, thirds, fourths, half of, third of, fourth of, quarter and quarter of.

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 Lesson 7.2, the Connect to Vocabulary box includes Nickel: “A nickel has a value of 5¢.” “A quarter has a value of 25¢.”
• In Module 18, five new terms are introduced. New terms are used consistently throughout the module. In Lesson 5, 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 6, Step It Out, Item 1, Part A states, “How many feet long is each ribbon?”, and in Check Understanding, Item 1 states, “Find a table. Estimate. About how many yards long is the table?”.
• In Lesson 18.2, Connect Math Ideas, Reasoning, and Language Compare and Connect states, “Remind children they understand how to measure and estimate length in units. Before beginning the task, have children use their own words to explain the terms measure and inch ruler. Then, have partners share their work and discuss their responses compare and contrast.” 
• In Lesson 21.1, 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.) 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.”