1st Grade - Gateway 2

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

• In Lesson 1.2, 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.3.5)
• In Lesson 6.2, students represent the number of animal cards in two different ways using counters, connecting cubes, or pictures and an equation. (1.OA.1.1)
• In Lesson 7.3, students select and draw a number of counters that are more than three and fewer than 10. Students then show three fewer counters. Students use pictures, counters, and equations to deepen their understanding of unknown problems. (1.OA.1.1)
• In Lesson 9.1, students choose a tool to represent a teens number using tens and ones. Students also use a ten frame and answer, “How many one cubes fill a ten frame?” “How do you know the value of each digit?”. (1.NBT.2.2)
• In Lesson 10.1, students work with partners to represent a two-digit number and use that drawing/representation (using a picture or base ten blocks) to help them write the number of tens and ones in the number. (1.NBT.2.2)
• In Lesson 18.2, students compare the length of objects to the length of a string and determine if the string is longer or shorter. Students answer, “How does the length of the object on the first table compare to the length of the string?” Students also compare the lengths of objects and draw to show which objects are longer or shorter. (1.MD.1.1)

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

• In Lesson 1.4, 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. The Teacher’s Edition states, “Select children with a clear understanding of making a ten to explain how they solved the problem. Select children with a clear understanding of addition to explain how they found different ways. Encourage children to ask questions of their classmates. Discuss any patterns children notice such as adding in any order to make 10.” (1.OA.3.6)

The instructional materials for Into Math Florida 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, 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 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.1.1)
• In Lesson 2.6, Step It Out, students use subtraction strategies to solve problems, “Another way to solve 9 - 2 is to count on.” (1.OA.1.1, 1.OA.3.6)
• In Lesson 3.1, students use pictures and equations to practice representing addition in other ways using the commutative property of addition throughout the lesson. 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.2.3)
• In Lesson 8.7, Spark Your Learning, students use information in a chart to solve a problem using tally marks to show how many flowers are in different vases. “What does each of Lian’s tallies mean?” Students use the tallies to create an addition equation to show the total number of flowers in the vase. (1.MD.3.4)
• In Lesson 12.5, Spark Your Learning, students build numbers using tens and ones and then add more ones to find a total. In Build Understanding, students continue to show the problem using tens and ones and writing an addition equation to solve the problem. (1.NBT.3.4)

Unit 1 addresses 1.OA.3.6. The lessons address addition and subtraction within 20 and demonstrating fluency within 10. Students build fluency through adding 10 and more, making a 10 to add, adding doubles, and using known sums to add. Specific examples include:

• In Lesson 1.3, On My Own, Numbers 3-6, students practice adding more to 10 and practice counting using the strategy 10 and more. Students use ten frames to count on from 10 and to add and subtract within 20.
• In Lesson 1.4, On My Own, Numbers 4-10, students “make ten” to add within 20. Students fluently build 10 and add the remaining value.
• In Lesson 2.2, 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 by counting back.
• In Lesson 3.7, 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 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 3.3, students use mental math to add within 20 and solve addition equations (1.OA.3.6).

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

• In Lesson 1.7, Independent Practice, students draw and solve, “Julian has 7 green marbles. His brother gives him 4 blue marbles. How many marbles does Julian have? Choose a strategy to add.” (1.OA.1.1)
• In Lesson 2.6, Check Understanding, students write an equation and explain the strategy they used to solve a word problem: “Nathaniel has 14 apples. He gave 5 away. How many apples does he have now?” (1.OA.1.1)
• In Lesson 7.6, students solve “There are 5 cherries. There are 8 more blueberries than cherries. How many blueberries are there? Use a strategy to solve the problem. There are ______ blueberries.” (1.OA.3.6)
• In Lesson 8.2, On My Own, students use colored marbles to create a graph, interpret the data, and reason about the data. (1.MD.3.4)
• In Lesson 12.8, 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.3.5)

Examples of non-routine application of the mathematics include: 

• In Lesson 3.5, On My Own, Problem 8, students write and solve a word problem with three addends. (1.OA.1.2)
• In Lesson 5.4, students “Write a story problem for the equation. Solve the problem. 17-8=?”. (1.OA.1)
• In Lesson 6.7, Independent Practice, students respond to “Cindy sees 18 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 for Into Math Florida Grade 1 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 Lesson 1.3, students develop procedural skill and fluency of addition within 20. (1.OA.3.6)
• In Lesson 5.1, 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.1.11)
• In Lesson 5.4, students solve Add to and Take from 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.1.1)

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 Lesson 4.3, 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.3.6)
• In Lesson 7.4, students use procedural skills and visual models to solve difference unknown word problems. (1.OA.1.1)
• In Lesson 7.7, students build conceptual understanding using a bar model to represent word problems. (1.OA.1.1)

The instructional materials reviewed for Into Math Florida Grade 1 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 7.4, Build Understanding, Task 1, “What information in the problems help you know if you should add or subtract?”. Directions continue with, “How is this word problem different from the first problem on this same page?”.
• MP.2.1: In Lesson 3.5, On my Own, Problem 7, “Reason: Complete the equation to find the unknown addend. 1 + 3 + ? = 5.”
• MP.4.1: In Lesson 6.7, Step It Out, Task 2, “Lia has some animal books. She gives 9 books to Max. Now she has 7 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.
• MP.5.1: In Lesson 14.1, Build Understanding, Task 1, “Use tools to solve the problem. How can you use the place value chart to show your work? “There are ten children at the park. Then 26 more children come. How many children are at the park now?” Teachers are guided to ask, “What do you know about place value charts and how to use them?” What are some tools you could use with the place value chart to solve the problem? How will you use the place value chart and tools to add the numbers?”.
• MP.6.1: In Lesson 12.5, Step It Out, Task 2 prompts teachers to “Guide students through the steps to solve an unknown addend.” “Eli has 6 paint jars. He gets some more paint jars. Now he has 46 paint jars. How many paint jars did Eli get?”
• MP.7.1: In Lesson 16.1, Build Understanding, Turn and Talk, “How is a rectangle and a square the same? How are they different?”.
• MP.8.1: In Lesson 2.2, Build Understanding, Task 2, “The kangaroo jumps down 3 steps. What step is the kangaroo on now?... A. 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 Into Math Florida 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 Lesson 6.6, Step It Out, the Turn and Talk prompts students with “What is another equation you could write to show the answer to this problem?”
• In Lesson 8.3, Build Understanding, Problem 1C states, “Did more friends wear blue or red shirts? Circle your answer. How do you know?”
• In Lesson 8.5, On My Own, Problem 3 states, “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 Lesson 14.3, Build Understanding, the Turn and Talk states, “How can you solve a subtraction problem? Explain the steps you used to subtract 90-40.”
• In Lesson 16.2, On My Own, Problem 4, students “....construct arguments. Explain how to draw a triangle. Write your answer.”
• In Lesson 17.2, Independent Practice, Problem 6, students “....construct arguments. How do you know that the shares are equal?” “Jake has a square patio. How can he use tape to make 2 equal shares?”
• In Lesson 17.3, On My Own, Problem 7, students “....construct arguments. How do you know that you colored half of each shape?”.
• In Lesson 18.1, On My Own, students “....construct arguments. Mercer thinks she circled the longest pencil. What is her mistake?”.

The instructional materials reviewed for Into Math Florida 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 Lesson 4.3, Learn Together, Turn and Talk states, “Have children share their Turn and Talk responses with a partner. Remind children to ask questions of each other that focus on understanding how to order the numbers used in an addition fact to write a related subtraction fact. Then, have them refine their answers.”
• In Lesson 4.7, Step It Out, What to Watch For, the teacher asks students, “Is there only 1 possible answer to this problem? Explain.”
• In Lesson 11.1, Spark, Build Shared Understanding Let’s Talk states, “Have a pair of students show each number. Have the class use the agree sign to determine which number is greater by comparing the number of tens and ones.” 
• In Lesson 13.1, Spark, Build Shared Understanding Let’s Talk states, “After a child has walked through their logic about how they ordered the coins, have children use an agree sign to critique the reasoning of the child who shared.” 
• In Lesson 17.2, 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?”.
• In Lesson 18.1, Build Understanding, Task 1 prompts, “Before beginning the task, have children describe how to identify the longest and the shortest object of the three. Have partners share their work and discuss how their descriptions compare and contrast.” The Sample Guided Discussion includes, “How do you know which carrot is the longest? I look for the carrot that is longer than the other 2 carrots. What do you need to show?”.

The instructional materials reviewed for Into Math Florida 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. 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 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 start, change, result, and difference unknown problem types.” Module 6 includes Review Vocabulary: related facts, addends, total, unknown, difference. 
• Module 7 includes Key Academic Vocabulary: more, fewer, and how many.
• 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?”.
• In Module 18, the Linguistic Note states, “The 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, Think It Through, “What makes a fact related to the fact that you used to solve the problem?”.

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 2.2, Count Back is highlighted in yellow and a visual model of counting back with counters and an equation is represented. 
• In Lesson 3.6, students draw pictures to define equal and unequal. 
• In Lesson 8.3, Connect Math Ideas, Reasoning, and Language Compare and Connect states, “Remind children they are familiar with using symbols to represent numbers. Before beginning the task, show children samples of tally charts. Have partners share their work and discuss how each tally mark represents 1 unit of data.”
• In Lesson 15.1, Build Understanding, Connect Math Ideas, Reasoning, and Language Compare and Connect states, “Remind children they can describe shapes using real-world knowledge. Before beginning the task, have children describe in their own words the shape of a cone, cylinder, and 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.) 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.”