Kindergarten - Gateway 2

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

• In Lesson 2.3, Read and Write 4 and 5, Spark Your Learning includes the teacher modeling the number four with counters, and students show four with crayons (to correspond to counters). Students answer, “How many crayons did you use? What items do we have four of in the classroom?” 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 five swimmers and draw what they know about four and five. (K.CC.1.3)
• In Lesson 11.3, students solve addition problems within 10 by counting the number of objects in the picture. Spark Your Learning includes questions to further develop understanding depending on if students modeled adding two groups, showed one group, and common errors students might make. (K.OA.1.1, K.OA.1.2)
• In Lesson 11.5, students count how many objects are in each group in a picture and write a corresponding addition equation. (K.OA.1.1, K.OA.1.2)
• In Lesson 12.3, 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.1.2)
• In Lesson 17.1, Problem 1, students use 10 cubes and draw some more ones to show the number. Students write the number to show how many more ones, and a picture of 10 cubes at the top of the page represents ten ones. (K.NBT.1.1)
• In Lesson 17.2, Build Understanding, there are ten connecting cubes at the top of the page. Students answer “How many more cubes do you need to make fifteen?” and “describe the number 15” using tens and ones. Students draw a picture and fill in the blank to tell how many ones are needed to make 14 and 15. (K.NBT.1.1)

Every lesson also has an optional Sharpen Skills activity to build conceptual understanding. For example, Lesson 5.1, Act Out Addition Problems Within 5, students act out and tell a story about a picture shown and write numbers to show how many children are in each group.  

The instructional materials for Into Math Florida 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, 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 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 six, draw six, and then talk about a part-part-whole relationship to make the connections about part-whole relationships. (K.OA.1.5)
• Activating Prior Knowledge is included in lessons. For example, in Lesson 5.1 students use 5 frames and determine the number shown and “how many more” needed to make five. This supports fluency with addition within 5. (K.OA.1.5)

Students develop fluency with K.OA.1.5 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:

• In Lesson 5.3, ReTeach includes independent practice for students to build fluency of addition within five. Additionally, the More Practice/Homework and Test Prep sections provide students with independent practice of adding within five.
• In Lesson 5.4, Step It Out, On My Own, and More Practice/Homework, students independently practice take from problems within five. Test Prep extends independent practice by having students use pictures to demonstrate fluency of take from problems within five.
• In Lesson 5.6, Step It Out, students use pictures to create matching subtraction equations to practice fluency within five. During On My Own, students independently practice within five. The More Practice/Homework provides students with additional practice opportunities to build subtraction skills.
• In Lesson 6.3, Practice and Homework Journal, students write subtraction equations and solve based off a story problem, “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 skill and fluency. For example, in Module 5 Lesson 3, students use connecting cubes of two different colors while articulating sentences using addition vocabulary. “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.1.5)

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

• In Lesson 5.1, students “act out and tell a story about the picture. Write the numbers that show how many children are 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? What groups are you adding?” (K.OA.1.2).
• 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?”
• In Lesson 11.1, Spark Your Learning begins with a story problem and picture to show how many backpacks are in a truck. Students consider, “How can you act out the backpacks in the truck? How many backpacks does the girl bring? How can you act out the backpacks that girl brings? Now how many backpacks are there altogether?” During Learn Together, students use counters, cubes, or pictures and write an addition problem for story problems. During On My Own, students act out an addition story about camping by a lake within 10, draw to show the story, and write a matching equation. (K.OA.1.5a)
• In Lesson 12.5, Step It Out and On My Own, students create picture models and equations to solve addition word problems, and in More Practice/Homework, students solve word problems within 10 in a variety of contexts. (K.OA.1.2)
• In Lesson 13.3, students listen to a story problem and use objects and drawing to solve addition equations within 10. (K.OA.1.1)

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

• In Lesson 5.6, students “Tell a subtraction word problem about the picture. Mark an X to show what is being taken away from the group. Write an equation to model the problem.” (K.OA.1.2)
• In Lesson 6.1, students represent addition problems within five using objects and drawings. During Spark Your Learning, students respond to, “Look at the picture. What addition story can you tell about the cat and dogs? Write how many total pets?” Students create their own story problem based on the picture of 4 pets. (K.OA.1.2)
• In Lesson 6.7, students “Tell your own addition word problem within 5. Draw to represent your story. Write the equation to model your story” (Both addends unknown). (K.OA.1.2)
• In Lesson 12.1, Spark Your Learning, students look at a picture of dogs. “What addition story can you tell about the small dogs and the big dogs? Complete the equation to show how many dogs in total.” 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 small (red counters) and large 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, equal sign, and find the total sum. (K.OA.1.1)

The instructional materials for Into Math Florida Kindergarten 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:

• Lessons 1.1-1.3 address developing conceptual understanding of numbers to five where students use counters to represent numbers. Students listen to stories and show the number using counters. “Maria went to the park. She saw three squirrels in the tree. There were also four ducks under the tree.” 
• In Lesson 2.4, 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.1.3)
• In Lesson 5.2, students use pictorial representations to demonstrate story problems to show subtraction within five. (K.OA.1.5)

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 Lesson 9.3, students apply what they know about counting to understand how to count on from any given number in a counting sequence. (K.CC.1.2)
• In Lesson 10.1, 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. Students draw a group greater than the picture shown and circle the group that has a number of objects greater than the other group. (K.CC.3.6)

The instructional materials reviewed for Into Math Florida Kindergarten 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 Guides include 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 11.1 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?”
• MP.2.1: In Lesson 12.2, students solve, “There are ten plates on the picnic table. Six are blue. The rest are red. How many are red?”. Students draw a picture to represent the problem and write an equation.
• MP.4.1: In Lesson 6.4, On My Own, students “Make up your own take apart story problem using numbers within 5. Draw to show your problem. Write the equation that matches your story.”
• MP.5.1: In Lesson 10.1 students show which group has a greater number of objects. Students can choose to use counters, connecting cubes, or drawings.
• MP.6.1: In Lesson 19.1, Learn Together, “Leo builds two castles with blocks. He wants to put one of the castles in his bookcase. Use the words taller than, shorter than, or longer than to describe the heights and lengths of the castles. Use red to show the height of each castle. Use blue to show the length of each castle. Circle the castle Leo places on the bookcase.”
• MP.7.1: In Lesson 7.4, 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.
• MP.8.1: In Lesson 9.2, Step It Out, Task 2, “Have children listen as you read the questions aloud. Explain the pattern: each row of the hundred chart goes up by ten.” “What number should you start with if you want to count by ten? Start with ten and count ten more. How many do you count? Point to the numbers on the chart. Ten more is 20, The one changes to a 2 and the 0 stays the same. Now count ten more. How many do you count? How has the number changed? How do you count to one hundred by tens?”

The instructional materials reviewed for Into Math Florida 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 such as explaining their thinking or justifying steps, and the materials prompt them to analyze the arguments of others. Examples include:

• In Lesson 5.4, Build Shared Understanding, students explain how they solved a problem, and students are encouraged to ask their classmates what they did first, next, and last.
• In Lesson 8.1, Put It In Writing, students explain how to find whether there are eight or nine objects in each group. 
• In Lesson 9.2, On My Own, students “place rows of numbers as you count by tens to 100. Tell a classmate about counting by tens to 100. What is alike about the numbers? What is different about the numbers?”.
• In Lesson 12.5, 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 cartons of each kind of milk to draw.” 
• In Lesson 14.3, Learning Mindset, Try Again, students discuss, “Which strategies they used? Which they like best? Which were most challenging? Would they try any of the strategies their classmates mentioned?”.
• In Lesson 19.2, On My Own, Problem 3, the materials state, “Look at the rugs. Compare the lengths of the two rugs. Use the words longer than or shorter than to describe the lengths. Draw a circle around to 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 Lesson 19.3, 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 Into Math Florida 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 Lesson 3.5, Step It Out, the Turn and Talk states, “Have students share their drawings and strategies and speak sentences about Problems 4 and 5 using the words less than, fewer than, greater than and more.”
• In Lesson 4.1, 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 color, classify, and category. Have students discuss their examples. Prompt discussion by asking: Do you agree or disagree? Why?”.
• In Optimize Output of Lesson 5.6, Stronger and Clearer, the teacher directs students to share their Turn and Talk responses with a partner, encouraging them to describe mathematical ideas clearly. Students are asked to discuss other possible solution strategies and refine their answers if necessary throughout the conversation.
• In Lesson 7.2, the Turn and Talk states, “Have children explain to partners how they chose what counters to use. Did they use all red or all yellow? Did they use one color for each number? Why?”.
• In Lesson 16.4, the Learn Together, Connect Math Ideas, Reasoning, and Language, Compare and Connect states, “Help children identify the relationship between a square and a rectangle. Ask children to discuss whether all squares are rectangles and whether all rectangles are squares.”
• In Lesson 19.3, the Turn and Talk states, “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 they could or could not compare the heights?”.
• In Lesson 20.3, Optimize Output, the materials assist teachers by having them ask students to identify the error, correct the placement of the label, and explain their reasoning.

The instructional materials reviewed for Into Math Florida 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. 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:

• 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 number “in all.” 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: Lesson 14.4, Task 1 states, “What does a cone look like? What do curved surfaces look like? What do flat surfaces look like? Which objects have all the qualities of 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 Lesson 3.2, Connect Math Ideas, Reasoning, and Language Compare and Connect states, “Remind children that they have learned the vocabulary term greater than. Have children compare the meanings of the terms greater than and less than. As children discuss, ask questions to help them clarify their reasoning.”
• In Lesson 4.2, 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 Lesson 5.7, students use visual models and equations to help review the module’s vocabulary for addition and subtraction.
• In Lesson 16.1, 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.”