Kindergarten - Gateway 2

The materials reviewed for Reveal Math Kindergarten meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings.

The materials develop conceptual understanding throughout the grade level, with teacher guidance, through discussion questions and conceptual problems with low computational difficulty. Examples include:

• In Lesson 3-7, Numbers to 10, Explore & Develop, Activity-Based Exploration, students draw 5 circles, then discuss, “How can you find one more?” Math is Thinking, “What does it mean to have one more?” Work Together, “How can you show each number? Draw counters to show each number.” Students represent numbers 0-10. At the conclusion of the lesson, Assess, Exit Ticket, Exercise 2, “How many carrots will there be if you add one more? Circle a number to show how many?” Students see 5 carrots and the choices “5, 6, 7, 8”. These activities support conceptual development of K.CC.4, understand the relationship between numbers and quantities; connect counting to cardinality.

• In Lesson 6-1, Represent and Solve Add To Problems, Launch, Notice & Wonder, students use Think-Pair-Share to respond to questions about the number of soccer players they see in a picture. Students see a group of 5 players standing in a goal, and another group of 2 players running towards them. “What do you see in the picture? Have you seen something like this before? What can you say about the two groups of players? Why might there be two groups of players? What do you think might be happening in the picture? Explain your thoughts to a friend.” The teacher guides the discussion towards the concept of putting two different groups together into one group to show the total number of objects. In Explore & Develop, Bring it Together, Work Together, “Use counters or drawings to represent the addition story. Then write the total. Four backpacks were on the bench. Two more backpacks were placed there. How many backpacks are on the bench now?” These activities support conceptual development of K.OA.1, represent addition and subtraction with objects, fingers, mental images, drawings, sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations.

• In Lesson 9-4, Represent 14 and 15, Launch, Notice & Wonder, students are shown a picture with a building with 15 windows and 15 bushes. “What groups of objects do you see? How can you find the number of windows? Bushes? What do you know about counting beyond 10 that can help you count these objects?” This exercise supports conceptual development of K.NBT.A, compose and decompose numbers from 11 to 19 into ten ones and some further ones.

The materials provide opportunities for students to independently demonstrate conceptual understanding through concrete, semi-concrete, verbal, and written representations. Examples include:

• In Lesson 3-3, Count 8 and 9, Practice & Reflect, Exercise 4, ”How many tablets? Color the counters to show how many.” Students are shown pictures of tablets with blank counters for students to color. This supports conceptual understanding of K.CC.4, understand the relationship between numbers and quantities; connect counting to cardinality.

• In Unit 6, Understand Addition, Math Probe, Exercise 1, students solve an addition word problem and circle the number that represents the answer. “Pat has 2 pet (picture of a dog) and 3 pet (picture of a cat). Circle the number of pets Pat has in all.” Answer choices include, “2, 3, 4, 5, 6, 7”. In the column to the right of the problem, students “Tell or show why” to justify their reasoning. This activity supports conceptual understanding of K.OA.1, represent addition and subtraction with objects, fingers, mental images, drawings, sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations.)

• In Lesson 8-7, Ways to Make 10, Explore & Develop, Activity-Based Exploration, students use ten-frames to practice composing groups of 10. “Give student-pairs a group of small objects and the Ten-Frame Teaching Resource. Have one student pick a number of objects that is less than 10 and place them on the ten-frame. Ask the other student to find the number of objects needed to make 10 and place those on the ten-frame. Have students repeat the activity, trading roles each time. Ask them to record equations that represent each of their groupings of 10.” Activity Debrief, “Invite student pairs to share one example of a grouping of 10 they made. Have them explain how the number in their equation represent the numbers of objects they used.” Practice & Reflect, On My Own, Reflect, students are shown a ten frame with 4 red counters and 6 yellow counters. “What is a different way to make 10 than what is shown?”  These activities support conceptual understanding of K.OA.4, for any number from 1 to 9, find the number that makes 10 when added to the given number. 

• In Lesson 10-2, Make 16 and 17, Practice & Reflect, On My Own, Exercise 2, “How can you make 16? Draw counters to show a group of ten ones and some more ones. Complete the equation to match.” Students are given two ten frames to draw their counters. This activity supports conceptual understanding of K.NBT.1, compose and decompose numbers from 11 to 19 into ten ones and some further ones.

The materials reviewed for Reveal Math Kindergarten meet expectations for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency. 

The materials develop procedural skill and fluency throughout the grade with teacher guidance, within standards and clusters that specifically relate to procedural skill and fluency, and build fluency from conceptual understanding. Examples include:

• Fluency Practice exercises are provided at the end of each unit. Each Fluency Practice includes: Fluency Strategy, Fluency Flash, Fluency Check, and Fluency Talk. “Fluency practice helps students develop procedural fluency, that is, the ‘ability to apply procedures accurately, efficiently, and flexibly.’ Because there is no expectation of speed, students should not be timed when completing the practice activity.” Fluency Practice exercises in Kindergarten progress toward K.OA.5, fluently add and subtract within 5. 

• In Lesson 6-5, Represent and Solve More Addition Problems, Practice & Reflect, On My Own, Exercise 1, “Use counters or drawings to represent the problem. Then write an equation to match. Four balls were in the basket. Three more balls were tossed in the basket. How many balls are in the basket now? ____ + ____ =  ____.” Students build fluency from conceptual understanding as they use counters or drawings to represent a problem. This exercise supports the development of procedural skill and fluency of K.OA.1, represent addition and subtraction with objects, fingers, mental images, drawings, sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations.

• In Unit 6, Understand Addition, Fluency Practice, Fluency Strategy, “You can count objects up to 10 in different arrangements.” Students see 3 groups of paper clips arranged in different ways. Group 1 has 10 paper clips, group 2 has 6 paper clips, group 3 has 8 paper clips. Exercise 1, “How many dog bones? Write the number of dog bones in each group.” Students see a picture of 7 dog bones and a picture of 9 dog bones in different configurations. This activity provides an opportunity to develop procedural and fluency of K.CC.5, count to answer “how many?” questions about as many as 20 things.

• In Lesson 7-1, Represent Take Apart Problems, Practice & Reflect, On My Own, Reflect, “How can you take apart the group of baseballs?” Students are given 8 baseballs to decompose into whatever arrangement of 8 students choose. Math is Mindset: “How did you picture a problem in your mind to help you find the answer?” Students build fluency from conceptual understanding of subtraction to support the development of procedural skill and fluency of K.OA.5, fluently add and subtract within 5.

• In Unit 7, Understand Subtraction, Math Probe, “Use models to show the situation.” Exercise 3, “Which equation shows this problem? There are 6 birds on a fence. Two fly away. How many birds are there now? Circle the equation. Tell or show why.” Students circle “6 - 2 = 4.” This activity shows the development of the cluster K.OA.A, understand addition as putting together and adding to, and understand subtraction as taking apart and taking from, relating to the procedural skill and fluency of K.OA.5, fluently add and subtract within 5.

• In Lesson 8-2, Subtract within 5, Explore & Develop, Pose the Problem, students act out a subtraction scenario, guided by the teacher. “There are four frogs on a lily pad. One frog hops off. How many frogs are left on the lily pad?” Pose Purposeful Questions, “How could you find the number of frogs left on the lily pad? How would you show this problem on a number path?” This activity supports the development of  procedural skill and fluency of K.OA.5, fluently add and subtract within 5.

The materials provide opportunities for students to independently demonstrate procedural skill and fluency. Examples include:

• In Lesson 8-2, Subtract within 5, Practice & Reflect, On My Own, Exercise 2, students use a provided number path with the numbers 1 to 5 to solve a subtraction problem. “How can you count back to find 5 - 2? Color the starting number. Count back 2. Circle the stopping number. What is the difference?” This exercise provides an opportunity for students to demonstrate procedural skill and fluency of K.OA.5, fluently add and subtract within 5.

• In Unit 8, Addition and Subtraction Strategies, Unit Assessment, Form A, Item 6, “Which of these make 4? Circle all the correct answers.” Students see 3 dominos. The first domino has 5 circles split into groups of  2 and 3, the second domino has 4 circles split into groups of 1 and 3, and the third domino has 4 circles split into 2 groups of 2. Students circle the second and third dominos which show different ways to make 4. This provides an opportunity for students to demonstrate procedural skill and fluency of K.OA.5, fluently add and subtract within 5.

• In Unit 8, Performance Task, Part E, students see a picture of 5 bowls with blank lines underneath for writing an equation and an answer. “Kenzo has 5 bowls. He put berries in 4 bowls. How many bowls are left? Draw an X on the bowls Kenzo used for berries. Write a subtraction equation for the problem.” This provides an opportunity for students to demonstrate procedural skill and fluency of K.OA5, fluently add and subtract within 5.

• In Unit 13, Analyze, Compare, and Compose Shapes, Fluency Practice, Fluency Flash, Exercise 2, “What is the sum of 4 and 0? Count on from 4. Write the sum” This activity provides an opportunity for students to demonstrate procedural skill and fluency of K.OA.5, fluently add and subtract within 5.

The materials reviewed for Reveal Math Kindergarten meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics. Additionally, the materials provide students with the opportunity to independently demonstrate multiple routine and non-routine applications of the mathematics throughout the grade level. 

The materials provide specific opportunities within each unit for students to engage with both routine and non-routine application problems. In the Digital Teacher Center, Program Overview: Learning & Support Resources, Implementation Guide, Focus, Coherence, Rigor, Application, “Students encounter real-world problems throughout each lesson. The On My Own exercises include rich, application-based question types, such as ‘Find the Error’ and ‘Extend Thinking.’ Daily differentiation provides opportunities for application through the Application Station Cards, STEM Adventures, and WebSketch Explorations. The unit performance task found in the Student Edition offers another opportunity for students to solve non-routine application problems.” 

The materials develop application throughout the grade as students solve routine problems in a variety of contexts and model the contexts mathematically within standards and clusters that specifically relate to application, both dependently and independently. Examples include:

• In Lesson 5-5, Position of 2-Dimensional Shapes, Launch, “Students notice different real- world 2-dimensional shapes and their locations by looking at a picture composed of different shapes.” Notice & Wonder, “What do you notice? What do you wonder?” In the Teaching Tip, teachers “may give clues by asking questions using the terms above, behind, below, inside, in front of, or next to. Although students are not expected to use these terms during the Notice and Wonder, using them yourself will introduce students to the terms.” This exercise allows students to develop and apply mathematics of K.G.1, describe objects in the environment using names of shapes, and describe the relative positions of these objects. In Practice & Reflect, On My Own, Exercise 3, Error Analysis, “Julian says the object above the clipboard is shaped like a triangle. Is he correct? Circle the object that is above the clipboard. Describe its shape.” This exercise allows students to independently apply the mathematics of K.G.1, describe objects in the environment using names of shapes, and describe the relative positions of these objects.

• In Unit 6, Understand Addition, Performance Task, Part A, “Roman finds 7 bees on flowers.  Some are on pink flowers and some are on red flowers. How many bees can be on each type of flower? Draw to show the problem. Then write an addition equation for your drawing.” This allows students to independently apply mathematics of K.OA.2, solve addition and subtraction word problems, and add and subtract within 10.

• In Lesson 11-6, Describe Solids, Explore & Develop, Learn, “How can you use names of shapes to describe positions of objects in the classroom?” Students are shown a classroom scene with items such as round tables, chairs, a window, an aquarium, teacher desk, chalkboard, clock, etc. Pose Purposeful Questions, “Which shape do you notice first? Where is it? Is there a shape that is located at the top of the picture? The bottom? Explain.” This exercise allows students to develop and apply mathematics of K.G.1, describe objects in the environment using names of shapes, and describe the relative positions of these objects using terms such as above, below, beside, in front of, behind and next to.

The materials develop application throughout the grade as students solve non-routine problems in a variety of contexts and model the contexts mathematically within standards and clusters that specifically relate to application, both dependently and independently. Examples include:

• In Lesson 7-2, Represent and Solve Take from Problems, Explore & Develop, Activity Based Exploration, Activity Debrief, after students have used counters to model and solve teacher provided word problems, “Have groups share how they represented the word problems with counters. Encourage students to make up their own subtraction word problem and share it with the class.” This exercise allows students to develop the mathematics of K.OA.1, represent addition and subtraction with objects, fingers, mental images, drawings, sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations.

• In Lesson 7-4, Represent and Solve Subtraction Problems, Explore & Develop, Activity- Based Exploration, “Have one student draw 6 objects and roll the number cube. Then, have that same student cross out the number of objects based on the number he or she rolled, showing that many objects were subtracted from the original group. Ask the other student to write the subtraction equation below the drawing. Have students tell a word problem that matches the drawing and equation. Repeat using starting numbers of 7, 8, 9, and 10.” This exercise allows students to develop the mathematics of K.OA.2, solve addition and subtraction word problems, and add and subtract within 10. In Practice & Reflect, On My Own, Exercise 4, Extend Your Thinking, “Create your own subtraction story. Draw a picture to represent your story. Then write numbers and trace symbols to represent your story.” This exercise allows students to independently apply K.OA.2, solve addition and subtraction word problems, and add and subtract within 10.

• In Lesson 7-5, Represent and Solve Addition and Subtraction Problems, Practice & Reflect, On My Own, Exercise 5, Extend Your Thinking, Draw or use counters to represent the problem. Then write and trace to complete the equation, “Seven boats are in the water. Two of the boats are sailboats and the rest are tugboats. How many tugboats are in the water?” This exercise allows students to independently apply K.OA.2, solve addition and subtraction word problems, and add and subtract within 10.

• In Lesson 8-1, Add within 5, Explore & Develop, Pose the Problem, “Marcella has three red stickers. She gets two pink stickers from her friend. How many stickers does she have now?” Pose Purposeful Questions, “What is one way to find how many stickers Marcella has? Can anybody else describe a different way? How could you check if your answer is correct?” This exercise allows students to develop the mathematics of K.OA.2, solve addition and subtraction word problems, and add and subtract within 10. In Differentiate, Extend Thinking, Differentiation Resource Book, Add within 5, Exercise 2, “Draw groups of rockets to match your equation. Tell a story for your equation. Write the total.” The equation is, “3 + 1 = ___.” This exercise allows students to independently apply K.OA.2, solve addition and subtraction word problems, and add and subtract within 10.

The materials reviewed for Reveal Math Kindergarten meet expectations in that the three aspects of rigor are not always treated together and are not always treated separately. There is a balance of the three aspects of rigor within the grade. Additionally, multiple aspects of rigor are engaged simultaneously to develop students’ mathematical understanding of a single topic/unit of study throughout each grade level. 

All three aspects of rigor (conceptual understanding, procedural skill & fluency, and application) are present independently throughout the grade level. Examples include:

• In Unit 7, Represent and Solve Addition and Subtraction Problems, Unit Review, Performance Task, Part A, “An animal trainer throws 7 toys to train her dog to catch. Some fall on the grass. The rest her dog catches. How many toys could fall on the grass and how many could be caught? Part A. Draw circles to show one way to take apart the group of toys.” This exercise provides an opportunity for students to demonstrate conceptual understanding of K.OA.1, represent addition and subtraction with objects, fingers, mental images, drawings, sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations.

• In Lesson 9-6, Decompose 14 and 15, Differentiate, Extend Thinking, Use It! Digital Application Station: Numbers Podcast, “Directions: Write a script for a podcast. Teach others how to write, compose, and decompose numbers from 11 to 15. Practice your podcast. If possible, record it.” This exercise provides an opportunity for  students to apply the mathematics of K.NBT.1, compose and decompose numbers from 11 to 19 into ten ones and some further ones.

• In Unit 11, 3-Dimensional Shapes, Fluency Practice, Fluency Flash, Exercise 2, “How can you count on to find 2 + 2? Write the sum.” This activity provides an opportunity for students to independently develop procedural skill and fluency of K.OA.5, fluently add and subtract within 5.

The materials provide a balance of the three aspects of rigor as multiple aspects of rigor are engaged simultaneously to develop students’ mathematical understanding of a single topic/unit of study throughout the grade level. Examples include:

• In Unit 6, Understand Addition, students are provided opportunities to develop conceptual understanding, procedural skill and fluency, and apply mathematics of solving addition and subtraction word problems within 10. In Lesson 6-2, Represent and Solve More Add to Problems, Explore & Develop, Activity-Based Exploration, “Directions: Distribute cards and connecting cubes to student pairs. Have one student create a word problem using the connecting cubes such as, Lilly has 2 cubes and Bella gives her 5 more. How many cubes does Lilly have? Ask the other student to create an equation to show the problem. Have students work together to see if the equation is correct for the situation. Repeat as time allows.” In Lesson 6-4, Represent and Solve Addition Problems, Practice & Reflect, On My Own, Exercise 2, “Use counters or drawings to show one way to solve the problem. Then write an equation to match. Sally has 3 kickballs. How many can she put in the red bin and how many in the green bin.” Unit Review, Performance Task, “A paramedic has 8 bins for organizing her first-aid supplies. Some bins are orange. Some are purple. How many of each color bin could the paramedic have?  Part A Color the bins to show one way. Part B Color the bins a different way. Part C How many bins does the paramedic have in all? Write an equation to match one way you colored the bins.” Unit 6 balances the three aspects of rigor for K.OA.2, solve addition and subtraction word problems, and add and subtract within 10.

• In Lesson 7-3, Represent and Solve More Take From Problems, Practice & Reflect, On My Own, Exercise 5, Extend Your Thinking, “Write numbers and trace symbols to complete the equation. Anthony threw the bowling ball and 4 pins were still standing. How many pins did Anthony knock down?” Students see six bowling pins and the equation 6 - ___ = 4. This exercise provides an opportunity for students to develop procedural skill and fluency and conceptual understanding, while applying the mathematics of K.OA.2, solve addition and subtraction word problems, and add and subtract within 10.

• In Lesson 11-3, Spheres, Explore & Develop, Activity-Based Exploration, students sort wooden geometric solids and classroom objects. “Have students lay all objects in the middle of the table. Ask students to take turns choosing the spheres and the objects shaped like spheres. Once all of the spheres are chosen, ask students how they are alike. Discuss that spheres have a rounded surface, no flat faces, and can roll. Ask students to explain why the other shapes are not spheres or shaped like spheres. Discuss the attributes of the other objects and how they are different than the attributes of a sphere.” Math is...Generalizations, “How are all spheres the same?” Practice & Reflect, On My Own, Exercise 3, “Which shapes are spheres? Circle all the spheres.” Students see several three dimensional shapes (sphere, cube and cylinder) of different sizes and colors. These exercises provide students opportunities to develop conceptual understanding and procedural skill and fluency of K.G.2, correctly name shapes regardless of their orientations or overall size.

The materials reviewed for Reveal Math Kindergarten meet expectations for supporting the intentional development of MP1: Make sense of problems and persevere in solving them; and MP2: Reason abstractly and quantitatively, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

Each Unit Overview, Math Practices and Processes section, identifies one mathematical practice that is prevalent in the unit, and gives an overview of its use within the unit. In the Standards section of each lesson, mathematical practices for the lesson are also identified; however, in both sections, the mathematical practice is labeled as MPP Reason abstractly and quantitatively, rather than MP1 or MP2. Within each of the lesson components, mathematical practices are not labeled or identified, leaving where they are specifically addressed up for interpretation and possible misidentification.

The materials provide intentional development of MP1: Make sense of problems and persevere in solving them, in connection to grade-level content. Examples include:

• In Lesson 8-3, Ways to Make 6 and 7, Explore & Develop, Work Together, “How can you make 6? Use two colors to show a way to make 6. Write the equation to match.” Students engage with the full intent of MP1 as they use a variety of strategies to work with a peer to show a way to make 6 using the provided images.

• In Lesson 11-6, Describe Solids, Explore & Develop, Activity-Based Exploration, “Directions: Have students scatter the shapes (wooden geometric cube, sphere, cone, and cylinder) and connecting cubes in front of the group. Ask the first student to describe the location of a solid shape in relation to the other shapes or cubes. For example, a student may say that the sphere is beside the 4 cubes. Encourage the students to discuss the description and make sure everyone is in agreement as to the name of the shape and its location. Have each student take a turn naming a shape and describing its location. After all students have had a turn, ask students to rearrange the shapes and cubes, and repeat as time allows.” Students engage with the full intent of  MP1, as they analyze and make sense of problems and determine if their answers make sense as they use positional words to describe the location of 3-dimensional objects in relation to one another.

• In Lesson 14-3, Compare Heights, Explore & Develop, Pose the Problem, a picture of one short plant and one tall plant is shown, “How can we compare these objects by height? What is height? What do you think you need to do before you compare the heights of two objects?” 

Students engage with the full intent of MP1 as they analyze and make sense of problems as they explore height comparison by examining and comparing two different sized plants.

The materials provide intentional development of MP2: Reason abstractly and quantitatively, in connection to grade-level content. Examples include:

• In Lesson 2-9, Compare Numbers to 5, Practice & Reflect, On my Own, Exercise 3, “How can you compare the groups? Count. Circe the group that is less.” Students see 3 ladybugs with the number 3 below the picture, and 4 flies with the number 4 below the picture. Students engage with the full intent of MP2 as they attend to the meaning of quantities and compare numbers.

• In Unit 3, Numbers to 10, Unit Overview, Math Practices and Processes, Reason Abstractly and Quantitatively, “Comparable problem analysis: The concepts taught in this unit are similar to what was taught in the previous unit. Students can use prior knowledge as they solve similar problems in this unit. Symbolic representation: Using symbols to represent problems correctly is part of the quantitative reasoning process. Working with numbers up to 10 numerically at this level is the basis of what students will need to do in the future mathematical operations.” This mathematical content attends to the full intent of MP2, as students count, compare, and write numbers throughout the unit.

• In Lesson 10-4, Represent 18 and 19, Explore & Develop, Pose the Problem, students are shown a picture of a little boy with 18 red trucks and 19 blue trucks. “What is in the picture? (red trucks, blue trucks) How many red/blue trucks are there? (18,19) Point to the group of 10 red/blue trucks. How many more red/blue trucks are there? (8, 9) How can we talk about what we see in the picture?” Pose Purposeful Questions, “How can you show the number of trucks in each group?” Bring It Together, “How can you find the number of objects in a group? How can counters help to show the number of objects in a group? What does a number tell you about objects in a group?” Students engage with the full intent of MP2 as they represent situations symbolically by representing the number of objects in a group.with a written numeral.

• In Lesson 12-3, Count By 10s to 100, Practice & Reflect, On My Own, Exercise 4, STEM Connection, “A landscape architect is buying flowers. The flowers come in groups of 10. Each row of cubes represents 10 flowers. How many flowers does the landscape architect have? Count by 10s. Write how many.” Students engage with the full intent of MP2 as they represent their work symbolically and understand the relationships between problem scenarios and mathematical representations.

The materials reviewed for Reveal Math Kindergarten meet expectations for supporting the intentional development of MP3: Construct viable arguments and critique the reasoning of others, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

Each Unit Overview, Math Practices and Processes section, identifies one mathematical practice that is prevalent in the unit and gives an overview of its use within the unit. In the Standards section of each lesson, mathematical practices for the lesson are also identified; however, the mathematical practice is labeled MPP: Construct viable arguments and critique the reasoning of others, rather than MP3: Construct viable arguments and critique the reasoning of others. Additionally, the math practices are not identified within the lesson sections, therefore leaving where they are specifically addressed up for interpretation and possible misidentification.

Examples of intentional development of students constructing viable arguments in connection to grade-level content, including guidance for teachers to engage students in MP3 include:

• In Lesson 4-1, Alike and Different, Practice & Reflect, On My Own, Reflect, students construct viable arguments as they discuss similarities and differences about a group of three triangles of different colors, “How are the objects in the group alike? How are they different? Draw a shape that is different from the other three. Ask students to share their reflections with their classmates.”

• In lesson 5-4, Circles, Launch, Which Doesn’t Belong?, students are shown images of one small orange circle, one small blue circle, one small orange square, and one large orange circle. Students construct viable arguments as they identify one of the shapes in the group and justify why it doesn’t belong with the others based on its attributes, “Which doesn’t belong? Who can explain why a different shape doesn’t belong?”

• In Lesson 10-6, Decompose 18 and 19, Explore & Develop, Work Together, students construct viable arguments as they work together to decompose a number, 18 pennies are shown, “How can you decompose 18? Circle groups to decompose 18 into ten ones and some more ones. Complete the equation to match. 18 = 10 + ____.” 

• In Unit 14, Compare Measurable Attributes, Unit Overview, Math Practices and Processes, Construct Viable Arguments and Critique the Reasoning of Others, provides guidance to teachers to engage students in MP3, “Remind students to be polite and respectful during discussions. Encourage students to use objects or drawings to construct their arguments. Encourage students to ask useful questions to clarify or improve the arguments being made.” 

Examples of intentional development of students critiquing the reasoning of others in connection to grade-level content, including guidance for teachers to engage students in MP3 include:

• In Lesson 5-4, Circles, Explore & Develop, Activity-Based Exploration, students critique the reasoning of others as they agree or disagree with how classmates sorted shapes. “Students discover attributes of circles by completing a shape sort.” Activity Debrief: “Ask each group to share how they sorted. Encourage classmates to state whether they agree or disagree with the group's choices.” 

• In Lesson 10-6, Decompose 18 and 19, Practice & Reflect, Exercise 4, Error Analysis, students critique the reasoning of others when they are shown an equation and asked, “Nate is decomposing 18. He says the equation is 18 = 10 + 9. Do you agree? Complete the equation to correct his thinking.”  

• In Unit 13, Analyze, Compare, and Compose Shapes, Unit Overview, Math Practices and Processes, Construct Viable Arguments and Critique the Reasoning of Others, provides guidance for teachers in engaging students in MP3, “Encourage students to be specific in their answers and ask deliberate questions of their classmates. You may need to spend time modeling effective communication and helpful feedback. A key goal of this mathematical practice is to be able to justify one's answers by articulating how and why a given conclusion is reached. When comparing shapes, students should be able to explain how they identified each shape and the process they used to understand the comparison. Encourage students to use drawings and objects to help them be specific in their explanations. When students disagree, have them practice asking questions to help others clarify their thinking.”

• In Lesson 13-5, Build 3-Dimensional Shapes, Practice & Reflect, Exercise 4, Error Analysis, students critique the reasoning of others when they are shown three shapes and asked, “Lily was asked to use clay to build three different models of a cone. She built two models correctly. Circle the model that is not correct.”

The materials reviewed for Reveal Math Kindergarten meet expectations for supporting the intentional development of MP4: Model with mathematics; and MP5: Use appropriate tools strategically for students, in connection to the grade-level content standards, as expected by the Mathematical Practice Standards.

 Each Unit Overview, Math Practices and Processes section, identifies one mathematical practice that is prevalent in the unit, and gives an overview of its use within the unit. In the Standards section of each lesson, mathematical practices for the lesson are also identified; however, the mathematical practice is labeled MPP Model with mathematics, rather than MP4: Model with mathematics. Additionally, the math practices are not identified within the lesson sections, therefore leaving the location of where they are specifically addressed up for interpretation and possible misidentification. 

Examples of intentional development of students modeling with mathematics in connection to grade-level content, including guidance for teachers to engage students in MP4 include:

• In Lesson 5-1, Triangles, Explore & Develop, Activity-Based Exploration, students work in small groups as they describe triangles of different shapes and sizes using a variety of provided objects including real world objects, pattern blocks, and paper cut-outs. “How are your shapes the same? What can you say about the sides of the objects? Think about something in your house that looks like these shapes. Describe it.” 

• In Unit 5, 2-Dimensional Shapes, Performance Task, students use what they know about 2-dimensional shapes to describe and model the correct position of a triangle, Stimulus, “Listen carefully. Bailey is creating a picture with shapes. She wants to make a house, a sun, and a tree.” Directions, Part A, “Bailey draws a house. Draw a triangle above the square. Write the number of sides and vertices a triangle has.” By putting the triangle above the square, a picture of a house will appear. 

• In Lesson 8-6, Ways to Decompose 8 and 9, Explore & Develop, Digital Guided Exploration: Ways to Decompose 8 and 9, Develop the Math, presentation slide 2.12, students are shown a blank number bond and given digital cubes to represent the problem, “9 = 3 + 6.” Math is...Modeling, “How does a number bond show how to break apart a number?” 

• In Lesson 10-2, Make 16 and 17, Differentiate, Build Proficiency, Digital Additional Practice Book: Make 16 and 17, Exercise 2, students model the composition of a number with a ten-frame and represent the composition with an equation, “How can you make 17? Draw counters to show a group of ten ones and some more tens. Write the equation to match.” Students complete the blank ten-frame, and also write an equation on the lines provided. 

The materials identify MP5 in seven lessons and in the Math Practices and Processes section of the Unit Overview in two out of fourteen units. Examples of intentional development of students using appropriate tools strategically in connection to grade-level content, including guidance for teachers to engage students in MP5 include:

• In Lesson 8-6, Ways to Decompose 8 and 9, Launch, Notice & Wonder, students recognize the benefits of using certain tools to show their work, Math is...Mindset, students see a picture of 8 pencils in no particular pattern, laying outside of 2 pencil boxes and explore ways they could be placed in the boxes, “How can tools help you show your work?” As students complete the Notice & Wonder exercise, “Invite students to discuss the tools they may use to organize their work while finding ways to decompose 8 and 9, such as using connecting cubes or making lists of equations. Encourage them to think about why this tool may be helpful for their work with breaking apart numbers.” In Practice & Reflect, Digital On My Own: Ways to Decompose 8 and 9, students use the writing tool to circle groups of 8 or 9 and write in an equation showing their decomposition, Exercise 1, students see 8 turtles and the equation ___ = ___ + ___, “Circle groups to show a way to decompose 8. Write the equation to match. How can you decompose 8?” Exercise 2, students see 8 snails and the equation ___ = ___ + ___, “Circle groups to show a different way to decompose 8. Write the equation to match. How can you decompose 8?” 

• In Unit 9, Numbers 11 to 15, Unit Review, Performance Task, Reflect, students use the tools and strategies they have learned to represent numbers, “What are different ways you can count, read, and show numbers 11 to 15?” 

• In Unit 10, Numbers 16 to 19, Unit Overview, Math Practices and Processes, “In Kindergarten, students are introduced to a variety of tools that they can use to make sense of and solve problems. In this unit, students use counters and ten-frames to compose numbers, as well as connecting cubes and number bonds to decompose numbers. Practicing with these tools and the numerical representations that correspond with them helps students deepen their understanding of the concepts behind teen numbers.”

• In Lesson 10-1, Represent 16 and 17, Explore & Develop, Digital Guided Exploration, slide 2, Develop the Math, students are shown 16 leaves.  Math is...Precision, “Why is it helpful to use counters when you count objects?” Reveal, “There are sixteen leaves.  How can you use counters to show sixteen leaves?” Students are given digital counters to complete the task.

The materials reviewed for Reveal Math Kindergarten meet expectations that there is intentional development of MP6: Attend to precision; and attend to the specialized language of mathematics, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

Each Unit Overview, Math Practices and Processes section, identifies one mathematical practice that is prevalent in the unit, and gives an overview of its use within the unit. In the Standards section of each lesson, mathematical practices for the lesson are also identified; however, the mathematical practice is labeled MPP Attend to precision, rather than MP6: Attend to precision. Additionally, the math practices are not identified within the lesson sections, therefore leaving where they are specifically addressed up for interpretation and possible misidentification. 

The instructional materials address MP6 in the following components:

• In the Digital Teacher Center, Program Overview: Learning & Support Resources, Implementation Guide, Language of Math, Unit-level Features, “The Language of Math feature highlights math terms that students will use during the unit. New terms are highlighted in yellow. Terms that have a math meaning different from everyday means are also explained.” Math Language Development, “This feature targets one of four language skills - reading, writing, listening, speaking - and offers suggestions for helping students build proficiency with these skills in the math classroom.” Lesson Level Features, “The Language of Math feature promotes the development of key vocabulary terms that support how we talk about and think about math in the context of the lesson content.” Each Unit Review also includes a vocabulary review component which references specific lessons within the unit.

Examples of intentional development of MP6: attend to precision, in connection to the grade-level content standards as expected by the mathematical practice standards, including guidance for teachers to engage students in MP6 include:

• In Lesson 4-3, Count Objects in Groups, Explore & Develop, Work Together, students attend to precision as they count and write the number of birds in each group. “Common Misconception: Students may still be struggling to count objects accurately. Remind them to say only one number as they touch each object, and not to count one object twice. Also remind them that the last number they say is the number of objects in the group.” Students see two bird cages. One bird cage has 5 blue birds in the cage. The other bird cage has 4 blue birds in the cage. “How many birds in each group? Write a number to show how many.” 

• In Lesson 5-3, Hexagons, Practice & Reflect, On My Own, Exercise 3, students attend to the precision of mathematics as they count the sides of shapes to determine which shapes have 6 sides, “Which shapes have 6 sides? Circle the shapes with 6 sides.”

• In Lesson 7-2, Represent and Solve Take From Problems, Explore & Develop, Activity-Based Exploration, students attend to precision as they count correctly to subtract, “Distribute one set of counters to each pair or small group of students. Read various take from word problems aloud to the students. For example, ‘Maria has 8 strawberries. She gives 3 strawberries to her sister. How many strawberries does Maria have now?’ Have students use their counters to model each situation and find the difference. Repeat as time allows.” Support Productive Struggle, “How can you use counters to show a take from problem? How do you know how many counters you need to start with? How do you know how many counters to take from the group?” Math is...Precision, “Why is it important to count correctly when you subtract?”

• In Lesson 14-2, Compare Lengths, Practice & Reflect, On My Own, Exercise 3, students attend to precision as they compare the lengths of provided objects and identify which object is longer. “Which is longer? Circle the longer object. Underline the objects if they are the same length.” 

Specialized language stands alone with vocabulary presentations in each lesson. When MP6 is identified for a lesson, MP6 specifically refers to precision with mathematics. Examples where the instructional materials attend to the specialized language of mathematics, including guidance for teachers to engage students in MP6 include:

• In Unit 5, 2-Dimensional Shapes, Unit Overview, Math Practices and Processes, Attend to Precision, “Attending to precision is important in geometry. Describing attributes and positions of shapes accurately provides an opportunity to introduce students to description words and to strengthen their vocabulary. Students should be given many opportunities to use these words throughout this and future units. As students become more proficient in using these words, encourage them to use as many as possible when describing a shape, including words that describe the shape’s position. Some suggestions for building precision include: 

  • Students provide detailed descriptions of shapes to partners who try to draw the shape.  

  • Students discuss the similarities and differences of two or more shapes. 

  • Students describe a shape in the classroom, adding more detail until a classmate is able to identify the shape.  

  • Students create a word wall of different description words that can be used to describe shapes.”  

• In Lesson 7-2, Represent and Solve Take From Problems, Explore & Develop, Language of Math, “The term subtract is a verb. Talk to students about how to subtract also means to find the difference. Encourage students to use subtract or difference in a sentence as they discuss it with other students.” This guidance for teachers engages students in the full intent of MP6, attending to the specialized language of mathematics.

• In Lesson 13-6, Describe 3-Dimensional Shapes in the World, Explore & Develop, Activity- Based Exploration, students attend to the specialized language of mathematics as they discuss the attributes of geometric solids. Directions, “Discuss the attributes of each wooden geometric solid. Remind students about the faces, bases, apex, and vertices. Also, discuss if the shapes can roll or stack. Show the real-world objects and discuss what they are shaped like. Then, ask students to think of other objects that they have seen at home, at school, on the playground, in the store, etc. that are shaped like 3-dimensional shapes that they know. After naming a shape, discuss how to describe its relative position.” 

• In Unit 14, Compare Measurable Attributes, Unit Review, Performance Task, Reflect, students attend to the specialized language of mathematics as they compare a small pot of water to a large pot of water, “How can you compare objects? Explain how you can compare the length, weight, height, and capacity of the objects.”

The materials reviewed for Reveal Math Kindergarten meet expectations for supporting the intentional development of MP7: Look for and make use of structure; and MP8: Look for and express regularity in repeated reasoning, for students, in connection to grade-level content standards, as expected by the mathematical practice standards.

Each Unit Overview identifies one mathematical practice that is prevalent in the unit and gives an overview of its use within the unit. In the Standards section of each lesson, mathematical practices for the lesson are also identified; however, the mathematical practice is labeled MPP Look for and make use of structure, rather than MP7: Look for and make use of structure; and MP8: Look for and express regularity in repeated reasoning. Additionally, the math practices are not identified within the lesson sections, therefore leaving where they are specifically addressed up for interpretation and possible misidentification. However, upon review, it was found that the materials provide additional opportunities for students to engage in the full intent of MP8 that were not identified for teachers. 

Examples of intentional development of students looking for and making use of structure, to meet its full intent in connection to grade-level content, including guidance for teachers to engage students in MP7 include:

• In Unit 2, Numbers to 5, Unit Overview, Math Practices and Processes, Look for and Make Use of Structure, “This unit introduces students to numerals 0 through 5 and makes the connection between numerals and number names. Students will have opportunities to count different objects, which will help them connect a number name with an amount. Students will also see that the arrangement of the objects does not change the number in the group. In addition, students will use the structure of matching objects in two groups to determine if the two groups are equal or if one is greater. Some suggestions to help students identify these connections and structure are: 

  • Show representations of numbers using different objects and in differing arrangements. Have students identify how the representations are connected and how they relate back to the number being discussed.

  • Allow students time to represent each number using different objects.  Students can then discuss how the representations are alike and different.  

  • Encourage students to use a strategy such as matching when comparing numbers, and have them justify their answer.”  

• In Lesson 2-7, Equal Groups to 5, Practice & Reflect, On My Own, Exercise 2, students look for structure as they draw lines to match objects to determine if groups are equal. “Are the two groups equal? Circle the groups if they are equal. Draw an X on the groups if they are not equal.” 

• In Lesson 4-4, Describe Groups of Objects, Explore & Develop, Activity-Based Exploration, teachers engage students to look for and explain structures as they use a 3-part sorting mat to have students sort, count, and compare the number of pattern blocks in the groups. “Directions: Provide each student or group of students with a collection of pattern blocks and a 3-Part Sorting Mat Teaching Resource. Each collection of blocks should contain only 3 different shapes with no more than 10 of each shape. Have students sort the pattern blocks into groups. Then have them count the number of objects in each group. Ask students to compare the number of objects in each group, encouraging them to use comparison terms such as more and fewer. Math is ... Structure: What can you look at to describe and compare sorted groups?” 

• In Lesson 13-1, Compare and Contrast 2-Dimensional Shapes, Practice & Reflect, On My Own, Reflect, students look for structure by looking at the number of sides and vertices (corners) to make generalizations to describe and compare shapes, “How can you compare the shapes?” Students see 5 different colored shapes in row:  hexagon, triangle, circle, triangle, square. 

Examples of intentional development of students looking for and expressing regularity in repeated reasoning, including guidance for teachers to engage students in MP8 include:

• In Lesson 4-2, Sort Objects into Groups, Explore & Develop, Guided Exploration, students use regularity in repeated reasoning to explain how objects are sorted. “Students apply the skills they used to describe attributes of objects to sorting objects by attribute. Students will become aware that objects can be sorted according to more than one attribute, which helps them develop an understanding that multiple solution strategies are acceptable in mathematics. Discuss that the group of buttons can be sorted in different ways. Guide students in seeing that some of the ways that the buttons could be sorted are by size, color, and shape.” Digital Guided Exploration, Presentation slide slide 2.4, students are shown 4 heart shaped buttons, 5 small buttons, and 3 red buttons, Math is … Generalizations, “How can you describe sorted groups of objects?”

• In Lesson 5-2, Squares and Rectangles, Explore & Develop, Activity-Based Exploration, teachers engage students with MP8 as they support them in sorting squares and rectangles of different sizes and colors and make generalizations about the shapes. “Directions: Students work in small groups. Provide each group with objects shaped like squares and rectangles of different sizes and colors. Groups should sort the shapes according to a common attribute.” Math is...Generalizations, “How are all the rectangles the same? How are all squares the same? Why are all squares also rectangles? Explain whether or not all rectangles are squares.” Within the Practice & Reflect, On My Own, Reflect, students look for and use repeated reasoning to explain how they know a shape is a rectangle, “How do you know if a shape is a rectangle?” Students see a row of 5 different 2-dimensional shapes: rectangle, circle, rectangle, triangle, rectangle. 

• In Lesson 7-3, Represent and Solve More Take From Problems, Practice & Reflect, Reflect, students use repeated reasoning as they create an equation to represent a subtraction situation, “How can you use an equation to represent subtraction?” Students see 5 shirts with 3 crossed out.