Kindergarten - Gateway 2

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

Materials develop conceptual understanding throughout the grade level. According to the Teacher’s Edition’s Program Overview, “conceptual understanding and problem solving are crucial aspects of the curriculum.” In the Topic Overview, Math Background: Rigor, “Conceptual Understanding Background information is provided so you can help students make sense of the fundamental concepts in the topic and understand why procedures work.” Each Topic Overview includes a description of key conceptual understandings developed throughout the topic. The 3-Act Math Task Overview indicates the conceptual understandings that students will use to complete the task. At the lesson level, Lesson Overview, Rigor, the materials indicate the Conceptual Understanding students will develop during the lesson.

Materials provide opportunities for students to develop conceptual understanding throughout the grade level. The Visual Learning Bridge and Guided Practice consistently provide these opportunities. Examples include: 

• Topic 4, Lesson 4-1, Lesson Overview, Conceptual Understanding states, “Students further their understanding of comparison as they compare larger groups to determine which is greater or less in number.” In the Visual Learning Bridge, the materials show two adjacent frames: A) contains scattered groups of yellow chicks and black chicks and B) contains a row of yellow chicks and another row of black chicks. Classroom Conversation asks students the following questions: “A) What do you see? Can you tell which group of chicks is greater in number than the other group just by looking at them? How can you compare the two groups? B) Construct Arguments Let’s count the chicks together. Is a group of 7 chicks greater in number or less in number than a group of 10 chicks? How do you know?” In Guided Practice, Problem 2, students compare two groups of chicks. Directions: “Have students compare the groups, draw a line from each chick in the top group to a chick in the bottom group, and then draw a circle around the group that is greater in number than the other group.” The image for Problem 2 shows the two rows of chicks: three yellow and eight yellow. Students develop conceptual understanding by using strategies to identify whether one group is greater than, less than, or equal to another. (K.CC.6)

• Topic 6, Lesson 6-3, Lesson Overview, Conceptual Understanding states, “Students explore addition as putting together. The addition problems continue in the format of addition sentences as students build understanding.” In the Visual Learning Bridge, the materials show three frames: A) has a column of two red tomatoes and a column of four yellow tomatoes, B) has a column of two red counters and a column of four yellow counters, and C) shows a boy drawing a circle around the two groups of tomatoes to put them together, indicating “2 and 4 is 6.” Classroom Conversation asks students the following questions: “A) What number story can you make up about the tomatoes in the box? B) What can you show with the counters? How many tomatoes are in the first group? In the second group? C) Reasoning What does the drawing show? What is one way to put together the 2 groups? What does the sentence tell? How many are there in all when you put together 2 and 4?”  In Guided Practice, Problem 3, students use counters to model putting together groups. Directions: “Have students use counters to model putting together the groups, draw a circle around the groups to put them together, and then write an addition sentence to tell how many in all.” The materials show a group of five yellow corns and a group of four blue corns with the sentence frame: and   is __. Students develop conceptual understanding as they represent addition using counters. (K.OA.1)

• Topic 13, Lesson 13-2, Lesson Overview, Conceptual Understanding states, “Students learn to identify 3-D shapes based on common attributes. These include the attributes that allow solid figures to roll, stack, or slide.” In the Visual Learning Bridge, the materials show four frames: A) shows a cube, sphere, cone, and cylinder; students discuss which solid figure has two or more vertices and which figures have flat surfaces. B) shows three solid figures in motion, and students discuss what a solid object needs to look like to roll. C) focuses on student reasoning about why a cube and a cylinder can be stacked. D) shows three solids in motion, and students discuss what a solid object needs to look like to slide. Classroom Conversation asks students the following questions: “A) What solid figures do you see? Which solid figure has 2 or more vertices? Which solid figures have flat surfaces? B) Which solid figures do you see? What movement is each figure doing? What does an object have to look like to roll? C) Reasoning What do you see? Why can these solid figures be stacked? Can you stack a cone? D) Which solid figures slide? Which can stack, slide, and roll?” Students build on prior and emerging understandings as they identify which 3-D shapes can roll, stack, and slide. In Guided Practice, Problem 2, students compare shapes based on their attributes. Directions: “2 look at the rolling solid figure on the left, and then draw a circle around the other solid figures that roll.” The materials show a circle and the indication that it can roll, with three shapes next to it, a cube, a cone, and a cylinder. Students develop conceptual understanding as they analyze and compare three-dimensional shapes using their attributes. (K.G.4)

Materials provide opportunities for students to independently demonstrate conceptual understanding throughout the grade level. The Practice problems consistently provide these opportunities. Examples include:

• Topic 1, Lesson 1-5, Lesson Overview, Conceptual Understanding states, “Students’ understanding of counting is deepened as they realize the arrangement of objects does not affect the number of objects.” In Independent Practice, Problem 11, students draw a circle around the group that matches a given number. Directions: “11 count the groups, and then draw a circle around the groups that show 4”. The materials show three groups of bees: a row of four bees, a random arrangement of four bees, and a random arrangement of three bees. Students independently demonstrate conceptual understanding by recognizing that the number of objects is the same regardless of their arrangement or the order in which they were counted. (K.CC.4b) 

• Topic 10, Lesson 10-5, Lesson Overview, Conceptual Understanding states, “Students will build on the concept that a number can be shown as two parts. They focus on decomposing the numbers into a group of 10 ones and some more ones. Students expand their knowledge of an equation representing a quantitative relationship. They continue to establish a basic understanding of place value in our base-ten numeration system.” In Independent Practice, Problem 5, students draw counters to match an equation. Directions: “5 draw counters to match the equation. Then have them tell how the picture and equation show 10 ones and some more ones.” Students draw counters in two ten frames to match the equation 16 = 10 + 6. Students independently demonstrate conceptual understanding by composing and decomposing numbers from 11 to 19 into ten ones and some further ones. (K.NBT.1)  

• Topic 14, Lesson 14-4, Lesson Overview, Conceptual Understanding states, “Previously students have been thinking of length, height, weight, and capacity individually when comparing. They now think of objects as being described by more than one of these attributes. This is an important step in understanding how attributes describe objects, focusing on what defines the object and understanding that not every object can necessarily be described by every attribute.” In Independent Practice, Problem 7, students identify the attributes that can be measured of an object. Directions: “Have students look at the object on the left, identify the attributes that can be measured, and then draw a circle around the tools that could be used to tell about those attributes.” A picture of a mug is on the left and the options of tools are the following measure length (cubes), weight (scale), and capacity (measuring cup). Students independently demonstrate conceptual understanding by describing measurable attributes of an object. (K.MD.1) 

The materials reviewed for enVision Mathematics 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 skills and fluency throughout the grade level within various portions of lessons. The Teacher’s Edition Program Overview indicates, “Students perform better on procedural skills when the procedures make sense to them. So procedural skills are developed with conceptual understanding through careful learning progressions. … A wealth of resources is provided to ensure all students achieve success on the fluency expectations of Grades K-5.” Various portions of lessons that allow students to develop procedural skills include Solve & Share, Visual Learning Bridge, Convince Me!, Guided Practice, and 3-ACT MATH; in addition, the materials include Fluency Practice Activities. Examples include: 

• Topic 1, Lesson 1-3, Lesson Overview, Procedural Skill states, “Students practice how to write 1, 2, and 3 to tell how many are in a group.” In Guided Practice, the materials show one star, then two stars, and finally three stars. In Problems 1-3, students develop procedural skills and fluency by writing the number of stars. “Directions. “Have students count the stars, and then write the number to tell how many.” Students write numerals 1, 2, and 3. (K.CC.3)

• Topic 6, Lesson 6-4, Lesson Overview, Procedural Skill states, “Students write addition equations to show adding two groups to find a sum.” In the Visual Learning Bridge, the materials show three adjacent frames: A) includes a boy and five scattered drums; B) depicts the boy assigning the numbers 4 and 1 to groups of 4 and 1 drum, respectively; and in C) the boy translates “4 and 1 is 5” to the equation “4 + 1 = 5.” In Guided Practice, Problem 1, students develop procedural skills and fluency by translating “2 and 6 is 8” to the equation “2 + 6 = 8.” Directions: “Have students add the groups to find the sum, and then write an equation to show the addition.” The image for Problem 1 shows a group of 2 drums and a group of 6 drums. (K.OA.1)

• Topic 13, Lesson 13-1, Lesson Overview, Procedural Skill, "Students identify shapes when given attributes as clues." In the Visual Learning Bridge, the materials show a triangle, a square, and a rectangle; students compare their attributes. In Convince Me!, students develop procedural skills and fluency by knowing that the number of sides and/or the number of vertices can help them identify the shape. “Which shape has 4 sides and 4 vertices: squares, rectangles, circles, or triangles?” (K.G.4)

Materials provide opportunities for students to independently demonstrate procedural skills and fluency throughout the grade level. Independent Practice and Problem Solving consistently include these opportunities. When appropriate, teachers may use other portions of lessons for independent demonstration of procedural skill and fluency. Examples include:

• Topic 3, Lesson 3-3, Lesson Overview, Procedural Skill states, “Students’ knowledge of the counting sequence builds as they extend their counting to groups of 8 and 9 objects.” In Independent Practice, Problem 8, students independently demonstrate procedural skills and fluency when they connect that the last number counted tells how many pieces of fruit there are. “Directions: “Have students count the pieces of fruit, and then draw counters to show how many.” The materials show eight strawberries and include a ten-frame. (K.CC.4a)

• Topic 8, Lesson 8-1, Lesson Overview, Procedural Skill states, “Students work on the procedural skill of showing parts of a number and representing those parts in an equation as they solve word problems.” In Independent Practice, Problem 5, students independently demonstrate procedural skills and fluency when they draw pictures to create a representation of 5. “Directions: Higher Order Thinking Have students draw another way to break apart 4 with flowers, and then write an equation to match the story and show the parts that equal 5.” The materials show “5 = ___ + ___.” (K.OA.2)

• Topic 14, Lesson 14-1, Lesson Overview, Procedural Skill states, “Students identify the longer/taller and shorter objects (or objects that are the same length) as they make comparisons throughout this lesson.” In Independent Practice, Problems 7 and 8, students independently demonstrate procedural skills and fluency as they tell which object is longer or shorter by sight. Directions: “Have students mark an X on the shorter object and draw a circle around the longer object, or underline the objects if they are the same length.” For Problem 7, the materials show a long wrench above a short wrench; for Problem 8, the materials show two sneakers that are the same length. (K.MD.2)

The materials for enVision Mathematics Kindergarten meet expectations that the materials are designed so that teachers and students spend sufficient time working with engaging applications of the mathematics.

Engaging applications—which include single and multi-step, routine and non-routine applications of the mathematics—appear throughout the grade level and allow for students to work with teacher support and independently. In each Topic Overview, Math Background: Rigor provides descriptions of the concepts and skills that students will apply to real-world situations. Each Topic is introduced with a STEM Project, whose theme is revisited in activities and practice problems in the lessons. Within each lesson, Application is previewed in the Lesson Overview. Practice & Problem Solving sections provide students with opportunities to apply new learning and prior knowledge.

Examples of routine applications of the math include:

• In Topic 2, Lesson 2-5, Independent Practice, Problem 3, students independently represent a number of objects with a written numeral 0–20. “Directions Say: Carlos has 4 red blocks and 3 blue blocks. Which group of blocks is less in number than the other group? How can you use numbers to show your answer? Have students use a number to show and explain their answer.” (K.CC.3 and K.CC.6)

• In Topic 8, Lesson 8-9, Independent Practice, Problem 10, students independently find the number that makes 10 when added to a given number. “Directions Higher Order Thinking Say: A child is holding up 3 fingers to show how old she is. What part of 10 is she showing? Use that number to write the missing numbers in the equation to tell the parts of 10. __ + __ = 10” (K.OA.4)

• In Topic 10, Lesson 10-5, Solve & Share, students use counters and write an equation by decomposing 14 into two parts. “Directions Say: 14 students go to the zoo. The first bus takes 10 students. The rest of the students go on the second bus. Use counters to describe this situation. Then complete the equation to match the counters and tell how the counters and equation show 10 ones and some more ones. 14 = __ + __ ” The materials show two ten-frame counters shaped as buses. (K.NBT.1 and K.CC.5)

Examples of non-routine applications of the math include:

• In Topic 5, Topic Performance Task, students classify objects into given categories as well as count the number of objects in each category and sort the categories by count. “Directions Works of Art Say: A kindergarten class uses paintbrushes and paint to draw pictures. Have students: 1 draw a circle around the little paintbrush, and then mark an X over the paintbrushes that are NOT little; 2 draw lines in the first chart as they count the paintbrushes that are little and the paintbrushes that are NOT little. Then have them write the number to tell how many are in each group in the second chart, and draw a circle around the number of the group that is less than the number of the other group. 3 Have students show one way to organize the jars of paint they see on the page before, and then explain how they sorted them. 4 Say: Tina says that the number of jars of paint is equal to the number of large jars of paint. Does her answer make sense? Have students look at the paint on the page before, draw a circle around yes or no, and then use the sorting and counting of each category to explain their reasoning.” The materials include a picture of a table with small and large paintbrushes and small and large jars of paint of various numbers on it. (K.MD.3)

• In Topic 7, enVision STEM Project: Animal Needs, students independently add and subtract within 10 using drawings to represent the context of a non-routine problem. “Directions Read the character speech bubbles to students. Find Out! Have students find out about how plants, animals, and humans use their environment to meet basic needs such as food, water, nutrients, sunlight, space, and shelter. Say: Different organisms need different things. Talk to friends and relatives about the different needs of plants, animals, and humans, and how different organisms meet those needs. Journal: Make a Poster Have students make a poster. Ask them to draw as many as 5 pictures of a human’s needs and as many as 5 pictures of an animal’s needs. Have then cross out the needs that are the same for humans and animals, and then write how many are left.” One speech bubble says “Food” and the other says “Animals need food and water.” (K.OA.2)

• In Topic 11, Topic Performance Task, Problem 5, students independently solve a non-routine problem by counting forward beginning from a given number within the number sequence. “Directions Say: Ty has 64 raisins in one bag. He has 18 raisins in another bag. Help Ty count his raisins. Have students start at 64 on the number chart and they make a path to show how to count up 18 in any way they choose. Then have them draw a circle around the number where they stopped, and then explain how they counted up.” Students are provided a number chart that ranges from 51 - 100. (K.CC.2)

The materials for enVision Mathematics Kindergarten meet expectations that the three aspects of rigor are not always treated together and are not always treated separately. 

Each Topic Overview contains Math Background: Rigor, where the components of Rigor are addressed. Every lesson within a topic contains opportunities for students to build conceptual understanding, procedural skills and fluency, and/or application. During Solve and Share and Guided Practice, students explore alternative solution pathways to master procedural fluency and develop conceptual understanding. During Independent Practice, students apply the content in real-world applications, use procedural skills and/or conceptual understanding to solve problems with multiple solutions, and explain/compare their solutions.

The three aspects of rigor are present independently throughout the grade. For example:

• Topic 2, Lesson 2-1, Solve & Share, students attend to conceptual understanding, explaining how they know whether the number of objects in one group is greater than, less than, or equal to the number of objects in another group. “Directions Say: Marta has some toy cars. Are there the same number of red cars as there are yellow cars on the rug? How do you know? Use counters to show your work.” The materials show Marta lying on a rug playing with cars: four yellow and four red. (K.CC.6)

• Topic 8, Fluency Practice Activity, students attend to procedural skills and fluency as they practice fluently adding and subtracting within 5. “Directions Have students: 1 color each box that has a sum or difference that is equal to 3; 2 write the letter that they see.” The materials present two frames for student engagement: (1) a 5 by 5 worksheet of addition and subtraction problems (such as 1 + 2 and 5 - 2) and (2) blank space where the students write the letter. (K.OA.5)

• Topic 14, Lesson 14-1, Solve & Share, students attend to application as they compare measurable attributes of two objects. “Directions Say: Marta makes a cube train with 4 cubes. Is her cube train bigger or smaller than the crayon? Is her cube train bigger or smaller than the pencil? How can you find out?” The materials show images of a crayon and a pencil. (K.MD.2)

Multiple aspects of Rigor are engaged simultaneously to develop students’ mathematical understanding of a single topic/unit of study throughout the grade. For example:

• Topic 3, Lesson 3-6, Independent Practice, Problems 6 and 7, students attend to conceptual understanding and procedural skills and fluency as they see that 10 can be made in different ways and practice how to write 10 to tell how many are in a group. “Directions Number Sense 6 and 7 Have students count the shells, and then write the number to tell how many.” The materials show for problem 6, 10 blue shells situated in one row and for problem 7, 10 red shells divided evenly into two rows of five and provide lines for students to write their answers. (K.CC.3 and K.CC.5)

• Topic 6, Lesson 6-5, Guided Practice, Problem 5, students attend to conceptual understanding and application as they solve an addition word problem using drawings and an equation. “Directions Have students listen to the story, use counters to show the addition, look at or draw a picture, and then write an equation to tell how many in all. 5. 2 turtles swim in the water. 6 more join them. How many turtles are swimming in all?” Students draw in the provided space and write the equation 2 + 6 = 8. (K.OA.2)

• Topic 11, Lesson 11-5, Independent Practice, Problem 5, students attend to procedural skills and application as they count forward from a beginning number by ones and complete the sequence by counting by tens. “Directions Have students count forward, and then draw a circle around the row that shows the missing set of numbers.” The materials show a three-row table: the first row shows numbers 31-39 with a blank cell for 40. The second and third rows similarly show 41-49 and 51-59, respectively, with blank cells for the 50 and 60. The materials provide students with three options for the missing set of numbers: 40, 50, 60 or 40, 41, 42 or 38, 39, 40. (K.CC.1 and K.CC.2)

The materials reviewed for enVision Mathematics 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. 

Students have opportunities to engage with the Math Practices across the year, and they are explicitly identified for teachers within the Program Overview within the Topic Contents at the Lesson-level. The Math Practices and Problem Solving Handbook introduces each of the Math Practices with specific emphasis on making connections among representations to develop meaning and corresponding Thinking Habits. The Teacher’s Edition provides support for developing, connecting, and assessing each math practice. Topic Planners include the Math Practices at the lesson level; relevant practices are specified in Lesson Overviews. 

MP1 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students make sense of problems and persevere in solving them as they work with the support of the teacher and independently throughout the Topics. Examples include:

• Topic 2, Lesson 2-5, Problem Solving, Performance Task, Problem 4, students make sense of problems and persevere in solving them as they apply matching and counting strategies to identify whether a number of objects in one group is greater than a number of objects in another group. “Directions Read the problem aloud. Then have students use multiple problem-solving methods to solve the problem. Say: Marta has 2 stickers. Emily has a greater number of stickers than Marta. How many stickers could Emily have? Make Sense. What do you know about the problem? Can Emily have 1 sticker? Tell a partner why or why not?” 

• Topic 8, Lesson 8-2, Convince Me!, students make sense of problems and persevere in solving them as they make sense of situations that involve addition and subtraction within five to solve problems. “Give students 2 red and 2 yellow cubes. Have them tell and act out an addition and subtraction story. Have students compare the stories (both use the same numbers but the actions are different).” 

• Topic 14, Lesson 14-4, Solve & Share, students make sense of problems and persevere in solving them as they describe measurable attributes of objects, such as length or weight, or describe several measurable attributes of a single object. “Directions Say: These are 2 tools for measuring. What can you measure with the cup? What can you measure with the cube train? Draw an object you can measure with each tool.” The materials show a cup and a train of six counting cubes. 

MP2 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students reason abstractly and quantitatively as they work with the support of the teacher and independently throughout the Topics. Examples include:

• Topic 4, Lesson 4-2, Guided Practice, Problem 2, students reason abstractly and quantitatively as they compare groups that contain different objects and represent quantities with written numerals. “Directions Have students count the vegetables in each group, write the number to tell how many, draw a line from each vegetable in the top group to each vegetable in the bottom group, and then mark an X on the number that is less than the other number.” The materials show a top row of four peppers and a bottom row of five ears of corn. 

• Topic 9, Lesson 9-7, Solve & Share, students reason abstractly and quantitatively as they count forward beginning from a given number and write numbers 10 to the number 20 and identify possible answers to word problems that have more than one potential answer. “Directions Say: Carlos wants to put some or all of the eggs in the carton. Draw a circle around the numbers that tell how many eggs he could put in the carton. Explain why there could be more than one answer.” The materials show an empty egg carton, 14 scattered eggs, and the numbers 10 through 14. 

• Topic 13, Lesson 13-3, Solve & Share, students reason abstractly and quantitatively as they analyze and compare two- and three-dimensional shapes, in different sizes and orientations, and relate the two-dimensional shapes to the shapes of the flat surfaces in the three-dimensional shapes, and vice versa. “Directions Say: Jackson needs to find a circle that is a flat surface of a solid figure. Which of these solids has a flat circle as part of the figure? Draw a circle around each solid figure that has a flat circle part. Mark an X on the solid figures that do NOT have a flat circle part. How many shapes in all are there on the page? How many shapes did you circle? Without counting, how many shapes have an X? Count the shapes with an X to check your answer.” The materials show scattered shapes: pyramid, shipping box, cone, tennis ball, sphere, cylindrical clock, tube of tennis balls, log, and cylinder. 

The materials reviewed for enVision Mathematics 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. 

Students have opportunities to engage with the Math Practices across the year, and they are explicitly identified for teachers within the Program Overview within the Topic Contents at the Lesson-level. The Math Practices and Problem Solving Handbook introduces each of the Math Practices with specific emphasis on making connections among representations to develop meaning and corresponding Thinking Habits. The Teacher’s Edition provides support for developing, connecting, and assessing each math practice. Topic Planners include the Math Practices at the lesson level; relevant practices are specified in Lesson Overviews.

Students construct viable arguments and critique the reasoning of others in connection to grade-level content as they work with the support of the teacher and independently throughout the Topics. Examples include:

• Topic 1, Lesson 1-2, Solve & Share, students construct viable arguments and critique the reasoning of others as they explain solutions to problems where they have to count and consider other students’ work. “Directions, Say: “Redbird and Bluebird each have 2 babies. Redbird and Bluebird get worms for their babies and put them in their nests. Bluebird’s worms moved around in the nest. Show and count how many worms with your counters. Color the boxes to show the worms in each nest. Tell how you know you are correct.” The teacher is prompted to, “choose which solutions to have students share and in what order. Focus on the idea that no extra counters were placed or taken away. If needed, show and discuss the student work at the right.” The student work at the right shows two worms in each nest accompanied by the teacher's statement, “Marlon says each bird found a different number of worms. Why might Marlon have thought the birds found a different number of worms?”

• Topic 5, Lesson 5-4, Solve & Share, students construct viable arguments and critique the reasoning of others as they tell whether a given statement makes sense—providing reasons for their choice using numbers, pictures, or words to explain—and compare their answer to other student work.  “Directions Say: Carlos says that the number of blue cubes is equal to the number of cubes that are NOT blue. Does his answer make sense? Use numbers, pictures, or words to explain your answer.” The materials show ten blue cubes, five yellow cubes, and four green cubes as well as the statement, “I can tell whether the way objects have been sorted, counted, and compared makes sense. I can explain how I know.” Teachers are prompted to use questions and additional work to help students construct viable arguments and critique the reasoning of others such as: “Based on your [teacher] observations, choose which solutions to have students share and in what order…If needed, show and discuss the student work at the right.” There are two pieces of work displayed at the right one is labeled Kirsty’s Work and the other is labeled Tim’s Work. The following questions are asked: Kirsty says that Carlos’s answer does not make sense because 10 is greater than 9. Do you agree with Kirsty? Why? How did Kirsty show this? What mistake did Tim make? Why might this mistake make him think Carlos was correct?

• Topic 9, Lesson 9-7, Problem Solving, Performance Task, Problems 6 and 7, students construct viable arguments and critique the reasoning of others as they identify possible answers to word problems that have more than one possible answer and compare their work with another student’s work. “Directions Read the problem to students. Then have them use multiple problem-solving methods to solve the problem. Say: Alex lives on a farm with so many cats that they are hard to count. Sometimes the cats are outside and sometimes they hide in the shed. Alex knows that the number of cats is greater than 11. There are less than 15 cats on the farm. How can Alex find out the number of cats that could be on his farm? 6. Model How can you show a word problem using pictures? Draw a picture of the cats on Alex’s farm. Remember that some may hide inside the shed. 7. Explain Is your drawing complete? Tell a friend how your drawing shows the number of cats on Alex’s farm.” The materials suggest to teachers, “Students should explain how their drawing represents each part of the word problem.” Teachers say, “Think about how the drawing shows the problem, not whether your drawings are the same.”

• Topic 12, Lesson 12-1, Guided Practice, Problem 2, students construct viable arguments and critique the reasoning of others as they identify an object as flat or solid, explaining how they distinguish between the two. “Directions “Have students: draw a circle around the objects that are flat, and mark an X on the objects that are solid.” The materials suggest “Have students compare their answers and explain their choices, telling why they think a particular shape was or was not flat or solid. Encourage students to state if they agree and tell why.”

The materials reviewed for enVision Mathematics 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. 

Students have opportunities to engage with the Math Practices across the year, and they are explicitly identified for teachers within the Program Overview within the Topic Contents at the Lesson-level. The Math Practices and Problem Solving Handbook introduces each of the Math Practices with specific emphasis on making connections among representations to develop meaning and corresponding Thinking Habits. The Teacher’s Edition provides support for developing, connecting, and assessing each math practice. Topic Planners include the Math Practices at the lesson level; relevant practices are specified in Lesson Overviews. 

MP4 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students model with mathematics as they work with the support of the teacher and independently throughout the Topics. Examples include:

• Topic 2, Lesson 2-5, Solve & Share, students model with mathematics as they compare groups of objects, using models to show how they know those which are greater in number, less in number, and equal in number. “Directions Say: Work with your partner and take turns. Take 1 cube at a time from the bag and place it on your mat. Keep taking cubes until all the cubes are gone. Do you have a greater number of red cubes or blue cubes? How can you show your answer? Explain and show your work.”

• Topic 6, Lesson 6-6, Independent Practice, Practice 9, students model with mathematics as they represent real-world word problems involving addition and subtraction in different ways and add and subtract within 10. “Directions Higher Order Thinking Have students listen to the story, circle the connecting cubes that show the story and tell why the other cubes do not show the story, and then write the number to tell how many in all. Say: Jimmy pics 5 raspberries. Then he picks 3 more. How many raspberries does he have in all?” A picture of two sets of connecting cubes is shown: the first set has five red cubes and three purple cubes; the second set has four red cubes and three purple cubes.

• Topic 12, Lesson 12-5, Independent Practice, Problem 12, students model with mathematics as they use knowledge about specified shapes to draw objects to match the given shapes.   “Directions Higher Order Thinking Have students name the solid figure on the left, and they draw 2 more objects that look like that shape.” The materials show a sphere.

MP5 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students use appropriate tools strategically as they work with the support of the teacher and independently throughout the Topics. Examples include:

• Topic 3, Lesson 3-4, Solve & Share, students use appropriate tools strategically as they use counters and draw pictures to make a group of 8 or 9 in different ways. “Directions Say: Jackson sees some turtle eggs. Draw a number card to tell how many. Count out that many counters and place them across the top of the workmat. What are some different ways to make the number? Draw two ways on the turtle shells. Are there different ways to count the number? Tell how you know.” The materials show a picture of two large turtles.

• Topic 9, Lesson 9-4, Convince Me!, students use appropriate tools strategically as they use counters and a double ten-frame to represent and visualize numbers 18, 19, and 20. Teacher guidance: “Show a double ten-frame with 18 counters on it. Which number card shows how many? Have students hold up the number card for 18 and say the number. Repeat, placing additional counters in the double ten-frame for the numbers 19 and 20. Have students explain how the double ten-frame and counters help to show each number.”

• Topic 14, Lesson 14-5, Visual Learning Bridge, students use appropriate tools strategically as they identify tools that can be used to tell about measurable attributes of objects. Teacher guidance: “Essential Question Ask How can you describe and compare attributes of objects?” In (B) “Use Appropriate Tools StrategicallyMarta is thinking about three tools she can use to describe attributes of the vases. Which tool would Marta use to tell about their heights? Their weights? How much do they hold? ” The materials show a girl with thought clouds: measuring cup, scale, and counting cubes.

The materials reviewed for enVision Mathematics Kindergarten meet expectations for supporting the 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. 

Students have opportunities to engage with MP6 across the year, and they are explicitly identified for teachers within the Program Overview within the Topic Contents at the Lesson-level. The Math Practices and Problem Solving Handbook introduces each of the Math Practices with specific emphasis on making connections among representations to develop meaning and corresponding Thinking Habits. Topic Planners include the Math Practices at the lesson level; relevant practices are specified in Lesson Overviews.

Students attend to precision in mathematics in connection to grade-level content as they work with the support of the teacher and independently throughout the Topics. Examples include:

• Topic 3, Lesson 3-2, Guided Practice, Problem 3, students attend to precision by counting objects and showing how to use a specific symbol to represent the total quantities.   identifying and writing the numbers 6. “Directions Have students count the objects, and then practice writing the number that tells how many.” The materials show 6 flip flops and space for a student to write the number 6 three times. 

• Topic 8, Lesson 8-2, Guided Practice, Problem 3, students attend to precision by writing equations that distinguish between the appropriate use of a plus sign and a minus sign. “Directions Have students use cubes for these facts with 4. Have them decide whether the cubes show addition or subtraction. Encourage students to make up their own stories to match the cubes. Then have them write equations to tell the related facts.” Teacher guidance: “Be Precise … First, have students tell a take away story. Will you use a plus or minus sign for a take away story. Then have them tell an add to story. Will you use a plus or minus sign for an add to story? ” The materials show (i) two orange cubes adjoined to two blue cubes and (ii) two orange cubes separate from two blue cubes. Students write the equations 4 - 2 = 2 and 2 + 2 = 4.

• Topic 13, Lesson 13-1, Solve & Share, students attend to precision by identifying examples of shapes and nonexamples of shapes, based on given clues. “Directions Say: “Emily wants to figure out which shapes are behind the door. The mystery shapes that are behind the door have only 4 vertices (corners). Use the shapes shown above the door to help you decide which shapes are behind the door. Draw the shapes that match the clue on the door. How many shapes did you draw? Write that number next to the door. Now mark an X on the shapes that are NOT behind the door. Count those shapes and write the number. Look at the two numbers you wrote. Circle the number that is greater than the other number. If the numbers are the same, circle both numbers. Name the shapes that are behind the door.” The materials show the following shapes above the door: circle, equilateral triangle, square, rectangle, isosceles right triangle, isosceles trapezoid, regular hexagon, and regular octagon. 

Students attend to the specialized language of mathematics in connection to grade-level content as they work with the support of the teacher and independently throughout the Topics. Examples include:

• Topic 2, Lesson 2-4, Guided Practice, Problem 4, students use specialized language when they count to compare two groups of objects. “Directions Have students count the stickers, write the numbers to tell how many, and then draw a circle around the number that is greater than the other number and mark an X on the number that is less than the other number, or draw a circle around the numbers if they are equal.” The materials show four beers and four caves. Teacher guidance: “Remind students that they can use the totals they have counted for the two groups to compare how many are in each group. If the numbers are the same, then the number of objects in the two groups is equal.”

• Topic 5, Lesson 5-3, Independent Practice, Problem 4, students use specialized language when they count items in two groups and determine which group has the greater number of objects. “Directions Have Students: sort the balls into balls that are yellow and balls that are NOT yellow, count them and then write numbers in the chart to tell how many. Then have students draw a circle around the category that is greater in number than the other category and tell how they know.” The materials show two soccer balls, two baseballs, four basketballs, and five tennis balls. The chart asks students to quantify tennis balls and not tennis balls.

• Topic 14, Lesson 14-4, Visual Learning Bridge, students use specialized language when they identify attributes to describe an object. Visual Learning Bridge,“Essential Question Ask “What attributes can you use to describe objects?” Teacher guidance: (A) “What words can you use to describe this water bottle? Which attributes do these words describe? What tools do you see?”

The materials reviewed for enVision Mathematics 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 the grade-level content standards, as expected by the mathematical practice standards. 

Students have opportunities to engage with the Math Practices across the year, and they are explicitly identified for teachers within the Program Overview within the Topic Contents at the Lesson-level. The Math Practices and Problem Solving Handbook introduces each of the Math Practices with specific emphasis on making connections among representations to develop meaning as well as corresponding Thinking Habits. The Teacher’s Edition provides support for developing, connecting, and assessing each math practice. Topic Planners include the Math Practices at the lesson level; relevant practices are specified in Lesson Overviews. 

MP7 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students model with mathematics as they work with the support of the teacher and independently throughout the Topics. Examples include:

• Topic 1, Lesson 1-9, Guided Practice, Problem 1, students look for and make use of the structure as they work within the order of numbers 0-5, using objects to help them see the pattern that each number represents 1 more each time. “Directions Have students write the number that comes just before 1 and the number that comes just after 1. Then have them write the number that comes just before 4 when counting, and the number that comes just after 4 when counting. Have them say the numbers in order from 0 to 5.” The materials show ___ 1 ___ ___ 4 ___ and cube(s) above the spaces that correspond with the number.

• Topic 9, Lesson 9-4, Independent Practice, Problem 7, students look for and make use of structure as they look at the arrangement of a group to help find out how many are in the group. “Directions Have students count the stickers in each group, and then practice writing the number that tells how many.” The materials show two rows of ten dog’s faces each.

• Topic 13, Lesson 13-2, Solve & Share, students look for and make use of structure as they analyze 3-D shapes to determine if they will fit a given criteria. “Directions Say: Jackson wants to find a solid figure. The solid figure has more than one flat side and it rolls. Color the solid figures that match the description. Then count them. How many are there? How many shapes do you see in all?” The materials show twelve 3-D shapes: spheres, cones, cylinders, and prisms.

MP8 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students model with mathematics as they work with the support of the teacher and independently throughout the Topics. Examples include:

• Topic 4, Lesson 4-5, Independent Practice, Problem 2, students look for and express regularity in repeated reasoning when they find 1 more by counting on to find the total number rather than beginning counting from 1 each time, noticing how this is a general method that can be applied to different numbers in the same way. “Directions Say: Alex sees frogs at the pond. Then he sees 1 more. How many frogs are there now? Have students use repeated reasoning to find the number that is 1 greater than the number of frogs shown. Draw counters to show the answer, and then write the number. Have students explain their reasoning.” The materials show seven frogs. 

• Topic 6, Lesson 6-3, Guided Practice, Problems 4-6, students look for and express regularity in repeated reasoning when they notice that they can approach problems with similar steps even though quantities and objects may be different. “Directions Have students use counters to model putting together the groups, draw a circle around the groups to put them together, and then write an addition sentence to tell how many in all. The materials show for Problem 4, six radishes and one cabbage, Problem 5, eight cabbages and one radish, and for Problem 6, two radishes and eight carrots. Students complete sentences such as “___ and ___ is ___.”

• Topic 11, Lesson 11-4, Guided Practice, Problem 3, students look for and express regularity in repeated reasoning as they use a hundred chart to count forward by ones from a given number. “Directions Have students color the boxes of the numbers as they count aloud, starting at the yellow box and ending at the red box.” The materials show a hundred chart: its yellow box is at 34 and the red box is at 55.