The materials reviewed for enVision Mathematics Grade 1 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 1, Lesson 1-2, Lesson Overview, Conceptual Understanding states, “Students develop one meaning of addition, put together, as they solve addition word problems.” In the Visual Learning Bridge, the materials show a sequence of four frames: A) establishes the question, “4 red fish and 2 blue fish How many fish in all?” and shows an aquarium holding 4 red fish and an aquarium holding 2 blue fish. B) establishes that there are 2 parts: “The parts are 4 and 2” and shows a mat with four red cubes and two blue cubes; the student writes the numbers 4 and 2. C) reasons, “Add the parts to find the whole,” and the student writes 4 + 2. An image of a girl says, “The whole is also the sum.” D) prompts, “Write an addition equation”; the student writes 4 + 2 = 6. The girl says, “There are 6 fish in all.” Classroom Conversation asks students the following questions:  “A) How many red fish are there? How many blue fish are there? What are you asked to find? B) What do the red cubes show from the problem? What do the blue cubes show from the problem? What is another word for a part of something? C) Reasoning When you add the two parts together, what do you find? What number tells how many are in the red part of the whole? The blue part? D) What number tells how many cubes there are in all? How many fish are there in all?” Students develop conceptual understanding as they answer questions about putting two parts together with numbers within 20. (1.OA.1)

• Topic 8, Lesson 8-4, Lesson Overview, Conceptual Understanding states, “Using models to compose numbers establishes a foundation for place-value concepts with three- and four-digit numbers as well as for addition with greater numbers.” In Guided Practice, Problem 2, “Use cubes. Count the tens and ones. Then write the numbers.” The materials show four groups of tens and one leftover cube on the place-value mat with the sentence frame: tens and one is __.  Students develop conceptual understanding as they represent the two digits of a two-digit number as the amounts of tens and ones. (1.NBT.2)

• Topic 12, Lesson 12-3, Lesson Overview, Conceptual Understanding states, “Students deepen their understanding of measurement as they use identical units such as cubes or paper clips to measure objects and to express their length.” In Guided Practice, Problem 2, “Use cubes to measure the length.” The materials show a picture of a robot and the cube to use. Students develop conceptual understanding as they use multiple cubes to express the length of an object. (1.MD.2) 

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

• Topic 4, Lesson 4-4, Lesson Overview, Conceptual Understanding states, “A key goal of this topic is understanding the relationship between addition and subtraction. This lesson promotes this understanding and helps students begin to develop fluency with mathematical facts.” In Independent Practice, Problem 6, Number Sense, “Are the following equations a fact family? Explain your answer.” The materials show four equations: 9 + 5 = 14, 15 - 5 = 10, 4 + 4 = 8, 15 = 6 + 9 and the image of a little boy saying, “What is the whole? What are the parts?” Students independently demonstrate conceptual understanding by using the understanding of subtraction as an unknown-added problem to determine if equations are part of a fact family. (1.OA.4)

• Topic 10, Lesson 10-7, Lesson Overview, Conceptual Understanding states, “Students continue to develop an understanding of the place-value concepts behind formal addition algorithms and strengthen their understanding of our number system.” In Convince Me!, students determine if they need to make a ten to solve an addition problem. “Do you need to make a 10 to add 23 + 15? How do you know? ” Students independently demonstrate conceptual understanding by when it is necessary to compose a ten when adding two-digit numbers.  (1.NBT.4) 

• Topic 14, Lesson 14-3, Lesson Overview, Conceptual Understanding states, “Students use their conceptual understanding of attributes, such as the number of straight sides and vertices, to build or draw two-dimensional shapes, such as triangles, rectangles, squares, and hexagons.” In Independent Practice, Problems 2 and 3, students use materials supplied by their teacher to make a circle and a rectangle and explain how they know the shape is correct. In Problem 2, students make a circle and in Problem 3, students make a rectangle.  Students independently demonstrate conceptual understanding by building shapes based on their defining attributes and explaining those attributes. (1.G.1)

The materials reviewed for enVision Mathematics Grade 1 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 2, Lesson 2-1, Lesson Overview, Fluency states, “As students use the strategy of counting on to solve problems involving addition, they begin to develop fluency for addition facts within 10.” In Guided Practice, Problem 1, the materials show a bucket labeled with the number “3” and two carrots. Students develop procedural skills and fluency by counting on to find a sum. (1.OA.5)

• Topic 5, Lesson 5-5, Lesson Overview, Procedural Skill states, “By using these properties [associative property of addition], students develop procedural skills to record partial sums.” In Solve & Share, the materials show one of three children saying, “I have 6 oranges, Alex has 2 pears, and Jada has 4 apples. How many pieces of fruit do we have in all? Write 2 different addition equations to solve the problem.” Students develop procedural skills and fluency as they apply different strategies to solve a word problem with three addends. (1.OA.2)

• Topic 12, Lesson 12-1, Lesson Overview, Procedural Skill states, “Students align objects to visualize their relative lengths in order to make comparison statements. They order the objects from longest to shortest, and vice versa." In Solve & Share, the materials show a paintbrush, a marker, and a pencil. “Can you put these objects in order from longest to shortest? How can you tell if one object is longer than another object?" Students develop procedural skills and fluency by ordering three objects by length and by comparing the lengths of two objects indirectly using a third object. (1.MD.1)

Materials provide opportunities for students to independently demonstrate procedural skill 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 2, Lesson 2-4, Lesson Overview, Procedural Skill states, “Students develop procedural skills as they use facts with 5 to create a fact with a sum of 10” and  Fluency states, “As students use this model for facts with 5 as an addend, they continue to develop their fluency with addition facts to 10.” In Independent Practice, Problem 5, students independently demonstrate procedural skills and fluency when they look at a ten-frame, write an addition fact with 5, and then write an addition fact for 10. The materials show a ten-frame that consists of 9 counters. The materials show: 5 + ___ =  ___ and  ___ +  ___ = 10 .  (1.OA.6)

• Topic 7, Lesson 7-6, Lesson Overview, Procedural Skill states, “As students count the number of objects in groups, they combine their ability to count by 10s with their ability to count by 1s.” In Independent Practice, Problem 6, students independently demonstrate procedural skills and fluency when they recognize that a ten-rod is made up of 10 ones and can be counted by 10s and then count units by ones to find how many in all. Directions: “Use place-value blocks to count the tens and ones. Then write how many in all.” The materials show eight ten-rods and 3 unit cubes.  Students fill in the blanks, “___ tens ___ ones ___ in all.” (1.NBT.1)

• Topic 11, Lesson 11-3, Lesson Overview, Procedural Skill states, “students learn that they can start with a number and then move hops to the left to subtract tens. Each hop means 1 ten.” In Problem Solving, Problem 8, Higher Order Thinking, students independently demonstrate procedural skills and fluency as they write an equation to represent “hops” on a number line. The materials show a number line bearing the labels 20, 30, 40, 50; a series of three 10-unit hops begins at 50 and ends at 20. Students fill in the blanks to complete the equation: “___ - ___ = ___.” (1.NBT.6)

The materials for enVision Mathematics Grade 1 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 1, Lesson 1-9, Solve & Share, students construct an argument to explain how they use addition or subtraction within 20 to solve word problems. Directions: “Do you add or subtract to solve the problem? Tell why. Show how to solve. Use pictures, numbers, or words.” The materials prompt, “7 rabbits 3 turtles How many more rabbits than turtles?” (1.OA.1)

• In Topic 10, Lesson 10-8, Problem Solving, Problem 7, students independently solve a routine word problem by adding within 100 involving a two-digit number and a multiple of 10. “Reasoning Lilly makes necklaces. She has 43 blue beads. She has 20 pink beads. How many beads does Lilly have in all? ___ beads.” (1.NBT.4)

• In Topic 12, Lesson 12-1, Problem Solving, Problem 8, students independently order three objects by length. “Higher Order Thinking, Draw 3 lines with different lengths in order from longest to shortest. Label the longest and shortest lines.” (1.MD.1)

Examples of non-routine applications of the math include:

• In Topic 3, Lesson 3-8, Problem Solving, Problem 6, students use addition and subtraction within 20 to independently solve a non-routine word problem. “Model Leland cuts out 12 flowers. How many can he color red and how many can he color yellow? Draw a picture and write an equation to help solve the problem.” Students fill in blanks, “___ red flowers ___ yellow flowers” and write an equation such as “12 = 8 + 4.” (1.OA.A) 

• In Topic 5, Topic Performance Task, Problem 5, students understand the meaning of the equal sign and determine if an equation involving subtraction is true or false. “Terry says that if there were 2 fewer lilies, then the number of lilies would be equal to the number of daisies. He writes the equation below. Is this equation true or false? Explain how you know. 8 - 2 = 5” (1.OA.1 and 1.OA.7)

• In Topic 11, Lesson 11-6, Problem Solving, Problem 14, students independently respond to a non-routine problem that involves subtracting multiples of 10 in the range 10–90 from multiples of 10 in the range 10–90 using the relationship between addition and subtraction. “Higher Order Thinking Write a subtraction problem for which you would think addition to subtract. Explain why this would be a good strategy to use to solve this problem.” (1.NBT.6)

The materials for enVision Mathematics Grade 1 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 2-5, Independent Practice, Problem 3, students attend to procedural skills and fluency as they apply the commutative property of addition to add within 10. “Write the sum. Then change the order of the addends. Write the new addition equation.” The materials show the equation 2 + 3 = _____  and _____ + _____ = ______. (1.OA.3)

• Topic 7, Lesson 7-1, Solve & Share, students attend to conceptual understanding as they understand that the two digits of a two-digit number represent tens and ones, relating a number such as 40 as four tens and 0 ones. “Alex put counters in some ten-frames. How can you find out how many counters there are without counting each one? Write the number.” The materials show ten ten-frames. Students complete the sentence, “___ counters in all.” (1.NBT.2c)

• Topic 15, Lesson 15-4, Guided Practice, Problem 1, students attend to application as they use their understanding of fourths to partition a rectangle and describe the results as equal shares. “Draw a picture to solve the problem. Then complete the sentence. 1. Pete makes a purple and yellow flag. The flag is divided into fourths. 2 shares are yellow. The rest of the flag is purple. How many of the shares are purple?” Students complete the sentence, “___ out of ___ equal shares are purple.” (1.G.3)

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

• Topic 2, Lesson 2-8, Problem Solving, Problem 5, students attend to application and procedural skills and fluency as they use addition within 20 to solve word problems, demonstrating fluency for addition and subtraction within 10. “Solve each problem. Draw pictures and write equations to help. 5. Make Sense Charlie draws 9 stars. Joey draws 4 stars. How many fewer stars did Joey draw than Charlie.” Students complete the statement “___ fewer stars” and write an equation. (1.OA.1 and 1.OA.6)

• Topic 5, Lesson 5-7, Guided Practice, Problem 2, students attend to conceptual understanding and procedural skills and fluency as they understand the meaning of the equal sign and determine if an equation involving addition is true or false. “Write the symbol (+, -, or =) or number to make the equation true. Then tell how you know you found the correct symbol or number.” Students fill in the blank within the equation “4 + 3 + ___ = 13” and explain how they found their answer. (1.OA.7)

• Topic 11, Lesson 11-7, Problem Solving, Performance Task, Problems 6-8, students attend to application and conceptual understanding as students subtract multiples of 10 from multiples of 10 within the range 10-90 using concrete models or drawings and strategies. “Dog Walking James, Emily, and Simon walk dogs after school. On Monday, they have 40 dogs to walk. James and Emily take 20 of the dogs for a walk. How many dogs are left for Simon to walk? 6. Make Sense What problem do you need to solve? 7. Use Tools What tool or tools can you use to solve this problem? 8. Model Write an equation to show the problem. Then use pictures, words, or symbols to solve.” The materials show a girl walking three dogs. (1.NBT.6)

The materials reviewed for enVision Mathematics Grade 1 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 5, Lesson 5-6, Convince Me!, students make sense of problems and persevere in solving them as they make sense of comparison word problems and persevere to find unknown smaller amounts within 10. “Tom made 8 fewer sandcastles than Tina. Tina made 10 sandcastles. How many sandcastles did Tom make?” 

• Topic 7, Lesson 7-7, Problem Solving, Performance Task, Problem 7, students make sense of problems and persevere in solving them as they work to find out the total number of students. “Students and Snowmen 62 students stay inside at recess. The rest each build a snowman outside. How can you count to find the number of students in all? Make Sense What do you know about the students? What do you need to find?” 

• Topic 13, Lesson 13-1, Solve & Share, students make sense of problems and persevere in solving them as they solve word problems involving splitting coins in a fair way. “Jennifer has 8 coins. She wants to share them with her friend. She says, ‘We each get 4 coins like this.’ Do you think this is a fair way to share the coins? Explain.” The materials show four nickels and four dimes in two separate groups. 

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 1, Lesson 1-2, Solve & Share, students reason abstractly and quantitatively as they solve word problems about putting parts together. “4 red apples and 4 green apples. How many apples in all? Show how you solve. Use cubes to help.” The materials show an empty red plate and an empty green plate. Students complete the sentence, “___ apples in all.” 

• Topic 8, Lesson 8-6, Independent Practice, Problem 5, students reason abstractly and quantitatively as they show and explain how two representations of a two-digit number can show the same quantity. Write each number in two different ways. Use cubes to help if needed. Show two ways to break apart 25.” Students fill in the blanks, “25 is ___ tens and ___ ones. 25 is ___ tens and ___ ones.” 

• Topic 11, Lesson 11-4, Solve & Share, students reason abstractly and quantitatively as they write both an addition and subtraction equation to help solve a problem. “Mia has 70 stickers. Jack has 30 stickers. How many more stickers does Mia have than Jack has?”The materials show a boy who states, “Can you write both an addition equation and a subtraction equation to help solve the problem?” Students fill in the blanks  ___ + ___ = ___  and ___ - ___ = ___ and complete the sentence “Mia has ___ more stickers.”

The materials reviewed for enVision Mathematics Grade 1 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 1-9, Problem Solving, Problem 5, students construct viable arguments and critique the reasoning of others as they discuss someone's work and explain if it is correct. “Explain Alex sells 3 cups. Mark sells 5 cups. How many cups do they sell in all? Here is Alex’s work. Is his work correct? Tell why.” Alex’s work consists of a drawing of eight red cups, and the equation 3 + 5 = 8 cups. 

• Topic 8, Lesson 8-6, Problem Solving, Problem 6, students construct viable arguments and critique the reasoning of others as they identify and discuss a student error. “Explain Nate says 5 tens and 3 ones shows the same number as 3 tens and 13 ones. Do you agree? Explain.” The materials suggest to teachers, “Critique Reasoning Discuss why Nate is wrong without using cubes. One way to show a number is 5 tens and 3 ones. What number is this? The other way Nates says to show the number is 3 tens and 13 ones. How many tens are in 13 ones? So, could you make 5 tens and 3 ones with 3 tens and 13 ones?”

• Topic 10, Lesson 10-5, Convince Me!, students construct viable arguments and critique the reasoning of others as they apply a strategy to count on and consider other approaches. “Could you count on by 10s to add 21 + 20?” The materials suggest to teachers, “Construct Arguments Remind students that they have used a hundred chart to count on by 10s and ask a volunteer to show how they would solve this problem using a hundred chart. Then ask another volunteer to solve this problem by using place-value blocks to count on by 10s.”  Students explain how the blocks help them add tens to tens and ones to ones.

• Topic 14, Lesson 14-2, Solve & Share, students construct viable arguments and critique the reasoning of others as they use attributes to describe shapes and compare their explanations to those of other students. “Tell how the 5 shapes are alike. Tell how the 5 shapes are different. Use a tool to help.” The materials show a red square, orange rectangle, blue parallelogram, purple rhombus, and green trapezoid. 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 Juan’s Work and the other is labeled Erin’s Work. The following questions are asked: “Choose one the ways Juan describes the shapes. Do you agree? Explain. Erin says the shapes are different and cannot be alike. Do you agree? Explain.”

The materials reviewed for enVision Mathematics Grade 1 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 MP4 and MP5 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 3, Lesson 3-2, Visual Learning Bridge and Guided Practice, Problem 2, students model with mathematics as they use a specific tool, an open number line, to create a model of addition problems. Teacher guidance: “Essential Question Ask How can you use an open number line to count on to add?” In (C) “Model with Math How is this number line different from the previous one? Why does this number line show that you start at 7, add 3, and then add 3 again? Do you get the same answer as you did by starting at 7 and counting on 6? Explain.” The materials include the text, “You can also break apart the 6. Adding 3 and 3 is one way to add 6 more.” It also shows a number line with jumps from 7 to 10 to 13 and the equation 7 + 3 + 3 = 13. Guided Practice, Problem 2, students use a number line to model the steps in the equation and solve. “Use the open number line to solve. Show your work. 6 + 2 =__” 

• Topic 8, Lesson 8-1, Solve & Share, students model with mathematics as they model numbers 11-19 with counters and ten-frames. “Use counters and ten-frames to show 12, then 15, and then 18. Draw your counters in the ten-frames below. Tell what is the same and different about each number you show.” The materials show the indicated numbers with two ten-frames, of different colors, below the number.

• Topic 13, Lesson 13-6, Problem Solving, Performance Task, Problem 7, students model with mathematics as they draw a minute hand to show when the family arrives at an activity. “Visiting the City Andrew’s family takes a day trip to the city. Help him solve the problems below using the Family Schedule. 7. Model The minute hand fell off this clock. What time should the clock show when Andrew’s family arrives at the Aquarium? Draw the minute hand and write the time shown.” The materials include a table that lists the time of day and corresponding activity. The clock includes the hour hand. 

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 1, Lesson 1-4, Visual Learning Bridge, students use appropriate tools strategically as they discuss different tools that they can use to solve problems involving taking form and how they can use tools to show the action of taking away. Teacher guidance: “Essential Question Ask How can you use a subtraction equation to show a situation in which one part is taken from the whole?” In (B) “Use Appropriate Tools Strategically How do the cubes show the story? What do you need to do to find how many ducks are still in the pond?” The materials show “Use cubes.” Within a frame are seven counting cubes; the number 7 appears above. An image of a girl states, “7 is the whole.”

• Topic 5, Lesson 5-1, Visual Learning Bridge, students use appropriate tools strategically as they use counters to solve for the unknown part in subtraction problems. Teacher guidance: “Essential Question Ask How can you use models or the relationship between addition and subtraction to solve equations with an unknown part?”  In (B) “Use Appropriate Tools Strategically How could you counters to solve this problem?” What is the missing number in the equation? So what is the equation with the missing number filled in? What is the missing number in the equation? So what is the equation with the missing number filled in?” The materials state, “You can use counters to find the missing number.” Shown are three circles and nine circles with an X through them as well as the equation 12 - ___ = 3.

• Topic 14, Lesson 14-2, Problem Solving, Problem 5, students use appropriate tools strategically as they discuss the different tools available to differentiate length. “5. Use Tools Do all rectangles have equal sides? Circle Yes or No. Choose a tool to show how you know.” Teacher guidance: “Discuss the different tools available to students, including grid paper, ones cubes, straws, and counters. Which tool can help you show not all rectangles have equal sides? How can you show that the sides are not equal?”

The materials reviewed for enVision Mathematics Grade 1 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 2, Lesson 2-8, Convince Me!, students attend to precision by writing an equation to solve a word problem. “7 cubes are on a table. Some cubes fall off. Now 3 cubes are on the table. How many cubes fell off the table?” Teacher guidance: “Be Precise Have a volunteer draw a picture to solve the problem and explain how he or she used the drawing to find the solution. Have a second volunteer write an equation for the problem, identify the known whole and the part, and then describe how he or she can use the equation to find the solution.”

• Topic 7, Lesson 7-2, Visual Learning Bridge, students attend to precision as they show how numbers change when one counts by 1s, especially numbers when numbers change from two-digit to three-digit numbers. “Essential Question Ask How is counting forward from 100 to 120 like counting forward to a two-digit number? How is it different?” The materials show a hundred block, labeled, “100,” and including the text, “This block shows 100. You say one hundred for this number.” Teacher guidance: (A) “Be Precise How is this number different from numbers such as 97, 98, and 99? If we counted all of the little green blocks in this larger block, how many little green blocks do you think there would be?”

• Topic 13, Lesson 13-3, Independent Practice, Problem 10, students attend to precision as they differentiate between the hour and minute hands on analog clocks when telling time by the hour. “Draw the hour and minute hands to show the time. 10. 5 o’clock” The materials show an analog clock, without hands. Teacher guidance: “Be Precise Remind students that the minute hand should be pointing to 12 when showing time on the hour.”

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 5, Lesson 5-7, Problem Solving, Performance Task, Problem 9, students use specialized language when they use math symbols (+, -, or =) correctly and explain their reasoning about problems clearly. “Balloon Party Dani has 7 green and 4 yellow balloons. Gene has 15 blue balloons. 9. Be Precise If Gene keeps all 15 blue balloons, how many balloons would Dani need to buy to have the same number as Gene? Complete the equation to find the answer. 7 ___ 4 ___ ___ ___ 15  Did you use numbers and symbols correctly? Explain how you know.” The materials show seven green balloons (arranged in a row of four and a row of three), four yellow balloons (arranged in one row of four), and fifteen blue balloons (arranged in three rows of five each).

• Topic 9, Lesson 9-3, Problem Solving, Problem 13, students use specialized language as they explain the process of comparing two-digit numbers. “Assessment Practice Ann has 46 shells. Ben has 43 shells. Compare the numbers. Circle is greater than or is less than.” Teacher guidance: “How do you compare 46 and 43 to find out which number is greater and which number is less than? Who has the greater number of shells, Ann or Ben? Explain.”

• Topic 12, Lesson 12-1, Problem Solving, Problem 6, students use specialized language as they make accurate comparisons of the lengths of objects.“Be Precise Tomaz paints a line that is longer than the blue line. What color line did he paint? Use the picture to solve.” The materials show blue, yellow, and red lines of various lengths. Teacher guidance: “Be Precise In thai problem, students must identify the line that is longer than the blue line. Which line is shorter than the blue line? So, which line is longer than the blue line?”

The materials reviewed for enVision Mathematics Grade 1 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 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.

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 2, Lesson 2-4, Convince Me!, students use and look for structure when they explain how to use a ten-frame to add an addition problem. “How does a ten-frame help you add 5 + 4?” Teacher guidance: “Use Structure Ask students to show you a number from 6 through 10 on a ten-frame. Then have them identify an addition fact with 5 that adds the number in the top row with the number in the bottom row of the ten-frame.” 

• Topic 8, Lesson 8-7, Guided Practice, Problem 2, students use and look for structure when they use place-value knowledge and structure to list all the ways to make a two-digit number with ones and tens. “Make a list to solve. You can use cubes to help you. Talk to a partner about patterns you see in your list. Andy wants to show 31 as tens and ones. What are all the ways?” The materials show a table with a Tens column and an Ones column. Students use structure to write combinations of tens and ones that add to 31.

• Topic 12, Lesson 12-2, Solve & Share, students use and look for structure when they apply a process to compare the length of two objects that are not lined up next to each other. “How can you find out whether the shoe or the pencil is longer without putting them next to each other? What can you use? Circle the longer object and explain how you found out?” The materials show an image of a shoe and a pencil. Students measure the objects using a third object and compare the length of that structure to determine which of the first two objects is longer.

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 5, Lesson 5-5, Guided Practice, Problem 2, students use regularity in repeated reasoning as they discover and apply different ways to approach solving a three-addend word problem. “Write an equation to solve each problem. Choose a way to group the addends. Tom sees some birds. He sees 4 red birds, 2 blue birds, and 6 black birds. How many birds does Tom see in all?” Students complete the equation “___ + ___ + ___ = ___.”

• Topic 10, Lesson 10-1, Convince Me!, students use regularity in repeated reasoning by adding groups of ten similarly as they added numbers less than ten. “How is adding 6 + 3 like adding 60 + 30?” Teacher guidance: “Have students use a scrap piece of paper to cover the ones digits in 60 and 30 and find the sum of 6 + 3. Then have them uncover the ones digits, find the sum, and compare.”

• Topic 14, Lesson 14-7, Independent Practice, Problem 2, students use regularity in repeated reasoning when they contrast attributes that define a three-dimensional shape with ones that do not, such as color, size, and position. “Circle the words that are true for each shape. All cubes: have 12 edges. have 8 vertices. cannot roll. are blue.” The materials show a picture of a blue cube.

The materials reviewed for enVision Mathematics Grade 1 meet expectations for providing teacher guidance with useful annotations and suggestions for how to enact the student materials and ancillary materials, with specific attention to engaging students in order to guide their mathematical development. 

The Teacher’s Edition Program Overview provides comprehensive guidance to assist teachers in presenting the student and ancillary materials. It contains four major components: Overview of enVision Mathematics, User’s Guide, Correlation and Content Guide.

• The Overview provides the table of contents for the course as well as a pacing guide for a traditional year long course as well as block/half year course. The authors provide the Program Goal and Organization, in addition to information about their attention to Focus, Coherence, Rigor, the Math Practices, and Assessment.

• The User’s Guide introduces the components of the program and then proceeds to illustrate how to use a ‘lesson’: Lesson Overview, Problem-Based Learning, Visual Learning, and Assess and Differentiate. In this section, there is additional information that addresses more specific areas such as STEM, Building Mathematical Literacy, Routines, and Supporting English Language Learners.

• The Correlation provides the correlation for the grade.

• The Content Guide portion directs teachers to resources such as the Big Ideas in Mathematics, Scope and Sequence, Glossary, and Index.

Within the Teacher’s Edition, each Lesson is presented in a consistent format that opens with a  Lesson Overview, followed by probing questions to provide multiple entry points to the content, error intervention, supports for English Language Learners and ends with multiple Response to Intervention (RtI) differentiated instruction.

Materials include sufficient and useful annotations and suggestions that are presented within the context of the specific learning objectives. The Teacher’s Edition includes numerous brief annotations and suggestions at the topic and lesson level organized around multiple mathematics education strategies and initiatives, including the CCSSM Shifts in Instructional Practice (i.e., focus, coherence, rigor), CCSSM practices, STEM projects, and 3-ACT Math Tasks, and Problem-Based Learning. Examples of these annotations and suggestions from the Teacher’s Edition include:

• Topic 1, Lesson 1-1, Visual Learning Bridge, Essential Question, “Ask How can you use an addition equation to solve a problem about adding to one part?” Teachers begin the Classroom Conversation by saying the following: “How many cats are there at the beginning? [5] How many cats join? [2] What are you asked to find? [How many cats now?]” 

• Topic 5, Lesson 5-5, Independent Practice, Problem 5, “Write the missing numbers for each problem. 16 = 7 + ___ + 6.” Teacher guidance: “Algebra Have students use connecting cubes to represent the two addends. Then have them add on cubes of a different color until they get to the sum to find the missing addend.”

• Topic 11, Lesson 11-4, Problem Solving, Problem 7, “Write an equation and solve the problems below. Reasoning Mr. Andrews collects 90 papers from his students. He has already graded 40 papers. How many papers does Mr. Andrews have left to grade?” Teacher guidance: “Reasoning If students write an addition (subtraction) equation to solve the problem, have them write a subtraction (addition) equation to check that their answers make sense. Discuss with students how the problems could be solved by using either addition or subtraction.”

The materials reviewed for enVision Mathematics Grade 1 meet expectations for containing adult-level explanations and examples of the more complex grade concepts and concepts beyond the current course so that teachers can improve their own knowledge of the subject. 

The materials provide professional development videos at two levels to help teachers improve their knowledge of the grade they are teaching.

• “Professional development topic videos are at SavvasRealize.com. In these Topic Overview Videos, an author highlights and gives helpful perspectives on important mathematics concepts and skills in the topic. The video is a quick, focused ‘Watch me first’ experience as you start your planning for the topic.

• Professional development lesson videos are at SavvasRealize.com. These Listen and Look for Lesson Videos provide important information about the lesson.

An example of the content of a Professional development video:

• Topic 7: Professional Development (topic) Video, “Numbers can be used for different purposes and numbers can be used in different ways…. Our numeration system, the Hindu Arabic system, has the following attributes:

1. It uses digits 0-9.

2. It uses groups of ten.

3. Position tells the value of a digit.

4. Each position to the left is ten times more than the one to its right.

Connecting these concepts by making groups of ten is an important step for students. ...  Time spent helping students recognize patterns on the number chart, not only helps develop a deeper understanding of remuneration but lays a foundation for adding and subtraction two-digit numbers.”

The Math Background: Coherence, Look Ahead section, provides adult-level explanations and examples of concepts beyond the current grade as it relates what students are learning currently to future learning.

An example of how the materials support teachers to develop their own knowledge beyond the current grade:

• Topic 15, Math Background: Coherence, Look Ahead, the materials state, “Grade 2 Partition Shapes In Topic 13, students will partition circles and rectangles into two, three, or four equal shares and describe the shares using the words halves, thirds, fourths, half of, a third of, etc. They will also describe the whole as two halves, three thirds, and four fourths. Grade 3 Fractions as Numbers In Topic 12, students will partition shapes into parts with equal areas. They will express the area of each part as a unit fraction of the whole.” An example is shown of a shape divided into sixth.

The materials reviewed for enVision Mathematics Grade 1 meet expectations for including standards correlation information that explains the role of the standards in the context of the overall series. 

Standards correlation information is indicated in the Teacher’s Edition Program Overview, the Topic Planner, the Lesson Overview, and throughout each lesson. Examples include:

• The Teacher’s Edition Program Overview, Grade 1 Correlation to Standards For Mathematical Content organizes standards by their Domain and  Major Cluster and indicates those lessons and activities within the Student’s Edition and Teacher’s Edition that align with the standard. Lessons and activities with the most in-depth coverage of a standard are distinguished by boldface. The Correlation document also includes the Mathematical Practices. Although the application of the mathematical practices can be found throughout the program, the document indicates examples of lessons and activities within the Student’s Edition and Teacher’s Edition that align with each math practice.

• The Teacher’s Edition Program Overview, Scope & Sequence organizes standards by their Domain, Major Cluster, and specific component. The document indicates those topics that align with the specific component of the standard.

• The Teacher’s Edition, Topic Planner indicates the standards and Mathematical Practices that align to each lesson.

The Teacher’s Edition, Math Background: Coherence, provides information that summarizes the content connections across grades. Examples of where explanations of the role of the specific grade-level mathematics are present in the context of the series include:

• Topic 3, Math Background: Coherence, the materials highlights six of the learnings within the topics: “Addition Strategies, Doubles and Near Doubles, Make 10, Explain Strategies, Solve Word Problems, and Properties and Relationships” with a description provided for each learning, including which lesson(s) cover the learnings. The “Look Ahead” section asks the question, “How does Topic 3 connect to what students will learn later?” and provides Grade 2 connections, “Fluency with Facts to 20 In Topic 1, students will develop fluency with addition and subtraction within 20. By the end of Grade 2, they will be expected to know from memory all sums of two 1-digit numbers. Add and Subtract Within 1,000 In Topics 4 and 6, students will develop fluency with addition and subtraction within 100. In Topics 10 and 11, students will add and subtract within 1,000.” 

• Topic 6, Math Background: Coherence, the materials highlights four of the learnings within the topics: “Data Analysis, Addition and Subtraction Situations, Represent and Solve Addition and Subtraction Problems, and Add and Subtract Within 20” with a description provided for each learning, including which lesson(s) cover the learnings. The “Look Ahead” section asks the question, “How does Topic 6 connect to what students will learn later?” and provides a Grade 2 connection, “Data and Graphs In Topic 15, students will construct line plots that represent measurement data. They will make picture graphs and bar graphs that represent up to four categories of data. They will continue to interpret these data by solving ‘put together,’ ‘take apart,’ and  ‘compare’ word problems.”

• Topic 10, Math Background: Coherence, the materials highlight five of the learnings within the topics: “Connect Operations Using Models, Connect Operations Using Mental Math, Connect Operations Using a Hundred Chart, Connect Operations Using an Open Number Line, and Connect Strategies” with a description provided for each learning, including which lesson(s) cover the learnings. The “Look Ahead” section asks the question, “How does Topic 10 connect to what students will learn later?” and provides Grade 2 connections, “Add Within 100 Using Strategies In Topic 3, students will add using a hundred chart and an open number line. Students will also learn the strategies of breaking apart one or both addends and using compensation. Fluently Add Within 100 In Topic 4, students will add using partial sums. They will also add by drawing tens and ones and writing numerals in place-value charts. Students will fluently add two or more numbers within 100.”

The materials reviewed for enVision Mathematics Grade 1 meet expectations for providing explanations of the instructional approaches of the program and identification of the research-based strategies. The Teacher’s Edition Program Overview provides detailed explanations behind the instructional approaches of the program and cites research-based strategies for the layout of the program. Unless otherwise noted all examples are found in the Teacher’s Edition Program Overview.

Examples where materials explain the instructional approaches of the program and describe research-based strategies include:

• The Program Goal section states the following: “The major goal in developing enVision Mathematics was to create a program for which we can promise student success and higher achievement. We have achieved this goal. We know this for two reasons. 1. EFFICACY RESEARCH First, the development of enVision Mathematics started with a curriculum that research has shown to be highly effective: the original enVisionMATH program (PRES Associates, 2009; What Works Clearinghouse, 2013). 2. RESEARCH PRINCIPLES FOR TEACHING WITH UNDERSTANDING The second reason we can promise success is that enVision Mathematics fully embraces time-proven research principles for teaching mathematics with understanding. One understands an idea in mathematics when one can connect that idea to previously learned ideas (Hiebert et al., 1997). So, understanding is based on making connections, and enVision Mathematics was developed on this principle.”

• The Instructional Model section states the following: “There has been more research in the past fifteen years showing the effectiveness of problem-based teaching and learning, part of the core instructional approach used in enVision Mathematics, than any other area of teaching and learning mathematics (see e.g., Lester and Charles, 2003). Furthermore, rigor in mathematics curriculum and instruction begins with problem-based teaching and learning. … there are two key steps to the core instructional model in enVision Mathematics. STEP 1 PROBLEM-BASED LEARNING Introduce concepts and procedures with a problem-solving experience. Research shows that conceptual understanding is developed when new mathematics is introduced in the context of solving a real problem in which ideas related to the new content are embedded (Kapur, 2010; Lester and Charles, 2003; Scott, 2014)... STEP 2 VISUAL LEARNING Make the important mathematics explicit with enhanced direct instruction connected to Step 1. The important mathematics is the new concept or procedure students should understand (Hiebert, 2003; Rasmussen, Yackel, and King, 2003). Quite often the important mathematics will come naturally from the classroom discussion around students’ thinking and solutions from the Solve and Share task…”

• Other research includes the following:

  • Hiebert, J.; T. Carpenter; E. Fennema; K. Fuson; D. Wearne; H. Murray; A. Olivier; and P.Human. Making Sense: Teaching and Learning Mathematics with Understanding. Portsmouth, NH: Heinemann, 1997.

  • Hiebert, J. (2003). Signposts for teaching mathematics through problem solving. In F. Lester, Jr. and R. Charles, eds. Teaching mathematics through problem solving: Grades Pre-K–6 (pp. 53–61). Reston, VA: National Council of Teachers of Mathematics.

• Throughout the Teacher’s Edition Program Overview references to research-based strategies are cited with some reference pages included at the end of some authors' work.

The materials reviewed for enVision Mathematics Grade 1 meet expectations for providing a comprehensive list of supplies needed to support instructional activities.

In the online Teacher Resources for each grade, a Materials List is provided in table format identifying the required materials and the topic(s) where they will be used. Additionally, the materials needed for each lesson can be found in the Topic Planner and the Lesson Overview. Example includes:

• Topic 1, Topic Planner, Lesson 1-2, Materials, “Connecting cubes (or Teaching Tool 7)”

• Topic 6, Lesson 6-1, Lesson Resources, Materials, “Library books”

• Teacher Resources, Grade 1: Materials List, the table indicates that Topic 13 will require the following materials: “$1 Bills (or Teaching Tool 45), Analog clock (or Teaching Tool 30), Chart paper, ...”

The materials reviewed for enVision Mathematics Grade 1 meet expectations for including an assessment system that provides multiple opportunities throughout the grade  to determine students’ learning and sufficient guidance to teachers for interpreting student performance and suggestions for follow-up. 

The assessment system provides multiple opportunities to determine student’s learning throughout the lessons and topics. Answer keys and scoring guides are provided. In addition, teachers are given recommendations for Math Diagnosis and Intervention System (MDIS) lessons based on student scores. If assessments are given on the digital platform, students are automatically placed into intervention based on their responses.

Examples include:

• Topic 2, Lesson 2-4, Independent Practice, Evaluate, Quick Check, Problems 3, 9, and 10, “Check mark indicates items for prescribing differentiation on the next page. Items 10: 1 point. Items 3 and 9: each worth 2 points.” “Directions 3 Look at the ten-frames. Write an addition fact with 5. Then write an addition fact for 10.  9 Higher Order Thinking Write a story with an addition fact for 10. Then write an equation for your story. 10 Assessment Practice Which sums equal 10? Choose two that apply.” The following page, Step 3: Assess and Differentiate states, “Use the Quick Check on the previous page to prescribe differentiated instruction. I Intervention 0-3 points, O On-Level 4 points, A Advanced 5 points.” The materials provide follow-up activities—to be assigned at the teacher’s discretion—to students at each indicated level: Intervention Activity I, Technology Center I O A, Reteach to Build Understanding I,  Build Mathematical Literacy I O, Enrichment O A, Activity Centers I O A, and Additional Practice Leveled Assignment I Items 1-5, 9, O Items 3-5, 7-9, and A Items 4-9.

• Topic 4, Topic Performance Task, Problem 1, “Maria’s Stickers Maria collects stickers. The chart shows the different stickers she has.” The materials show a chart, “Maria’s Stickers”; it shows an image of each type of sticker and number of each sticker. “How many more moon stickers than sun stickers does Maria have? Count, make 10, think addition to solve. ___ more moon stickers.” Item Analysis for Diagnosis and Intervention indicates: DOK 1, MDIS B39, Standard 1.OA.C.6. Scoring Guide indicates: 1 point “Difference is correct.”

• Topic 6, Topic Assessment Masters, Problem 1, shows a picture graph “Favorite Instruments.”  Each picture represents one vote for either piano or trumpet. “A. Which set of tally marks shows the number of trumpets in the picture graph?  B. How many more trumpets need to be added to have more trumpets than pianos?” Item Analysis for Diagnosis and Intervention indicates: DOK 3; MDIS D48, D50, D55, and E37; Standards 1.MD.C.4 and 1.OA.A.1. Scoring Guide indicates: 1A 1 point “Correct choice selected”; 1B 1 point “Correct choice selected.”

• Topics 1-15, Cumulative/Benchmark Assessment, Problem 15, “Which 3-D shape does NOT have a flat surface?” Answer choices are images of A sphere, B cone, C cylinder, D cube. Item Analysis for Diagnosis and Intervention indicates: DOK 2, MDIS D35, Standard 1.G.A.1.  Scoring Guide indicates:

The materials reviewed for enVision Mathematics Grade 1 meet expectations for providing assessments that include opportunities for students to demonstrate the full intent of grade-level standards and practices across the series.

The materials provide formative and summative assessments throughout the grade as print and digital resources. As detailed in the Assessment Sourcebook, the formative assessments—observational tools, Convince Me!, Guided Practice, and Quick Checks—occur during and/or at the end of a lesson. The summative assessments—Topic Assessment, Topic Performance Task, and Cumulative/Benchmark Assessments—occur at the end of a topic, group of topics, and at the end of the year.  The four Cumulative/Benchmark Assessments address Topics 1-4, 1-8, 1-11, and 1-15. 

• Observational Assessment Tools “Use Realize Scout Observational Assessment and/or the Solve & Share Observation Tool blackline master.”

• Convince Me! “Assess students’ understanding of concepts and skills presented in each example; results can be used to modify instruction as needed.”

• Guided Practice “Assess students’ conceptual understanding and procedural fluency with lesson content; results can be used to review or revisit content.”

• Quick Check “Assess students’ conceptual understanding and procedural fluency with lesson content; results can be used to prescribe differentiated instruction.”

• Topic Assessment “Assess students’ conceptual understanding and procedural fluency with topic content.” Additional Topic Assessments are available with ExamView.

• Topic Performance Task “Assess students’ ability to apply concepts learned and proficiency with math practices.

• Cumulative/Benchmark Assessments “Assess students’ understanding of and proficiency with concepts and skills taught throughout the school year.”

The formative and summative assessments allow students to demonstrate their conceptual understanding, procedural fluency, and ability to make application through a variety of item types. Examples include: 

• Order; Categorize

• Matching

• Graphing

• Yes or No; True or False

• Number line

• True or False

• Multiple choice

• Fill-in-the-blank

• Technology-enhanced responses (e.g., drag and drop)

• Constructed response (i.e., short and extended responses)