The materials reviewed for enVision Mathematics Grade 3 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 explore the relationship between equal groups on a number line and multiplication.” In Solve & Share, students use a number line to model multiplication and solve a word problem. “Harvey the Hop Toad starts at 0 and jumps 7 times in the same direction. Each time he jumps 3 inches farther. How can you show how far Harvey goes on a number line?” An image of a girl points to a number line and states, “Model with math. A number line can be used to record and count equal groups.” Students develop conceptual understanding as they use repeated addition, skip counting, and number lines as ways to multiply. (3.OA.1)

• Topic 7, Lesson 7-2, Lesson Overview, Conceptual Understanding states, “Students learn to interpret data as they become familiar with picture graphs and bar graphs. The focus is on visual representation of the data as well as the data itself.” In Guided Practice, Do You Know How?, Problem 3, students use a frequency table to complete a picture graph. The materials show a frequency table of three “Favorite School Lunch” options and their respective number of votes: taco (2), pizza (8), and salad (3). Students develop conceptual understanding as they draw a scaled picture graph to represent three-category data. (3.MD.3)

• Topic 13, Lesson 13-6, Lesson Overview, Conceptual Understanding states, “Students use what they know about fractions and number lines to continue comparing fractions.” In Solve & Share, students use what they know about number lines and fractions to locate and compare fractions on a number line. The materials state, “Tanya, Ria, and Ryan each used a bag of flour to make modeling clay. The bags were labeled $\frac{3}{4}$lb, $\frac{1}{4}$lb, and $\frac{2}{4}$lb. Show these fractions on a number line. How can you use the number line to compare two of these fractions?” An image of a girl points to a number line and states, “You can use reasoning to compare fractions. Think about the size of the fractions. You can also use models such as a number line.” The materials show a number line with four intervals, beginning at 0 and ending at 1. Students use what they know about number lines and fractions to locate and compare fractions on a number line. Students develop conceptual understanding as they compare fractions by reasoning about their size. (3.NF.3) 

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

• Topic 3, Lesson 3-1, Lesson Overview, Conceptual Understanding states, “As students progress in their work with multiplication, they can apply more strategies to complex problems. By learning the Distributive Property, they gain a stronger conceptual understanding of number sense and the operation of multiplication, giving them greater recall with known facts.” In Independent Practice, Problem 8, students use the Distributive Property to find a missing factor. “8. 10 $\times$ 3 = (___ $\times$ 3) + (2 $\times$ 3).” Students independently demonstrate conceptual understanding by applying properties of operations as strategies to multiply. (3.OA.5)

• Topic 9, Lesson 9-4, Lesson Overview, Conceptual Understanding states, “Students learn to subtract multi-digit numbers using the expanded algorithm, and, by doing so, they use steps similar to those in the addition of multi-digit numbers with an expanded algorithm, but in reverse.” In Independent Practice, Problem 9, students estimate and then use partial sums to subtract. The materials provide an open number line with 865 positioned to the far right. “9. 865 - 506” Students independently demonstrate conceptual understanding by subtracting within 100 using algorithms based on place value and/or the relationship between addition and subtraction. (3.NBT.2)

• Topic 16, Lesson 16-4, Lesson Overview, Conceptual Understanding states, “The formulas for perimeter and area can be easily confused, which makes this lesson an especially rigorous and challenging one for many students.” In Independent Practice, Problem 12, students write the dimensions of a different rectangle with the same perimeter as the rectangle shown and then tell which rectangle has the greater area. The materials show a 3 ft 4 ft yellow triangle with grid lines. Students independently demonstrate conceptual understanding by solving mathematical problems involving rectangles with the same perimeter. (3.MD.8)

The materials reviewed for enVision Mathematics Grade 3 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 4, Lesson 4-2, Lesson Overview, Procedural Skill states, “Using multiplication facts to solve division equations, students see the relationship between the two operations.” In Guided Practice, Problem 3, students develop procedural skills and fluency as they apply knowledge of fact families to solve a word problem involving multiplication and division. The materials prompt, “Complete each fact family.” Students apply knowledge of the Commutative Property to complete the rewrite the equations "$3 \times 6 = 18$” and "$18 \div 3 = 6$.”  (3.OA.6)

• Topic 10, Lesson 10-1, Lesson Overview, Procedural Skill states, “Place-value blocks and open number lines are familiar tools that create strong visuals as students tackle more complex multiplication.” In Guided Practice, Problem 4, students develop procedural skills and fluency as they use a number line to multiply $4 \times 60$. The materials show an open number line that begins at 0. Students use the number line to show four jumps of 60 to arrive at a product of 240. (3.NBT.3)

• Topic 14, Lesson 14-5, Lesson Overview, Procedural Skill states, “Students use the knowledge they acquired in estimating capacity in the previous lesson as a basis for measuring capacity in this lesson. Because capacity and volume are challenging concepts, this lesson consistently uses marked 1-liter containers to show students that a 1-liter container can be used as the reference point for measurement. After many practice items and word problems, students become adept at measuring capacity.” In Guided Practice, Problem 4, students develop procedural skills and fluency as they add two liquid measurements. “Find the total capacity represented in each picture.” The materials show two 1-liter containers: one at full capacity, the second at half capacity. (3.MD.2)

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 2-5, Lesson Overview, Procedural Skill states, “Students use their understanding of basic multiplication facts to develop and understand procedural skills for multiplying 1-digit numbers by 0, 1, 2, 5, 9, and 10.” In Problem Solving, Problem 32, students independently demonstrate procedural skill and fluency by using multiplication within 100 to solve a word problem involving equal groups. The materials state, “Higher Order Thinking Brendan shot 3 arrows in the 10-point section, 4 arrows in the 9-point section, 9 arrows in the 5-point section, 8 arrows in the 2-point section, and 7 arrows in the 1-point section. What is the total number of points Brendan scored for all his arrows? (3.OA.3)

• Topic 7, Lesson 7-3, Lesson Overview, Procedural Skill states, “Students continue to work with scaled bar graphs using information from data tables. By showing that they can both interpret the data in a bar graph and create a bar graph, students show increasing facility in representing a set of data.” In Independent Practice, Problem 5, students independently demonstrate procedural skills and fluency as they draw a scaled bar graph to represent a data set with several categories. The materials state, “In 5, use the table at the right. Complete the bar graph to show the data.” The materials show a tally chart, “Favorite Store for Clothes” with columns indicating Store, Tally, and Number of Votes for categories Deal Mart, Jane’s, Parker’s, and Trends. The given bar graph includes labels and values on the appropriate axes.  (3.MD.3)

• Topic 10, Lesson 10-1, Lesson Overview, Procedural Skill states, “Place-value blocks and open number lines are familiar tools that create strong visuals as students tackle more complex multiplication.” In Independent Practice, Problem 5, students demonstrate procedural skill and fluency by multiplying one-digit whole numbers by multiples of 10 in the range 10–90. The materials prompt students to “use an open number line or draw place-value blocks to find each product” and provide an open number line on which students may illustrate $3 \times 70$. (3.OA.8)

The materials for enVision Mathematics Grade 3 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 3, Lesson 3-3, Assessment Practice, Problem 22, students independently apply properties of operations to multiply. “Select numbers to create a different expression that is equal to $7 \times 8$.” The materials show the numbers 1, 2, 3, 5, 7, 8 enclosed in a frame and an equation for students to complete: ".” (3.OA.5) 

• In Topic 11, Lesson 11-3, Solve & Share, students solve two-step word problems using multiplication and addition as well as writing equations with a letter standing for the unknown quantity. “An aquarium has 75 clownfish in a large water tank. The clownfish represented in the graph were added to this tank. How many clownfish are in the tank now? Write and explain how you found the answer.” The picture graph “Recent Arrivals at the Aquarium” shows a collection of triangles for Clownfish, Sea Stars, and Crabs; each triangle represents 5 animals. An image of a boy says, “Make Sense and Persevere Think about the information you need to solve the problem.” (3.OA.8)

• In Topic 16, Lesson 16-1, Assessment Practice, Problem 14, students solve a real-world problem involving the perimeter of a polygon given the side lengths. “Mr. Karas needs to find the perimeter of the patio shown at the right. What is the perimeter of the patio?” The materials include an image of a polygon with side lengths, 7 yd., 14 yd., 9 yd., 14 yd., and 10 yd. Students select an answer from amongst the following choices: 48 yards, 50 yards, 52 yards, and 54 yards. (3.MD.8)

Examples of non-routine applications of the math include:

• In Topic 2, Lesson 2-4, Assessment Practice, Problem 20, students use multiplication within 100 to solve word problems in situations involving equal groups. “Kinsey arranges her buttons in 4 equal groups of 10. Seth arranges his buttons in 3 equal groups of 10. Select numbers to complete the equations that represent the button arrangements.”  The materials show the numbers 3, 4, 9, 10, 40, 90 enclosed in a frame and three equations for students to complete: ",  , .” (3.OA.3)

• In Topic 7, Topic Performance Task, Problem 5, students draw a scaled picture graph to represent a data set with several categories. “Miles plans to use all of the balloons that are left and wants to make at least one of each balloon animal. Make a picture graph to show one way Miles can finish using the balloons. Part A. Circle the key you will use.” Students select one key: 1 balloon represents 1, 2, 3, or 4 balloon animals. The materials show a chart “Balloon Shapes and Colors”; parrot - 2 blue, monkey - 1 brown 1 yellow, frog - 2 green, dolphin - 1 blue. “Part B. Complete the picture graph and explain how you solved the problem.” The materials provide the shell of a picture graph “Balloon Animals”; students add an appropriate number of balloons for Parrot, Monkey, Frog, and Dolphin. (3.MD.3)

• In Topic 14, Lesson 14-3, Problem Solving, Problem 7, students independently solve a non-routine word problem involving the addition of time intervals in minutes. “Reasoning Ms. Merrill spends 55 minutes washing all the windows in her two-story house. How much time could she have spent on each floor? Complete the chart to show three different ways.” The materials show a chart, “Time Spent Washing Windows,” with a column for 1st floor and a column for 2nd floor. The first entry in the 1st floor column is 25 min. Students complete the chart. (3.MD.1)

​The materials for enVision Mathematics Grade 3 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 5, Lesson 5-3, Independent Practice, Problem 24, students attend to procedural skills and fluency as they choose different strategies when multiplying within 100. “Use strategies to find the product. What is $5 \times 8$?” (3.OA.7)

• In Topic 6, Lesson 6-6, Problem Solving, Problem 9, students attend to application as they recognize area as an additive and find areas of rectilinear figures by decomposing them into non-overlapping rectangles and adding the areas of the non-overlapping parts. “Reasoning Mrs. Kendel is making a model house. The footprint for the house is shown at the right. What is the total area? Explain your reasoning." The materials provide dimensions on a Model House blueprint. (3.MD.7d)

• Topic 12, Lesson 12-1, Independent Practice, Problem 8, students attend to conceptual understanding as they attend to partitioning shapes into parts with equal areas. “Draw lines to divide the shape into the given number of equal parts. Then write the fraction that represents one part. 8. 6 equal parts”  Provided for the students is a 6 by 6 square put on a unit grid. (3.G.2)

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 4, Lesson 4-2, Guided Practice, Problem 2, students attend to application and procedural skills and fluency as they apply knowledge of fact families to solve a word problem involving division. “Mr. Dean has 3 children. He buys 30 pencils to share equally among his children for the school year. How many pencils will each child get? Write the answer and the fact family you used.” (3.OA.6)

• Topic 10, Lesson 10-1, Solve & Share, students attend to conceptual understanding and procedural skills and fluency as they use multiplication facts or a number line to show repeated addition. “Companies package their goods in a variety of ways. One company packages a case of water as 2 rows of 10 bottles. How many bottles are in the number of cases shown in the table below? Explain your thinking.” The materials show two sets of 10 place-value blocks captioned “2 rows of 10 bottles = 1 case,” a number line that begins at zero with a looped arrow pointing to ? bottles 1 case, and a table representing the Number of Cases (1, 2, 3, 4) and Number of Bottles. (3.NBT.3)

• Topic 14, Lesson 14-8, Problem Solving, Problem 9, students attend to application and procedural skills and fluency as they solve a word problem about calculating time left. “Elijah has 2 hours before dinner. He spends the first 37 minutes practicing his guitar and the next 48 minutes doing his homework. How much time is left until dinner?” Students solve a word problem by adding minutes and subtracting from hours. (3.MD.1)

The materials reviewed for enVision Mathematics Grade 3 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 4, Lesson 4-1, Problem Solving, Problem 19, students make sense of problems and persevere in solving them as they write a multiplication and division equation based on a given story. “Lisa, Bret, and Gary harvested apples. Lisa filled 3 carts with apples. Bret also filled 3 carts with apples. Gary filled another 3 carts with apples. Write a multiplication equation and a division equation for this story.” 

• Topic 8, Lesson 8-7, Problem Solving, Problem 22, students make sense of problems and persevere in solving them as they use rounding and compatible numbers to estimate how much a goal was exceeded. “Higher Order Thinking One week Mrs. Runyan earned $486, and the next week she earned $254. If Mrs. Runyan’s goal was to earn $545, by about how much did she exceed her goal? Show how you used estimation to find your answer.” 

• Topic 12, Lesson 12-1, Independent Practice, Problem 9, students make sense of problems and persevere in solving them as they partition shapes into parts of equal area and name the area of each part as a unit fraction of the whole. “Draw lines to divide the shape into the given number of equal parts. Then write the fraction that represents one part. 9. 3 equal parts.” The materials show a complex figure consisting of a 2 by 2 rectangle and a 1 by 4 rectangle positioned on a grid.

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 4-5, Convince Me!, students reason abstractly and quantitatively as they identify multiplication patterns with even and odd numbers. “Generalize Does multiplying by 8 also always result in an even product? Explain.”  

• In Topic 12, Lesson 12-3, Independent Practice, Problem 6, students reason abstractly and quantitatively as they recognize a fraction that is equivalent to a whole number and find the whole given a fractional part. “Draw a picture of the whole and write a fraction to represent the whole.” The materials show a circle and the fraction 34.  

• Topic 16, Lesson 16-6, Problem Solving, Performance Task, Problem 7, students reason abstractly and quantitatively as they try to understand dimensional relationships within a real world mathematical context and solve problems involving perimeter of polygons, specifically finding an unknown side length. “A Wedding Cake The Cakery Bakery makes tiered wedding cakes in various shapes. Maria buys ribbon to decorate three squares of a cake. The ribbon costs 50¢ a foot. Reasoning How many inches of ribbon does Maria need for the middle layer and top layer? Use reasoning to decide.” The materials show a three-tiered cake, indicating side lengths of the bottom tier are 10 inches and the side lengths of the middle and top tiers are successively 1 inch shorter than the previous tier. An image of a girl is quoted saying, “Drawing a diagram can help your reasoning when solving a problem.”

The materials reviewed for enVision Mathematics Grade 3 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 1-5, Solve & Share, students construct viable arguments about how many tacos each friend can receive and critique the reasoning of other students’ work. “Li made 12 tacos. He wants to give some of his friends 2 tacos each. If Li does not get any of the tacos, how many of his friends will get tacos?” An image states, “You can use reasoning. How can what you know about sharing help you solve the problem?” 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 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 Sophia’s Work and the other is labeled Jose’s Work. The following questions are asked: “How did Sophia show 2 tacos being given to each friend? What error did Jose make when he found the number of friends that would each get 2 tacos?”

• Topic 8, Lesson 8-4, Problem Solving, Problem 19, students construct viable arguments and critique the reasoning of others as they explain if students different strategies to subtract will yield the same results. “Higher Order Thinking To find 357 - 216, Tom added 4 to each number and then subtracted. Saul added 3 to each number and then subtracted. Will both ways work to find the correct answer? Explain.”

• Topic 10, Lesson 10-3, Solve & Share, students construct viable arguments and critique the reasoning of others as they explain which strategies of the three displayed gives the correct solution. “Three students found 5 30 in different ways. Which student is correct? Explain.” The strategies of the three students (Janice, Earl and Clara) are displayed each strategy also has an explanation written by the student.

• Topic 13, Lesson 13-8, Independent Practice, Problems 3-5, students construct viable arguments and critique the reasoning of others as they construct arguments involving fractions to justify a conjecture. “Construct Arguments Reyna has a blue ribbon that is 1 yard long and a red ribbon that is 2 yards long. She uses $\frac{2}{4}$ of the red ribbon and $\frac{2}{4}$ of the blue ribbon. Conjecture: Reyna uses the same amount of red and blue ribbon. 3. Draw a diagram to help justify the conjecture. 4. Is the conjecture correct? Construct an argument to justify your answer. 5. Explain another way you could justify the conjecture.”

The materials reviewed for enVision Mathematics Grade 3 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 1, Lesson 1-1, Problem Solving, Problem 13, students model with mathematics as they write equations to represent and solve problems. “Model with Math Salvatore gets 50 trading cards for his birthday. He gives 22 cards to Madison, and Madison gives 18 cards to Salvatore. Then Salvatore’s sister gives him 14 cards. How many trading cards does Salvatore have now? Use math to represent the problem.” Teacher guidance: “Model with Math Remind students that when doing a multi-step problem, using representations, such as equations, helps to break the problem into simpler steps. What equation can you write to find how many cards Salvatore has after giving some cards to Madison?” 

• Topic 6, Lesson 6-5, Independent Practice, Problem 4, students model with mathematics as they represent the Distributive Property using area models. “Complete the equation that represents the picture.” The materials show an area model with a height of 5 units and partial widths of 4 units and 3 units for an overall width of 7 units. Students fill in the blanks within the equation . 

• Topic 13, Lesson 13-2, Independent Practice, Problem 7, students model with mathematics as they use number lines to name equivalent fractions. “Find the missing equivalent fractions on the number line. Then write the equivalent fractions below.” The materials show a number line with one-eight intervals. Students write the equivalent fractions for $\frac{2}{8}$ and $\frac{6}{8}$ , fill in blanks at $\frac{4}{8}$, $\frac{2}{4}$ and $\frac{ }{ }$ = $\frac{ }{ }$.

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 6, Lesson 6-1, Independent Practice, Problem 6, students use appropriate tools strategically as they use unit squares to find the area of regular and irregular shapes. “Count to find the area. Tell if the area is an estimate.” For Problem 6, the materials show a yellow balloon superimposed on a unit grid.

• Topic 13, Lesson 13-3, Problem Solving, Problem 16, students use appropriate tools strategically when they use models to compare two fractions that are part of the same whole and have the same denominator. “Use the pictures of the strips that have been partly shaded. Do the yellow strips show $\frac{2}{4}>\frac{3}{4}$? Explain.” The materials show two fraction strips: one showing three of four boxes shaded and the other two of four boxes shaded.

• Topic 16, Lesson 16-5, Independent Practice, Problem 10, students use appropriate tools strategically to find the relationship between shapes with the same area but different perimeters. “Use grid paper to draw two different rectangles with the given area. Write the dimensions and perimeter of each rectangle. Circle the rectangle that has the smaller perimeter. 10. 32 square centimeters.”

The materials reviewed for enVision Mathematics Grade 3 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 2-3, Guided Practice, Do You Know How?, Problems 5 and 6, students attend to precision as they apply properties when they multiply by 0 or 1. “Find each product. 5.   6. ”. The materials include an image of a boy who states, “You can use the Identity and Zero Properties of Multiplication to find these products.”

• Topic 6, Lesson 6-3, Convince Me!, students attend to precision as they explain how changing the units would change the solution to a problem. “Be Precise If square inches rather than square centimeters were used for the problem above, would more unit squares or fewer unit squares be needed to cover the shape? Explain.” The referenced “problem above” is the Visual Learning Bridge that asks students to find the area of a sticker in square centimeters.

• Topic 12, Lesson 12-1, Problem Solving, Problem 13, students attend to precision when they name the flag that satisfies the criteria in the question. “Higher Order Thinking The flag of this nation has more than 3 equal parts. Which nation is it, and what fraction represents 1 part of its flag?” The materials show flags of four countries, all four of which are partitioned into two to five equal parts.

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 1, Lesson 1-3, Visual Learning Bridge and Problem Solving, Problem 6, students use specialized language when they engage with the definition of array. Visual Learning Bridge (A), “Dana keeps her swimming medal collection in a display on the wall. The display has 4 rows. Each row has 5 medals. How many medals are in Dana’s collection?” The materials show the array of medals and a girl who states, “The medals are in an array. An array shows objects in equal rows and columns.” Problem Solving, Problem 6, “Liza draws these two arrays. How are the arrays alike? How are they different?” The materials show an array that consists of five rows of three circles and an array that consists of three rows of five circles. Students will use the term “array” in their answers.

• Topic 6, Lesson 6-1, Visual Learning Bridge and Convince Me!, students use specialized language when they differentiate between unit square and square units when finding area. Visual Learning Bridge (A), “Area is the number of unit squares needed to cover a region without gaps or overlaps. A unit square is a square with sides that are each 1 unit long. It has an area of 1 square unit.” Convince Me!, “Construct Arguments Karen says these shapes each have an area of 12 square units. Do you agree with Karen? Explain.” The materials show two shapes superimposed on a grid.

• Topic 13, Lesson 13-3, Independent Practice, Problem 8, students use specialized language when they use math symbols >, <, or = to compare two fractions with the same denominator. “Compare. Write <, >, or =. Use or draw fraction strips to help.The fractions refer to the same whole. 8. $\frac{5}{8}$ ___ $\frac{7}{8}$”

The materials reviewed for enVision Mathematics Grade 3 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 2-1, Problem Solving, Problem 13, students use and look for structure as they look at the place value of the total to answer a question .”Eric has some nickels. He says they are worth exactly 34 cents. Can you tell if Eric is correct or not? Why or why not?” 

• Topic 5, Lesson 5-1, Solve & Share, students use and look for structure when they find patterns in factors and products using known facts and the Distributive Property. “Max found $6 \times 8 = 48$. He noticed that $(6 \times 4) + (6 \times 4)$ also equals 48. Use the multiplication table to find two other facts whose sum is 48. Use facts that have a 6 or 8 as a factor. What pattern do you see?” The materials show an 8 by 8 times table with the following labels: “These are the factors. These are the products.” 

• Topic 10, Lesson 10-1, Problem Solving, Problem 12, students use and look for structure as they compare the structure of two number lines to answer how two multiplication problems are similar or different. “Higher Order Thinking On one open number line, show $4 \times 30$. On the other open number line, show $3 \times 40$. How are the problems alike? How are they different?” The materials show two open number lines that begin at zero.

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 3, Lesson 3-6, Problem Solving, Problem 22, students look for and express regularity in repeated reasoning when they generalize strategies to multiply using the Associative Property of Multiplication. “Anita has 2 arrays. Each array has 3 rows of 3 counters. Explain why Anita can use the Associative Property to find the total number of counters in two different ways.”

• Topic 6, Lesson 6-4, Assessment Practice, Problem 12, students look for and express regularity in repeated reasoning when they find the area of rectangles using two different methods. “Marla makes maps of state preserves. Two of her maps of the same preserve are shown. Select all the true statements about Marla’s maps.” Students select from the following statements: “You can find the area of Map A by counting the unit squares. You can find the area of Map B by multiplying the side lengths. The area of Map A is 18 square feet. The area of Map B is 18 square feet. The areas of Maps A and B are NOT equivalent.” The materials show Map A superimposed on square grid paper and Map B is labeled with a height of 9 feet and a width of 2 feet.

• Topic 9, Lesson 9-3, Independent Practice, Problem 9, students look for and express regularity in repeated reasoning when they add three or more numbers using addition strategies. “Estimate and then find each sum: $602 + 125 + 231$” 

The materials reviewed for enVision Mathematics Grade 3 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 1-4, Visual Learning Bridge, Teachers begin the Classroom Conversation by saying the following, “Why do you need 3 equal groups? [There are 3 friends who want to share the toys equally.]” 

• Topic 6, Lesson 6-6, Problem Solving, Problem 10, “Vocabulary Fill in the blanks. Mandy finds the ___ of this shape by dividing it into rectangles. Phil gets the same answer by counting ___.” The materials shows the outline of a shape on a unit square grid. Teacher guidance: “Vocabulary If students struggle to fill in the blanks, have them work with a partner and review the vocabulary terms for the topic by using the Topic 6 My Word Cards available on SavvasRealize.com.”

• Topic 15, Lesson 15-3, Problem Solving, Problem 11, “Reasoning Explain which of the shapes at the right you can cover with whole unit squares and not have any gaps or overlaps.” The materials show a parallelogram and a rectangle. Teacher guidance: “Reason Abstractly Help students to understand that to cover one of the shapes with unit squares and not have any gaps or overlaps, the unit squares need to fit tightly into all the corners of the shape. Can you fit unit squares tightly into all the corners of the shape on the left? Why or why not? [No; Sample answer: The corners are not right angles.]”

The materials reviewed for enVision Mathematics Grade 3 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 10: Professional Development (topic) Video, “Multiplying by multiples of ten is an important component in computing products of multi-digit factors. Students can combine their knowledge of place-value patterns, properties of multiplication, and basic facts in several ways to develop strategies that lead to fluency in multiplying by multi-digit numbers … Students can use what they know about patterns and skip counting and place value to skip count by multiples of ten on an open number line ... This pattern is the result of the structure of our base-ten numeration system … the fact that each place-value position is ten times the position to its right … Using these types of patterns to find products of multiples of ten helps students build confidence and accuracy in multiple-digit computation.”

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 14, Math Background: Coherence, Look Ahead, the materials state, “Grade 4 Time In Topic 10, students will use the four operations to solve word problems involving intervals of time, including problems with simple fractions.” An example word problem is given where teachers are tasked with figuring out how many hours a runner trains for a race. “Equivalence In Units of Measure In Topic 13, students will extend their understanding of customary and metric units of length, weight, mass, and capacity (liquid volume). They will learn the relative sizes of various measurement units and solve problems involving converting a measurement from a larger unit to a smaller one.” A word problem about making gallons of punch given a recipe is provided.

The materials reviewed for enVision Mathematics Grade 3 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 3 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 2, Math Background: Coherence, the materials highlight three of the learnings within the topics: “Patterns, Skip Counting, and Equal Groups” with a description provided for each learning, including which lesson(s) cover the learnings. The “Look Ahead” section asks the question, “How does Topic 2 connect to what students will learn later?” and provides a Grade 4 connection, “Multiplication with Greater Numbers In Topics 3 and 4, students will multiply greater numbers using strategies and properties.”

• Topic 6, Math Background: Coherence, the materials highlight four of the learnings within the topics: “Area as Covering, Relate Area to Multiplication and Addition, Distributive Property, and Area of Irregular Rectilinear Figures” 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 4 connection, “Area Formulas In Topic 13, students will apply the area formula for a rectangle to solve real-world and mathematical problems.”

• Topic 12, Math Background: Coherence, the materials highlight three of the learnings within the topics: “Fractions, Fraction Representations, and Line Plots” with a description provided for each learning, including which lesson(s) cover the learnings. The “Look Ahead” section asks the question, “How does Topic 12 connect to what students will learn later?” and provides Grade 4 connections, “Fraction Equivalence and Comparison In Topic 8, students will generate equivalent fractions by multiplying or dividing the numerator and denominator by the same nonzero number. They will also, compare fractions with different numerators and different denominators. Operations with Fractions In Topic 9, students will add and subtract fractions and mixed numbers with like denominators. In Topic 10, students will multiply a whole number and a fraction. Solve Problems Involving Fractions and Line Plots In Topic 11, students will solve problems involving line plots that display measurements in fractions of a unit ($\frac{1}{2}$, $\frac{1}{4}$, $\frac{1}{8}$).”

The materials reviewed for enVision Mathematics Grade 3 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 3 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, “Number lines  (or TT 7), Colored pencils”

• Topic 6, Lesson 6-1, Lesson Resources, Materials, “Two-color tiles (or Teaching Tool 8), Area of shapes (or Teaching Tool 12), Centimeter grid paper (or Teaching Tool 13)”

• Teacher Resources, Grade 3: Materials List, the table indicates that Topic 14 will require the following materials: “1-liter bottles, 1-liter beaker, Blank Clock Faces (Teaching Tool 20), ...”

The materials reviewed for enVision Mathematics Grade 3 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 3, Lesson 3-2, Independent Practice, Evaluate, Quick Check, Problem 11, “A check mark indicates items for prescribing differentiation on the next page. Items 11 and 18: each 1 point. Item 17: up to 3 points.” For example, Directions: “Leveled Practice Multiply. You may use counters or pictures to help. 11. ” 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, 11-12, 15, 17, 19-21, O Items 3-4, 6-7, 12-13, 15-16, 18-21 and A Items 8-14, 16, 18-21.

• Topic 5, Topic Assessment, Problem 4, “Find the product. $5 \times 7$” Students choose amongst answer options (A) 28 (B) 30 (C) 35 (D) 42. Item Analysis for Diagnosis and Intervention indicates: DOK 1; MDIS B52, and B53; Standard 3.OA.C.7. Scoring Guide indicates: 4 1 point “Correct choice selected.”

• Topic 8, Topic Performance Task, Problem 5, “Vacation Trip Mia is planning a vacation in Orlando, FL. The Mia’s Route table shows her route and the miles she will drive. … Mia has to book a hotel and buy theme park tickets. The Hotel Prices and Theme Park Prices tables show the total prices for Mia’s stay. The Mia’s Options list shows two plans that Mia can choose from. … 5. One theme park has a special offer. For each ticket Mia buys, she gets another ticket free. Shade the squares in the table at the right to show this pattern. Explain why the pattern is true.” The materials show a 5 by 5 addition table. Item Analysis for Diagnosis and Intervention indicates: DOK 3, MDIS C23, Standard 3.OA.D.9, MP.8. Scoring Guide indicates: 2 points “Pattern identified with explanation” and 1 point “Pattern identified without explanation.”

• Topics 1-12, Cumulative/Benchmark Assessment, Problem 15, “Lenny’s goal is to collect 500 tabs from cans to recycle. One month he collects 217 tabs and the next month he collects 186 tabs. How many more tabs does Lenny need to collect? Write equations to represent the number of tabs he still needs to collect.” Item Analysis for Diagnosis and Intervention indicates: DOK 2, MDIS E3, Standard 3.OA.D.8. Scoring Guide indicates: 2 points “Correct answer and equations” and 1 point “Correct answer or equations.”

The materials reviewed for enVision Mathematics Grade 3 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 1-16. 

• 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)