4th Grade - Gateway 2

The materials reviewed for enVision Mathematics Grade 4 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 2, Lesson 2-7, Lesson Overview, Conceptual Understanding states, “Students must understand that when subtracting across the zeros, they may need to regroup in more than one place.” In Solve & Share, students use place value or the standard algorithm to subtract across zeros. The materials state, “London, England, is 15,710 kilometers from the South Pole. Tokyo, Japan is 13,953 kilometers from the South Pole. How much farther is London than Tokyo from the South Pole? Solve this problem any way you choose.” An image of a girl states, “you can use reasoning to identify the operation you use to compare two distances. Show your work in the space below!” Students develop conceptual understanding as they subtract multi-digit whole numbers using the standard algorithm. (4.NBT.4)

• Topic 8, Lesson 8-1, Lesson Overview, Conceptual Understanding states, “Students use an area model to demonstrate that two fractions are equivalent when they name the same part of the same whole.” In the Visual Learning Bridge, students use different area models to represent the same part of a whole. The materials show three frames: A) “James ate part of the pizza shown in the picture at the right.  He said \frac{5}{6} of the pizza is left.  Cardell said \frac{10}{12} of the pizza is left.  Who is correct?” The materials show a rectangular pizza cut into 6 pieces with 1 piece missing. In the fraction \frac{5}{6}, 5 is identified as the numerator and 6 as the denominator. A girl states, “Equivalent fractions name the same part of the same whole. B)  “One Way Use an area model. Draw a rectangle and divide it into sixths. Shade \frac{5}{6}. Then divide the rectangle into twelfths. The number and size of parts differ, but the shaded part of each rectangle is the same. \frac{5}{6} and \frac{10}{12} are equivalent fractions. The materials show rectangular area models and circular area models that illustrate the equivalent fractions. C) “Use a different area model. Draw a circle and divide it into sixths. Shade \frac{5}{6}. Then divide the circle into twelfths. The number and size of parts differ, but the shaded part of each circle is the same. \frac{5}{6} and \frac{10}{12} are equivalent fractions. A girl states, “Both James and Cardell are correct because \frac{5}{6} = \frac{10}{12}.”  Classroom Conversation asks students the following questions: “A) How much of the pizza is left according to James? according to Cardell? What do you need to do? What does the denominator of a fraction tell you? What does the numerator tell you? How can you tell from the picture that \frac{5}{6} of the pizza is left? B) Use Appropriate Tools Strategically Why is the first area model labeled \frac{5}{6}? Why is the second area model labeled \frac{10}{12}? Why are \frac{5}{6} and \frac{10}{12} equivalent? C) Does it matter what shape is used to show each of the two fractions?” Students develop conceptual understanding as they use visual fraction models to recognize and generate equivalent fractions. (4.NF.1)

• Topic 14, Lesson 14-3, Lesson Overview, Conceptual Understanding states, “Students extend repeating patterns of shapes.” In Guided Practice, Problem 2, students extend a pattern consisting of green triangles and orange circles. “What is the 20th shape? The rule is ‘Triangle, Circle, Circle.”The materials show two iterations of a pattern. Students develop conceptual understanding as they generate a shape pattern that follows a given rule.  (4.OA.5)

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

• Topic 1, Lesson 1-3, Lesson Overview, Conceptual Understanding states, “Students comparing the values of numbers use symbols of relation (>, =, and <).” In Independent Practice, Problem 12, students complete a comparison of sums by writing <, =, or <. “12. 40,000 + 2,000 + 600 + 6 ___ 40,000 + 3,000 + 10.” Students independently demonstrate conceptual understanding by comparing two multi-digit numbers using the symbols >, =, >  to record the results of the comparisons. (4.NBT.2)

• Topic 7, Lesson 7-4, Lesson Overview, Conceptual Understanding states, “Students use arrays to find all the factors of a number and to decide if the number is prime or composite.” In Independent Practice, Problem 11, students use given arrays to tell whether the number 10 is prime or composite. The materials show the following arrays: one row of 10, two rows of 5, five rows of 2, and one column of 10. Students independently demonstrate conceptual understanding by determining whether a given whole number in the range 1–100 is prime or composite. (4.OA.4)

• Topic 10, Lesson 10-2, Lesson Overview, Conceptual Understanding states, “Students use models to multiply a fraction by a whole number.” In Independent Practice, Problem 5, students write and solve a multiplication equation using drawings or number lines. The materials show three pictures of a fraction of a circular block of cheese; each is labeled \frac{2}{10}. Students independently demonstrate conceptual understanding by extending previous understandings of multiplication to multiply a fraction by a whole number. (4.NF.4b)

The materials reviewed for enVision Mathematics Grade 4 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, Procedural Skill states, “Students will apply the mental-math strategies [i.e. breaking apart, compensation, counting on…] taught in this lesson.” In Guided Practice, Problem 4, “Use mental math strategies to solve. 9,100 + 2,130 + 900.” Students develop procedural skills and fluency as they apply mental-math strategies to perform multi-digit arithmetic. (4.NBT.4)

• Topic 6, Lesson 6-4, Lesson Overview, Procedural Skill states, “Students continue to develop procedural skills in writing and solving equations to solve multi-step problems.” In Independent Practice, Problem 4, “Five toymakers each carved 28 blocks and 17 airplanes. Three other toymakers each carved the same number of airplanes and twice as many blocks. How many toys did the eight carve in all?”  Students develop procedural skills and fluency as they use variables to represent unknown quantities in equations and solve multistep word problems posed with whole numbers and having whole-number answers.  (4.OA.3)

• Topic 12, Lesson 12-4, Lesson Overview, Procedural Skill states, “Students learn how to rename fractions to have common denominators in order to find sums.” In Independent Practice, Problem 21, “Leveled Practice Add the fractions. \frac{44}{100}+\frac{34}{100}+\frac{9}{10}.  Students develop procedural skill and fluency as they express a fraction with denominator 10 as an equivalent fraction with denominator 100 and use this technique to add two fractions with respective denominators 10 and 100. (4.NF.5)

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 3, Lesson 3-3, Lesson Overview, Procedural Skill states, “Students practice finding the products using partial products.” In Guided Practice, Do You Know How?, Problems 2 and 3, students independently demonstrate procedural skill and fluency as they multiply a whole number of up to four digits by a one-digit whole number. “2. 2 \times 24 3. 3 \times 218”. (4.NBT.5)

• Topic 9, Lesson 9-6, Lesson Overview, Procedural Skill states, “Students’ proficiency with adding and subtracting fractions builds as they practice a different method.” In Independent Practice, Problem 7, students independently demonstrate procedural skills and fluency as they write an equation showing the addition of fractions as joining parts shown by a given number line. Directions: write the equation shown by each number line. The materials show a number line that begins at 0 and ends at 1 using \frac{1}{4} unit intervals. The number line includes a curved arrow that starts at \frac{2}{4} and ends at \frac{3}{4}.  (4.NF.3a)

• Topic 13, Lesson 13-6, Lesson Overview, Procedural Skill states, “Students learn and use perimeter and area formulas to find missing side lengths of rectangles.” In Independent Practice, Problem 6, students independently demonstrate procedural skills and fluency as they find the missing dimension in a quadrilateral. “6. Find n.” An image of a rectangle is shown with the width labeled as 6 ft, and the length labeled n. The area of the rectangle equals 60 sq ft. (4.MD.3)

The materials for enVision Mathematics Grade 4 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 6, Lesson 6-3, Assessment Practice, Problem 7, students independently solve a routine multi-step word problem using the four operations and a letter to represent the unknown quantity. “The gym teacher has $250 to spend on volleyball equipment. She buys 4 volleyball nets for $28 each. Volleyballs cost $7 each. How many volleyballs can she buy? Explain how you solve. Use one or more equations and bar diagrams in your explanation. Tell what your variables represent.” (4.OA.3)

• In Topic 9, Lesson 9-3, Problem Solving, problem 20, students solve a word problem involving addition of fractions referring to the same whole and having like denominators. “Write and solve an equation to find what fraction, f, of the set is either circles or rectangles.”(4.NF.3d)

• In Topic 13, Lesson 13-6, Problem Solving, Problem 11, students apply the area formula in a real-world problem. “Greg covered the back of the picture with a piece of felt. The picture is 1\frac{1}{4} inches shorter than the frame and 1 inch less in width. What is the area of the felt?” The materials show the framed picture with the dimension "l = 15\frac{1}{4} in.” (4.MD.3)

Examples of non-routine applications of the math include:

• In Topic 2, Lesson 2-3, Solve & Share, students use place-value understanding to add 3-digit numbers in a multi-step word problem. “Students collect empty plastic water bottles to recycle. How many bottles were collected in the first two months? How many bottles were collected in all three months? Solve this problem using any strategy you choose.” A picture of a girl says, “You can use appropriate tools. such as drawing or place-value blocks, to help you add.” The materials show a data table that consists of the two columns Month and Water Bottle. The information on the table is the following: “September 325, October 243, November 468.” (4.OA.3 and 4.NBT.4)

• In Topic 10, Lesson 10-2, Solve & Share, Look Back!, students use their understanding that \frac{a}{b} is a multiple of \frac{1}{b} to multiply a fraction by a whole number as they independently solve a non-routine word problem. “Use Structure How does finding the total juice for 4 people with \frac{2}{3} cup servings compare to finding it for \frac{1}{3} cup servings? Why?” (4.NF.4b)

• In Topic 14, Lesson 14-2, Problem Solving, Problem 12, students generate a number pattern that follows a given rule in solving a non-routine problem. “12. Higher Order Thinking How many more batteries do 20 flashlights need than 15 flashlights? Explain.” The materials show a data table “Batteries for Flashlights” that consists of two columns, Number of Flashlights and Number of Batteries. The three rows indicate 1, 3; 2, 6; and 3, 9. (4.OA.5)

The materials for enVision Mathematics Grade 4 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 1, Lesson 1-2, Problem Solving, Problem 11, students attend to conceptual understanding as they recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in a place to its right. “Higher Order Thinking In 448,244, how is the relationship between the first pair of 4s the same as the relationship between the second pair of 4s?” (4.NBT.1)

• Topic 5, Lesson 5-9, Problem Solving, Problem 16, students attend to procedural skills and fluency as they find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors. “Ron’s Tires has 1,767 tires for heavy-duty trucks. Each heavy-duty truck needs 6 tires. How many heavy-duty trucks can get all new tires at Ron’s? (4.NBT.6)

• Topic 16, Lesson 16-4, Problem Solving, Problem 20, students attend to application as they use a picture of real-world location to determine if it is line symmetric. “The Thomas Jefferson Memorial is located in Washington, D.C. Use the picture of the memorial at the right to decide whether the building is line symmetric. If so, the building is line symmetric. If so, describe where the line of symmetry is.” The materials show an image of the Thomas Jefferson Memorial. (4.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, Solve & Share, students attend to application and procedural skills and fluency as they apply problem-solving methods to solve a word problem about toy donation involving addition and subtraction of multi-digit numbers. “A group of students collected donations for a toy drive. They collected a total of 3,288 toys one week and 1,022 toys the next week. They donated 1,560 toys to the Coal City Charity and the rest were donated to Hartville Charity. How many toys were donated to Hartville Charity? Use reasoning about numbers to show and explain how the two quantities of toys given to charity are related.” The materials include, “Thinking Habits Be a good thinker! These questions can help you. What do the numbers and symbols in the problem mean? How are the numbers or quantities related? How can I represent a word problem using pictures, numbers, or equations?” (4.OA.3)

• Topic 6, Lesson 6-2, Problem Solving, Problem 9, students attend to conceptual understanding and application as they extend their understanding of multiplicative comparisons to solve real-world problems. “Model with Math Dave is making soup that includes 12 cups of water and 3 cups of broth. How many times as much water as broth will be in the soup? Draw a bar diagram and write and solve an equation. (4.OA.2)

• Topic 13, Lesson 13-7, Problem Solving, Performance Task, Problem 10, students attend to conceptual understanding and procedural skills and fluency as they apply the perimeter formula to find out how much ribbon is needed for a card. “Making Thank You Cards Tanesha is making cards by gluing 1 ounce of glitter on the front of the card and then making a border out of ribbon.  She makes each card the dimensions shown.  How much ribbon does Tanesha need? 10. Be Precise  How much ribbon does Tanesha need? Use math language and symbols to explain how you solved the problem and computed accurately.” The materials show a Thank You card with width 85 mm and height 9 cm and a boy who states, “When you are precise, you specify and use units of measure appropriately.”  (4.MD.3)

The materials reviewed for enVision Mathematics Grade 4 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 4-7, Solve & Share, students make sense of problems and persevere in solving them as they solve multi-step problems that involve multi-digit multiplication. “Five students set a goal to raise $500 from their charity walk. Sponsors donated $25 for each mile walked. By how much did these students exceed or miss their goal? Solve this problem using any strategy you choose.” The materials show a data table that lists five students names and their individual “Miles Walked”. 

• Topic 11, Lesson 11-2, Problem Solving, Problem 6, students make sense of problems and persevere in solving them as they analyze information presented in a line plot that displays lengths in fourths to solve a problem involving addition and subtraction of fractions. “Nora weighed each of the 7 beefsteak tomatoes she picked from her garden. The total weight of the 7 tomatoes was 10\frac{3}{4} pounds. Her line plot shows only 6 dots. What was the weight of the missing tomato?” The materials show a line plot that indicates two dots at 1, one dot at 1\frac{1}{4}, one dot at 1\frac{2}{4}, one dot at 2, and one dot at 2\frac{1}{4}. 

• Topic 15, Lesson 15-6, Problem Solving, Performance Task, Problem 7, students make sense of problems and persevere in solving them as they measure angles in whole-number degrees.  “Mural Before Nadia paints a mural, she plans what she is going to paint. She sketches the diagram shown and wants to know the measures of ∠WVX, ∠WVY, ∠XVY, and ∠YVZ. Make Sense and Persevere  What do you need to find?” The materials show an image of two kites (X and Y) such that they form angles with the ground (W, Y, and Z) and share the common vertex Y.  

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

• Topic 4, Lesson 4-2, Guided Practice, Do You Understand?, Problem 1, students reason abstractly and quantitatively as they create an area model to help them understand the product of a two-digit number and a multiple of ten. “Draw an area model to show 20 \times 26. Then find the product.” 

• Topic 11, Lesson 11-1, Problem Solving, Problem 11, students reason abstractly and quantitatively as they use data shown in a line plot to determine which time occurred the most. “Use the line plot at the right. Reasoning Mr. Dixon recorded the times it took students in his class to complete a project. Which time was most often needed to complete the project?” The materials show the line plot, “Time Spent Completing Project,” which reflects student times in hours: four dots are above 2\frac{2}{4}, three dots above 2\frac{3}{4}, one dot above 3, three dots above 3\frac{1}{4}, and two dots above 3\frac{2}{4}. 

• Topic 16, Lesson 16-6, Problem Solving, Performance Task, Problem 7, students reason abstractly and quantitatively as they explain what the quantities given in the problem mean. “Dog Pen Caleb is designing a dog pen for the animal shelter.  He has 16 feet of fence, including the gate. His designs and explanation are shown. Critique Caleb’s reasoning. Reasoning What quantities are given in the problem and what do the numbers mean?” The materials show Caleb’s designs and reasoning using a sketch and notes: “Dog pens usually have right angles, so I just used rectangles. Both my plans used 16 feet of fence. I think the square one is better because it has more area.” The designs are a 4 ✕ 4 square and a 2  ✕ 5 rectangle. 

The materials reviewed for enVision Mathematics Grade 4 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. 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, Look Back!, students construct viable arguments and critique the reasoning of others as they examine a student claim about land area. “Construct Arguments Mary said Georgia’s land area is about 10 times greater than Hawaii’s land area. Is Mary correct? Construct a math argument to support your answer.” Students use the table in the Solve & Share to obtain information about the land area (in square miles) of Georgia and Hawaii.

• Topic 3, Lesson 3-6, Problem Solving, Problem 14, students construct viable arguments and critique the reasoning of others as they perform error analysis of provided student work and justify their thinking by solving the problem correctly. “Critique Reasoning Quinn used compensation to find the product of 4 \times 307. First, she found 4 \times 300 = 1,200. Then she adjusted the product by subtracting 4 groups of 7 to get her final answer of 1,172. Explain Quinn’s mistake and find the correct answer.” 

• Topic 7, Lesson 7-1, Convince Me!, students construct viable arguments and critique the reasoning of others as they evaluate a statement made about factors and begin to explore properties of factors. “Critique Reasoning Blake says, ‘Greater numbers will always have more factors.’ Do you agree? Explain.” 

• Topic 15, Lesson 15-2, Visual Learning Bridge and Convince Me!, students construct viable arguments and critique the reasoning of others as they explain their thinking about the partitioning of two circles and explain their argument.  Visual Learning Bridge (D), “Add to find the measure of an angle that turns through \frac{2}{6} of a circle. Remember \frac{2}{6}=\frac{1}{6}+\frac{1}{6}. Add to calculate the measure of \frac{2}{6} of a circle. 60° + 60° = 120° The angle measure of \frac{2}{6} of a circle is 120°. The materials show two circles: (i) a circle with a central angle labeled \frac{1}{6}=60° and (ii) a circle with a central angle labeled \frac{2}{6}=?. The materials suggest to teachers, “Construct Arguments How could you show that the angle turns through \frac{2}{6} of the circle?” Convince Me!,  “Critique Reasoning Susan thinks the measure of angle B is greater than the measure of angle A. Do you agree? Explain.” The materials show two circles: a small and large circle partitioned into sixths with a central angle equal to a sixth of the circle.

The materials reviewed for enVision Mathematics Grade 4 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 2, Lesson 2-8, Problem Solving, Performance Task, Problem 9, students model with mathematics as they use bar diagrams to represent real-world multistep problems involving addition, subtraction, and unknowns. “Ornithology is the scientific study of birds. Every year, some birds travel great distances, or migrate, to find food and start families. The table shows the distances five species of birds flew over one year, as observed by an ornithologist. How much farther did the Arctic Tern fly than the Pectoral Sandpiper and the Pied Wheatear combined? Model with Math Complete the bar diagrams to show how to find the answer to the hidden question and the main questions.  Write and solve equations.” The materials show a data table, “Distances Traveled by Birds,” which specifies the distance in miles of five species of birds. The materials also provide, for student use, two configurations of bar diagrams: one with which to model addition and the other to model subtraction.

• Topic 8, Lesson 8-6, Independent Practice, Problem 7, students model with mathematics as they use number lines to compare fractions. “Find equivalent fractions to compare. Then, write >, <, or =.” The materials show \frac{5}{6} ___ \frac{10}{12} and provide a number line representing 0 to 1 with one-twelfth intervals for students to use in their modeling.

• Topic 12, Lesson 12-1, Problem Solving, Problem 12, students model with mathematics as they use drawings, decimal grids, and decimals to represent fractions with denominators of 10 and 100. “Higher Order Thinking The diagram models the plants in a vegetable garden. Write a fraction and a decimal for each vegetable in the garden.” The materials show a diagram of a hundreds chart with red squares representing radishes, orange squares representing carrots, yellow squares representing corn, and green squares representing lettuce.

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 6-6, Problem Solving, Performance Task, Problem 7, students use appropriate tools strategically as they solve word problems using bar diagrams to visualize equations and represent the hidden questions. “Designing a Flag Rainey’s group designed the flag shown for a class project. They used 234 square inches of green fabric. After making one flag, Rainey’s group has 35 square inches of yellow fabric left.  How can Rainey’s group determine the total area of the flag?” The materials show a rectangular flag that consists of two green and one yellow horizontal stripes as well as one vertical orange rectangular region. Boxed text indicates, “Twice as much green as orange” and “3 times as much green as yellow.” “7. Appropriate Tools Draw diagrams and write equations to represent the hidden question(s). Be sure to tell what each variable represents.” Teacher guidance:  “Use Appropriate Tools What do you know that can help you draw a bar diagram to find out how much orange fabric is in the flag? What do you know that can help you draw a bar diagram to find how much yellow fabric is in the flag?”

• Topic 10, Lesson 10-4, Solve & Share, students use appropriate tools strategically when they use number lines and clock faces to solve problems involving time. “The Big Sur International Marathon is run on the California coast each spring. Sean’s mother was the women’s overall winner. How much faster was Sean’s mother than the women’s winner in the Ages 65-69 group? Tell how you decided. Solve this problem any way you choose.” The materials show a data table that specifies the times of men and women in three categories: overall and two age ranges. A boy states, “You can use appropriate tools such as bar diagrams or number lines to solve problems involving time.” Teacher guidance: “What tools do students use to visualize the difference in time? How can you use a clock or number line to show the amount of time between the shorter time and the longer time?” 

• Topic 15, Lesson 15-4, Problem Solving, Problem 20, students use appropriate tools strategically as they use a protractor to measure and draw angles.“Use a protractor to find the measure of the angle, then use one of the angle’s rays to draw a right angle. Find the measure of the angle that is NOT a right angle.” The materials show an obtuse angle.

The materials reviewed for enVision Mathematics Grade 4 meet expectations for supporting the intentional development of MP6: Attend to precision; and attend to the specialized language of mathematics, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

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

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

• Topic 3, Lesson 3-5, Problem Solving, Problem 9, students attend to precision when they accurate calculate the number of days in a eight year period. “Be Precise There are usually 365 days in each year. Every fourth year is called a leap year and has one extra day in February. How many days are there in 8 years if 2 of the years are leap years?” Teacher guidance: “Remind students that in order to find the answer they will have to complete two calculations: the number of days in 6 regular years lus the number of days in 2 leap years, or they can find the number of days in 8 years plus a day for each of the leap years.”

• Topic 7, Lesson 7-3, Problem Solving, Performance Task, Problem 8, students attend to precision when they find all the factor pairs for a whole number. “Store Displays A pet store needs 3 displays with the products shown. The boxes of kitty litter need to be stacked with the same number of boxes in each row. There needs to be at least 3 rows with at least 3 boxes in each row. What are all the ways the boxes of kitty litter could be stacked?” The materials show three products and their quantity: 50 fish bowls, 48 boxes of kitty litter, and 88 bags of dog food. “8. Be Precise What are all the ways the boxes of kitty litter can be stacked with at least 3 rows with at least 3 boxes in each row?”

• Topic 12, Lesson 12-6, Problem Solving, Performance Task, Problem 11, students attend to precision when they use appropriate numbers, units, and symbols to represent money. “Watching Savings Grow Tomas deposits money in his savings account every month. If he continues to save $3.50 each month, how much money will he have at the end of 6 months? 12 months? Use the table and Exercises 6-11 to help solve. 11. Be Precise Use the answers from the table to find how much money Tomas will have at the end of 12 months. Show your work.”  The materials show a table that represents the money in a savings account for months 0 to 3. Teacher guidance: “Attend to Precision Are you using numbers, units, and symbols appropriately? Does your answer have the correct units? Explain.”

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 5-4, Visual Learning Bridge and Guided Practice, Problem 2, students use specialized language to analyze solutions that have a remainder in terms of a problem situation. Visual Learning Bridge (A), “When you divide with whole numbers, any whole number that remains after the division is complete is called the remainder. Ned has 27 soccer cards in an album.  He puts 6 cards on each page. He knows 27 \div 6 = 4 with 3 left over, because 6 \times 4 = 24 and 24 + 3 = 27. Use an R to represent the remainder: 27 \div 6 = 4 R3. How do you use the remainder to answer questions about division?” The materials show an image of a  girl who states, “The remainder must be less than the divisor.” Guided Practice, Do You Understand?, Problem 2, “Dave is packing 23 sweaters into boxes. Each box will hold 3 sweaters. How many boxes will he need? Explain how the remainder affects your answer.”

• Topic 11, Review What You Know/Vocabulary Cards and Activity, Topic 11 Vocabulary, students use specialized language using terms such as line plot and scale. Teacher guidance: “Have students use Teaching Tool 25 (Vocabulary: Frayer Model) to display information about each vocabulary word [Line plot and Scale]. For example, have students complete the Frayer Model by writing “line plot” in the center, and then writing characteristics of a line plot, drawing an example and non-example, and defining a line plot in the boxes around the outside.” 

• Topic 16, Vocabulary Review, Use Vocabulary in Writing, Problem 11, students use specialized language as they use mathematics vocabulary to describe a shape. “Rebecca drew a figure. Describe Rebecca’s figure. Use at least 3 terms from the Word List in your description.” The materials show a blue parallelogram.

The materials reviewed for enVision Mathematics Grade 4 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 4, Lesson 4-1, Solve & Share, students look for and make use of structure when they use multiplication facts and place value to multiply by multiples of 10. “The principal of a school needs to order supplies for 20 new classrooms. Each classroom needs the following items: 20 desks, 30 chairs, and 40 pencils. How many of each item does the principal need to order? Solve these problems using any strategy you choose.” The materials show the image of a boy who states, “You can use structure. What basic facts can you use to help solve these problems? How are they related? Show your work in the space below!”

• Topic 8, Lesson 8-4, Problem Solving, Problem 28, students look for and make use of the structure of fractions to determine who ate more of a sandwich. “Use Structure Ethan ate \frac{4}{8} of his sandwich. Andy ate \frac{1}{2} of his sandwich. The sandwiches were the same size. a. Whose sandwich had more equal parts? b. Whose sandwich had larger equal parts? c. Who ate more? Explain.”

• Topic 12, Lesson 12-6, Convince Me!, students look for and make use of the structure of a drawing when they use a part of the drawing to determine where another point will be placed. when they use the decimal place-value system to solve real-world problems. “Use Structure Use the drawing of the trail shown. Where is the 1.5-mile mark on the trail? How did you decide?” The materials show a wavy trail that has labels, start and 0.5. Teacher Guidance: Look for and Make Use of Structure Students use knowledge of decimal meanings to locate a point on a number line beyond the points given instead of between given points.

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 1, Lesson 1-2, Convince Me!, students look for and express regularity in repeated reasoning when they use place-value blocks to reinforce how place-value positions are related. “Generalize Use place-value blocks to model 1 and 10, 10 and 100, 100 and 1,000.  What pattern do you see?” 

• Topic 9, Lesson 9-8, Solve & Share, students look for and express regularity in repeated reasoning when students apply their knowledge about adding fractions to add mixed numbers as they solve real-world problems. “Joaquin used 1\frac{3}{6} cups of apple juice and 1\frac{4}{6} cups of orange juice in a recipe for punch. How much juice did Joaquin use? Solve this problem any way you choose.” The materials show two measuring cups — 1\frac{3}{6} cups of apple juice and 1\frac{4}{6} cups of orange juice—and a girl who states, “Generalize. You can use what you know about adding fractions to solve this problem.”  

• Topic 16, Lesson 16-2, Problem Solving, Problem 12, students look for and express regularity in repeated reasoning when they look for attributes that all triangles have in common and that groups of triangles have in common and use these generalizations to categorize triangles by their sides and angles. “enVision STEM A rabbit’s field of vision is so wide that it can see predators that approach from behind. The diagram shows the field of vision of one rabbit and the field where the rabbit cannot see. Classify the triangle by its sides and its angles.” The materials present a diagram of the rabbit’s field of vision; it includes the labels “Seen by the left eye, Seen by both eyes, Seen by the right eye.”