The instructional materials for enVisionMATH California Common Core Grade 5 partially meet expectations that the materials develop conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings.

The instructional materials present a Problem-Based Interactive Learning activity (PBIL) and a Visual Learning Bridge (VLB) within each lesson to develop conceptual understanding. However, the PBIL and VLB are teacher-directed and do not offer students the opportunity to practice conceptual understanding independently through the use of pictures, manipulatives, and models.

Overall, the instructional materials do not consistently provide students opportunities to independently demonstrate conceptual understanding throughout the grade level.

• In Topic 1 Lesson 1-5, the Overview of PBIL states, “Students will learn to compare decimals through the thousandths place.” In the teacher-directed PBIL activity, students use a place-value chart to compare decimals to the thousandths place. The Develop the Concept: Visual section of the lesson shows three separate steps to compare decimals without place-value charts. Step 1 states, “Write the numbers, lining up the decimal points. Start at the left. Compare digits of the same place value.” The directions for the Independent Practice state, “Copy and complete. Write $$\gt$$, $$\lt$$, or = for each circle.” Students do not demonstrate the conceptual understanding of comparing decimals to the thousandths place independently as $$\gt$$, $$\lt$$, and = are shown as sample answers in the 12 problems in the Independent Practice.
• In Topic 3 Lesson 3-3, the Overview of PBIL states, “Students extend the addition of partial products to the standard algorithm to multiply two-digit numbers by multiples of ten.” In the teacher-directed PBIL activity, students use the area model to represent the product of 90 x 23. The directions for the Independent Practice state, “In 10 through 30, multiply to find each product.” Students do not demonstrate the conceptual understanding of multiplying two-digit numbers independently as products are shown as sample answers in the 21 problems in the Independent Practice.
• In Topic 9 Lesson 9-5, the Overview of PBIL states, “Students formulate a method for adding fractions with unlike denominators.” In the teacher-directed PBIL activity, students use tape diagrams to represent the sum of 1/4 and 3/8. The directions for the Independent Practice state, “In 7 through 22, find each sum. Simplify, if necessary.” Students do not demonstrate the conceptual understanding of adding fractions with unlike denominators independently as sums are shown as sample answers in the 16 problems in the Independent Practice.
• In Topic 11 Lesson 11-4, the Overview of PBIL states, “Students find a fraction of a fraction.” In the teacher-directed PBIL activity, students fold paper to demonstrate finding 1/4 of 1/2. The Develop the Concept: Visual section of the lesson describes the procedural steps of multiplying fractions. The directions for the Independent Practice state, “In 7 through 31, find each product. Simplify if necessary.” Students do not demonstrate the conceptual understanding of multiplying fractions independently as products are shown as sample answers in the 25 problems in the Independent Practice.

The instructional materials for enVisionMATH California Common Core Grade 5 partially meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics. Engaging applications include single and multi-step problems, routine and non-routine, presented in a context in which the mathematics is applied.

Each topic includes at least one Problem Solving lesson that can be found at the end of the topic. These lessons offer students opportunities to integrate and apply concepts and skills learned from earlier lessons. Within each individual lesson, there is a section titled, Problem Solving, where students practice the application of the mathematical concept of the lesson. However, the applications of mathematics in Problem Solving are routine problems.

The instructional materials have few opportunities for students to engage in non-routine application throughout the grade level. Examples of routine applications, where a solution path is readily available, are:

• In Topic 2 Lesson 2-7, students use addition and subtraction to solve a multi-step word problem. Independent Practice problem 4 states, “Elias saved $30 in July, $21.50 in August, and $50 in September. He spent $18 on movies and $26.83 on gas. How much money does Elias have left?”
• In Topic 4, Stop and Practice, students determine if a statement involving decimals is true or false and explain their reasoning. Number Sense problem 24 states, “The difference of 15.9 and 4.2 is closer to 11 than 12.”
• In Topic 9 Lesson 9-7, students solve word problems involving addition and subtraction of fractions with unlike denominators. Problem Solving problem 26 states, “Tara made a snack mix with 3/4 cup of rice crackers and 2/3 cup of pretzels. She then ate 5/8 cup of the mix for lunch. How much snack mix is left?”
• In Topic 11 Lesson 11-6, students solve real-world problems involving multiplication of fractions and mixed numbers. Problem Solving problem 25 states, “The city plans to extend the Wildflower Trail 2 1/2 times its current length in the next 5 years. How long will the Wildflower Trail be at the end of 5 years?”
• In Topic 11 Lesson 11-12, students use multiplication of mixed numbers to solve a word problem. Independent Practice problem 7 states, “Tina is making a sign to advertise the school play. The width of the sign is 2 2/3 feet. If the length is 4 1/2 times as much, then what is the length of the sign?”
• In Topic 14 Lesson 14-5, students use division and addition to solve a multi-step word problem. Independent Practice problem 14 states, “Students at Gifford Elementary collected stamps from various countries. The students collected 546 stamps from Africa, 132 from Europe, and 321 from North and South America. If a stamp album can hold 24 stamps on each page, how many pages will the stamps completely fill?”

The instructional materials reviewed for enVision Grade 5 partially meet expectations that the Standards for Mathematical Practice are identified and used to enrich mathematics content within and throughout the grade level.

Mathematical Practice standards are identified in three places within the Teacher Edition: Problem Based Interactive Learning activity, Guided Practice exercises, and Problem-Solving exercises. Throughout the teacher and student editions, there is a symbol that indicates that one or more MP is being used. Key phrases such as “Look for Patterns,” “Use Tools,” and “Reason” identify which practice is being highlighted. At the beginning of each lesson, all eight mathematical practices are listed. A check mark is placed beside each practice that is to be addressed within the lesson.

An example of an MP that is identified but does not enrich the mathematical content includes:

• In Topic 7 Lesson 7-1, MP1 is identified with the icon and the key word “Persevere." Question 8 states, “If Shandra wanted to cut the cloth into 100 strips, how wide would each strip be?”

An example of MPs that are identified and enrich the mathematical content include:

• In Topic 9 Lesson 9-1, MP6 is identified with the icon and the key word “Be Precise.” Question 22 states, “Which is a prime number?” Four answer choices are given for the students. Teachers are given information that reads, “Remind students to review the definition of a prime number.”
• In Topic 9 Lesson 9-2, MP4 is identified with the icon and the key word “Model.” Question 26 states, “Draw a diagram that could be used to build a cube.” Teachers are given information that reads, “Remind students that, when looking at the cube in Exercise 26, there are faces that they cannot see in the drawing.”

An example where the MPs are incorrectly labeled:

• In Topic 9 Lesson 9-6, MP7 is identified with the icon and the key word “Use Structure.” Question 26 states, “ Find the perimeter of the figure below.” A triangle with side lengths is shown.

Overall, all eight math practices are included within the curriculum and are not treated as separate standards. However, the standards are not used to enrich the content. They are aligned to some of the problems as an explanation to what math practice students might need to use to solve the problem.

The instructional materials reviewed for enVisionMATH California Common Core Grade 5 partially meet expectations for assisting teachers in engaging students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics.

The Teacher Edition contains a Mathematical Practice Handbook which defines each math practice and includes question stems for each MP to help the teacher engage students. MP3 offers the following questions stems: “How can I use math to explain why my work is right?”, “How can I use math to explain why other people’s work is right or wrong?”, and “What questions can I ask to understand other people’s thinking?”

The materials label multiple questions as MP3 or parts of MP3; however, those labeled have little information assisting teachers to engage students in constructing viable arguments or to critique the reasoning of others. The information that the materials provide is not specific and are often hints or reminders to give students while they are solving a problem.

There are some missed opportunities where the materials could assist teachers in engaging students in both constructing viable arguments and analyzing the arguments of others. For example:

• In Topic 1 Lesson 1-1, Problem Solving Question 31 states, “Critique Reasoning. Paul says that in the number 6,367, one 6 is 10 times as great as the other 6. Is he correct? Explain why or why not.” No teacher guidance is given for this question.

Examples where teachers are supported, although generally, to assist students in constructing viable arguments and analyzing the arguments of others include:

• In Topic 2 Lesson 2-5, Problem Solving Question 28 “Critique Reasoning. Juan adds 3.8 + 4.6 and gets a sum of 84. Is his answer correct? Tell how you know.” Teacher guidance for this MP is “If students have difficulty understanding how Juan’s answer is not correct, ask: If you add two numbers less than 5, will their sum be greater than 80?”
• In Topic 6 Lesson 6-6, Problem Solving Question 33 “Construct Arguments. Mary Ann ordered 3 pens and a box of paper on the Internet. Each pen cost $1.65 and the paper cost $3.95 per box. How much did she spend?” Teacher guidance for this MP is “Remind students that they may need to obtain information that is not given explicitly in the problem.”