5th Grade - Gateway 2

The instructional materials for Common Core Coach Suite 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.

Materials present concrete visual models and lessons are scaffolded and redundant in problem structure, thus not meeting the full intent of conceptual understanding standards. When the materials present conceptual understanding, it is generally as part of class instruction. Students are typically shown a problem and solution using a representation or strategy. Later in the lesson, before students have any independent practice using the representation or strategy, they are given the formal mathematics rule. For example:

• In Common Core Coach Lesson 16 Multiplying Fractions, students use multiple shaded-fraction models to represent multiplication of a fraction by a whole number and how to shade models to represent multiplying a fraction by a fraction. However, in the practice portion of the lesson, students are given the models and find the product. Students do not use models to show the connection between their understanding of multiplying whole numbers with multiplying fractions. (5.NF.4)
• In Common Core Support Coach Lesson 6 Dividing Whole Numbers, during the “Plug In” portion of the lesson, students use place value to divide but then are provided the traditional algorithm for division, which is a Grade 6 standard. Students are given six practice problems to solve, using the standard algorithm. During the “Power Up,” students use the traditional algorithm for multiplication which is followed by four practice problems using the standard algorithm to find the product. During the “Ready to Go” portion of the lesson, students use the traditional algorithm for division. and the problems are scaffolded so students are filling in digits to long division problems. Students do not have the opportunity in this lesson to use a strategy based on place value to find quotients. (5.NBT.6)
• Common Core Performance Coach Lesson 2 Evaluating Numerical Expressions addresses order of operations (5.OA.1); however, students are given the mnemonic “PEMDAS” and do not have opportunities to learn how the operations are related to one another.

Students are given few opportunities to demonstrate conceptual understanding independently. During independent practice, students solve problems similar to those examples from class instruction, with slight difference in context and/or numbers. Students rarely create visual representations on their own or explain concepts to demonstrate understanding. There are Practice questions with labels such as “Write,” “Draw,” or “Prove,” where students explain mathematical concepts. The questions elicit students' ability to restate the mathematics ideas addressed by the teacher during class instruction. The materials address conceptual understanding standards in a proceduralized way and do not enhance the student's ability to form a conceptual understanding of major work within the grade.

• Common Core Support Coach Lesson 10 Multiplying Fractions, students both model multiplication and use the algorithm. The Teacher Edition states, “Guide students through shading the model. Monitor that students have correctly translated from the model to the equation. DO B: Show students the multiplication of 9 x 1/8 by renaming 9 as 9/1 then multiplying the numerators and multiplying the denominators.” In the practice problems of the “Power Up” section, students complete one model and then use the algorithm for four additional problems. Practice problems in the “Ready to Go” section direct students to “Shade the models and then multiply the fractions,” thus combining conceptual understanding with a procedure for multiplying fractions. (5.NF.4a and 5.NF.4b)
• In Common Core Performance Coach Lesson 8 Rounding Decimals, students are taught to use place- value charts to round decimals. Students are then given scaffolded close sentences to fill in with numbers/digits to walk them through the process of rounding decimals. In Practice problems, students answer true false/questions to tell if numbers are correctly rounded. Students do not have to independently create and fill out place value charts to help them round numbers. (5.NBT.A.4)

The instructional materials for Common Core Coach Suite Grade 5 partially meet expectations that they attend to those standards that set an expectation of procedural skill and fluency.

Specific lessons in the suite address the fluency standards in the CCSSM. All lessons in the suite provide students opportunities to use computation skills. Common Core Coach lessons conclude with two pages of Practice problems, Common Core Support Coach lessons conclude with three practice problems, and Common Core Performance Coach lessons conclude with independent practice problems. Additional fluency practice pages are found in Appendix A of the Common Core Coach Teacher’s Guide.

Specific lessons in the suite addressing fluency standards include:

• 5.NBT.5: Common Core Coach Lesson 8, Common Core Support Coach Lessons 5 and 6, Common Core Performance Coach Lesson 9
• 5.NF.1: Common Core Coach Lesson 13, Common Core Support Coach Lesson 8, Common Core Performance Coach Lesson 14
• 5.NBT.5: Common Core Coach Lesson 8

Since there are very few lessons specifically identified as addressing fluency standards in the suite, there are few opportunities for students to practice fluency skills throughout the entire year. Common Core Support Coach and Common Core Performance Coach do not identify specific fluency components; lessons in these components are developed around the fluency standards. However, additional procedural practice is not provided outside of those specific lessons.

• Lessons in all three components of the suite allow students to practice procedural skill. However, the lessons do not allow for students to demonstrate fluency since many scaffold thinking. Students fill in spaces, but are not provided opportunities to demonstrate fluency independently. Examples of this include:
  • Common Core Coach Lesson 8 Multiplying Whole Numbers
  • Common Core Support Coach Lesson 17 Measuring Volume of Rectangular Prisms
  • In the Common Core Coach Teacher’s Guide, teachers are instructed to assign various pages for fluency practice throughout the year. No further instructions are given for demonstrating mastery of procedural skill and/or fluency on these pages.

The instructional materials for Common Core Coach Suite Grade 5 partially meet expectations that the materials are designed so that teachers and students spend sufficient time working with engaging applications of the mathematics, without losing focus on the major work of the grade. Engaging applications include single and multi-step problems, routine and non-routine, presented in a context in which the mathematics is applied.

In the Common Core Coach Teacher’s Edition, the Table of Contents denotes lessons that apply skills to real-world problems. Common Core Support Coach does not label specific lessons as application or provide performance tasks that apply skills to real-world situations. In Common Core Performance Coach, there is one Performance Task at the end of each domain that applies concepts and skills to real-world problems. Non-routine problems are addressed in the Performance Tasks, and there are five Performance Tasks throughout the year.

Examples where the instructional materials do not provide opportunities for students to demonstrate the use of mathematics independently and flexibly in a variety of contexts include:

• Common Core Coach contains 28 lessons; 11 are identified as “Problem Solving.” Each “Problem Solving” lesson follows a specific, scaffolded procedure: “Read, Plan, Solve, & Check.” Students are then given five practice problems to solve, each with diminishing scaffolds.
  • In Lesson 7 Rounding Decimals Using Place Value, students use number lines and place value to round. There is one practice problem and students are given a place-value chart along with blank spaces for students to fill in to determine how to round the number. During independent practice students solve 22 procedural fluency problems related to rounding and four routine story problems (5.NBT.5).
• In Common Core Support Coach, lessons are scaffolded for students and include a checklist for students to follow when solving problems. Application standards and clusters are therefore not presented appropriately. For example, Lesson 14 Dividing Unit Fractions and Whole Numbers (5.NF.7.a, 5.NF.7.b, & 5.NF.7.c) contains nine questions within the main lesson, four of which are scaffolded and do not provide students opportunities to apply an understanding of multiplication. There are an additional three “Ready to Go” problems to solve that contain an example of a procedural process: “Read, Plan, Solve, and Check,” with a checklist for students to reference throughout the procedure.
• Common Core Performance Coach provides three to five worked problems as examples to solve the subsequent routine problems.
  • In Lesson 22 Solving Problems with Division of Unit Fractions and Whole Numbers Question 3, students determine if Julio or Dana used the correct equation to determine how many cashews they can each buy. In Question 5, students sort quantities of cereal into two groups: “Less than 7 servings” or “7 servings or more.” (5.NF.6)

The instructional materials for Common Core Coach Suite Grade 5 partially meet expectations that the three aspects of rigor are not always treated together and are not always treated separately. All three aspects of rigor are present in the program; however, they are mostly treated separately, and there is emphasis on procedural skill and fluency over the other aspects of rigor.

Common Core Coach designates lessons that are specifically identified as fluency, concept, or problem solving (application) lessons. However, the majority of the materials present the mathematics procedurally.

Although all three aspects of rigor are addressed in the materials, lessons throughout the suite quickly move from conceptual understanding or application to procedural skill. For example:

• In Common Core Coach Lesson 10 Adding and Subtracting Decimals, students use models and place- value charts to add decimals. As the lesson progresses, students use the traditional algorithm for adding decimals. Students then subtract decimals using place value and the traditional algorithm, but no models are given. Students use the standard algorithm to solve practice problems.
• In Common Core Performance Coach Lesson 14 Adding and Subtracting Fractions and Mixed numbers, students use models to add fractions, then use the algorithm for finding common denominators, creating equivalent fractions, and adding them. The lesson encourages students to use the algorithm and reinforces this in practice problems.

The instructional materials reviewed for Common Core Coach Suite 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.

The Standards for Mathematical Practice (MPs) are identified in the Teacher Editions of all three components of the suite. The MPs are identified throughout the “teacher notes” and are mostly found during the discussion portion of the lessons. The MPs are identified for each lesson and guidance is given to teachers as to where they are woven into lessons; however, the meaningful connections to the mathematical content are missing.

• Common Core Coach Teacher Edition page 4 states that “MP standards are woven throughout the curriculum, aligning to interactive questions, practice questions, and assessment items.” Within the Teacher Edition answer key, MPs are identified for each lesson and attached to specific story problems. For example, Lesson 8 Multiplying Whole Numbers addresses MP1 in Question 19: “Write a multiplication problem that has a product of 48,000.”
• Common Core Support Coach Teacher Edition includes a “Spotlight on Mathematical Practices” section in each lesson, providing the teacher with more detail on where the MPs are woven into the lesson and “notes that support teachers at point-of-use to develop strong mathematical behaviors.” (page xi)
• Common Core Performance Coach Teacher Edition identifies between two to four MPs in each lesson, linked to the discussion questions and journal prompts at the start of the lesson. For example:
  • Lesson 13 Dividing Decimals discussion question: “How are models for multiplication problems similar to models for division problems? How are they different?” (MP4)
  • Lesson 14 Adding and Subtracting Fractions and Mixed Numbers journal prompt: “When adding two fractions with unlike denominators, what should you do first?” (MP7)

Although MPs are identified throughout the suite, they do not serve to enrich the mathematical content. Since they are identified primarily in discussion questions, the materials lack guidance for teachers on how the highlighted MPs connect to the mathematics students are engaged in. Thus, the treatment of the MPs is fragmented across the suite, and do not provide opportunities for students to make connections and interact with the MPs in a meaningful way. For example:

Common Core Coach:

• Lesson 2 Writing and Interpreting Numerical Expressions and Lesson 10 Adding and Subtracting Decimals are the two lessons identified as addressing MP2. These two lessons contain three example problems that align to MP2. MP2 is not identified in any other lesson during the year.
• No MPs are identified for Lesson 14 Problem Solving: Adding and Subtracting Fractions and Mixed Numbers, Lesson 15 Problem Solving: Interpreting Fractions as Division, Lesson 18 Problem Solving: Multiplying Fractions and Mixed Numbers, and Lesson 20 Problem Solving: Dividing with Unit Fractions.

Common Core Support Coach addresses multiple MPs within lessons. For example:

• MP3 and MP6 are identified in every lesson.
• In Lesson 3 Reading and Writing Decimals, MPs 1-7 are identified as being present in the lesson. There is no guidance for teachers as to how the seven MPs meaningfully connect to the content within this lesson.

The instructional materials reviewed for Common Core Coach Suite Grade 5 partially meet expectations that the instructional materials prompt students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics.

While many of the lessons throughout all three components of the suite identify opportunities for students to discuss problem solving and mathematics, the materials provide few opportunities for students to construct arguments using mathematics or to analyze the reasoning and mathematics in others’ arguments. For example, the following lessons identify MP3, but they do not present opportunities for students to critique the work of other students or to construct an argument:

• In Common Core Coach Lesson 12 Dividing Decimals, after completing Example A, teachers are instructed “To reinforce the relationship between multiplication and division, have students use the multiplication sentence provided to check their work.” While students are encouraged to discuss their own work, using inverse operation is not an example of constructing an argument or critiquing the work of other students.
• In Common Core Coach Lesson 25 Recognizing Volume as Additive, teachers are directed to “Explain that there may sometimes be more than one way to separate a composite solid figure into smaller rectangular prisms. Stress that however the figure is divided, the total volume of the figure will be the same.”
• In Common Core Support Coach Lesson 7 Multiplying Decimals, teachers are instructed to “Have partners discuss briefly before group discussion. If needed, encourage students to represent $$3\times2.4$$ as repeated addition.”
• In Common Core Support Coach Lesson 12 Interpreting Multiplication of Fractions, teachers are directed to “Encourage students to see how comparing one factor to 1 helps them interpret the product. Have partners complete the following sentences, then share and compare their answers: If one factor is less than 1, then the product will be ___ the other factor. If one factor is greater than 1, then the product will be greater than ___. If one factor is equal to 1, then ____.”
• In Common Core Performance Coach Lesson 7 Comparing Decimals, students are given the following journal prompt: “Give a real-world example of when you might need to compare decimals. Then write an explanation for how you would find the decimal with the greater value.”
• Common Core Performance Coach Lesson 8 Rounding Decimals has students explain, “How can you tell when the rounded decimal will be less than or greater than the given decimal?”

The instructional materials reviewed for Common Core Coach Suite Grade 5 partially meet expectations that the instructional materials assist teachers in engaging students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics. There is little teacher guidance on how to lead discussions beyond the provided discussion questions, and there are missed opportunities to guide students in analyzing the arguments of others.

In Common Core Coach, items marked as addressing MP3 are often related to the teaching of the content with little or no assistance to teachers to engage students in both constructing viable arguments and analyzing the reasoning of others. Most often the materials prompt a discussion about “the topic” to assist students, but there are limited questions or prompts for teachers to support students’ development of arguments. Often suggestions for teachers regarding MP3 focus on students checking their work. Teachers are not provided with strategies for students to analyze the work of others in any of the lessons. For example:

• In Lesson 10 Adding and Subtracting Decimals, the teacher directions state, “Since addition is the opposite of subtraction, students can use the given addition problem to check their answer.”
• In Lesson 12 Dividing Decimals, the teacher directions state, “To reinforce the relationship between multiplication and division, have students use the multiplication sentence provided to check their work. Ask: Why are you multiplying 10.3 by 1.2 instead of by 12, which was the divisor you used to divide? Be sure students understand the importance of using the numbers in the original problem to check the answer.”
• In Lesson 17 Interpreting Multiplication of Fractions, the teacher directions state, “Students should use the products they find as examples to support the conclusion that the answer to the question is yes.”

Common Core Support Coach provides limited assistance to teachers in engaging students in both constructing viable arguments and analyzing the arguments of others. Most often when MP3 is identified, teachers are directed to “Have partners discuss briefly before group discussion.” Some lessons contain a section titled “Spotlight on Mathematics” that offers additional support for teachers in developing critical thinking by offering probing questions to use with students. In addition, teachers are frequently provided a prompt and sentence starter to assist students. However, these probing questions and prompts do not allow for students to construct arguments or critique the reasoning of others.

• Lesson 4 Comparing Decimals directs teachers to “Have partners discuss briefly before group discussion. As needed, have students look back at the Instruction box to review expanded form of decimal numbers.” This does not explicitly instruct teachers how to support students in constructing arguments about number sense or in analyzing the arguments of others.
• In Lesson 5 Multiplying Whole Numbers, students discuss with each other before group discussion. In addition, the Teacher Edition prompts, “What steps do you take to find the product by using partial products and place value?” and a sentence starter “I find the product by….”. However, there are no instructions to have student analyze the reasoning of others.

In Common Core Performance Coach, there are no directions to assist teachers in engaging students in constructing arguments or analyzing the arguments of others. Although discussion questions and journal prompts are provided, there are no prompts for teachers, or example student answers to guide the teacher. MP3 is addressed within the discussion questions at the beginning of lessons and within the journal prompt that accompanies most lessons. Additional support for the teacher related to MP3 is not present within the lessons.

• Lesson 8 Rounding Decimals addresses MP3 with the discussion question, “How can you tell when the rounded decimal will be less than or greater than the given decimal?” No additional support is offered for the teacher and students do not need to construct an argument or analyze others’ arguments based on this prompt.
• Lesson 19 Comparing Products to Factors, teachers are prompted to ask, “How can multiplication be used for scaling or resizing?” This question serves as a prompt, and students do not need to construct an argument or analyze the reasoning of others to provide an answer.

The instructional materials reviewed for Common Core Coach Suite Grade 5 partially meet expectations that materials use accurate mathematical terminology.

A glossary is available in the student edition of the materials, allowing students to look up definitions for highlighted terms they find in lessons, but these terms are not part of the lesson practice nor do they appear on assessments in any component of the materials.

The Common Core Coach Teacher Edition has a vocabulary box at the beginning of each lesson with mathematical terms and their definitions; however, the lessons do not specifically address the teaching of these terms. Common Core Support Coach does not address specific vocabulary and there is no instruction regarding mathematical language in this component of the suite. Key terms in Common Core Performance Coach are identified at the beginning of each lesson. There is little to no instruction on how to use the language of mathematics within any of the components of the suite. For example:

• Common Core Coach Lesson 16 Multiplying Fractions, teachers are instructed, “Show how to plot (3, 5) and (5, 3) on the same grid. This will help students understand that the two ordered pairs represent different locations on the coordinate plane.” It is unclear how the student demonstrates precise language. Other than highlighting key vocabulary and a glossary, there is little additional instruction on using accurate mathematical vocabulary.
• Common Core Coach Lesson 28 Extending Classification of Two-Dimensional Figures Example A introduces the concept that figures can be classified and sorted according to the types of sides they have. Students are instructed to use the glossary to help review terms such as perpendicular and parallel, but no instruction is given for students to use these mathematical terms throughout the lesson.
• Common Core Support Coach Lesson 7 Dividing Decimals directs teachers to “Have partners discuss briefly before group discussion. As needed, direct students’ attention to the models in [example] DO A. Have students compare the two sets of models.” There is no direction on how students are to use precise language during the discussion.
• While Common Core Performance Coach lessons never specifically identify key vocabulary, the ELL Support component does suggest students keep dictionaries and use the Frye Method to define terms, use them in a sentence, and give examples and non-examples. Sentence frames are also provided for teachers so they can assist students in understanding the concepts of mathematical terms. However, there is little to no instruction for teachers on how any student should use the language of mathematics.
  • Lesson 12 Multiplying Decimals directs the teacher to ask, “How can you make sure you have placed the decimal point in the correct location of a product when multiplying decimal factors?” This does not support students’ use of precise language around number sense, but rather students could provide a rote response.