5th Grade - Gateway 2

The instructional materials reviewed for Ready Grade 5 meet the expectation for developing conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings. Students use pictures, manipulatives, and models to demonstrate conceptual understanding.

The Teacher Resource Book contains a section called Concept Extension that provides teachers with additional ways to support building students’ conceptual understanding. Examples of how the Concept Extension supports teachers in building conceptual understanding include:

• Unit 1 Lesson 3 is focused on how to deepen students’ understanding of the relationship between fractions and decimals (5.NBT.3) through representations of decimals and fractions on place value charts, number lines, and double number lines.
• Unit 5 Lesson 29 provides guidance to explore coordinates with decimal values (5.G.2). Students are asked to graph points on a coordinate plane with decimal values, i.e. (1, 6.50), (3, 3.50), etc.

Clusters 5.NBT.A and 5.NBT.B focus on understanding the place value system and performing operations with multi-digit whole numbers and decimals to hundredths.

• In Unit 1 Lesson 3 Read and Write Decimals (5.NBT.3) students use a meter stick in pairs to measure pieces of strings given in tenths and hundredths to write down and read decimals correctly. During this lesson, groups of students are also given cards to build a model of expanded from. Students also draw a place-value chart for the number 1.016 and deconstruct sixteen thousandths to create the expanded form. Finally, students create place-value charts and use counters to model decimal numbers given by the teacher.
• In Unit 1 Lesson 6 Divide Whole Numbers (5.NBT.6) students use base ten blocks to estimate quotients and build their understanding of place value by finding partial quotients.

Cluster 5.MD.C addresses concepts of volume and how volume relates to multiplication and addition.

• In Unit 4 Lesson 25 Finding Volume Using Unit Cubes (5.MD.4) students draw models to connect finding the area of a rectangle to finding the volume of a prism. Students show how they can determine the number of cubes in the bottom layer of a rectangular prism without counting them. Student groups get unit cubes and create the bottom layer of a prism and take measurements of the length and width. They continue adding layers to the prism and find its height. Once the prism is built, students calculate the volume then check their work by counting the cubes.

The instructional materials reviewed for Ready Grade 5 meet the expectation for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency. The materials provide opportunities to attend to procedural skill and fluency throughout the course, including fluency of multiplying multi-digit whole numbers using the standard algorithm (5.NBT.5).

All lessons provide an opportunity for students to use computation skills. Each lesson contains a Building Fluency section which is designed to be used twice during a lesson. The Student Practice and Problem-Solving Book contain fluency practice pages, and Math Center Activities are included in the Teacher Toolbox which include activities for both procedural skill and fluency.

• Unit 1 Lesson 5 focuses on multiplying multi-digit whole numbers using the standard algorithm, which is the fifth grade fluency. The lesson connects partial products to the standard algorithm. In addition, the Practice and Problem Solving Book for this lesson has students practice using the standard algorithm.
• Units 1 and 5 Math in Action integrates multi-digit multiplication (5.NBT.5).
• In Unit 2 Lessons 10 and 11 students add and subtract fractions with different numerators and denominators (5.NF.A).
• In Unit 3 Lesson 19 students write and evaluate expressions containing parentheses, using whole numbers, fractions, and decimals (5.OA.A).

iReady Door 24 Plus is a free iPad app for fact fluency practice and is only available on the Apple platform. The game includes the fluencies for Grade 5.

The instructional materials reviewed for Ready Grade 5 meet the expectation for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics without losing focus on the major work of each grade.

Opportunities to work with engaging applications are provided throughout the instructional materials. Each unit contains a lesson called Math in Action where students are exposed to non-routine problems that contain many points of entry, have more than one possible solution, integrate multiple standards, and include a Persevere on Your Own section. During guided practice Try It and independent practice Practice by Myself, students apply what they have learned to solve real-world problems. Most lessons also have an online interactive tutorial for students which features real-world applications. In addition, there are Problem Solving Connection problems where students use multiplication of fractions and mixed numbers (5.NF.6) and use division of unit fractions to solve real-world problems (5.NF.7c). The following are examples of the applications included in the instructional materials:

• In Unit 2 Lesson 16 students are presented with several opportunities to solve real-world problems that involve multiplication of fractions and mixed numbers.
• In Unit 2 Lesson 18 students are presented with several opportunities to solve real-world problems that involve dividing a unit fraction by a whole number and dividing a whole number by a unit fraction.
• In Unit 3 Math in Action Expressions, Patterns and Relationships students analyze and try different approaches to price snacks at a theater, reasoning about whether to use whole dollars or dollars and cents, and evaluate how different entry points yield information. Next, students write and evaluate expressions for different snack combinations, resulting in multiple solutions dependent on the number of combinations. Finally, they estimate the cost for two different program formats.

The instructional materials reviewed for Ready Grade 5 meet the expectations for balancing the three aspects of rigor. Overall, the three aspects of rigor are not always treated together and are not always treated separately within the materials.

Each lesson contains opportunities for students to build conceptual understanding, procedural skills and fluency, and apply their learning in real-world problems. Lessons are designed so students engage with all three components of rigor at different points in the lesson. During Guided Instruction and Guided Practice, students explore alternative solution pathways to master procedural fluency. During Independent Practice, students apply the concept in real world applications where they need to use both the procedural skills and their understanding of the concept to solve problems with multiple solutions and explain/compare their solutions.

For example, in Unit 1 Lesson 4 Compare and Round Decimals students use place value to estimate and round decimals. They are shown models of number lines, inequality statements ($$<$$, =, $$>$$), and tables to order numbers. They use these same models to determine how best to round numbers and to what value. Students apply their procedural skills to problems with various constraints and use their conceptual understanding of decimal order and place value to explain how their solutions represent the given situation. The Unit 1 Lesson 4 Quiz also includes problems where students demonstrate procedural fluency, a task where students must compare statements and determine their veracity based on underlying concepts of place value and decimals, and a two-part application problem.

Math in Action lessons occur at the end of most units. These lessons focus on application problems where students apply procedural fluency and conceptual understanding to solve problems in a non-routine, real-world context.

The instructional materials reviewed for Ready Grade 5 meet the expectations for identifying the Mathematical Practices (MPs) and using them to enrich the mathematics content within and throughout the grade.

The MPs for each lesson are identified in the CCSS Focus section as part of the Lesson Overview. SMP TIPs are found in the Teacher Resource Book throughout the lessons, and these tips highlight the integration of particular MPs within the lessons. The MPs are also identified for each lesson in the Table of Contents for the Teacher Resource Book on pages A4-A7.

Some examples of where the MPs are identified and used to enrich the mathematics content include:

• Unit 1 Lesson 4: MPs 1, 2, 4, 5, 6, and 7 are identified in the Lesson Overview. The SMP TIP in the lesson for MP1 states, “Students might not realize it, but the Model It activities are helping them make sense of the problem by presenting a variety of entry points for understanding what the numbers mean and how they are related. Ask students how these models add to their understanding of the problem and ways they could solve it.” The SMP TIP in the lesson for MP4 states, “Support students’ efforts to use mathematical models when they solve problems. Encourage students to represent the quantities in Try It on a number line or in a place-value chart.”
• Unit 2 Lesson 11: MPs 1, 2, 3, 4, 5, 6, 7, and 8 are identified in the Lesson Overview. The SMP TIP for MP5 states, “Take time to demonstrate how to draw a number line with benchmark fractions and give students an opportunity to practice. This number line is a practical tool that students can use in many situations to help the estimate a solution. When students are comfortable constructing this number line, they will use it often.” The SMP TIP for MP6 states, “Encourage students to attend to precision in their pictures and models. Point out the way that the number line and the fraction strips are labeled. Ask students to suggest additional ways to label the models.”
• Unit 4 Lesson 21: MPs 1, 2, 5, 6, and 7 are identified in the Lesson Overview. The SMP TIP in the lesson for MP7 states, “Students are looking for and making use of structure as they examine numerical patterns and relate the pattern to a rule. Encourage students to use a table to help them understand how to convert units of measure.” The SMP TIP in the lesson for MP2 states, “Students use quantitative reasoning to convert measurements. When converting units of measurement, they need to decide whether they should multiply or divide by comparing the relative sizes of the units they are converting.”

The instructional materials reviewed for Ready Grade 5 meet the expectation for carefully attending to the full meaning of each practice standard. Overall, the materials attend to aspects of the mathematical practices (MPs) during different lessons throughout the grade, so when taken as a whole, the instructional materials attend to the full meaning of each MP.

Examples of where the instructional materials attend to each of the MPs include:

• MP1: In the Unit 1 Lesson 8 Try It teachers are prompted to encourage students to verbalize their understanding of the problem before solving the problem. In Unit 2 Lesson 16 students complete problems that involve multiplying fractions. Students have to persevere in completing the problems as they come to recognize that one strategy for fraction multiplication might be more efficient than another.
• MP2: In Unit 4 Lesson 23 students are asked to generate questions to be answered using a line plot. Students then work with partners to answer each other’s questions. In Unit 4 Lesson 25 students reason about how the volume of a box would change if the height of the box changes.
• MP4: In Unit 2 Lesson 10 students model fractions visually to help them determine same-size parts of the whole when adding and subtracting fractions that have different denominators.
• MP5: In Unit 3 Lesson 20 students choose from creating tables, arrays, graphs, or pictures to help them analyze patterns and determine relationships among variables in several problems.
• MP7: In Unit 2 Lesson 15 students look at and use the structure of products greater than one of the factors and products less than one of the factors and predict what will happen with a different factor. In Unit 5 Lesson 21 students look for and make use of structure as they convert between metric units.
• MP8: In Unit 1 Lesson 2 students describe shortcuts that come from reasoning about the patterns they are investigating and state a “rule” for multiplying or dividing by powers of 10. In Unit 3 Math in Action students look for patterns and use repeated reasoning to solve problems.

The instructional materials reviewed for Ready Grade 5 meet the expectation for prompting students to construct viable arguments and analyze the arguments of others. Overall, the materials offer students multiple opportunities to construct viable arguments and/or analyze the arguments of others throughout the materials.

Examples where students are prompted to construct viable arguments and analyze the arguments of others include:

• Unit 1 Lesson 3 Question 19 students choose which of the four answers correctly represents 4.082. The student then answers questions about a fictional student: “Rachel chose D as the correct answer. How did she get the answer? Does Rachel's answer make sense?”
• Unit 1 Lesson 4 Question 4 states: “How does rewriting the fractional part or decimal part of 3.24 make it easier to compare 3.24 to 3.275?”
• Unit 3 Interim Assessment Performance Task students analyze 6 equations from Sophie, who omitted parentheses from the equations to determine which equations are correct and which ones need parentheses. Students also write a letter to Sophie to explain why parentheses were needed in some of the equations.

The instructional materials reviewed for Ready Grade 5 meet the expectation for assisting teachers to engage students in constructing viable arguments and analyzing the arguments of others concerning key grade level mathematics detailed in the content standards. The materials provide teachers with SMP TIPs to help facilitate students to construct arguments and/or analyze the arguments of others.

Examples where teachers are supported to help students construct viable arguments and analyze the arguments of others include:

• Unit 1 Math In Action Mathematical Discourse 1 has a note for teachers that states, “Invite students to discuss how they might begin to solve this problem. Allow them to describe different approaches, but don’t yet carry through an actual solution.” This is followed by a note for Mathematical Discourse 2 that states, “Explain that students will look at the sample solution on the next page to see one way the problem could be solved. Then they will read it again and discuss what makes it a good solution by using the Problem-Solving Checklist.”
• Unit 2 Lesson 13 SMP TIP states: “Encourage students to share their thinking or solution. This provides an opportunity for students to practice constructing viable arguments and critiquing the reasoning of others. Model and explain how students can explain their reasoning step by step, rephrase each other’s explanations, ask for clarification, or point out the misconception.”

The instructional materials reviewed for Ready Grade 5 meet the expectations for explicitly attending to the specialized language of mathematics. Overall, the materials for both students and teachers have multiple ways for students to engage with the vocabulary of Mathematics.

• The Student Practice and Problem Solving Book has notes at the bottom of the pages where mathematical vocabulary is defined.
• The Teacher Resource Book has Lesson Vocabulary for each lesson with mathematical terms and their definitions.
• Teachers are prompted in the Teacher Resource Book to have students use precise mathematical language. For example, Unit 4 Lesson 24 Let’s Talk About It states, “Remind students that when finding volume, they should label their answer as cubic units.”
• Each lesson has an “English Language Learners” section in the Teacher Resource Book that contains some ways to support vocabulary development for all students.
• Lessons contain language objectives. For example, Unit 1 Lesson 2 states, “Use language of equivalent fractions to describe equivalent decimals.”