6th Grade - Gateway 2

The instructional materials for Ready Grade 6 meet expectations that the materials develop conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings. The materials include problems and questions that develop conceptual understanding throughout the grade level. The materials provide opportunities for students to independently demonstrate conceptual understanding throughout the grade through the use of visual models, real-world connections, mathematical discourse prompts, concept extensions, and hands-on activities.

• In Unit 1 Lesson 4 Solve Problems with Unit Rate, students use tables and double number lines to solve problems with unit rates. Additionally, students model finding a unit rate and extending that concept to solve a problem. The concept extension provided has students use multiple strategies to solve a problem. In Lesson 5, students use double number lines to solve percent problems.
• In Unit 2 Lesson 12 Understand Positive and Negative Numbers, students explain what they know from the position of a point in relation to zero. Students use variables to represent integers and explain how they know the positions of the variables.
• In Unit 3 Lesson 17 Equivalent Expressions, the Teacher Resource section provides questions to promote mathematical discourse to support building student conceptual understanding. Students relate the model to the expression. Furthermore, students explain different ways to show how the problem can lead to different equivalent expressions.
• In Unit 3 Lesson 18 Understand Solutions to Equations, students draw a pan balance for each of three, varied situations, write an equation, and explain their reasoning.

The instructional materials for Ready Grade 6 meet the expectation for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency. The instructional materials attend to procedural skills and fluency in lessons that address specific standards, activities, and problem sets. Many of these opportunities are provided by the Additional Fluency Practice book for Grade 6. In the Practice and Problem Solving book, the development of procedural skill and fluency happen daily through Fluency Skills Practice, and Fluency Repeated Reasoning Practice. In addition, the resources include math games for centers, and unit practice is included in the units.

• Fluency standard 6.NS.2 is explicitly addressed in Unit 2 Lessons 8-11.
• Unit 2 Lesson 9 Add and Subtract Decimals specifically addresses adding and subtracting decimals (6.NS.3).
• In Unit 4 Lesson 25 Volume, students evaluate expressions at specific values of variables which include expressions from formulas. Students perform arithmetic operations to develop procedural skills with evaluating expressions (6.EE.2).
• Unit 3, Lesson 16, Algebraic Expressions specifically addresses evaluating expressions (6.EE.2).

The instructional materials reviewed for Ready mathematics Grade 6 meet the expectation that teachers and students spend sufficient time working with engaging applications of mathematics without losing focus on the major work of the grade. Overall, the materials have opportunities for students to apply mathematical knowledge and/or skills in a real-world context.

During Independent Practice students often engage with problems that include real-world context and present opportunities for application. The Practice and Problem Solving workbook contains additional routine application problems, and Mid-Unit, Interim, and Unit Assessments often include problems that are contextual. For example:

• Unit 1 Lesson 4 Independent Practice Problem 6 states, “Ivan and Jeff buy a package of eight pens for $4.00. Ivan wants five of the pens, and Jeff wants three. How much should each student pay?” Students find the unit rate and apply it in the context of the problem.
• Unit 4 Lesson 24 Practice and Problem Solving Problem 6 states, “Susana is making a small box. The 20-cm by 20-cm front of the box will be glass. The other faces will be wood. How much wood does Susana need to make the box?” Students are given the dimensions for the depth of the box.
• Unit 2 Interim Assessment Performance Task is an opportunity for students to apply dividing fractions to a real-world situation. Students find the cost of a number of snack items, set a selling price for each item, account for ease of making change, and explain their reasoning (6.NS.3).

The instructional materials reviewed for Ready Mathematics Grade 6 meet the expectations for balance. Overall, the three aspects of rigor are not always treated together and are not always treated separately.

The Teacher’s Resource Book Program Overview Built for Rigor and Engagement outlines how the materials balance conceptual understanding, procedural skills and fluency, and application. The consistent structure of the lessons presents opportunities to “Understand,” where they develop conceptual understanding, followed by the sections Think It Through, Connect, and Apply, where they explore different procedural skills and application of the mathematics of the lesson. In addition, the Practice and Problem Solving book includes problems that present further opportunities to engage with the three aspects of rigor.

Students engage in the three aspects of rigor independent of each other. For example:

• Understand lessons focus on building conceptual understanding of mathematical content. Unit 3 Lesson 18 Understand Solutions to Equations begins with a question: “What does it mean to solve an equation?” The focus of the lesson is to build an understanding of equations and move students from concrete representations (balance) to algebraic representations.
• Skills and Strategy lessons help students acquire and apply efficient procedures for calculations. In Unit 4 Lesson 25, students are introduced to the formulas for the area of a triangle and a rectangle. During independent practice, students use these formulas to find the areas of different polygons.
• Application problems are found throughout the materials. In Unit 2 Lesson 11 Guided Practice Problem 16, students solve a routine application problem: “Yari wants to make egg biscuits to sell at a fundraiser. Eggs come in packs of 12 and biscuits come in packs of 8. What is the least number of packs of eggs and biscuits Yari can buy to have an equal number of each?”

Balance is displayed in each unit with multiple lessons where two or three aspects of rigor are interwoven.

• Unit 1 Lesson 5 Practice and Problem Solving Problem 7 states: “Kelly drove 440 miles in 8 hours. Alberto drove 468 miles in 9 hours. Both drove at a constant speed. Who drove further in 1 hour? How many miles further?” Students apply their understanding of unit rate and proportional reasoning to find how far each drove, who drove furthest, and by how much.
• In Unit 2 Lesson 14 The Coordinate Plane, students count units between points to find distance and then explain how to use absolute value to find the distance between two points.
• In Unit 5 Lesson 29 Analyze Numerical Data Independent Practice Problem 3, students are given a table that “represents daily attendance at two movie theaters for one week.” In Part A, students calculate the mean and median for each theater’s attendance. In Part B, students are asked: “Which is a better measure of center for Theater 1, mean or median? Explain.” In Part C, students determine the better measure of center for Theater 2. In this problem, students use both procedural skill and their conceptual understanding of measures of center.

The instructional materials reviewed for Ready Grade 6 meet expectations that the Standards for Mathematical Practice (MPs) are identified and used to enrich mathematics content within and throughout the grade level.

The Teacher Resource Book identifies the MPs for each lesson in the Table of Contents and 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 lesson. Some examples of where the MPs are identified and used to enrich the mathematics content include:

• Unit 3 Lesson 15 the SMP Tip for MP 7 states: “Recognizing the structure of $$10^3$$ as 10 X 10 X 10 prepares students to make use of the same structure in determining the value of $$6^3$$.”
• Unit 4 Lesson 25 Volume identifies MPs 1 and 4 as being present in the lesson. The teacher uses a diagram to help students make sense of problems. The SMP Tip states, “Regularly ask students to draw a sketch or diagram to understand the situation in word problems or to find a solution strategy (SMP 1).” An additional SMP Tip states, “...they have interpreted their mathematical results in the context of the model and reflected on whether these results make sense... (SMP 4)”
• In the Unit 1 End-of-Unit Performance Task, students solve a multi-step problem of purchasing items from a given amount of money that involves addition, subtraction, multiplication, and finding the percent of numbers. Students make sense of the problem by basing their purchasing decisions on the actual discounted prices. Students persevere in solving the task by finding the actual price, deciding what items to buy, how many to buy, and the amount of money left over. (MP 1)

The Mathematical Practices Handbook describes each of the MPs for students and provides students with questions to elicit thinking and discuss with a partner.

The instructional materials reviewed for Ready Grade 6 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.

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

• Unit 3 Lesson 15 Modeled and Guided Instruction-Evaluation Expressions with Exponents: A table contains the expressions that three fictional students wrote when asked to write and simplify the expression “6 plus 4 cubed.” Each fictional student wrote a different expression. The materials include the following questions below the table, “Which students will get the correct answer? Which, if any, of the expressions are equivalent? Explain.” Students analyze the expressions in the table and explain why the expressions are incorrect or correct.
• In Unit 3 Lesson 16, students agree or disagree with the statement: “A student wrote ½ to represent Evan’s share of the prize money.” Students explain why the answer makes sense and write another expression to represent Evan’s share of the prize money.
• In Unit 4 Lesson 22, students engage in four pair/share prompts: “How could you check to see if your answer makes sense? How could you help Manuel answer the question? How is this problem different than the ones before this lesson? Could you solve this problem another way?”

The instructional materials reviewed for Ready Grade 6 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 facilitate students in constructing arguments and/or analyzing the arguments of others.

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

• In Unit 1 Lesson 5, teachers find ways to encourage students to find a variety of ways to solve the problem and critique the reasoning of others to evaluate the solutions to determine which is most efficient.
• In Unit 2 Lesson 6, students construct arguments with the aid of models. Teachers give students chances to enhance mathematical communication skills through discussions in which students evaluate their own thinking and the thinking of other students.
• In Unit 5 Lesson 28, students compare two histograms and give an explanation of which histogram better represents the spread of the given data. The SMP Tip states, “When students discuss which histogram shows the data most effectively, they must construct viable arguments and critique the reasoning of others. Remind students to give reasons for their choices and relate their reasons to the types of information gained from each histogram. (SMP 3)”

The instructional materials reviewed for Ready Grade 6 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.

• Lesson Vocabulary is identified at the beginning of every lesson and used correctly to refer to mathematical terms and topics. For example, in Unit 4 Lesson 23, the definition of polygon is in both the teacher materials and the student materials and is defined as, “a closed plane figure whose sides are line segments that intersect only at their endpoints.”
• Each lesson has an English Language Learners section in the Teacher Resource Book that contains some ways to support vocabulary development for all students. Some examples are
• The Unit 1 Lesson 1 English Language Learners tip states, “Write part and whole on the board. Have volunteers explain the difference between the two terms. Show 3 red pens and 2 blue pens. Say: One part is 3 red pens. Another part is 2 blue pens. The whole is 5 pens. Ask students to use other classroom objects to model the terms part and whole.”
• The Unit 2 Lesson 11 English Language Learner tip states, “Write the word common on the board. Explain if two people have something in common, it is the same for both of them, Identify two students who have white shoes. Say: They have white shoes in common. Repeat with other characteristics.”
• Teachers pose questions to students and attend to precision using appropriate terminology. For example, in Unit 4 Lesson 24, teachers pose the following question, “How does the net of a square pyramid differ from the net of a triangular prism?”
• In the Teacher Instruction Book, mathematical vocabulary is defined in the Find Out More section.
• Teachers are prompted in the Teacher Resource Book to have students use precise mathematical language. For example, Unit 3 Lesson 17 Connect It states, “Richard says that 5h and 2h3 are like terms because they both have the variable h. Is Richard correct? Explain.”
• In the Student Practice and Problem Solving Book, mathematical terms are defined, along with a picture example in small square sections labeled, “Vocabulary.”