6th Grade - Gateway 3

The instructional materials reviewed for enVision Mathematics Common Core Grade 6 meet expectations that the underlying design of the materials distinguishes between problems and exercises. In essence, the difference is that in solving problems, students learn new mathematics, whereas, in working exercises, students apply what they have already learned to build mastery. Each problem or exercise has a purpose.

Materials engage students in both problems and exercises through the grade level. Problems where students learn new mathematics are typically found in the Lesson’s Visual Learning Bridge. This portion of the lessons consists of visual examples that formalize the mathematics of the lesson by providing guided instruction of the math concepts with one example stepped-out. Examples from the Teacher Resource include:

• Lesson 1-3, Multiply Fractions, Visual Learning, Example 1, students model multiplication of fractions using two methods, “There was $$\frac{1}{4}$$ of a pan of lasagna left. Tom ate $$\frac{1}{3}$$ of this amount. What fraction of a whole pan of lasagna did Tom eat? One Way: Divide one half into fourths. Another way: Shade 1 of the 3 rows yellow to represent $$\frac{1}{3}$$. Shade 1 of the 4 columns red to represent $$\frac{1}{4}$$.”
• Lesson 4-6, Understand and Write Inequalities, Visual Learning, Example 1 students model inequalities to present more than one value, “How can you write an inequality to describe the ages of the children who must be accompanied by an adult at this sledding hill? One model: You can show some of the ages on a number line. Second model: Let a represent the ages of children who must be accompanied by an adult. Use the less than symbol (<) to write the inequality.”
• Lesson 7-3, Find Areas of Trapezoids and Kites, Visual Learning, Example 1, students use properties of a trapezoid to find its area, “The pasture is in the shape of a trapezoid. What is the area of the pasture? One strategy: Decompose the trapezoid into a rectangle and two right triangles. Second strategy: Find the area of each shape and then add the areas. The triangles are identical.”

Exercises, where students apply learning to build mastery, are typically found in the Practice and Problem Solving section. These exercises build independent proficiency, challenge higher-order thinking, and simulate high-stakes testing questions. Examples from the Teacher Resource include:

• Lesson 2-2, Represent Rational Numbers on the Number Line, Practice & Problem Solving, Item 30, students explain how to represent rational numbers using a number line, “Suppose $$\frac{a}{b}$$, $$\frac{c}{d}$$, and $$\frac{e}{f}$$ represent three rational numbers. If$$\frac{a}{b}$$ is less than$$\frac{c}{d}$$ and$$\frac{c}{d}$$ is less than$$\frac{e}{f}$$, compare$$\frac{a}{b}$$ and$$\frac{e}{f}$$. Explain.”
• Lesson 3-3, Write and Evaluate Numerical Expressions, Practice & Problem Solving, Item 27, students evaluate numerical expressions, “Frederick evaluates the numerical expression [(53.7 + 37.2) - (3$$^3$$+ 3.8)]- 8.6 and records the answer as 51.5. Lana evaluates the numerical expression 53.7 + 37.2 - 3$$^3$$+ 3.8- 8.6 and records the answer as 59.1. The expressions have the same numbers and operations. Explain how Frederick and Lana can both be correct.”
• Lesson 5-4, Represent and Graph Ratios, Practice & Problem Solving, Item 12, students explain how to use a table or graph to represent equivalent fractions, “Ishwar can read 5 pages in 15 minutes. Anne can read 15 pages in 1 hour. Explain How you could use a table or graph to find how much longer it would take Anne to read a 300-page book than Ishwar.”

The instructional materials reviewed for enVision Mathematics Common Core Grade 6 meet expectations that the design of assignments is not haphazard: exercises are given in intentional sequences.

Lesson activities within each Topic are intentionally sequenced developing student understanding and leading towards mastery of the content. Students are introduced to concepts and procedures with a problem-solving experience, Solve & Discuss it. The Visual Learning Bridge provides direct instruction that makes the important mathematics explicit through class discussion of student thinking and solutions. Examples from the Teacher Resource include:

• Lesson 6-5, Find the Percent of a Number, Solve & Discuss It!, Example 2, students use models to find the percent of a number in order to find the part in a percent problem, “D’wayne plans to wallpaper 72.5% of a 60-square foot wall. How many square feet of the wall does D’wayne plan to wallpaper? Find 72.5% of 60.”
• Lesson 8-3, Display Data in Box Plots, Solve & Discuss It!, Example 1, students describe data by finding the minimum value, maximum value, median and connect it to making a box plot, “Helen wants to display the lengths of 15 fish she caught this year to compare to the lengths of fish she caught last year. How can she use the data to make a box plot? Find the minimum, median, and maximum values of the data. Find the median for each half. Draw the box plot. Show a number line with an appropriate scale, a box between the first and third quartiles, and a vertical segment that show the median.”
• Lesson 8-6, Choose Appropriate Statistical Measures, Solve & Discuss It!, Example 1, students determine whether the mean, median, or mode best describes the data in a set and select the best measure of center to describe a data set, “Gary reviews the scores on his weekly quizzes. What measure should Gary use to get the best sense of how well he is doing on his weekly quizzes? Display Gary's scores on a dot plot. Describe the shape of the data. Then find the mean, median, and mode of the data set.”

The instructional materials reviewed for enVision Mathematics Common Core Grade 6 meet expectations that there is variety in what students are asked to produce. For example, students are asked to produce answers and solutions; but also, in a grade-appropriate way, arguments and explanations, diagrams, mathematical models, etc. Examples from the Teacher Resource include:

• Lesson 2-1, Understand Integers, Do You Understand?, Item 4, students use integers in real-world contexts, “Which amount represents a debt of two hundred fifty dollars, $250 or -$250? Explain.”
• Lesson 4-1, Understand Equations and Solutions, Practice & Problem Solving, Item 25, students determine whether a given number makes an equation true, “Alisa’s family planted 7 palm trees in their yard. The park down the street has 147 palm trees. Alisa guessed that the park has either 11 or 31 times as many palm trees as her yard has. Is either of Alisa’s guesses correct? Use the equation 7n = 147 to justify your answer.” 
• Lesson 8-1, Recognize Statistical Questions, Practice & Problem Solving, Item 11 students use data to make a frequency table to display answers to statistical questions, “Sergei asked his classmates, Will you take Spanish or French next year? He collected these responses: 15 classmates chose Spanish and 13 chose French. Make a frequency table to display the data.”

The instructional materials reviewed for enVision Mathematics Common Core Grade 6 meet expectations that manipulatives are faithful representations of the mathematical objects they represent and when appropriate are connected to written methods. 

Students have access to Anytime Math Tools powered by Desmos to build understanding and are accessible from the Tools panel online. Desmos tools include a graphing calculator, a scientific calculator, and a geometry construction tool. In addition, students have access to digital math tools such as algebra tiles, integer chips, area models, and bar diagrams. Students see an icon with a wrench when tools are suggested for use during examples and questions. Examples from the Teacher Resource include:

• Lesson 4-2, Apply Properties of Equality, Visual Learning, Example 1, students use a balance to define properties of equality, “What do the Addition and Subtraction Properties of Equality state? What does the Multiplication Property of Equality state? How does the Division Property of Equality differ from the other three properties of equality? Explain.”
• Lesson 6-2, Relate Fractions, Decimals, and Percents, Solve and Discuss It!, students use grid paper or colored tiles to model how to represent values out of 100 using fractions, decimals, and percents and make connections when converting between these forms, “The grid is shaded with blue, orange, and yellow? What part of the grid is shaded blue? What part is shaded orange? What part of the grid is shaded?”
• Lesson 7-7, Find Surface Areas of Pyramids, Visual Learning, Example 2, Try It!, students use a net to visualize the faces of square and triangular pyramids when finding the surface area, “Draw a net and find the surface area of the triangular pyramid. Find the Area (T) of each equilateral triangle. Find the surface area (SA) of the triangular pyramid.”

The instructional materials reviewed for enVision Mathematics Common Core Grade 6 meet expectations that materials support teachers in planning and providing effective learning experiences by providing quality questions to help guide students’ mathematical development.

Effective Mathematics Teaching Practices (ETP) side notes provide quality questions that are designed to promote reasoning and problem solving, support productive struggle, and engage students in mathematical discourse. Establish the Mathematical Goal provides questions related to the Essential Question. Use and Connect Mathematical Representations and Pose Purposeful Questions provide probing questions to enrich the mathematics. Elicit Student Thinking is an opportunity to formatively assess students to determine their understanding of concepts learned. Examples from the Teacher Resource include:

• Lesson 1-4, Understand Division with Fractions, Visual Learning, Key Concept, Pose Purposeful Questions, “How is dividing a whole number by a fraction similar to dividing a fraction by a whole number? Sample answer: You write the whole number as a fraction. You multiply by the reciprocal of the divisor.”
• Lesson 5-10, Relate Customary and Metric Units, Visual Learning, Do You Understand? Essential Question, “How can you use ratios to convert customary and metric units of measure? Students should know that to convert between customary and metric units they can find the conversion rate that relates the appropriate units and then find an equivalent rate or use dimensional analysis.” 
• Lesson 7-4, Find Areas of Polygons, Visual Learning, Example 3, Pose Purposeful Questions, “Why do you need to find the area of the polygon to find the amount of bamboo flooring needed? The polygon is decomposed into which shapes? How do you find the dimensions of the shapes? Why was addition used to find the area of the polygon instead of subtraction?”

The instructional materials reviewed for enVision Mathematics Common Core Grade 6 meet expectations that materials contain a teacher's edition with ample and useful annotations and suggestions on how to present the content in the student materials and in the ancillary materials. Where applicable, materials include teacher guidance for the use of embedded technology to support and enhance student learning. 

Effective Mathematics Teaching Practices (ETP) side notes provide Before, During, and After suggestions regarding lesson implementation. Examples from the Teacher Resource include:

• Lesson 3-1, Understand and Represent Exponents, Solve & Discuss It!, ETP: Before, “1. Introduce the Problem: Provide sheets of paper as needed. 2. Check for Understanding of the Problem: Engage students by asking them to fold a piece of paper in half in as many different ways as possible.”
• Lesson 7-2, Solve Triangle Area Problems, Solve & Discuss It!, ETP: During, “3. Observe Students at Work: Students might notice that both triangles have the same base and height, so the triangles have the same area. Students might calculate the area of the parallelogram and divide it by 2 to find the area of each triangle. If needed, ask, is there anything you learned in the previous lesson that can help you solve this problem? For Early Finishers, pose the question, How would the problem change if points B and C were moved 2 units up?”
• Lesson 8-5, Summarize Data Using Measures of Variability, Solve & Discuss It!, ETP: After, “4. Discuss Solution Strategies and Key Ideas: Have students share their work. Have them discuss the connections between the dot plot that varied a little; all values on the dot plots are close to the median. Have students discuss the connections between the dot plots that varied a lot; the values on the dot plot are spread out from the median. Have students discuss how their points are distributed and whether they need to be symmetrically distributed around 6; they do not, as long as they are the same number of points on either side of 6 the median will remain the same. 5. Consider Instructional Implications: After presenting Example 2, have students find the interquartile range for both of their dot plots from the Solve & Discuss It. Have students articulate what the interquartile ranges mean in the context of the problem; the interquartile ranges show the range of numbers of fruits eaten by the middle 50% of the students.”

The instructional materials reviewed for enVision Mathematics Common Core Grade 6 meet expectations that materials contain a teacher’s edition (in print or clearly distinguished/ accessible as a teacher’s edition in digital materials) that contains full, adult-level explanations and examples of the more advanced mathematics concepts in the lessons so that teachers can improve their own knowledge of the subject, as necessary.

Each Topic contains a Topic Opener, Math Background: Focus section that provides a discussion of the math content in the topic along with sample work and strategies that illustrate the underlying concepts to help teachers anticipate the works students will do. The Topic Opener also contains Advanced Concepts for the Teacher that provides examples and adult-level explanations of more advanced mathematical concepts related to the topic with explanations and examples to support teacher understanding of the underlying mathematical progressions. Examples from the Teacher Resource include:

• Topic 1, Use Positive Rational Numbers, Math Background, “Multiplying Decimals: The standard algorithm for multiplying decimals extends the algorithm for multiplying multi-digit whole numbers by combining it with properties of multiplication and exponents. However, the manner in which decimal multiplication is typically represented presents some apparent contradictions. Consider the product of 4.32 and 1.8. The partial products, in red, are integers. These results appear to be false. Likewise, the sum of the 3456 and 4300 appears to be 7.776, which is also false. These apparent contradictions arise because the original factors, 4.32 and 1.8, were each multiplied by the smallest integral power of 10 required to turn them into integers. This step is hidden in the standard algorithm above. The question has actually been changed so that integers are being multiplied. Now the partial products 3456 and 4320 make sense. But their sum 7776, is not the answer to the original problem.” Visual representations are provided.
• Topic 5, Understand and Use Ratio and Rate, Math Background, “Ratios as a Multiplicative Relationship: A ratio describes a multiplicative relationship between values. The comparison between the two values is not achieved by finding a difference, but rather by scaling. A ratio of 5 red to 1 blue does not describe relationships where there are 4 more red than blue, but rather that there are 5 times as many red as blue, or 5 red for every 1 blue. As the relationship is multiplicative, increasing just one quantity that is described by the ratio changes the ratio of the two quantities in a way that may be unintuitive. For example, you have 10 marbles with a ratio of 4 red to 1 blue. That means 8 marbles are red and 2 marbles are blue in the batch of 10. If 1 more red marble is added, the new ratio is not 5 red to 1 blue, but rather 9 red to 2 blue. 
• Topic 8, Display, Describe, and Summarize Data, Math Background, “Choosing a Data Displays. When choosing a data display, a number of key questions need to be considered: Is displaying individual data values important? Plots like dot plots and scatter plots include a point for every data value. In contrast, histograms show the data in given intervals and box-and-whisker plots show the range and interquartile range of data. Graphs such as dot plots quickly show clusters and gaps in the data, which can be hidden in a box-and-whisker plot. Is displaying the data as a part of a whole important? Graphs like pie charts and stacked bar charts show the proportion of a set of data that falls within a given category, unlike a regular bar graph which displays absolute but not relative amounts. Is displaying the distribution of the data important? Graphs such as histograms and dot plots help to show how the data is distributed. It makes the data range, outliers, and data skew more apparent. A box-and-whisker plot clearly shows the spread of the data and the median, but does not show the distribution as clearly as a histogram. Is identifying the relationship between sets of data important? Graphs like line plots and scatter plots may reveal how one variable relates to another. Scatterplots can be used to identify trend lines, which in turn can be used to estimate how the change in one variable affects a change in another.”

The instructional materials reviewed for enVision Mathematics Common Core Grade 6 meet expectations that materials contain a teacher’s edition (in print or clearly distinguished/ accessible as a teacher’s edition in digital materials) that explains the role of the specific grade-level mathematics in the context of the overall mathematics curriculum for kindergarten through grade twelve.

Each Topic Opener contains a section Math Background: Coherence that summarizes the content connections through the materials to prior and future grades. Look Back illustrates connections to previously taught concepts and skills include those within the grade, across content, or across grades. Look Ahead illustrates connections within or across grades. Examples from the Teacher Resource include:

• Topic 2, Integers and Rational Numbers, Math Background, Look Back, “Grade 5: Rational Numbers - In Grade 5, students extended their understanding of decimal place value to the thousandths place. They graphed decimals on a number line to help them compare and round decimals. They also extended their ability to do computations with rational numbers to include adding, subtracting, multiplying, and dividing decimals and fractions. Graph Points on the Coordinate Plane- In Grade 5, students learned about the coordinate plane and graphed points in the first quadrant to solve real-world and mathematical problems.”
• Topic 4, Represent and Solve Equations and Inequalities, Math Background, Look Back, “Earlier in Grade 6: Numerical and Algebraic Expressions- In Topic 3, students developed understanding of algebraic expressions, including evaluating algebraic expressions, generating equivalent expressions, and simplifying algebraic expressions. Graph Rational numbers on the Coordinate Place-In Topic 2, students graphed points with rational c\coordinates on a coordinate plane.”
• Topic 7, Solve Area, Surface Area, and Volume Problems, Math Background, Look Ahead, “Grade 7: Solve Measurement Problems -In Grade 7, students will solve real-world and mathematical problems involving area, surface, area, and volumen of two-and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prism.Scale Drawings, students will solve problems involving scale drawings, including finding the actual area of a figure given a scale drawing. Three-dimensional Figures, students will describe the two-dimensional figures that result from slicing three-dimensional figures.”

The instructional materials reviewed for enVision Mathematics Common Core Grade 6 meet expectations that materials provide strategies for gathering information about students’ prior knowledge within and across grade levels.

Materials provide strategies for gathering students’ prior knowledge. Examples include:

• Grade Level Readiness Test diagnoses students’ readiness for learning by assessing prerequisite content. This assessment is also available online and is autoscored. An Item Analysis is provided for diagnosis and remediation in the Teacher Resource. 
• Topic Readiness Assessment diagnoses students’ proficiency with Topic prerequisite concepts and skills. This assessment is available online and is autoscored. An Item Analysis is provided for diagnosis and remediation in the Teacher Resource. 
• Review What You Know, found at the beginning of each Topic, checks for understanding of key math concepts previously learned. An Item Analysis is provided for diagnosis and remediation in the Teacher Resource.

The instructional materials reviewed for enVision Mathematics Common Core Grade 6 meet expectations that materials provide strategies for teachers to identify common student errors and misconceptions. 

Materials provide strategies to identity student errors. Prevent Misconceptions are found in the Teacher Resource sidenotes for the Visual Learning portion of the lesson and Error Interventions are found in the Practice & Problem Solving Section. Examples from the Teacher Resource include:

• Lesson 3-1, Understand and Represent Exponents, Practice & Problem Solving, Error Intervention, Item 32, “Students may not realize that numbers can be written using exponents in more than one way. Q: What rule do you use to evalue 10$$^0$$ and 1 1.0$$^0$$? [Any nonzero number raised to an exponent of 0 has a value of 1.] Q: Write 16 in two different ways using exponents.”
• Lesson 5-3, Compare Ratios, Practice & Problem Solving, Error Intervention, Item 9, “Students may not know how to answer because the question does not include a second ratio table. Q: What is the ratio of minutes of news to minutes of music for radio station WILM? [4:25] Q: Do you need to find the number of minutes of news that each station plays in one hour to answer the question? Explain. [No; Whichever station has a greater ratio of news to music will broadcast more news each hour.]”
• Lesson 8-5, Summarize Data Using Measures of Variability, Do You Understand/Do You Know How?, Prevent Misconceptions, Item 3, “Explain that the mean absolute deviation shows the average distance each data value is from the mean. Q: What does MAD of 2 indicate? [On average, the data values are 2 units from the mean.] Q: What does MAD of 4 indicate? [On average, the data values are 4 units from the mean.] Q: Why does a greater MAD show greater variability? [It means that the data are more spread out from the mean.] ”

The instructional materials reviewed for enVision Mathematics Common Core Grade 6 meet expectations that materials provide opportunities for ongoing review and practice, with feedback, for students in learning both concepts and skills.

Materials provide opportunities for ongoing review of concepts and skills. Examples Include:

• Each Topic includes Review What You Know to activate prior knowledge and and review prerequisite skills needed for the Topic. Both vocabulary and practice problems are provided.
• The Cumulative/Benchmark Assessments are found at the end of Topics 2, 4, 6 and 8 assess students’ understanding and proficiency with concepts and skills taught throughout the year. An item analysis is provided for diagnosis and intervention. Students can take the assessment online, with differentiated intervention automatically assigned to students based on their scores. 
• The Math Diagnosis and Intervention System has practice pages which are specific to a skill or strategy (i.e. Markups and Markdowns and Mental Math). 
• There are multiple pages of extra practice available at Pearson Realize online that give students extra opportunities to review skills assigned by the teacher. Each of these pages is able to be customized by the teacher or used as is.
• Different games online at Pearson Realize support students in practice and review of skills, as well procedural fluency.

The instructional materials reviewed for enVision Mathematics Common Core Grade 6 meet expectations that materials offer ongoing formative and summative assessments, clearly denoting which standards are being emphasized. 

Formative and summative assessments clearly denote standards being assessed. Examples include:

• Try It! and Convince Me! are found following the Visual Learning Examples and assess students’s understanding of concepts and skills presented in each Example and results can be used to modify instruction. Standards assessed are listed in the Lesson Overview, Mathematics Overview, Common Core Standards, Content Standards.
• Do You Understand? And Do You Know How? are found after the Visual Learning instruction and assess students’ conceptual understanding and procedural fluency and results can be used to review content. Standards assessed are listed in the Lesson Overview, Mathematics Overview, Common Core Standards, Content Standards.
• Following each lesson is a Lesson Quiz that assesses students’ conceptual understanding and procedural fluency with the lesson content. Results can be used to determine differentiated instruction. Standards assessed are listed in the Lesson Overview, Mathematics Overview, Common Core Standards, Content Standards.
• At the end of each Topic there is a Topic Assessment with 2 forms, Form A and Form B, that assesses students’ conceptual understanding and procedural fluency with the topic content. Standards for these assessments are found in the teacher side matter under Item Analysis for Diagnosis and Remediation.
• At the end of each Topic there is a Performance Task with 2 forms, Form A and Form B, that assess students’ ability to apply concepts learned and proficiency with math practices. Standards for these assessments are found in the teacher side matter under Item Analysis for Diagnosis and Remediation.
• Cumulative/Benchmark Assessments found at the end of Topics 2, 4, 5, and 8 assess students’ understanding and proficiency with concepts and skills taught throughout the school year; results can be used to determine intervention. Standards for these assessments are found in the teacher side matter under Item Analysis for Diagnosis and Remediation.

The instructional materials reviewed for enVision Mathematics Common Core Grade 6 meet expectations that materials offer ongoing formative and summative assessments, which include aligned rubrics and scoring guidelines that provide sufficient guidance to teachers for interpreting student performance and suggestions for follow-up. 

Following Lesson Quizzes, Topic Assessments, Topic Performance Task and Cumulative/Benchmark Assessments Scoring Guides are provided. Teachers can also assign these assessments online where they are auto-scored and differentiated intervention is automatically assigned to students based on their scores. Examples from the Teacher Resource include:

• Lesson 1-7, Solve Problems with Rational Numbers, Lesson Quiz, “Use the student scores on the Lesson Quiz to prescribe differentiated assignments. Intervention 0-3 Points. On-Level 4 Points. Advanced 5 Points. You may opt to have students take the Lesson Quiz online. The Lesson Quiz will be automatically scored and appropriate remediation, practice, or enrichment will be assigned based on student performance.”
• Topic 3, Numeric and Algebraic Expressions, Topic Assessment, Form A, “Greater Than 85%: Assign the corresponding MDIS for items answered incorrectly. Use Enrichment activities with the student. 70% - 85%: Assign the corresponding MDIS for items answered incorrectly. You may also assign Reteach to Build Understanding and Virtual Nerd Video assets for the lessons correlated to the items the student answered incorrectly. Less Than 70%: Assign the corresponding MDIS items answered incorrectly. Assign appropriate intervention lessons available online. You may also assign Reteach to Build Understanding, Additional Vocabulary Support, Build Mathematical Literacy, and Virtual Nerd Video assets for the lessons correlated to the items the student answered incorrectly.”
• Topic 7, Solve Area, Surface Area, and Volume Problems, Performance Task, Form A, Item 3, “Rafael needs to find the area of each section of the design to determine how much paint he needs. Part A. Describe a strategy to find the total area of the sections painted with each color. Part B. Find the total area of the sections painted with each color. Complete the table. Show your work.” Two charts are provided for the teacher, Item Analysis for Diagnosis and Intervention and Scoring Rubric for forms A and B. The Item Analysis for Diagnosis and Intervention Chart contains information to help the teacher with RTI such as DOK, MDIS, and standard. The scoring rubric provides the teacher with solutions and scoring explanations. “Item 3, Form A 2 Points: Correct response. 1 Point: Partially correct response.”

The instructional materials reviewed for enVision Mathematics Common Core Grade 6 meet expectations that materials provide strategies to help teachers sequence or scaffold lessons so that the content is accessible to all learners.

The Topic Overview in the Teacher Resource provides a coherence section which enhances the opportunity to scaffold instruction by identifying prerequisite skills needed. All lessons include instructional notes and classroom strategies in the side matter labeled ETP, Effective Teaching Practices. ETP notes provide teachers with sample questions, differentiation strategies, discussion questions, possible misconceptions, and student “look fors” to assist in making content accessible to all learners. Additionally, the Solve and Discuss It! section provides teachers with Before, During, and After instruction notes to help scaffold learning for students. Examples from the Teacher Resource include:

• Lesson 4-2, Apply Properties of Equality, Example 3, ETP: Pose Purposeful Questions, “Q: Do you think adding 12 to one side and adding 12 to the other side would maintain equality? [Yes]. Q: What happens to the left side of the equation y - 12 + 12 = 30 + 12 when the expression is simplified? [The only thing left is y.] Q: What happens to the right side when the expression is true? Explain. [You get 42.] Q: How can you show that the equation is true? Explain. [Sample answer: Substitute 42 in for y in the original equation and then simplify. The equation is true because 30 = 30.]”
• Lesson 6-6, Find the Whole Given a Part and the Percent, Solve and Discuss It!, ETP: Before, “1. Introduce the Problem. Provide blank number lines, as needed. 2. Check for Understanding of the Problem. Ask students: How many games did your favorite team win last year?”
• Lesson 7-2, Solve Triangle Area Problems, Example 1, ETP: Use and Connect Mathematical Representations, “Q: How can you find the area of a parallelogram? [Multiply the base by the height]. Q: How do you know that the two triangles are identical? [Sample answer: One triangle can be rotated onto the other triangle. The heights, bases, and angles of the two triangles are the same.] Q: If the two different triangles have the same base and height, what do know about their areas? [The areas are the same.] Q: What relationship does the height of a triangle have to its base? [The height is perpendicular to the base.]”

The instructional materials reviewed for enVision Mathematics Common Core Grade 6 meet expectations that materials provide teachers with strategies for meeting the needs of a range of learners.

Each lesson contains Response to Intervention and Enrichment strategies in each lesson. Additional Examples and Additional Practice are provided if students need more support. At the end of each lesson Differentiated Intervention is provided for Intervention, On-Level, and Advanced learners. Examples from the Teacher Resource include:

• Lesson 3-4, Write Algebraic Expressions, Response to Intervention, “Use with Example 3: Some students may need additional help identifying parts of an expression. Provide students with strips of paper and scissors. Have them write expressions with at least three terms on the strips and then trade the strips with a partner. Q: How many terms does the expression on your strip have? Q: Make a cut in your strip after each operation to separate the terms. How many pieces do you have? Students should check this answer against their answer to the first question. Discuss any discrepancies.”
• Lesson 5-7, Solve Unit Rate Problems, Enrichment, “Use with Example 3 Try It!, Challenge advanced students to use the distance equation to solve the following problems. Ask them to show their work. Q: At the same rate, how far would the submarine travel in 5 hours? Q: At the same rate, how long would it take the submarine to travel 285 miles? Q: What would the submarine’s rate be if it traveled 60 miles in 2$$\frac{1}{2}$$ hours?”
• Lesson 7-7, Find Surface Area of Pyramids, Differentiate Intervention, Reteach to Build Understanding, Problem 1, “Find the surface area of the square pyramid. First, complete the net below with four identical ____faces and one____ face.”

The instructional materials reviewed for enVision Mathematics Common Core Grade 6 meet expectations that materials embed tasks with multiple entry-points that can be solved using a variety of solution strategies or representations.

Each lesson begins with a Problem-Based Learning activity, Solve & Discuss It, Explore It or Explain it! that offer multiple entry-points. 3-Act Mathematical Modeling tasks and Performance Tasks also include questions with multiple entry points that can be solved using a variety of representations. Examples from the Teacher Resource include:

• Topic 3, Numeric and Algebraic Expressions, Performance Task Form A, Item 1, “Ali is renting tables for the cookout and wants to seat an equal number of people at each table. She needs to decide how many tables to get. How could she arrange the seating so that a reasonable and equal number of people sit at each table? Explain.”
• Topic 5, Understand and Use Ratio and Rate, 3-Act Mathematical Modeling, Get in Line. Students are shown a video and then encouraged to consider the situation and ask any questions that come to mind. Teachers pose the Main Question, “How long will it take to get past the traffic light?” Teachers are given questions and tips to facilitate discussion about the 3-Act Mathematical Modeling activities. “Why do you think your prediction is the answer to the Main Question? Who had a similar prediction? How many agree with that prediction? Who has a different prediction?” 
• Lesson 7-6, Find Surface Area of Prisms, Solve & Discuss It!, “Marianne orders boxes to pack gifts. When they arrive, she finds flat pieces of cardboard as shown below. Marianne needs to cover each face of the boxes with green paper. What is the least amount of paper needed to cover the box? Explain. Suppose Melanie has only one large sheet of green paper that is 15 inches by 30 inches. Is the area of this sheet of paper great enough to cover all of the faces of one box? Explain.”

The instructional materials reviewed for enVision Mathematics Common Core Grade 6 meet expectations that materials suggest support, accommodations, and modifications for English Language Learners and other special populations that will support their regular and active participation in learning mathematics.

Each lesson contains instructional strategies for Emerging, Developing, and Expanding English Language Learners. Additionally, the Language Support Handbook provides Topic and Lesson instructional support and online academic vocabulary activities. Examples from the Teacher’s Resource include:

• Lesson 2-3, Absolute Values of Rational Numbers, English Language Learners, “Emerging: As students work through Examples 1-3, ask: Q: What words do you not know? Circle them. Write or display the words with simple definitions. Refer to the terms as you discuss the examples. Q: What words in your language have the same meaning as order? Write or display these words as appropriate. Be sure to point out that order is a homonym. Elicit only words that have the same meaning as the word used in the lesson.”
• Lesson 5-4, Represent and Graph Ratios, English Language Learners, “Developing: Have students complete the following sentences stems for each equivalent ratio in Example 1. The ratio ____ is equivalent to 3:2. It can be written as the ordered pair ______. [6:4; (6,4)] The point ____ shows that the cost of _____ balloons is ____. [(6,4); 6; $4]”
• Lesson 7-2, Solve Triangle Area Problems, English Language Learners, “Expanding: Have students read Example 1 with partners. Write parallelogram, decompose, area, triangle, base and height on the board. Ask each pair of students to write the vocabulary terms on index cards. Partners should take turns drawing a card and using the word in context.”

The instructional materials reviewed for enVision Mathematics Common Core Grade 6 meet expectations that materials provide opportunities for advanced students to investigate mathematics content at greater depth.

Each lesson provides an Enrichment side note with instructional strategies for advanced learners. The Problem-Based Learning activity provides instructional strategies During the lesson for Early Finishers. A Challenge question is presented in the teacher side notes for Practice & Problem Solving. Examples from the Teacher Resource include:

• Lesson 2-1, Understand Integers, Enrichment, “Use with Example 1 Try It!, Challenge advanced students to apply their answer to another situation. State the following question or write it on the board. May measures the temperatures at 5 A.M. Its is -4°F. At 9 A.M., the temperature is the opposite of its value at 5 A.M. What is the temperature at 9 A.M.? Q: What is the opposite of -4°F? Q: How much did the temperature rise from 5 A.M. to 9 A.M.?”
• Lesson 4-6, Understand and Write Inequalities, Solve & Discuss It!, ETP: During, “Early Finishers, Find some possible times Camilla would have to swim to break a record of 24.02 seconds.”
• Lesson 8-6 Choose Appropriate Statistical Measures, Practice & Problem Solving, Item 15, “Challenge: Reasoning, You can use this item to extend students’ understanding of mean, median, and mode. Q: If there were 7 data points could 12,500 occur three times?”

The instructional materials reviewed for enVision Mathematics Common Core Grade 6 meet expectations that materials provide a balanced portrayal of various demographic and personal characteristics.

Different cultural names and situations are represented. Role names are used instead of pronouns referencing gender. Objects, animals, and cartoon drawings are used in place of actual people. Examples from the Teacher Resource include:

• Lesson 2-1, Understand Integers, Additional Practice, Item 30, students compare integers, “Roberto and Jeanne played a difficult computer game. Roberto’s final score was -60 points, and Jeanne’s final score was -160 points. Use <, >, or = to compare the scores, then find the player who had the higher final score.” 
• Lesson 4-1, Understand Equations and Solutions, Visual Learning, Example 2, three students are shown guessing the correct number of marbles: one caucasion female, one caucasion male, and one African American female.
• Lesson 7-3, Find Areas of Trapezoids and Kites, Practice & Problem Solving, Item 15, students decompose a shape into a trapezoid and then find the area, “A craftsman wants to build this symmetrical fiddle. He needs to know the area of the face of the fiddle. How could he use the measurement shown to find the area? Use your strategy to find the area of the face of the fiddle.”