7th Grade - Gateway 3

The instructional materials reviewed for enVision Mathematics Common Core Grade 7 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 2-5, Graph Proportional Relationships, Visual Learning, Example 1, students learn how to recognize a proportional relationship using graphs, “Tanya exercise for 30 minutes. She noted the Calories burned at three times during her workout. How can Tanya use this information to find how many Calories she burned after 15 minutes of exercise?”
• Lesson 4-4, Expand Expressions, Visual Learning, Example 3, students learn two methods for expanding complex expressions, “Simplify the expression -$$\frac{1}{3}$$ (2 - 3x + 3). One way: Use the Distributive Property first to distribute the coefficient -$$\frac{1}{3}$$. Another way: Simplify within the parentheses first. Then distribute the coefficient -$$\frac{1}{3}$$.”
• Lesson 8-4, Solve Problems Using Angle Relationships, Visual Learning, Example 1, students learn to solve problems involving adjacent and vertical angles, “A skewed intersection has two roads that intersect at more than 20 degrees away from 90°. Determine whether the road intersection shown is skewed by finding the measures of ∠ABC and ∠DBE.”

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 1-3, Add Integers, Practice & Problem Solving, Item 18, students add two negative integers to find a solution, “A deep-sea diver dives 81 feet from the surface. He then dives 14 more feet. The diver’s depth can be represented by -81 + (-14). What is the diver’s present location?”
• Lesson 5-2, Solve Two-Step Equations, Practice & Problem Solving, Item 9, students write and solve two-step equations, “While shopping for clothes, Tracy spent $38 less than 3 times what Daniel spent. Write and solve an equation to find how much Daniel spent. Let x represent how much Daniel spent.”
• Lesson 7-6, Find Probabilities of Compound Events, Practice & Problem Solving, Item 10, students find the probability of a compound event using a table, tree diagram, or organized list, “Gary spins two game wheels at the carnival. He will win a prize if both of the wheels land on any red section. How does the chance of winning change if different game wheels are used with more sections that aren’t red?”

The instructional materials reviewed for enVision Mathematics Common Core Grade 7 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 2-2, Determine Unit Rates and Ratios of Fractions, Solve & Discuss It!, students are presented with the problem and work independently to solve using various strategies, after which discussion ensues to develop key concepts important in finding fractions and decimals in unit rates. “Allison and her classmates planted bean seeds at the same time as Yuki and her classmates in Tokyo did. Allison is video-chatting with Yuki about their class seedlings. Assume that both plants will continue to grow at the same rate. Who should expect to have the taller plant at the end of the school year?” The picture shown states that Allison’s class plant grew 2.5 inches in 5 days and Yuki’s class plant grew 5.5 centimeters in 4 days.
• Lesson 4-8, Analyze Equivalent Expressions, Solve & Discuss It!, students are presented with the problem and work independently to solve using various strategies, after which discussion ensues to develop key concepts important in using equivalent expressions. “How many toothpicks make a triangle? Two triangles? Write an expression that represents the number of toothpicks needed to make x triangles that appear side-by-side in a single row, as shown. Explain your reasoning.”
• Lesson 8-8, Solve Problems Involving Surface Area, Solve & Discuss It!, students are presented with a situation allowing them to compare the surface area of a box and connect it to finding the area of a rectangular prism. “Alaya will paint the outside of a box with three different colors. Decide how she could paint the box. What is the total area that each color will cover?”

The instructional materials reviewed for enVision Mathematics Common Core Grade 7 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 1-9, Divide Rational Numbers, Practice & Problem Solving, Item 17, students critique the work of another and provide an justify their thinking, “Kayla wants to find 2$$\frac{2}{3}$$ $$\div$$ (-1$$\frac{3}{7}$$). She first rewrites the division as (2$$\frac{2}{3}$$)(-1$$\frac{3}{7}$$). What is wrong with Kayla’s reasoning?” 
• Lesson 3-4, Solve Percent Change and Percent Error Problems, Practice & Problem Solving, Item 17, students visually represent an expression, “You have 20 quarters. You find 40% more in your room. Then you go shopping and spend 50% of the total number of quarters. Write an expression that represents the total number of quarters you take with you when you go shopping.”
• Lesson 7-7, Simulate Compound Events, Practice & Problem Solving, Item 11, students justify their understanding of simulated probability and theoretical probability events, “How is the difference between simulated probability and the theoretical probability of an actual event related to the number of simulated trials conducted?”

The instructional materials reviewed for enVision Mathematics Common Core Grade 7 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 1-6, Multiply Integers, Visual Learning, Example 1, students use a number line to represent integers, “While playing a board game, unlucky Lawrence had to move back 2 spaces for 4 turns in a row. What integer represents his change in position? Use a number line to represent the change in position on the gameboard.”
• Lesson 4-6, Add Expressions, Solve & Discuss It!, students use blank number lines as they add expressions, “The Smith family took a 2-day road trip. On the second day, they drove ¾ the distance they traveled on the first day. What is a possible distance they could have traveled over the 2 days? Is there more than one possible distance? Justify your response.”
• Lesson 7-1, Understand the Likelihood and Probability, Solve & Discuss It, students use grid paper or squares of paper to add the possibilities of winning. “For a game show, Jared has to choose 1 of 8 boxes to win a prize. One of the boxes has a big prize, 3 boxes have a medium prize, 3 boxes have smaller prizes, and 1 box is empty. How confident should Jared be that whatever box he chooses, he will win a prize? Support your response with a mathematical argument.”

The instructional materials reviewed for enVision Mathematics Common Core Grade 7 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 3-6, Solve Simple Interest Problems, Visual Learning, Example 2, Try It, Elicit and Use Evidence of Student Thinking, “How can you find the amount of interest paid in one year? What ratio gives the interest rate?”
• Lesson 5-2, Solve Two-Step Equations, Visual Leaning, Example 2, Pose Purposeful Questions, “What does Jon want to do with his gift card? What does he want to find out? What steps did he take to isolate the variable? Why is the variable positive when the expression and number on each side of the equal sign are negative?” 
• Lesson 8-4, Solve Problems Using Angle Relationships, Visual Learning, Example 1, Use and Connect Mathematical Representations, “What is another name for ∠ABE? What is the difference between adjacent and vertical angles formed by two intersecting lines? What are the possible measures of the acute angle and obtuse angle formed by a skewed intersection? Why might a civil engineer be concerned if the intersection of roads is skewed?”

The instructional materials reviewed for enVision Mathematics Common Core Grade 7 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 1-2, Understand Rational Numbers, Solve & Discuss It!, ETP: Before, “1. Introduce the Problem: Provide blank number lines, as needed. 2 Check for Understanding of the Problem: Ensure students understand the problem by asking: ‘Would you rather surf on a shorter narrower surfboard or longer wider one? Why?’”
• Lesson 4-2, Generate Equivalent Expressions Explore It!, ETP: During, “3. Observe Student Work: How do students represent the total number of eggs with a diagram? Students might draw a diagram that indicates an unknown number of cartons with 6 eggs each and unknown number of cartons with 12 eggs each. How do students represent the total number of eggs using an expression? Students might let x represent the number of cartons with a half-dozen eggs and y represent the number of cartons with a dozen eggs and write the expression 6x + 12y. If needed, ask, What is known about the situation? What is unknown?”
• Lesson 7-4, Use Probability Models, Explain It!, ETP: After, “4. Discuss Solution Strategies and Key Ideas: Have students present their analyses of the models. Encourage them to use mathematical terms, such as outcome, chance, and probability. Have students discuss if all the team members have the same probability of being chosen caption (they do). Then have them discuss if the probability would be the same or different if their caption was not drawn randomly; if the name is not drawn randomly, each team member would not have the same chance of being picked. 5. Consider Instructional Implications: When presenting Example 1, make connections between the random drawing of a name in the Explain It, and the random drawing of marbles. In the Explain It, each team member had only one instance of their name in the draw, so each probability was equal. In Example 1, however, there are multiple marbles of each color. Have students discuss how the chess team draw could be altered to have uneven outcomes for team members; for example, some team members may be able to enter a name more than once.”

The instructional materials reviewed for enVision Mathematics Common Core Grade 7 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 2, Analyze and Use Proportional Relationships, Math Background, “Solving Proportions by Inspection: Equivalent ratios are ratios that express the same relationship between numbers. When two equivalent ratios have the same first or second term, then the other terms are equal. For example, if $$\frac{z}{x}$$ is equivalent to $$\frac{z}{y}$$, then x must equal y. For some equivalent ratios, one term is a multiple of a corresponding term of the other ratio. For example, the ratio $$\frac{2}{5}$$ shows a relationship of 2 to 5, shown in the diagram as 2 green for every 5 blue. The ratio $$\frac{4}{10}$$ shows a relationship that is equivalent to 2 groups of 2 and 5 groups of 2. Therefore the ratio $$\frac{4}{10}$$ is equivalent to $$\frac{2}{5}$$. Any ratio in the form of $$\frac{2n}{5n}$$, where n $$\not=$$ 0 is equivalent to $$\frac{2}{5}$$. This is generalized as a proportion, $$\frac{a}{b}$$ = $$\frac{an}{bn}$$.” Visual representations are provided.
• Topic 4, Generate Equivalent Expressions, Math Background, “Equivalence of Algebraic Expressions: Two expressions are equivalent if, for any value in the domain of the expression, the expressions represent the same value. Expressions that are not equivalent for all rational numbers may be equivalent on more restricted domains. For example, consider a single-element domain of (1). The following expressions are equivalent on the domain. Restricted domains can also be infinite. For all positive rational numbers, the expressions x and |x| have equal values. Therefore on the domain of x $$\ge$$ 0 the expressions x and |x| are equivalent. These same expressions are not equivalent on a domain of all rational numbers. Disproving equivalence over a domain requires only a single counterexample. Proving equivalence over infinite domains cannot be done by example. The properties of real numbers and the properties for simplifying algebraic expressions are given as a set of rules that always produce or verify equivalent expression for the set of all rational numbers.” Visual representations also provided.
• Topic 7, Probability, Math Background, “Likelihood: In Lesson 7-1, students analyze how likely an event is to occur. They use probability to examine fairness. In Lesson 7-2, students determine theoretical probability and then make predictions. Experimental Probability: In Lesson 7-3 and 7-4, students compare and contrast theoretical and experimental probability and construct probability models. Students use experimental probability to make predictions, estimate and evaluate a situation.” An example of likelihood outcomes is provided.

The instructional materials reviewed for enVision Mathematics Common Core Grade 7 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 3, Analyze and Solve Percent Problems, Math Background, Look Back, “Grade 6: Ratio Reasoning - In Grade 6, students learned to reason about ratios by using equivalent ratios and tables of equivalent ratios, and used their understanding of ratios to work with a special type of ratio called a percent. Rates- In Grade 6, students learned about a special type of ratio called a rate.”
• Topic 5, Solve Problems Using Equations and Inequalities, Math Background, Look Back, “Earlier in Grade 7: Define and Evaluate Expressions - Students identified parts of an expression and learned to view the expression as one entity. They performed arithmetic operations and substituted letters for unknown quantities to create algebraic operations.”
• Topic 7, Probability, Math Background, Look Ahead, “Grade 8: Two-Way Frequency Tables - In Grade 8, students will continue to find probabilities of simple and compound events. They will extend this knowledge to finding probabilities and making inferences and predictions using a two-way frequency table.”

The instructional materials reviewed for enVision Mathematics Common Core Grade 7 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 7 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 2-1, Connect Ratios, Rates, and Unit Rates, Do You Understand/Do You Know How?, Prevent Misconceptions, Item 4, “If students start by comparing $134.97 for Plan 2 with $34.99 for Plan 1, remind them that when comparing Internet service plans, they must compare the cost for the same number of months for each plan. Q: What is the unit rate for Plan 1? For Plan 2?”
• Lesson 5-3, Solve Equations Using the Distributive Property, Practice & Problem Solving, Error Intervention, Item 14, “Students often make mistakes with the signs of terms. Q: State in your own words the rules for multiplying terms with like or opposite signs. [Sample answer: Multiplying numbers with like signs results in a positive number. Multiplying numbers with opposite signs results in a negative number.] Q: Why is it useful to first change the subtraction expression to addition? [So it is obvious to multiply -3 and -r. It is easy to overlook the sign of the term when this first step is omitted.]”
• Lesson 8-5, Solve Problems Involving Circumference of a Circle, Do You Understand/Do You Know How?, Prevent Misconceptions, Item 3, “Make sure students understand that $$\frac{22}{7}$$ is a rational approximation but is not equal to . Q: Is the rational number $$\frac{22}{7}$$ equal $$\pi$$? Explain. [No; Sample answer: is not a rational number. It never terminates or repeats. The rational number $$\frac{22}{7}$$ is a useful estimate but is only an approximation of $$\pi$$.”

The instructional materials reviewed for enVision Mathematics Common Core Grade 7 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 7 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 7 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 1-9, Divide 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, Analyze and Solve Percent Problems, 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, Probability, Performance Task, Form A, Item 1, “Luca decides to play Wheel of Letters. To play the game, contestants spin a wheel with 26 letters equal sections, lettered A through Z. If the pointer lands on any letter in the phrase “COUNTRY FAIR” the contestant wins a prize. Part A: What is the probability that Luca will win Wheel of Letters? Is your answer a theoretical probability or an experimental probability? Explain.” 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 1, Form A 2 Points:Correct probability and explanation. 1 Point: Correct probability or explanation.”

The instructional materials reviewed for enVision Mathematics Common Core Grade 7 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 3-1, Analyze Percents of Numbers, Solve & Discuss It!, ETP: Before, “How would you find the total cost of a bill plus a 15% tip?” During: “How do you find an equal share among four people?” After: “What did Caleb assume about the amount each person would pay? Did Caleb and Jackie use the same steps to find one share with a 15% tip? Explain.”
• Lesson 5-2, Solve Two-Step Equations, Example 1, ETP: Use and Connect Mathematical Representations, “Q: What does each block labeled m in the bar diagram represent? [The cost of one movie ticket]. Q: Why was 6 subtracted from both sides of the equation? [Sample answer: In order to write an equivalent equation, an operation performed on one side of an equation must also be performed on the other side of the equation.] Q: Why was each side divided by 3? [To find the cost of 1 ticket.]”
• Lesson 8-1, Solve Problems Involving Scale Drawings, Example 3, ETP: Pose Purposeful Questions, “Q: Why is 4 the constant of proportionality? [The scale factor is 1 in. = 4 ft. The value, 4, that relates the scale drawing to actual measures is constant.] Q: Why are the length and height of the landscape drawing each multiplied by 4? [To find the length and width of the actual landscape.] Q: What is another way to solve this problem? [Sample answer: Relate the measures of the original landscape drawing to the measures of the drawing on the mural. 10 in. = 80 in., so the scale factor is 8. The height of the drawing on the mural is 8 x 8 in. = 64 in.] Have students mark the dimensions for the mural on the wall.”

The instructional materials reviewed for enVision Mathematics Common Core Grade 7 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 3-1 Analyze Percents of Numbers, Response to Intervention, “Use with Example 2: Some students may need reinforcement on the concept of percents greater than 100%. Remind students that a fraction with a numerator greater than the denominator is greater than 1. Q: How is $$\frac{3}{2}$$ written as a decimal? Is this greater than or less than 1? Connect this idea to a percent greater than 100%. Q: How do you write a percent as a decimal? Q: How is 150% written as a decimal? Is this greater or less than 1?”
• Lesson 5-2 Solve Two-Step Equations, Enrichment, “Use with Example 3, Challenge students to write an algebraic and arithmetic solution to the following scenarios involving the band in Example 3. Q: The number of percussion players is 2 more than twice the number of students who play trumpet. How many percussion players are there? Q: The number of flute players is 3 fewer than $$\frac{1}{8}$$ of the total number of students in the band. How many flute players are there?”
• Lesson 7-6, Find the Probabilities of Compound Events, Differentiate Intervention, Reteach to Build Understanding, Problem 2, “List all the possible outcomes to describe the side of the coin that may be facing up when the coin lands in the container. Are the outcomes equally likely?”

The instructional materials reviewed for enVision Mathematics Common Core Grade 7 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 1, Rational Number Operations, 3-Act Mathematical Modeling, Win Some, Lose Some. 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, “Who will win the game? What is the final score?” 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 4-2, Generate Equivalent Expressions, Explore It!, “A shipment of eggs contains some cartons with a dozen eggs and some cartons with half-dozen eggs. A. How can you represent the total number of eggs in the shipment using diagrams or images? Explain your diagram. B. How can you represent the total number of eggs in the shipment using expressions? What variables do you use? What do they represent? How do the two representations compare? How are they different?”
• Topic 7, Probability, Performance Task Form A, Item 4, “In Toss-N-Turn, contestants flip a fair coin and then spin the pointer of a spinner with seven equal sections, numbered 1-7. Contestants win if their coin lands heads up and the pointer lands on an even number. If Paulina wants to win a prize, should she play Toss-N-Turn or Rubber Duckie? Explain.”

The instructional materials reviewed for enVision Mathematics Common Core Grade 7 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 Resource include:

• Lesson 2-4, Describe Proportional Relationships: Constant of Proportionality, English Language Learners, “Emerging: Complete Example 1. Write the equation y = 0.8x on the board. Q: Discuss with a partner what x, y, and 0.8 mean in this equation. Listen for students who use content vocabulary and build academic language proficiency. Q: What could you use the equation y = 0.8(5) to do? [Find the amount of water the sponge filters in 5 hours.] Q: Discuss with a partner how you could find the time it takes the sponge to filter 6 liters, using either equivalent ratios or an equation.”
• Lesson 5-6, Solve Two-Step Inequalities, English Language Learners, “Developing: Ask students questions as they work through Example 2, in order to make sure they understand the terms in the problem. Q: What does exceed mean? What symbol will you use to represent an amount that exceeds the amount raised last year? [Sample answer: Exceeds means greater than. Use the greater than symbol in the inequality.] Q: What does raised mean in this problem? [Sample answer: To collect.]”
• Lesson 7-3, Understand Experimental Probability, English Language Learners, “Expanding: Read Example 1. Have students work with a partner and discuss the following concepts. Then have a volunteer share the results with the class. Q: Why does the experimental probability value change each time the experiment is carried out? Q: Does the experimental probability equal the theoretical probability in every probability model? Explain.”

The instructional materials reviewed for enVision Mathematics Common Core Grade 7 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 2-2, Determine Unit Rates with Ratios of Fractions, Enrichment, “Use with Example 3, Challenge advanced students to recognize that a unit rate on a map is the scale of the map. Have them reason about the unit rate for the map in the Try It. Q: What is the unit rate? Q: What does the unit rate tell you about the map? Q: How could you use the unit rate, or scale of the map, to find any actual distance from the map?”
• Lesson 4-5, Factor Expressions, Explain It!, ETP: During, “Early Finishers, How would the problem change if Tasha has 54 bottles of bubbles? Explain.”
• Lesson 8-4, Solve Problems Using Angle Relationships, Practice & Problem Solving, Item 13, “Some students can be challenged to find angle measures of complementary angles. Q: What is the measure of ∠A, and ∠B if the two angles were complementary, not supplementary? Explain.”

The instructional materials reviewed for enVision Mathematics Common Core Grade 7 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 2-6, Apply Proportional Reasoning to Solve Problems, Visual Learning, Example 4, four kids are shown: two caucasion (female and male) and two African Americans (female and male). Students use the picture with the data to answer the question.
• Lesson 4-5, Factor Expressions, Visual Learning, Example 2, students factor expressions with negative coefficients, “Rodrigo and Jordan each factor the expression -2x - 6. Who factored the expression correctly?” 
• Lesson 7-3, Understand Experimental Probability, Practice & Problem Solving, Item 15, students identify similar and different frequencies, “A basketball player makes 65% of all free throws in her first 5 seasons. In her 6th season she makes 105 out of 150 free throws. How does the observed frequency of her 6th season compare to the expected frequency? Provide a possible explanation for any similarities or differences in the frequencies.”