The materials reviewed for enVision Mathematics Grade 7 meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings. 

Materials include problems and questions that develop conceptual understanding throughout the grade level. According to the Teacher Resource Program Overview, “Problem-Based Learning The Solve & Discuss It in Step 1 of the lesson helps students connect what they know to new ideas embedded in the problem. When students make these connections, conceptual understanding takes seed. Visual Learning In Step 2 of the instructional model, teachers use the Visual Learning Bridge, either in print or online, to make important lesson concepts explicit by connecting them to students’ thinking and solutions from Step 1.” Examples from the materials include:

• Topic 1, Lesson 1-4, Practice & Problem Solving, Problem 13, students write an equation based on a picture of a number line and use a number line to a different equation with the same difference. “Higher Order Thinking Use the number line at the right. a. What subtraction equation does the number line represent? b. Use the number line to represent a different subtraction equation that has the same difference shown in the number line. Write the subtraction equation.” (7.NS.1c)

• Topic 2, Lesson 2-2, Solve & Discuss It!, students extend their understanding of rates and ratios as they explore real-world problems. “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 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?” (7.RP.1 and 7.RP.3)

• Topic 4, Lesson 4-3, Solve & Discuss It!, students develop conceptual understanding by connecting sorting terms into categories to combining like terms. “How can the tiles be sorted?” Ten tiles are shown with various terms on them, some with and without variables. (7.EE.1)

Materials provide opportunities for students to independently demonstrate conceptual understanding throughout the grade. Practice & Problem Solving exercises found in the student materials provide opportunities for students to demonstrate conceptual understanding. Try It! provides problems that can be used as a formative assessment of conceptual understanding following Example problems. Do You Understand?/Do You Know How? Problems have students answer the Essential Question and determine students’ understanding of the concept. Examples from the materials include:

• Topic 3, Lesson 3-1, Do You Understand?, Problem 3, students develop conceptual understanding as they determine if two different procedures will yield the same results. “Construct Arguments Gene stated that finding 25% of a number is the same as dividing the number by $\frac{1}{4}$. Is Gene correct? Explain.” (7.RP.3)

• Topic 4, Lesson 4-6, Solve & Discuss It!, students develop conceptual understanding as they determine if a scenario has one solution or many solutions. “The Smith family took a 2-day road trip. On the second day, they drove $\frac{3}{4}$ 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.” (7.EE.1 and 7.EE.2)

• Topic 6, Lesson 6-1, Solve & Discuss It!, students analyze data from a sample and use it to gather information about the population. “The table shows the lunch items sold on one day at the middle school cafeteria. Use the given information to help the cafeteria manager complete his food supply order for next week.” A table is provided with two columns, one labeled “Lunch Item” and the other labeled “Number Sold”, the following information is in the table, Turkey Sandwich/43, Hot Dog/51, Veggie Burger/14, and Fish Taco/ 27. (7.SP.1)

The materials reviewed for enVision Mathematics Grade 7 meet expectations for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency. 

The materials develop procedural skill and fluency throughout the grade level. According to the Teacher Resource Program Overview, “Students develop skill fluency when the procedures make sense to them. Students develop these skills in conjunction with understanding through careful learning progressions.” Try It! And Do You Know How? Provide opportunities for students to build procedural fluency from conceptual understanding. Examples include:

• Topic 1, Lesson 1-2, Try It!, students use long division to convert a fraction to a decimal. “In the next several games, the pitcher threw a total of 384 pitches and used a fastball 240 times. What decimal should Janita use to update her report?”(7.NS.2d)

• Topic 5, Lesson 5-2, Practice & Problem Solving, Problem 10, students solve an equation that has decimals. “Solve the equation 0.5p - 3.45 = -1.2.” (7.EE.4a)

• Topic 8, Lesson 8-4, Practice & Problem Solving, Problem 10, students solve an equation to find the measurement of an unknown angle. “Find the value of x.” Two vertical angles are shown one measuring 125 and the other measuring (5x + 30). (7.G.5) 

The materials provide opportunities to independently demonstrate procedural skill and fluency throughout the grade level. Practice & Problem Solving exercises found in the student materials provide opportunities for students to independently demonstrate procedural skill and fluency. Additionally, at the end of each Topic is a Concepts and Skills Review which engages students in fluency activities. Examples include:

• Topic 2, Lesson 2-4, Practice & Problem Solving, Problem 7, students find the constant of proportionality of a given equation. “What is the constant of proportionality in the equation y = 5x?” (7.RP.2b)

• Topic 4, Lesson 4-5, Practice & Problem Solving, Problem 8, students factor the GCF from a given expression. “Factor the expression. 14x + 49” (7.EE.1 and 7.EE.2)

• Topic 8, Lesson 8-5, Practice & Problem Solving, Problem 7, students find the circumference of a circle in terms $\pi$ of given the diameter. “Find the circumference of the circle. Use $\pi$ as part of the answer.” An image of a circle is shown with a diameter of 7cm. (7.G.4)

The materials reviewed for enVision Mathematics Grade 7 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of mathematics. Engaging applications include single and multi-step problems, routine and non-routine, presented in a context in which mathematics is applied. 

The materials include multiple opportunities for students to independently engage in routine and non-routine application of mathematical skills and knowledge of the grade level. According to the Teacher Resource Program Overview, “3-Act Mathematical Modeling Lessons In each topic, students encounter a 3-Act Mathematical Modeling lesson, a rich, real-world situation for which students look to apply not just math content, but math practices to solve the problem presented.” Additionally, each Topic provides a STEM project that presents a situation that addresses real social, economic, and environmental issues, along with applied practice problems for each lesson. For example:

• Topic 1, 3-Act Mathematical Modeling: Win Some, Lose Some, Question 14, students predict the winner of a trivia game and the final score, “Construct Arguments If there were one final round where each contestant chooses how much to wager, how much should each person wager? Explain your reasoning." (7.NS.1 and 7.NS.3)

• Topic 5, STEM Project, Water is Life, students research filtration systems, decide which one they would purchase, and plan a fundraiser. Part of planning is writing an equation to represent the amount of money they will earn from a fundraiser to purchase the filtration system, "You have water to drink, to use to brush your teeth, and to bathe. You and your classmates will research the need for safe, clean water in developing countries. Based on your research, you will determine the type, size, and cost of a water filtration system needed to provide clean, safe water to a community. You will also develop a plan to raise money to purchase the needed filtration system.” (7.EE.3 and 7.EE.4)

• Topic 8, STEM Project, Upscale Design, students make scale drawings of existing paths or create plans for new walking paths or bikeways. "Review your survey results on the needs of walkers and bicyclists in your area. Choose an existing path or bikeway and make a scale drawing of the route. Add improvements or extensions to your drawing that enhance the trails and better meet the needs of users. If your area lacks a trail, choose a possible route and make a scale drawing that proposes a new path. How will your proposal enhance the quality of life and provide solutions for potential users?” (7.G.1 and 7.G.2)

The materials provide opportunities for students to independently demonstrate the use of mathematics flexibly in a variety of contexts. Pick a Project is found in each Topic and students select from a group of projects that provide open-ended rich tasks that enhance mathematical thinking and provide choice. Additionally, Practice & Problem Solving exercises found in the student materials provide opportunities for students to independently demonstrate mathematical flexibility in a variety of contexts. For example:

• Topic 2, Lesson 2-1, Practice & Practice Solving, Problem 9, students apply knowledge of solving multi-step problems with rational numbers to solving problems with ratios, rates, and unit rates. “Which package has the lowest cost per ounce of rice?” An image is provided of three bags of rice with various packaging sizes and prices, for example: One type of white rice is 12 punches and costs $4.56. (7.RP.1 and 7.RP.3)

• Topic 7, Pick a Project 7A, students design a game of chance and calculate theoretical probabilities of certain events happening. “Design and develop a game of chance. Find the theoretical probabilities of certain events (winning, losing, winning under certain conditions, losing under certain conditions). Test those probabilities. Have several people play your game. Write a report to accompany your game that compares the actual results to your theoretical results. Explain any inconsistencies.” (7.SP.5 and 7.SP.6)

• Topic 8, Lesson 8-9, Practice & Problem Solving, Problem 15, students solve real-world problems involving the volume of three-dimensional objects. “A cake has two layers. Each layer is a regular hexagonal prism. A slice removes one face of each prism, as shown. a. What is the volume of the slice? b. What is the volume of the remaining cake?” (7.NS.3 and 7.G.6)

The materials reviewed for enVision Mathematics Grade 7 meet expectations in that the three aspects of rigor are not always treated together and are not always treated separately. There is a balance of the three aspects of rigor within the grade.

All three aspects of rigor are present independently throughout the program materials. Examples, where materials attend to conceptual understanding, procedural skill and fluency, and application, include:

• Topic 1, Lesson 1-3, Explore It!, students extend their conceptual understanding of positive and negative numbers as they use number lines and absolute value to solve problems, “Rain increases the height of water in a kiddie pool, while evaporation decreases the height. The pool water level is currently 2 inches above the fill line. A. Look for patterns in the equations in the table so you can fill in the missing numbers. Describe any relationships you notice. B. Will the sum of 2 and (-6) be a positive or negative number? Explain.” (7.NS.1b and 7.NS.1d)

• Topic 4, Lesson 4-5, Practice & Problem Solving, Problem 14, students develop procedural skill and fluency in finding the GCF and factoring expressions. “You are given the expression 12x + 18y + 26. a. Make Sense and Persevere What is the first step in factoring the expression? b. Factor the expression.” (7.EE.1 and 7.EE.2)

• Topic 8, Lesson 8-8, Practice & Problem Solving, Problem 12, students use application of surface area knowledge to solve real-world problems, “A box has the shape of a rectangular prism. How much wrapping paper do you need to cover the box?” Illustration dimensions provided are h = 3 inches, w = 15 inches, and l = 16 inches. (7.G.6)

Multiple aspects of rigor are engaged simultaneously to develop students’ mathematical understanding of a single topic/unit of study throughout the materials. Examples include:

• Topic 1, Lesson 1-2, Practice & Problem Solving, Problem 18, students solve real-world problems while developing procedural skill and fluency with rational numbers. “Reasoning Aiden has one box that is 3$$\frac{3}{11}$$feet tall and a second box that is 3.27 feet tall. If he stacks the boxes, about how tall will the stack be?” (7.NS.2d)

• Topic 5, Mid-Topic Performance Task, students develop procedural skill and fluency as they apply their knowledge to writing and solving equations in a real-world scenario. “Marven and three friends are renting a car for a trip. Rental prices are shown in the table. Part A Marven has a coupon that discounts the rental of a full-size car by $25. They decide to buy insurance for each day. If the cost is $465, how many days, d, will they rent the car? Write and solve an equation.” A table is given with the “Item” in one column and the  “Price” in the next column. (7.EE.3 and 7.EE.4a)

• Topic 7, Lesson 7-3, Do You Know How?, Problem 4, students develop conceptual understanding and procedural skill and fluency as they find the theoretical probability for an event. “Kelly flips a coin 20 times. The results are shown in the table where ‘H’ represents the coin landing heads up and ‘T’ represents the coin landing tails up. 4. The theoretical probability that the coin will land heads up is ____.”. (7.SP.6 and 7.SP.7)

The materials reviewed for enVision Mathematics Grade 7 meet expectations for supporting the intentional development of MP1: Make sense of problems and persevere in solving them, and MP2: Reason abstractly and quantitatively, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

Some examples where the materials support the intentional development of MP1 are:

• Topic 3, Lesson 3-2, Practice & Problem Solving, Problem 11, students examine the relationships between the quantities and solve for the whole, “A restaurant customer left $3.50 as a tip. The tax on the meal was 7% and the tip was 20% of the cost including tax. a. What piece of information is not needed to compute the bill after tax and tip? b. Make Sense and Persevere What was the total bill?”

• Topic 4, Lesson 4-2, Practice & Problem Solving, Problem 17, students are asked to apply understanding of equivalent expressions and look at a group chat message in order to find the amount of money individuals put in. “Higher Order Thinking To rent a car for a trip, four friends are combining their money. The group chat shows the amount of money that each puts in. One expression for their total amount of money is 189 plus p plus 224 plus q. a. Use the Commutative Property to write two equivalent expressions. b. If they need $500 to rent a car, find at least two different pairs of numbers that p and q could be.”  

• Topic 5, Lesson 5-6, Practice & Problem Solving, Problem 11, students must make sense of the advertisement for car rental to ensure that they stay within budget. “Make Sense and Persevere Talia has a daily budget of $94 for a car rental. Write and solve an inequality to find the greatest distance Talia can drive each day while staying within her budget.” ( A chart that includes the rate per day and cost per mile for a car rental is included.) 

Some examples where the materials support the intentional development of MP2 are:

• Topic 6, Lesson 6-3, Practice & Problem Solving, Problem 9, students interpret and compare statistical measures and reason about data sets in both qualitative and quantitative forms. “Reasoning A family is comparing home prices in towns where they would like to live. The family learns that the median home price in Hometown is equal to the median home price in Plainfield and concludes that the homes in Hometown and Plainfield are similarly priced. What is another statistical measure that the family might consider when deciding where to purchase a home?”

• Topic 7, Lesson 7-3, Practice & Problem Solving, Problem 11, students reason about the difference between theoretical and experimental probability. “The theoretical probability of selecting a consonant at random from a list of letters in the alphabet is $\frac{21}{26}$. Wayne opens a book, randomly selects a letter on the page, and records the letter.  He repeats the experiment 200 times. He finds P(consonant)= 60%. How does the theoretical probability differ from the experimental probability? What are some possible sources for this discrepancy?”

• Topic 8, Lesson 8-1, Do You Understand? Problem 3, students reason about the difference between scaling on a map and in real life.“Reasoning Mikayla is determining the actual distance between Harrisville and Lake Town using a map. The scale on her map reads, 1 inch = 50 miles. She measures the distance to be 4.5 inches and writes the following proportion. $\frac{1 in}{4.5 in}$ = $\frac{50 mi}{x mi}$ Explain why her proportion is equivalent to $\frac{50 mi}{1 in}$ = $\frac{x mi}{4.5 in}$.

The materials reviewed for enVision Mathematics Grade 7 meet expectations for supporting the intentional development of MP3: Construct viable arguments and critique the reasoning of others, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

Student materials consistently prompt students to construct viable arguments. These opportunities are found in the following activities: Solve & Discuss It!, Explain It!, Explore It!, Practice & Problem Solving, Do You Understand?, and Performance Tasks. Examples include:

• Topic 5, Lesson 5-3, Explain It!, students use their understanding of the Distributive Property to construct arguments. “Six friends go jet skiing. The total cost for the adventure is $683.88, including a $12 fee per person to rent flotation vests. Marcella says they can use the equation 6r + 12 = 683.88 to find the jet ski rental cost, r, per person. Julia says they need to use equation 6(r + 12) = 683.33. A. Construct Arguments Whose equation accurately represents the situation? Construct an argument to support your response. B. What error in thinking might explain the inaccurate equation?”

• Topic 6, Lesson 6-4, Do You Understand?, Problem 3, students use their understanding of inferences to construct arguments. “Construct Arguments Two data sets have the same mean but one set has a much larger MAD than the other. Explain why you may want to use the median to compare the data sets rather than the mean.”

• Topic 8, Lesson 8-4, Explore It!, students analyze problems and use angle relationships to construct and justify arguments. “The intersecting skis form four angles. A. List all the pairs of angles that share a ray. B. Suppose the measure of $\angle$1 increases. What happens to the size of $\angle$2? $\angle$3?  C. How does the sum of the measures of $\angle$1 and $\angle$2 change when one ski moves? Explain. Focus on math practices Construct Arguments Why does the sum of all four angle measures stay the same when one of the skis moves? Explain.”

Student materials consistently prompt students to analyze the arguments of others. These opportunities are found in the following activities: Solve & Discuss It!, Explain It!, Explore It!, Practice & Problem Solving, Do You Understand?, and Performance Tasks. Examples include:

• Topic 2, Lesson 2-5, Do You Understand?, Problem 3, students analyze the arguments of others by interpreting if points contain a proportional relationship. “Construct Arguments Makayla plotted two points (0, 0) and (3, 33), on a coordinate grid. Noah says that she is graphing a proportional relationship. Is Noah correct? Explain.”

• Topic 6, Lesson 6-2, Do You Understand?, Problem 3, students analyze the arguments of others as they make inferences about a population from sample data. “Critique Reasoning Darrin surveyed a random sample of 10 students from his science class about their favorite types of TV shows. Five students like detective shows, 4 like comedy shows, and 1 likes game shows. Darrin concluded that the most popular type of TV shows among students in his school is likely detective shows. Explain why Darrin’s inference is not valid.” 

• Topic 7, Lesson 7-4,  Explain It!, students critique the reasoning of two members of the chess club about their chances of being captain, by using mathematical arguments to justify their answers. “The Chess Club has 8 members. A new captain will be chosen by randomly selecting the name of one of the members. Leah and Luke both want to be captain. Leah says the chance that she will be chosen as captain is $\frac{1}{2}$ because she is either chosen for captain or she is not. Luke says the chance that he is chosen is $\frac{1}{8}$. A. Construct Arguments Do you agree with Leah’s statement? Use a mathematical argument to justify your answer. B. Construct Arguments Do you agree with Luke’s statement? Use a mathematical argument to justify your answer.

The materials reviewed for enVision Mathematics Grade 7 meet expectations for supporting the intentional development of MP4: Model with mathematics; and MP5: Use appropriate tools strategically, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

The materials allow for the intentional development of MP4 to meet its full intent in connection to grade-level content. Examples of this include:

• Topic 3, Lesson 3-3, Practice & Problem Solving, Problem 15, students identify important quantities, use equations to represent their relationships, and interpret the results using mathematical models in a real-world situation. “Model with Math There are 4,000 books in the town’s library. Of these, 2,600 are fiction. Write a percent equation that you can use to find the percent of books that are fiction. Then solve your equation.” 

• Topic 5, Lesson 5-1, Practice & Problem Solving, Problem 14, students create an equation to represent the scenario of purchasing a pet iguana. “You want to buy a pet iguana. You already have $12 and plan to save $9 per week. a. Model with Math If w represents the number of weeks until you have enough money to buy the iguana, what equation represents your plan to afford the iguana? b. Explain how you could set up an equation to find the amount of money you should save each week to buy the iguana in 6 weeks.”

• Topic 7, Lesson 7-1, Practice & Problem Solving, Problem 15, students determine if a model created by another person is fair and explain how to make it fair. “Model with Math Henry is going to color a spinner with 10 equal-sized sections. Three of the sections will be orange and 7 of the sections will be purple. Is this spinner fair? If so, explain why. If not, explain how to make it a fair spinner.”

The materials allow for the intentional development of MP5 to meet its full intent in connection to grade-level content. Examples of this include:

• Topic 5, 3-ACT Mathematical Modeling: Digital Downloads, in Act 1, students watch a video on digital downloads and must determine how many songs a person can purchase using the balance on a gift card. In Act 2, Problem 7 asks students, “Use Appropriate Tools What tools can you use to solve the problem? Explain how you would use them strategically.“

• Topic 7, Lesson 7-7, Solve & Discuss It!, students explain how tools can be used to simulate events. “Jillian lands the beanbag on the board in about half of her attempts in a beanbag toss game. How can she predict the number of times she will get the beanbag in the hole in her next 5 attempts using a coin toss? Focus on math practices Use Appropriate Tools When might it be useful to model a scenario with a coin or other tool?”

• Topic 8, Lesson 8-2, Do You Understand?, Problem 2, students explain when it is appropriate to use technology strategically. “Use Appropriate Tools How can you decide whether to draw a shape freehand, with a ruler and protractor, or using technology?”

The materials reviewed for enVision Mathematics Grade 7 meet expectations for supporting the intentional development of MP6: Attend to precision; and attend to the specialized language of mathematics, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

Students are encouraged to attend to the specialized language of mathematics throughout the materials. A chart in the Topic Planner lists the vocabulary being introduced for each lesson in the Topic. As new words are introduced in a Lesson they are highlighted in yellow and students are encouraged to utilize the Vocabulary Glossary in the back of the text (with an animated version online in both English and Spanish) to find both definitions and examples where relevant. Lesson Practice includes questions that reinforce vocabulary comprehension and the teacher's side notes provide specific information about what math language and vocabulary are pertinent for each section.

Examples where students are attending to the full intent of MP6 and/or attend to the specialized language of mathematics include:

• Topic 1, Topic, Review, Use Vocabulary in Writing, students attend to the specialized language of mathematics and precision as they explain how they determined two decimals are equivalent. “Explain how you could determine whether${{\cfrac{21}{3}}\above{2pt}{\cfrac{120}{12}}}$and $\frac{7}{9}$ have the same decimal equivalent. Use vocabulary words in your explanation.”

• Topic 4, Mid-Topic Checkpoint, Problem 1, students attend to precision as they write an expression to represent a situation. “Vocabulary If you write an expression to represent the following situation, how can you determine which is the constant and which is the coefficient of the variable? The zoo charges the Garcia family an admission fee of $5.25 per person and a one-time fee of $3.50 to rent a wagon for their young children.” In the Mid-Topic Assessment, Problem 1, students attend to the specialized language of mathematics as they explain the difference between constants and variables. “Vocabulary How is a constant term different from a variable term for an expression that represents a real-world situation?” 

• Topic 6, Lesson 6-1, Try It!, students attend to the specialized language of mathematics as they fill in sentence frames with mathematical terminology. “Miguel thinks the science teachers in his school give more homework than the math teachers. He is researching the number of hours middle school students in his school spend doing math and science homework each night. The ______ includes all the students in Miguel’s middle school. A possible _________ is some of the students from each grade in the middle school.” 

• Topic 7, Get Ready!, Vocabulary, students attend to the specialized language of mathematics as they choose the best term from a box that fit the definition. “Choose the best term from the box to complete each definition. 1. A(n) ___ is a drawing that can used to visually represent information. 2. The number of times a specific value occurs is referred to as ______…” A box is given which has the following terms: equivalent, frequency, diagram, and ratio.

The materials reviewed for enVision Mathematics Grade 7 meet expectations for supporting the intentional development of MP7: Look for and make use of structure; and MP8: Look for and express regularity in repeated reasoning, for students, in connection to grade-level content standards, as expected by the mathematical practice standards.

Students are encouraged to look for and make use of structure as they work throughout the materials, both with the instructor's guidance and independently. Examples of where there is intentional development of MP7 include:

• Topic 1, Lesson 1-2, Practice & Problem Solving, Problem 14, students use the structure of a fraction to accurately find the decimal equivalent. “Use Structure Consider the rational number $\frac{3}{11}$. a. What are the values of a and b in a/b when you use division to find the decimal form? b. What is the decimal form for $\frac{3}{11}$?”

• Topic 3, Lesson 3-2, Practice & Problem Solving, Problem 15, students use structure to identify and align the part, whole, and percent to set up a proportion to solve real-world problems, “A school year has 4 quarters. What percent of a school year is 7 quarters?”

• Topic 4, Lesson 4-8, Practice & Problem Solving, Problem 13, students analyze relationships between quantities in real-world situations for equivalency. “Use Structure The area of a rectangular playground has been extended on one side. The total area of the playground, in square meters, can be written as 352 + 22x. Rewrite the expression to give a possible set of dimensions for the playground.” 

Students look for and express regularity in repeated reasoning as they are engaged in the course materials. Examples of intentional development of MP8 include: 

• Topic 3, Lesson 3-5, Do You Understand?, Problem 3, students use repeated reasoning to make a general statement about the price being similar or different after a price markdown and then markup. “Generalize When an item is marked up by a certain percent and then marked down by the same percent, is the sale price equal to the price before the markup and markdown?” 

• Topic 6, Lesson 6-3, Focus on math practices, students look for the regularity of real-world data in various display forms to draw a conclusion. “Reasoning Use your data display, what can you infer about the number of siblings that most seventh graders have? Explain?”

• Topic 8, Lesson 8-3, Practice & Problem Solving, Problem 12, students analyze triangles and generalize that its side and angle conditions determine if it results in one triangle, more than one triangle, or no triangle. “Given two side lengths of 15 units and 9.5 units, with a nonincluded angle of 75°, can you draw no triangles, only one triangle, or more than one triangle?”

The materials reviewed for enVision Mathematics Grade 7 meet expectations for providing teacher guidance with useful annotations and suggestions for how to enact the student materials and ancillary materials, with specific attention to engaging students in order to guide their mathematical development. 

The Teacher’s Edition Program Overview provides comprehensive guidance to assist teachers in presenting the student and ancillary materials. It contains four major components: Overview of enVision Mathematics, User’s Guide, Correlation, and Content Guide.

• The Overview provides the table of contents for the course as well as a pacing guide. The authors provide the Program Goal and Organization, in addition to information about their attention to Focus, Coherence, Rigor, the Math Practices, and Assessment.

• The User’s Guide introduces the components of the program and then proceeds to illustrate how to use a ‘lesson’: Lesson Overview, Problem-Based Learning, Visual Learning, and Assess and Differentiate. In this section, there is additional information that addresses more specific areas such as STEM, Pick a Project, Building Literacy in Mathematics, and Supporting English Language Learners.

• The Correlation provides the correlation for the grade.

• The Content Guide portion directs teachers to resources such as the Scope and Sequence, Glossary, and Index.

Within the Teacher’s Edition, each Lesson is presented in a consistent format that opens with a  Lesson Overview, followed by probing questions to provide multiple entry points to the content, error intervention, support for English Language Learners, Response to Intervention, Enrichment and ends with multiple Differentiated Interventions.

Materials include sufficient and useful annotations and suggestions that are presented within the context of the specific learning objectives. The Teacher’s Edition includes numerous brief annotations and suggestions at the topic and lesson level organized around multiple mathematics education strategies and initiatives, including the CCSSM Shifts in Instructional Practice (i.e., focus, coherence, rigor), CCSSM practices, STEM projects, and 3-ACT Math Tasks, and Problem-Based Learning. Examples of these annotations and suggestions from the Teacher’s Edition include:

• Topic 1, Lesson 1-1, Solve & Discuss It!, “Purpose Students engage in productive struggle to connect making sense of phrases to using integers to describe a real-world situation in the Visual Learning Bridge. Before Whole Class 1 Introduce the Problem Provide number lines, as needed. 2 Check for Understanding of the Problem Engage students with the problem by showing them footage of a rocket launch.”

• Topic 3, Lesson 3-3, Convince Me!, “How does the percent describe how the weights are related?” Teacher guidance: “Q: How does the definition of percent relate the proportional quantities? [Sample answer: The percent is the number of parts out of 100. So forever 100 pounds something weights on Earth, it would weigh about 17 pounds on the Moon.]”

• Topic 5, Lesson 5-2, Do You Understand?, Problem 1, “Essential Question How is solving a two-step equation similar to solving a one-step equation?” Teacher guidance: “Essential Question Students should understand that expressions on both sides of the equation must remain equal at all times. This is achieved by applying the properties of equality. Two-step equations require two properties of equality because they are solved using two operations.”

The materials reviewed for enVision Mathematics Grade 7 meet expectations for containing adult-level explanations and examples of the more complex grade concepts and concepts beyond the current course so that teachers can improve their own knowledge of the subject. 

The materials provide professional development videos at two levels to help teachers improve their knowledge of the grade they are teaching.

• “Topic-level Professional Development videos available online. In each Topic Overview Video, an author highlights and gives helpful perspectives on important mathematics concepts and skills in the topic. The video is a quick, focused ‘Watch me first’ experience as you start your planning for the topic.

• Lesson-level Professional Development videos available online. These Listen and Look For videos, available for some lessons in the topic, provide important information about the lesson.”

The Teacher’s Edition Program Overview, Professional Development section, states the “Advanced Concepts for the Teacher provides examples and adult-level explanations of more advanced mathematical concepts related to the topic. This professional development feature provides the teacher opportunities to improve his or her personal knowledge and build understanding of the mathematics in each topic. The explanations and examples in this section also support the teacher’s understanding of the underlying mathematical progressions.”

An example of an Advanced Concept for the Teacher:

• Topic 2, Topic Overview, Advanced Concepts for the Teacher, “Solving Proportions by Inspection Equivalent ratios are ratios that express the same multiplicative relationship between numbers. When two equivalent ratios have the same first or second term, then the other terms are equal. For example [an example is provided]…Understanding equivalency in ratios and solving proportional relationships is the foundation for slope of linear relationships, inverse relationships, and trigonometric ratios.”

The Topic Overview, Math Background Coherence,and Look Ahead sections, provide adult-level explanations and examples of concepts beyond the current grade as they relate what students are learning currently to future learning.

An example of how the materials support teachers to develop their own knowledge beyond the current grade:

• Topic 3, Topic Overview, Math Background Coherence, Look Ahead, the materials state, “Grade 8 … Similarity In Grade 8, students will use similar triangles to investigate slopes, and develop the equations y = mx and y = mx +b, for lines.”

The materials reviewed for enVision Mathematics Grade 7 meet expectations for including standards correlation information that explains the role of the standards in the context of the overall series. 

Standards correlation information is indicated in the Teacher’s Edition Program Overview, the Topic Planner, the Lesson Overview, and throughout each lesson. Examples include:

• The Teacher’s Edition Program Overview, Correlation to Grade 7 Common Core Standards, organizes standards by their Domain and Major Cluster and indicates those lessons and activities within the Student’s Edition and Teacher’s Edition that align with the standard. Lessons and activities with the most in-depth coverage of a standard are distinguished by boldface. The Correlation document also includes the Mathematical Practices. Although the application of the mathematical practices can be found throughout the program, the document indicates examples of lessons and activities within the Student’s Edition and Teacher’s Edition that align with each math practice.

• The Teacher’s Edition Program Overview, Scope & Sequence organizes standards by their Domain, Major Cluster, and specific component. The document indicates those topics that align with the specific component of the standard.

• The Teacher’s Edition, Topic Planner indicates the standards and Mathematical Practices that align to each lesson.

The Teacher’s Edition, Math Background: Coherence provides information that summarizes the content connections across grades. Examples of where explanations of the role of the specific grade-level mathematics are present in the context of the series include:

• Topic 4, Topic Overview, Math Background Coherence, the materials highlight two of the learnings within the topics: “Expressions” and “Equivalent Expressions” with a description provided for each learning including which lesson(s) cover the learnings. The “Look Back” section asks the question, “How does Topic 4 connect to what students will learn earlier?” and provides a Grade 6 connection, “Grade 6 Algebraic Expressions In Grade 6, students learned to read and interpret parts of an expression by using mathematical terms and viewing expressions as single entities. Students used elementary operations to write and evaluate expressions…” 

• Topic 6, Topic Overview, Math Background Coherence, the materials highlight three of the learnings within the topics: “Populations and Samples, Make Inferences” and “Compare Populations Informally” with a description provided for each learning including which lesson(s) cover the learnings. The “Look Ahead” section asks the question, “How does Topic 6 connect to what students will learn later?” and provides a Grade 8 and High School connection, “Grade 8 Display and Describe Numerical Data In Topic 4, students will extend their understanding of sample data sets and data displays to include bivariate data sets, and plot them on a scatter plot instead of a histogram, box plot, or dot plot. They will learn to apply the concepts of clusters, gaps, and outliers to bivariate data in a scatter plot…High School Understand and Evaluate Random Processes Students will extend the concepts of random representative samples as tools for making inferences about populations by using simulation models…”

• Topic 7, Topic Overview, Math Background Coherence, the materials highlight two of the learnings within the topics: “Simple Probability” and “Compound Probability” with a description provided for each learning including which lesson(s) cover the learnings. The “Look Ahead” section asks the question, “How does Topic 7 connect to what students will learn later?” and provides a Grade 8 connection, “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 materials reviewed for enVision Mathematics Grade 7 meet expectations for providing explanations of the instructional approaches of the program and identification of the research-based strategies. The Teacher’s Edition Program Overview provides detailed explanations behind the instructional approaches of the program and cites research-based strategies for the layout of the program. Unless otherwise noted all examples are found in the Teacher’s Edition Program Overview.

Examples where materials explain the instructional approaches of the program and describe research-based strategies include:

• The Program Goals section states the following: “The major goal in developing enVision Mathematics was to create a middle grades program that embodies the philosophy and pedagogy of the enVision series and was adapted for the middle school teacher and learner…enVision Mathematics embraces time-proven research principles for teaching mathematics with understanding. One understands an idea in mathematics when one can connect that idea to previously learned ideas (Hiebert et al., 1997). So, understanding is based on making connections, and enVision Mathematics was developed on this principle.”

• The Instructional Model section states the following: “Over the past twenty years, there have been numerous research studies measuring the effectiveness of problem-based learning, a key part of the core instructional approach used in enVision Mathematics. These studies have found that students taught partly or fully through problem-based learning showed greater gains in learning (Grant & Branch, 2005; Horton et al., 2006; Johnston, 2004; Jones & Kalinowski, 2007; Ljung & Blackwell, 1996; McMiller, Lee, Saroop, Green, & Johnson, 2006; Toolin, 2004). However, the interaction of problem-based learning, which fosters informal mathematical learning, and more explicit visual instruction that formalizes mathematical concepts with visual representations leads to the greatest gains for students (Barron et al., 1998; Boaler, 1997, 1998). The enVision Mathematics instructional model is built on the interaction between these two instructional approaches. STEP 1 PROBLEM-BASED LEARNING Introduce concepts and procedures with a problem-solving experience. Research shows that conceptual understanding is developed when new mathematics is introduced in the context of solving a real problem in which ideas related to the new content are embedded (Kapur, 2010; Lester and Charles, 2003; Scott, 2014). Conceptual understanding results because the process of solving a problem that involves a new concept or procedure requires students to make connections of prior knowledge to the new concept or procedure. The process of making connections between ideas builds understanding. In enVision Mathematics, this problem-solving experience is called Solve & Discuss It. STEP 2 VISUAL LEARNING Make the important mathematics explicit with enhanced direct instruction connected to Step 1. The important mathematics is the new concept or procedure students should understand. Quite often the important mathematics will come naturally from the classroom discussion around students’ thinking and solutions for the Solve & Discuss It! task. Regardless of whether the important mathematics comes from discussing students’ thinking and work, understanding the important mathematics is further enhanced when teachers use an engaging and purposeful classroom conversation to explicitly present and discuss an additional problem related to the new concept or procedure…”

• Other research includes the following:

  • Resendez, M.; M. Azin; and A. Strobel. A study on the effects of Pearson’s 2009 enVisionMATH program. PRES Associates, 2009. 

  • What Works Clearinghouse. enVisionMATH, Institute of Education Sciences, January 2013.

• Throughout the Teacher’s Edition Program Overview references to research-based strategies are cited with some reference pages included at the end of some authors' work.

The materials reviewed for enVision Mathematics Grade 7 meet expectations for providing a comprehensive list of supplies needed to support instructional activities.

In the online Teacher Resources for each grade, a Materials List is provided in table format identifying the required materials and the topic(s) where they will be used. Example includes:

• The table indicates that Topic 1 will require the following materials: “Base-ten blocks, Colored pencils (optional), Different lengths of nails (optional), Fraction bars...”

• The table indicates that Topic 4 will require the following materials: “Algebra tiles, Base-ten blocks, Colored pencils (optional), Index cards...”

• The table indicates that Topic 8 will require the following materials: “Anglegs, Colored pencils (optional), Compass (optional), Graph paper...”

The materials reviewed for enVision Mathematics Grade 7 meet expectations for including an assessment system that provides multiple opportunities throughout the grade to determine students’ learning and sufficient guidance to teachers for interpreting student performance and suggestions for follow-up. 

The assessment system provides multiple opportunities to determine student’s learning throughout the lessons and topics. Answer keys and scoring guides are provided. In addition, teachers are given recommendations for Math Diagnosis and Intervention System (MDIS) lessons based on student scores. If assessments are given on the digital platform, students are automatically placed into intervention based on their responses.

Examples include:

• Topic 1, Lesson 1-5, Lesson Quiz, “Use the student scores on the Lesson Quiz to prescribe differentiated assignments.” I Intervention 0-3 points, O On-Level 4 points, A Advanced 5 points.” The materials provide follow-up activities—to be assigned at the teacher’s discretion—to students at each indicated level: Reteach to Build Understanding I, Additional Vocabulary Support I O, Build Mathematical Literacy I O, Enrichment O A, Math Tools and Games I O A, and Pick a Project and STEM Project I O A. For example, Problem 1, “How would you subtract a negative fraction from a positive decimal? Explain.”

• Topic 4, Assessment Form A, Problem 7, “There are 11.5 ounces of cereal in a container. Additional cereal is poured into the container at a rate of 2.25 ounces per second. How many ounces are in the container after 4 seconds?” The accompanying Scoring Guide gives the following recommendations based on the score: 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 for 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. Item Analysis for Diagnosis and Intervention indicates Points 1, DOK 2, MDIS K20, Standard 7.EE.A.1.

• Topic 8, Performance Task Form A, Problem 1, “Dave wants to build a rectangular screened-in porch that is 20 feet long, and it will extend 6 feet out from the back side of his house. 1. Dave wants to sketch a scale drawing of the porch. Choose a reasonable scale and make a scale drawing of the porch. Label the dimensions of the model.” The Scoring Rubric indicates 2: Correct response, 1: Partially correct response. The Item Analysis for Diagnosis and Intervention indicates for DOK 2, MDIS M36, Standards 7.G.A.1, 7.G.A.2 and MP.4. 

• Topics 1-8, Cumulative/Benchmark Assessment, Problem 11, “Wyatt uses 3.15 cups of flour in a recipe that makes 9 shortcakes. Cora uses 2.4 cups of flour in a recipe that makes 8 shortcakes. How much more flower per shortcake is needed for Cora’s recipe? (A) 0.05 cup (B) 0.20 cup (C) 0.25 cup (D) 0.50 cup”  The accompanying Scoring Guide gives the following recommendations based on the score: Greater than 85% /Assign the corresponding MDIS for items answered incorrectly. 70% - 85% / Assign the corresponding MDIS for items answered incorrectly. Monitor the student during Step 1 and Try It! parts of the lessons for personalized remediation needs. Less Than 70% / Assign the corresponding MDIS for items answered incorrectly. Assign the appropriate remediation activities available online. Item Analysis for Diagnosis and Intervention indicates: DOK 2, MDIS M28 and M29, Standards 7.RP.A.1 and 7.RP.A.3.

The materials reviewed for enVision Mathematics Grade 7 meet expectations for providing assessments that include opportunities for students to demonstrate the full intent of grade-level standards and practices across the series.

The materials provide formative and summative assessments throughout the grade as print and digital resources. As detailed in the Assessment Sourcebook, the formative assessments—Try It! and Convince Me!, Do You Understand? and Do You Know How?, and Lesson Quiz—occur during and/or at the end of a lesson. The summative assessments—Topic Assessment (Form A and Form B), Topic Performance Task (Form A and Form B), and Cumulative/Benchmark Assessments—occur at the end of a topic, group of topics, and at the end of the year. The four Cumulative/Benchmark Assessments address Topics 1-2, 1-4, 1-6, and 1-8. 

• Try It! and Convince Me! “Assess students’ understanding of concepts and skills presented in each example; results can be used to modify instruction as needed.”

• Do You Understand? and Do You Know How? “Assess students’ conceptual understanding and procedural fluency with lesson content; results can be used to review or revisit content.”

• Lesson Quiz “Assess students’ conceptual understanding and procedural fluency with lesson content; results can be used to prescribe differentiated instruction.”

• Topic Assessment, Form A and Form B “Assess students’ conceptual understanding and procedural fluency with topic content. Additional Topic Assessments are available with ExamView CD-ROM.”

• Topic Performance Task, Form A and Form B “Assess students’ ability to apply concepts learned and proficiency with math practices.”

• Cumulative/Benchmark Assessments “Assess students’ understanding of and proficiency with concepts and skills taught throughout the school year.”

The formative and summative assessments allow students to demonstrate their conceptual understanding, procedural fluency, and ability to make applications through a variety of item types. Examples include: 

• Order; Categorize

• Graphing

• Multiple choice

• Fill-in-the-blank

• Multi-part items

• Selected response (e.g., single-response and multiple-response)

• Constructed response (i.e., short or extended responses)

• Technology-enhanced items (e.g., drag and drop, drop-down menus, matching)