7th Grade - Gateway 2

The instructional materials reviewed for enVision Mathematics Common Core Grade 7 meet expectations that the materials develop conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings.

Materials include problems and questions that develop conceptual understanding throughout the grade level. According to the Teacher Resource Program Overview, “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. 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 Teacher Resource include:

• Lesson 1-4, Subtract Integers, Visual Learning, Example 2, students use number lines to build their understanding of integers and writing matching equations. “Ian’s football team lost 2 yards on a running play. Then they received a 5-yard penalty. What is the team’s total change in yards? Write a subtraction expression to represent the change in yards. Write an equivalent addition expression.” (7.NS.1)
• Lesson 2-2, Determine Unit Rates with Ratios of Fractions, 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, 7.RP.3)
• Lesson 4-5, Factor Expressions, Visual Learning, Example 1, students develop conceptual understanding of factor expressions by using area models, “Kiana painted a rectangular wall blue to start an ocean mural. She used 3 cans of paint, each of which covered x square meters, and a different-sized can that covered 12 square meters. What are possible length and height dimensions of Kiana’s mural?” Teachers ask, “In the area model diagram, what does the green area labeled 12 represent? What does the blue area labeled 3x represent? How does the area model relate to the original and factored expressions?” (7.EE.1 and 7.EE.2)
• Lesson 6-1 Populations and Samples, Visual Learning, Example 2, students develop an understanding of the importance of random sampling when generalizing a population, “Morgan decides to survey a sample of the town’s voting population. How can she know that the survey results from the sample of voters represent the population of the entire town’s population? How much of the population should be sampled in a ‘representative’ sample? Explain. Why do you think a random sample is usually also a representative sample?” (7.SP.1) 
• Lesson 8-6, Solve Problems Involving Area of a Circle, Visual Learning, Example 3, students develop conceptual understanding as they use the formula for the area of the circle, "Elle needs new grass in the circular pen for her chickens. What is the area of the pen?" (7.G.4)

Materials provide opportunities for students to independently demonstrate conceptual understanding throughout the grade. Practice and 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 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 include:

• Lesson 1-1, Relate Integers and Their Opposites, Do You Know How?, Item 6, students independently demonstrate understanding of how positive and negative numbers relate to zero by using models and by combining opposite quantities, “The scores of players on a golf team are shown in the table. The team’s combined score was 0. What was Travis’s score?” (7.NS.1)
• Lesson 2-4, Describe Proportional Relationships: Constant Proportionality, Do You Know How?, Item 4, students independently demonstrate understanding of proportional relationships and their constant of proportionality, “Determine whether each equation represents a proportional relationship. If it does, identify the constant of proportionality. a. y = 0.5x - 2; b. y = 1,000x; c. y =  x + 1.”(7.RP.A.2)
• Lesson 4-6, Add Expressions, Practice & Problem Solving, Item 13, students independently demonstrate understanding of writing and simplifying expressions, “An art class is making a mural for the school that has a triangle drawn in the middle. The length of the bottom of the triangle is x. Another side is 1 more than three times the length of the bottom of the triangle. The last side is two more than the bottom of the triangle. Write and simplify an expression for the perimeter of the triangle.” (7.EE.2)
• Lesson 7-2, Understand Theoretical Probability, Practice & Problem Solving, Item 7, students independently demonstrate understanding of theoretical propropbality, “A spinner has 8 equal-sized sections.To win the game, the pointer must land on a yellow section.” (7.SP.6, 7.RP.2)
• Lesson 8-5, Solve Problems Involving Circumference of a Circle, Visual Learning, Example 1, Try It!, students independently demonstrate understanding of circumference of a circle as they use the formula, "What is the circumference of the rim of a basketball hoop with a radius of 9 inches? First, multiply the radius by ____ to get the diameter, ___ inches. Then multiply the diameter by 3.14 (an approximation for π) to get a circumference of about ____ inches." (7.G.4)

The instructional materials reviewed for enVision Mathematics Common Core Grade 7 meet expectations that they attend to those standards that set an expectation of procedural skill and fluency.

The instructional 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:

• Lesson 1-2, Understand Rational Numbers, Visual Learning, Example 2, Try It!, students write rational numbers in decimal form to develop and maintain fluency in dividing whole numbers and decimals, “What is the decimal form of $$\frac{100}{3}$$, $$\frac{100}{5}$$, and $$\frac{100}{6}$$? Determine whether each decimal repeats or terminates.” (7.NS.2)
• Lesson 2-3, Understand Proportional Relationships: Equivalent Ratios, Do You Know How?, Item 6, students determine whether quantities are proportional by testing for equivalent ratios, “Is the relationship between the number of tickets sold and the number of hours proportional? If so, how many tickets were sold in 8 hours?” Students are provided a table with hours (h) and tickets sold (t). (7.RP.2) 
• Lesson 4-3, Simplify Expressions, Visual Learning, Example 2, Try It!, students develop procedural skill when they simplify expressions, “Simplify each expression. a. 59.95m - 30 + 7.95m + 45 + 9.49m; b. -0.5p + $$\frac{1}{2}$$p- 2.75 + $$\frac{2}{3}$$p.” (7.EE.1) 
• Lesson 6-3, Make Comparative Inferences About Populations, Visual Learning, Example 1, Try It, students use box plots to compare and make inferences about populations, “Kono gathers the heights of a random sample of sixth graders and seventh graders and displays the data in box plots. What can he say about the two data sets? The median of the ___ grade is greater than the median of the ___ grade sample. The ____ grade sample has a greater variability.” Box Plots of 6th and 7th grade students’ heights are shown. (7.SP.3 and 7.SP.4)
• Lesson 8-3, Draw Triangles with Given Conditions, Do You Know How?, Item 4, students draw triangles when given information about their side lengths and angle measures, “How many triangles can be drawn with side lengths 4 centimenters, 4.5 centimenters, and 9 centemeters? Explain.” (7.G.2)

The instructional materials provide opportunities to independently demonstrate procedural skill and fluency throughout the grade level. Practice and 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 Fluency Practice page which engages students in fluency activities. Examples include:

• Lesson 1-5, Add and Subtract Rational Numbers, Practice & Problem Solving, Item 11, students use the same procedure for adding and subtracting signed rational numbers as they do when adding and subtracting integers, “Simplify each expression. a) 50$$\frac{1}{2}$$ + (-12.3) b) -50$$\frac{1}{2}$$ + (-12.3) c) -50$$\frac{1}{2}$$ + 12.3.” (7.NS.1)
• Lesson 2-2, Determine Unit Rates with Ratios of Fractions, Practice & Problem Solving, Item 12, students calculate unit rate with fractional measurements, “A box of cereal states that there are 90 Calories in a $$\frac{3}{4}$$ - cup serving. How many calories are there in 4 cups of the cereal?” (7.RP.1) 
• Lesson 4-5, Factor Expressions, Practice & Problem Solving, Item 8, students factor the GCF from expressions, “Factor the expression. 14x + 49.” (7.EE.1 and 7.EE.2)
• Lesson 7-6, Find Probabilities of Compound Events, Practice & Problem Solving, Item 9, students find the sample space and probability of compound events, “The organized list shows all the possible outcomes when three fair coins are flipped. The possible outcomes of each flip are heads (H) and tails (T). What is the probability that at least 2 fair coins land heads up when 3 are flipped?” (7.SP.8)
• Topic 8 Review, Fluency Practice, students use the percent equation to solve problems, “Pathfinder: Shade a path from START to FINISH. Follow the answers to the problems so that each answer is greater than the one before. You can only move up, down, right, or left.” The starting box states, “Amy deposits $360 in an account that pays 1.2% simple annual interest. How much interest will she earn in 6 years?” (7.RP.3)
• Lesson 8-5, Solve Problems Involving Circumference of a Circle, Practice & Problem Solving, Item 15, students calculate the circumference, radius, or diameter of a circle, “What is the radius of a circle with a circumference of 26.69 centimeters?” (7.G.4)

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

The instructional 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, “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. 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, “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 1, STEM Project, How Cold is Too Cold?, students collect and organize their data related to temperatures and plot their data on graphs, find the range in temperatures, and form conclusions, "There are many regions of the world with cold temperatures and extreme conditions. How do the inhabitants of these regions adapt and thrive? Do conditions exist that make regions too cold for human living? You and your classmates will explore and describe the habitability of regions with low temperatures.” (7.SP.1, 7.SP.2, and 7.SP.3)
• Topic 2, 3-Act Mathematical Modeling: Mixin' It Up, Question 15, students attempt to make the liquid in a water glass have the same flavor like that of a large water cooler, “A classmate usually adds 6 drops to 16 ounces of water. Use your updated model to predict the number she would use for the large container." (7.RP.1 and 7.RP.2)
• 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.” (7.EE.3 and 7.EE.4)
• Topic 5, 3-Act Mathematical Modeling: Digital Downloads, Question 14, students determine how many songs a person can purchase using the balance of a gift card, “If all single tracks were on sale for 10% off, how would your model change? How would the answer to the Main Question change?" (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, "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.” (7.G.1 and 7.G.2)

The instructional 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 and Problem Solving exercises found in the student materials provide opportunities for students to independently demonstrate mathematical flexibility in a variety of contexts. For example:

• Lesson 1-10, Solve Problems with Rational Numbers, Practice & Problem Solving, Item 15, students solve problems using rational numbers operations, “The table shows the relationship between a hedgehog's change in weight and the number of days of hibernation. a. What number represents the change in weight for each day of hibernation? b. What number represents the change in weight in ounces for the hedgehog in 115 days of hibernation?” (7.NS.3 and 7.EE.3)
• Lesson 2-1, Connect Ratios, Rates, Unit Rates, Practice & Practice Solving, Item 9, students apply knowledge of solving multi-step problems with rational numbers to solving problems with ratios, rates, and unit rates. Given 3 bags of rice, “Which package has the lowest cost per ounce of rice?” (7.RP.1 and 7.RP.3)
• Topic 3, Pick a Project 3A, students use coupons to calculate the best price for several items, “Select at least three items that you want to purchase from one or more stores. One should be something on sale or available at a discount. One should be something from the sporting goods section. One should be a toy or game. Research the selling price of each item. Use the coupons below to calculate the best price for each item. Use each coupon only once. Make a collage with pictures of the items and copies of the coupons. Write up your calculations, explain how you found the best price, and include this information with your collage.” (7.RP.3)
• Lesson 4-8, Analyze Equivalent Expressions, Practice & Problem Solving, Item 15, students apply the meaning of equivalent expressions to simplify a problem in a new way, “A customer at a clothing store is buying a pair of pants and a shirt. The customer can choose between a sale that offers a discount on pants, or a coupon for a discount on the entire purchase. Let n represent the original price of the pants and s represent the price of the shirt. a. Write two expressions that represent the ‘15% off sale on all pants’ option. b. Write two expressions that represent the ‘10% off her entire purchase’ option. c. If the original cost of the pants is $25 and the shirt is $10, which option should the customer choose? Explain.” (7.EE.2)
• Topic 6, Pick a Project 6A, students conduct a survey and analyze their results, “Conduct a survey on an improvement or change you want to see in your community. Write a letter to your representative or local council about the changes you would like to see in your community. In your letter, include data and conclusions from your survey to support your position. What might a better sample be?” (7.SP.1, 7.SP.2)
• Lesson 8-9, Solve Problems Involving Volume, Practice & Problem Solving, Item 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.G.6, 7.G.3, 7.EE.3, and 7.EE.4)

The instructional materials reviewed for enVision Mathematics Common Core Grade 7 meet expectations that the three aspects of rigor are not always treated together and are not always treated separately. 

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

• Lesson 1-3, Add Integers, 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.1)
• Topic 3 Review, Fluency Practice, students find unit rates with ratios of fractions, “Riddle Rearranging: Find the value of x in each unit rate. Then arrange the answers in order from least to greatest. The letters will spell out the answer to the riddle below.” Box K states, “$$\frac{\frac{3c}{4}}{\frac{1h}{3}} = \frac{xc}{1h}$$”. (7.RP.1)
• Lesson 8-8, Solve Problems Involving Surface Area, Practice & Problem Solving, Item 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:

• Lesson 1-2, Understand Rational Numbers, Practice & Problem Solving, Item 18, students solve real-world problems with rational numbers, “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?” This question develops conceptual understanding and application of 7.NS.2, apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.
• Lesson 4-1, Write and Evaluate Algebraic Expressions, Do You Know How?, Item 5, students use algebraic expressions to represent and solve problems with an unknown value, “Write an algebraic expression that Marshall can use to determine the total cost of buying a watermelon that weighs w pounds and some tomatoes that weigh t pounds. How much will it cost to buy a watermelon that weighs 18$$\frac{1}{2}$$pounds and 5 pounds of tomatoes?” This question develops procedural skill and fluency of 7.EE.3, solve real-life and mathematical problems using numerical and algebraic expressions and equations.
• Lesson 5-5, Solve Inequalities Using Multiplication or Division, Practice & Problem Solving, Item 12, students write inequalities to solve real-world problems, “Brittney can spend no more than $15 for new fish in her aquarium. a. Let f be the number of fish she can buy. What inequality represents this problem? b. How many fish can Brittney buy?” This question develops conceptual understanding and application of 7.EE.4, use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
• Lesson 7-3, Understand Theoretical Probability, Do You Know How?, Item 4, students 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.” This question develops conceptual understanding and procedural skill of 7.SP.6, approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability.
• Lesson 8-4, Solve Problems Using Angle Relationships, Do You Know How?, Item 6, students use understanding of angle relationships to find the value of a given angle, “Use diagram 4-6. If ∠1 and ∠3 are the same measure, what is the value of x? ” This question develops conceptual understanding and procedural skill of 7.G.5, use facts about supplementary, complementary, vertical, and adjacent angles to write and solve simple equations for an unknown angle in a figure.

The instructional materials reviewed for enVision Mathematics Common Core Grade 7 meet expectations that the Standards for Mathematical Practice are identified and used to enrich mathematics content within and throughout the grade level.

All 8 MPs are clearly identified throughout the materials, with few or no exceptions. Math Practices identification in this program according to the Teacher Resource Program Overview include:

• Materials provide a Math Practices and Problem Solving Handbook for students, “A great resource to help students build on and enhance their mathematical thinking and habits of mind.” This handbook explains math practices in student-friendly language and digital animation videos for each math practice are also available.
• Opportunities to apply math practices are found in the Explore It, Explain It, and Solve & Discuss It portions of the lesson. “The Solve & Discuss It calls on students to draw on nearly all of the math practices, but especially sense-making and solution formulation as well as abstract and quantitative reasoning. The Explore It focuses students on mathematical modeling, generalizations, and structure of mathematical models. The Explain It emphasizes mathematical reasoning and argumentation. Students construct arguments to defend a claim or critique an argument defending a claim.”
• The Math Practices and Problem Solving Handbook Teacher Pages, “provide overviews of the math practices, offer instructional strategies to help students refine and enhance their thinking habits, and include student behaviors to listen and look for for each standard.”
• Each Topic Overview contains Math Practices Teacher Pages which include, “Two highlighted math practices with student behaviors to look for, and questions to help students become more proficient with these thinking habits.” For example in Topic 7, mathematical reasoning and explanation questions state, “How would you describe the problem in your own words? What are some other strategies you might try in order to determine the different outcomes?”
• Math Practices boxes found in the student text provide, “Reminders to be thinking about the application of the math practices as they solve problems.”
• Math Practices Run-in Heads found in the Practice & Problem Solving questions, “Remind students to apply the math practices as they solve problems.”

The majority of the time the MPs are used to enrich the mathematical content and are not treated separately. Examples include:

• MP1: Make sense of problems and persevere in solving them. Lesson 3-2, Connect Percent and Proportion, Practice & Problem Solving, Item 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. What was the total bill?”
• MP2: Reason abstractly and quantitatively. Lesson 6-3, Make Comparative Inferences About Populations, Practice & Problem Solving, Item 9, students interpret and compare statistical measures and reason about data sets in both qualitative and quantitative forms, “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?”
• MP4: Model with mathematics. Lesson 7-1, Understand Likelihood and Probability, Practice & Problem Solving, Item 15, students use tools to determine the likelihood of an event occurring, “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.”
• MP5: Use appropriate tools strategically. Lesson 7-7, Simulate Compound Events, Practice & Problem Solving, Item 10, students choose an appropriate tool (e.g., spinner, coin, number cube) to stimulate the outcome of a compound event, “Julie used a number cube to simulate a flower seed sprouting, for which the success rate is 50%. She used even numbers to represent success and odd numbers to represent failure. The results of 8 trials that simulate the sprouting of five seeds are shown below. Based on the simulated results, what is the probability that none of the next five flower seeds will sprout successfully?”
• MP6: Attend to precision. Lesson 2-1, Connect Ratios, Rates, and Unit Rates, Practice & Problem Solving, Item 11, students use formal mathematical vocabulary to communicate ratio concepts and reasonings to solve problems, “An arts academy requires there to be 3 teachers for every 75 students and 6 tutors for every 72 students. How many tutors does the academy need if it has 120 students?” 
• MP7: Look for and make use of structure. Lesson 3-2, Connect Percent and Proportion, Practice & Problem Solving, Item 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?”
• MP8: Look for and express regularity in repeated reasoning. Lesson 8-3, Draw Triangles with Given Conditions, Practice & Problem Solving, Item 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 non included angle of 75°, can you draw no triangles, only one triangle, or more than one triangle?” 

The instructional materials reviewed for enVision Mathematics Common Core Grade 7 meet expectations that the instructional materials prompt students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics. 

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:

• Lesson 3-1, Analyze Percents of Numbers, Practice & Problem Solving, Item 18, students use their understanding of percents of numbers and construct arguments to support their response, “Brad says that if a second number is 125% of the first number, then the first number must be 75% of the second number. Is he correct? Justify your answer.”
• Lesson 4-2, Generate Equivalent Expressions, Explore It!, students construct arguments as they compare and contrast representations of real world situations, “How can you represent the total number of eggs in the shipment using diagrams or images? Explain your diagram. 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?” 
• Lesson 5-3, Solve Equations Using the Distributive Property, 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. Whose equation accurately represents the situation? Construct an argument to support your response. B. What error in thinking might explain the inaccurate equation?”
• Lesson 6-4, Make More Comparative Inferences About Populations, Do You Understand?, Item 3, students use their understanding of inferences to 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.”
• Lesson 8-4, Solve Problems Using Angle Relationships, Explore It!, students analyze problems and use angle relationships to construct and justify arguments, “C. How does the sum of the measures of ∠1 and ∠2 change when one ski moves? Explain. 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:

• Lesson 1-9, Divide Rational Numbers, Practice & Problem Solving, Item 17, students analyze the arguments of others as they explain the errors of dividing and multiplying rational numbers, “Kayla wants to find 2$$\frac{2}{3}$$ $$\div$$ (-1$$\frac{7}{3}$$). She first rewrites the division as (2$$\frac{2}{3}$$)(-1$$\frac{7}{3}$$). What is wrong with Kayla’s reasoning?”
• Lesson 2-5, Graph Proportional Relationships, Do You Understand?, Item 3, students analyze the arguments of others by interpreting if points contain a proportional relationship. “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.”
• Lesson 4-7, Subtract Expressions, Practice & Problem Solving, Item 16, students analyze the arguments of others using properties of operations to subtract expressions, “Tim incorrectly rewrote the expression 1/2p - (1/4p + 4) as 1/2p + 1/4p - 4. Rewrite the expression without the parenthesis. What was Tim’s error?”
• Lesson 5-7, Solve Multi-Step Inequalities, Practice & Problem Solving, Item 10, students analyze the arguments of others using inequalities. “Sierra says that she can simplify the left side of the inequality 2(-3 + 5) + 2 -4(x - 2) - 3 by combining the terms within the parentheses, but that she can’t do the same on the right side. Is Sierra correct? Explain.”
• Lesson 6-2, Draw Inferences from Data, Do You Understand?, Item 3, students analyze the arguments of others as they make inferences about a population from sample data, “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 like 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.”

The instructional materials reviewed for enVision Mathematics Common Core Grade 7 meet expectations that the instructional materials assist teachers in engaging students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics.

Teacher materials assist teachers in engaging students in constructing viable arguments frequently throughout the program. Examples include:

• Lesson 3-6, Solve Simple Interest Problems, Visual Learning, Example 1, “Victoria opens a savings account with a deposit of $300. She will earn 1.6% simple interest each year on her money. How much interest will she earn over 5 years (assuming she does not add or take out any money)?” ETP (Effective Teaching Practices) Use and Connect Mathematical Representations teacher prompt states, “Does Victoria earn the same amount of interest every year? Explain.”
• Lesson 5-5, Solve Inequalities Using Multiplication or Division, Example 1, Convince Me!, “Frances solved the inequality 5g $$\ge$$ 35. She says that 7 is a solution to the inequality. Is Frances correct? Explain.” ETP Elicit and Use Evidence of Student Thinking teacher prompt states, “What other values make the inequality true?”
• Lesson 7-3, Understand Experimental Probability, Visual Learning, Example 1, Try It!, “During the second day of the school fair, Talia and Yoshi recorded 43 winners out of a total of 324 players. How does the actual number of winners compare to the expected number of winners? ETP Convince Me! teacher prompt states, “Is it possible for the theoretical probability to be $$\frac{1}{2}$$ while the experimental probability is 1? Give an example.”

Teacher materials assist teachers in engaging students in analyzing the arguments of others frequently throughout the program. Examples include:

• Lesson 1-5, Add and Subtract Rational Numbers, Example 1, Try It!, and Convince Me!, “A dolphin is at the surface of the water and then descends to a depth of 4$$\frac{1}{2}$$ feet. Then the dolphin swims down another 2$$\frac{3}{4}$$ feet. What is the location of the dolphin relative to the surface of the water?” ETP Elicit and Use Evidence of Student Thinking teacher prompt states, “Why is the first number of the addition statement -4$$\frac{1}{2}$$? How is the subtraction of a positive number from a negative number changed to solve the problem?”
• Lesson 4-5, Factor Expressions, Explain It!, “Tasha is packing gift bags that include the same items. She has 72 glow sticks, 36 markers, and 24 bottles of bubbles. Tasha believes that she can pack no more than 6 bags using all of her supplies. Do you agree with Tasha? Explain. ETP Observe Students at Work teacher prompt states, “How do students decide whether or not they agree with Tasha? Students might find the GCF of the three numbers (12) and say that it is the greatest number of gift bags Tasha can pack.”
• Lesson 6-2, Draw Inferences From Data, Visual Learning, Example 3, “Margo and Ravi are also trying to get their parents to let them stay up later. They collect data about the number of hours of sleep a random sample of seventh graders get each night. The two box plots show their data. Do Margo’s and Ravi’s data support Sasha’s inference about the number of hours of sleep that seventh graders get?” ETP Pose Purposeful Questions teacher prompt states, “Why is it important that Ravi and Margo’s data corroborate, or support, Sasha’s data?”

Teacher materials assist teachers in engaging students in both the construction of viable arguments and analyzing the arguments or reasoning of others frequently throughout the program. Each Topic Overview highlights specific Math Practices and suggests look fors in student behavior and provides questioning strategies. Examples include:

• Topic 2, Analyze and Use Proportional Relationships, look fors, “Mathematically proficient students: Use what they have previously learned about ratios and rates in constructing arguments and explanations. Construct arguments by using accurate definitions of proportional quantities. Justify and support mathematical reasoning by using diagrams, tables, graphs, and equations. Ask questions to clarify others’ reasoning in order to decide whether arguments make sense or to improve the arguments.”
• Topic 2, Analyze and Use Proportional Relationships, questioning strategies, “How can you justify your answer? What mathematical language, models, or examples will help you support your answer? How could you improve this argument? How could you use counterexamples to disprove this argument? What do you think about this explanation? What questions would you ask about the reasoning used?”

The instructional materials reviewed for enVision Mathematics Common Core Grade 7 meet expectations that materials explicitly attend to the specialized language of mathematics.

The materials provide explicit instruction on how to communicate mathematical thinking using words, diagrams, and symbols. Each Topic Overview provides a chart in the Topic Planner that lists the vocabulary being introduced for each lesson in the Topic. As new words are introduced in a Lesson they are highlighted in yellow. Lesson practice includes questions to reinforce vocabulary comprehension and students write using math language to explain their thinking. Each Topic Review contains a Vocabulary Review section for students to review vocabulary taught in the Topic. Students have access to an Animated Glossary online in both English and Spanish. Examples include:

• Lesson 2-3, Understand Proportional Relationships: Equivalent Ratios, Visual Learning, Example 3, “A proportion is an equation that represents equal ratios.”
• Lesson 3-3, Represent and Use the Percent Equation, Visual Learning, Example 2, “Jane earns a 5.5% commission on the selling price of each home she sells. She earned $9,020 in commission on the sale of a home. What was the selling price of the home? Instead of a salary some workers earn a percent of the value of a transaction, called a commission.”
• Topic 4, Solve Problems involving Geometry, Mid-Topic Checkpoint, Question 1, “How are adjacent angles and vertical angles alike? How are they different?” 
• Lesson 6-1, Populations and Samples, Visual Learning, Example 1, “Morgan and her friends could ask every registered voter, or the entire population of voters in town, how they play to vote. Morgan and her friends could ask a subset, or a sample of the registered voters in town how they plan to vote.”
• Topic 7, Probability, Use Vocabulary in Writing, “A restaurant serves either skim milk or whole milk in glasses that are small, medium, or large. Use vocabulary words to explain how you could determine all the possible outcomes of milk choices at the restaurant. Use vocabulary words in your explanation.” Students are provided a word bank containing, “event, relative frequency, outcome, sample space, probability, and simulation.”

The materials use precise and accurate terminology and definitions when describing mathematics, and support students in using them. A Vocabulary Glossary is provided in the back of Volume 1 and lists all the vocabulary terms and examples. Teacher side notes, Elicit and Use Evidence of Student Thinking and Pose Purposeful Questions, provide specific information about the use of vocabulary and math language. Examples include:

• Lesson 1-9, Divide Rational Numbers, Visual Learning, Example 2, Pose Purposeful Questions, “Why is 3$$\frac{2}{3}$$ $$\div$$ (-$$\frac{2}{3}$$) equivalent to the given complex fraction? What do you notice about the signs of a fraction and its reciprocal? Explain why this is true.”
• Lesson 3-2, Connect Percent and Proportion, Visual Learning, Example 3, Pose Purposeful Questions, “Multiply the numerator and denominator of $$\frac{20}{100}$$ by an integer to get an equivalent fraction with a numerator of 260. What is the multiplier? What is the equivalent fraction?”
• Lesson 4-2, Generate Equivalent Expressions, Visual Learning, Example 2, Pose Purposeful Questions, “What does the Commutative Property of Addition mean? What does the Associative Property of Addition mean?”
• Lesson 5-4, Solve Inequalities Using Addition and Subtraction, Visual Learning, Example 1, Elicit and use Evidence of Student Thinking, ‘What does h represent in this scenario? Why is the $$\ge$$ inequality symbol used to represent this scenario? Explain.”
• Lesson 8-4, Solve Problems Using Angle Relationships, Visual Learning, Example 1, Convince me!, Elicit and Use Evidence of Student Thinking, “How can you use the definition of vertical angles to write an equation?”
• Student Edition, Glossary, “constant of proportionality: In a proportional relationship, one quantity y is a constant multiple of the other quantity x. The constant multiple is called the constant of proportionality. The constant of proportionality is equal to the ratio $$\frac{y}{x}$$. Example, In the equation y =4x, the constant of proportionality is 4.”