7th Grade - Gateway 2

The instructional materials for EdGems Math Grade 7 meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings.

The materials include problems and questions that develop conceptual understanding throughout the grade level. The instructional materials include Teacher Gems and Student Gems which provide links to activities that build conceptual understanding. Explore! activities provide students the opportunity to develop conceptual understanding at the beginning of each new lesson. In addition, Exercises, Online Practice, and Gem Challenges include problems to allow students to independently demonstrate conceptual understanding. Evidence includes:

• Lesson 4.3, Subtracting Integers, the Explore! activity uses number lines to build conceptual understanding of integer subtraction as adding the additive inverse. (7.NS.1c)
• In Lesson 6.3, Equivalent Expressions, the Teacher Gems, “Categories”, students connect verbal descriptions and different forms of algebraic models for expressions. Students use the cards to create their own categories using the sets of mathematical models and verbal descriptions. An example is, one card has the expression 1.16x, another has x + 0.16x, and a third has the statement, “Brandon buys lunch at a restaurant and leaves a 16% tip.” (7.EE.2)
• In Lesson 7.4, Linear Inequalities, the Teacher Gems include activities teachers can use with their students to develop conceptual understanding. An example is, “Four Corners” provides a sort activity with sections labeled Inequality, Graph, Situation, and Solution Set. Students match the cards given to each of the sections, making sure that all sections describe the same inequality. (7.EE.4b)
• Lesson 7.3, Rotations, Exercises, Problem 26, students demonstrate conceptual understanding by identifying the parts of an equation and explaining the context of the problem. An example is, “James went to Central Park one Saturday in October with his cross-country team for a 90 minute workout. He ran m minutes at a rate of 0.15 miles per minute. During the time he was not running, he walked at a rate of 0.06 miles per minute. He totaled 9.9 miles. This situation is represented by the equation: 0.15m + 0.06(90 − m) = 9.9. a. What does (90 − m) represent in this situation? b. Solve the equation to determine how many minutes James ran. Explain how you know your answer is correct.” (7.EE.2) 
• Lesson 6.2, The Distributive Property, Exercises, Problem 14, students demonstrate conceptual understanding of the distributive property by rewriting expressions in different forms within a problem context to help solve the problem. “Three friends went out to lunch. Each person bought a salad and sparkling water. The salads were $8.50 each and each sparkling water cost $1.25. Show how you could use the Distributive Property to find the total cost for the three lunches.” (7.EE.2)

The instructional materials for EdGems Math Grade 7 meet expectations for attending to those standards that set an expectation of procedural skill and fluency.

The materials include problems and questions that develop procedural skill and fluency and provide opportunities for students to independently demonstrate procedural skill and fluency throughout the grade. The materials develop procedural skills and fluencies in Student Gems, Lesson Examples, Student Exercises, and Teacher Gems. The materials provide opportunities for students to independently demonstrate procedural skills and fluencies in Proficient, Tiered, and Challenge Practice, Online Practice, Gem Challenges, and Exit Cards. Each unit provides additional practice with procedural skills in the Student Gems. Additional practice activities are specific to the standard(s) in each lesson. Included in each unit are links to: Khan Academy, IXL Practice, and Desmos Practice. Examples of developing procedural skill and fluency include:

• Lesson 6.2, Student Lesson, students use the distributive property to factor expressions, “Factor each expression using the greatest common factor. 15. 5x + 15, 16. 2x − 24, 17. 4x − 4” (7.EE.1) 
• Lesson 6.2, Student Gems, Khan Academy Quiz, students combine terms to simplify expressions, “Which expressions are equivalent to z + (z + 6)?” (7.EE.1)
• Lesson 4.2, Adding Rational Numbers, Explore!, students apply the Associative Property to add three or more rational numbers using an efficient strategy. “Step 1: How might you use the Associative Property to find the sum of 53 + 38 + 17 mentally (without writing anything down) ?  Step 2: Look at the expression below. Which two numbers would you group together to add first? Explain your reasoning. −2 2/3 + (−1 1/2)+2/3. Step 5: Do you think it is easier to group numbers with the same sign first or numbers with similar parts of whole (i.e. common denominators)? Explain your reasoning.” (7.NS.1d)
• Lesson 5.1, Multiplying and Dividing Integers, students multiply and divide integers. Example 1 states, “Find each product. a. 5(−4) b. −2(−3).” Example 2 states, “Find each quotient. a. 40 ÷ (−5) b. −20/-2” Exercises 4-27 contain multiplication and division with integers that have the same sign and different signs. (7.NS.2c)
• Lesson 5.4, Exit Card, students demonstrate procedural skill when solving multi-step problems with rational numbers, “Find the value of each expression. $$3(-5) + 5^2 - 1$$” (7.EE.4) 
• Lesson 7.3, Simplifying and Solving Equations, Exit Card, students independently demonstrate solving equations, “Solve each equation. Check each solution. 1. 4(x –2) = 32, 2. 3y + 5 + y + 10y = 19” (7.EE.1,4)

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

The instructional materials include multiple opportunities for students to engage in routine and non-routine application of mathematical skills and knowledge of the grade level. Students engage with materials that support non-routine and routine applications of mathematics in the Explore! activities, Teacher Gems, Performance Tasks, and Rich Tasks. Some of the Student Pages and Proficient, Tiered, and Challenge Practice allow students to engage with problems including real-world contexts and present multiple opportunities for students to independently demonstrate application of grade-level mathematics. Examples include:

• Lesson 1.2, Unit Rates, Explore! engages students in real-world application of representing proportional relationships between quantities. (7.RP.2) “Ethan and Priscilla bought cookies at the store. Ethan bought two dozen cookies for $6.20. Priscilla bought 5 dozen of the same cookies.” Students solve problems about the situation. For example, “Step 1: How much did Priscilla pay for five dozen cookies? Explain how you found your answer. Step 2: Write a ratio comparing the total cost of the cookies to the number of dozens purchased for Ethan. Write a similar ratio for Priscilla’s purchase. Step 4: One week later, Ethan went back to the store to buy eight dozen of the same cookies. The cookies were still the same price per dozen. Ethan used a proportion to determine the cost for eight dozen cookies. Look at the proportions below. Circle the proportion that could be used to find the cost of the eight dozen cookies. Why does the other proportion not work?”
• Lesson 5.1, Multiplying and Dividing Integers, Teacher Gems, Station activity, students solve real-world problems involving operations with rational numbers. (7.NS.3) The Stations activity includes a variety of tasks for students to complete as they move from station to station. Examples include, Station 4: “The price of a stock drops $4 each day for 8 days. If the stock was worth $120 before the drop began, how much is the stock worth now?” Station 6: “The width of a piece of land is changing at a rate of −13 inches per year. What integer represents the change after 8 years?” Station 7: “A loaf of bread requires 3-1/2 cups of flour. If Nathan plans to double the recipe but already has 2-1/4 cups of flour, how much more flour does he need?” 
• Lesson 10.3, Compound Probability, Explore! activity, students find the probability of events in a real-world setting.  An example is, “Each trimester in PE a student plays one sport. In the first trimester the possible sports are soccer, tennis or golf. Second trimester the possible sports are basketball or volleyball. For third trimester the possible sports are dodgeball or rugby. How many different possibilities are there of which three sports students will play? What is the probability a student will be enrolled in tennis, basketball and dodgeball if a schedule is assigned at random?” (7.SP.8)
• Unit 1, Ratios and Rates, Performance Task, students apply unit rates to solve problems. (7.RP.1) The task provides students with the situation, “Pedro and Ivan raced each other in a 24-mile bike race. After 20 minutes, Pedro had traveled 5 miles. Ivan had spent the 20 minutes traveling at a rate of 16 miles per hour.” The following questions are included: “1. Which person was in first place after 20 minutes? Use words and/or numbers to show how you determined your answer. 2. Pedro and Ivan continued bicycling at their original rates. After one hour, Pedro realized he needed to increase his speed to catch Ivan. Pedro assumed Ivan would continue to travel at a rate of 16 miles per hour. At what constant speed from now to the end of the race does Pedro need to travel to tie Ivan at the finish line? Write your answer using feet per minute."

The instructional materials for EdGems Math 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 include:

• In Lesson 1.3, students develop procedural skill by solving multiple problems to compute unit rates of complex fractions. One example is, “Find the unit rate. 1. 4/3 inches $$\div$$ 4/9 minutes” (7.RP.1)
• In Lesson 4.1, Adding Integers, Student Lesson, Example 2, students develop conceptual understanding by using a number line to model integer addition. An example is, “Find the value of each expression using a number line. a. 4 + (−2) b. −5 + (−3).” (7.NS.1c) 
• In Lesson 4.3, Subtracting Integers, students solve real-world problems by subtracting integers. An example is, “At 20,310 feet above sea level, Denali in Alaska is the highest point in the United States. The lowest point in the United States is in Death Valley, California. Death Valley is 282 feet below sea level. Write a subtraction expression to determine the difference in elevation between the highest and lowest points in the United States. Find the difference.” (7.NS.3)

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

• In Lesson 2.2, Problem Solving with Proportions, Proficient Practice, students develop procedural skill within application as they decide whether two quantities are in a proportional relationship. Problem 5 states, “Luis found a new text messaging plan which will charge him $2.00 for 80 messages. Using this plan, how much would he pay for 900 text messages in one month?” Problem 6 states, “A truck driver travels 93 miles in 1 hour and 30 minutes. At this rate, how far will he travel in 4 hours?”
• In Lesson 2.1, Proportional Relations, students develop conceptual understanding and procedural skills to solve proportions (7.RP.2). In the Explore!, Are You My Equal? activity, students make sense of proportions and find a strategy for solving. The Teacher Guide states, “Are You My Equal?” is an activity that is meant to be done prior to any instruction in this lesson. Students are given the definition of a proportion and then use this understanding to create proportions and learn about cross-products.” 
• In Lesson 8.4, Area of Polygons, Exercises, students develop procedural skill within and application problem by finding the area of polygons. (7.G.6) Exercises 1-9 provide students with diagrams of polygons (triangles, parallelograms, trapezoids) with dimensions and are instructed to, “Find the area of each polygon.” Exercises 10-18 provide students with diagrams of polygons (triangles, parallelograms, trapezoids) with one dimension and the area and are instructed to, “Find the missing measure.” There are multiple exercises where students must find area to solve real-world problems. For example, question 27 states, “The Owen family built a new house and wanted to put down sod in their rectangular front yard. The sod cost $0.32 per square foot plus $10 per square yard to lay the sod. The front yard is 30 feet by 50 feet. How much will it cost to purchase and lay the sod?”

The instructional materials reviewed for EdGems Math Grade 7 meet expectations for identifying the Standards for Mathematical Practice and using them to enrich mathematics content within and throughout the grade level.

All 8 MPs are identified throughout the materials. Each lesson includes a Lesson Guide with a section titled, Mathematical Practices - A Closer Look that explains a few of the MPs that will be used within that lesson. The MP is identified and an explanation of how to address the MP within the lesson is provided. At times, the identification is targeted, and gives a specific problem where the MP is included, but often it is broad and provides a general statement of how to include the MP within the lesson.

Examples of MPs that are identified and enrich the mathematical content include:

• Lesson 7.3, Simplifying and Solving Equations, “MP1: To start class, have students (with a partner or small group) choose either Exercise 14 or 15 and attempt to solve the problem before learning the process of solving an equation which requires simplifying prior to solving. Have students present their process to arrive at the solution and discuss, as a class, the method(s) that seem most effective.” 
• Lesson 5.3, Dividing Rational Numbers, “MP2: Write a division expression on the board. Have students contextualize it (write a story problem that matches the expression) and find the solution. Have students create a new expression using a different operation to give to a partner and repeat the process.”
• Lesson 1.1, Ratios, “MP3: Consider using Exercise 16 with students in pairs and have them share their work as a class, including their explanations. This will help students understand how to justify their thinking as well as help them begin to listen to the reasoning of others.”
• Lesson 2.4, Proportional Relationships Equations, “MP4: In this lesson, students are introduced to one more model (the equation) for a proportional relationship. Students can now model relationships with tables, graphs, equations and contexts. The Explore! allows students to practice this new model while connecting it to previously-learned models.”

The instructional materials reviewed for EdGems Math 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. 

Examples of the student materials prompting students to construct viable arguments and/or analyze the arguments of others include:

• In Lesson 2.1, Proportional Relationships, Exercise 27, students construct an argument to explain their understanding. “Describe two different ways you can determine if a pair of ratios have a proportional relationship.”
• Lesson 2.3, Tables and Graphs of Proportional Relationships, “Nina claimed that if you graph the ordered pairs from a table and they form a straight line, it is a proportional relationship. May-Lin said that is not always true. Sketch a graph that supports Nina’s claim and sketch a different graph that shows why a graph that forms a straight line is not always proportional.” Students provide evidence in the form of a graph to support a mathematical argument.
• One of the Teacher Gems activities is called “Always, Sometimes, Never.” The instructions for this type of activity state, “Always, Sometimes, Never is best used with concepts that allow for situations that create exceptions to the “rule” or require students to understand subcategories to fully understand the standard. Students create evidence to support whether a statement is always true, sometimes true or never true.” For example, in Lesson 4.4, Subtracting Rational Numbers, the student directions for the Always, Sometimes, Never Activity state, “Decide if the statement in the box below is always true, sometimes true, or never true. Use the remainder of the page to provide mathematical evidence that supports your decision. Statement 1: If you subtract a positive number from a negative number, you get a negative answer.”
• In Lesson 7.1, Solving One-Step Equations, Exercise 25, students write an equation and construct an argument to explain why their solution is correct. “Willis is thinking of two numbers. Their sum is −5. If one of the numbers is 19, what is the other number? Write an equation and solve it to find the other number. Explain how you know your answer is correct.”
• In Lesson 1.2, Unit Rates, Exercise 12 students analyze the solutions given by two students in which both students are correct, but the answers look different. “Levi walked 2 miles in 30 minutes. He and Sally found Levi’s unit rate as shown in the table. Explain why both of these rates are accurate but look different.”
• Lesson 3.1, Fractions, Decimals, and Percents, Exercise 7, “René needed to convert 334 to a decimal. She made an error in her work shown below. Describe her error and rewrite 334 as a decimal.” Students critique the work of others and provide the correct answer. 
• In Lesson 4.2, Adding Rational Numbers, problem 27, students analyze mathematical reasoning and correct mistakes made, “Alicia incorrectly added 34.6 and −5.7 as shown below. Explain what she did wrong and find the correct sum.”
• In Lesson 7.4, Linear Inequality, exercise 25, students critique the reasoning of others and justify their thinking. “Elijah claims that the solution set to −x ≤ 7 is the same as the solution set to x ≤ −7. Do you agree with Elijah? Explain your reasoning.”

The instructional materials reviewed for EdGems Math Grade 7 meet expectations for assisting teachers in engaging students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics.The Teacher/Lesson Guide and Teacher Gems within most lessons support teachers to engage students in constructing viable arguments and analyzing the reasoning of others. Examples include:

• In Lesson 1.1, Ratios, Teacher/Lesson Guide, teachers support students to analyze and critique the reasoning of others, “Consider using Exercise 16 with students in pairs and have them share their work as a class, including their explanations. This will help students understand how to justify their thinking as well as help them begin to listen to the reasoning of others.”
• In Lesson 3.3, Percent of Change, Teacher/Lesson Guide, teachers engage students in constructing a viable argument with the Explore! activity and analyzing others reasoning with a problem in the lesson. “Use the Explore! to lead students to a conjecture about how to find percent of change (that they will need to know the change in values and the original amount). Also, Exercise 15 asks students to critique the work of a student to fix an error that was made.”
• In Lesson 6.2, The Distributive Property, Teacher/Lesson Guide, teachers engage students in critiquing the reasoning of others “Refer to Exercise 13 for a common student mistake. Ask students to look over the problem and critique the reasoning shown in the exercise.”
• In Lesson 7.2, Solving Two-Step Equations, Teacher/Lesson Guide, teachers purposely make common errors that students make while solving equations. Students work in partners to explain the error and correct the error. 
• Lesson 8.2, Vertical and Adjacent Angles, Teacher Guide/Lesson Guide instructs teachers to “Use Exercise 21 for a turn and talk with partner sets. Ask one student in the partner set to complete the sentence starter: “Martin’s mistake was...” The partner should respond, “He can fix his mistake by…” In this manner, students are critiquing and analyzing the reasoning of others. 
• In Lesson 8.3, Drawing Triangles with Given Conditions, Teacher/Lesson Guide, teachers support students to make conjectures and analyze the reasoning of others. “In both parts of the Explore! students make conjectures. Have students share their conjectures and listen to the reasoning of others. Also, use Exercises 7–11 to have students discuss whether or not each statement is always, sometimes or never true. These can be difficult for students to answer and they will need to listen to the reasoning of others to learn how to reason through each statement on their own.”
• In Lesson 2.2, Problem-Solving with Proportions, Teacher Gem, Task Rotations, teachers support students in analyzing the reasoning of others. “At the fourth rotation, students do not complete the task, but rather read through the task card and examine the three team papers that have been left under the task card. In the fourth rotation, the team’s goal is to pick which paper they believe is the strongest paper. They mark this paper with a star and then each person in the group writes on their corresponding half-sheet using sentence starters like: “We chose Team __’s paper. Some reasons why we believe this paper is the best are _____.”
• In Lesson 10.2, Using Probability to Predict, Teacher Gem, Categories, teachers support students to conduct viable arguments. “Once sharing has been done informally in groups and through the use of the observers, the teacher may choose to ask students to share out what categories they created and how they knew what went in each category. Choosing a specific card and asking students which of their categories they would put it in and why allows students to construct arguments and attend to precision.”

The instructional materials reviewed for EdGems Math Grade 7 meet expectations for explicitly attending to the specialized language of mathematics. The materials provide explicit instruction in how to communicate mathematical thinking using words, diagrams, and symbols. Throughout the materials, precise terminology is used to describe mathematical concepts, and each lesson includes a visual lesson presentation. In most of the lesson presentations, there is at least one slide dedicated to explicit teaching of vocabulary. The Teacher/Lesson Guide, Student Lesson, and Parent Guide all contain information about mathematical language. Examples include:

• In the Student Lessons, the font for vocabulary words is red and a definition is included. 
• Each unit includes a Parent Guide which contains “Important Vocabulary” related to the unit. 
• In Lesson 4.1, Adding Integers, the Lesson Presentation includes a slide to introduce the terminology and definitions of the concepts, “Negative Numbers - A number less than 0. Positive Numbers - A number greater than 0. Opposites - Numbers that are the same distance from 0 on a number line but are on opposite sides of 0. Integers - The set of all whole numbers, their opposites, and 0. Absolute Value - The distance a number is from 0 on a number line.”
• In Lesson 7.4, Linear Inequalities, the Lesson Presentation includes a slide to introduce students to the vocabulary of the lesson and shows symbols related to the terminology. “Inequality - A mathematical sentence using <, >, ≤ or ≥ to compare two quantities. Solution Set - A set of numbers that make an equation or inequality statement true.”
• Lesson 2.3, Tables and Graphs of Proportional Relationships, Teacher/Lesson Guide, Teaching Tips includes information about mathematical language. “It is important to connect the idea that the rate of a proportional relationship is also called the constant of proportionality.”
• Lesson 5.4, Order of Operations, Teacher/Lesson Guide, Teaching Tips includes information about mathematical language. “When a power is 2 it is often read “squared”. Connect this with area and the fact that they write the units as square units or u2.”
• Lesson 6.3, Equivalent Expressions, Teacher/Lesson Guide, Teaching Tips includes practice with mathematical terms. “Practice with the vocabulary: term, constant, like terms and coefficient. These all appear multiple times in this text and future math courses. Perhaps have students demonstrate the vocabulary on whiteboards or with a partner. Be sure students use the vocabulary when talking about terms in problems.”
• In Lesson 8.4, Area of Polygons, Teacher/Lesson Guide, Communication Prompt, students describe and define math vocabulary. “Explain the difference between perimeter and area.” 
• In Lesson 8.2, Vertical Angles and Adjacent Angles, Teacher/Lesson Guide, students practice using correct mathematical terminology. “Emphasize the names of the special angle pairs. During instruction and work time, listen for students using correct names for each special angle pair.”
• In Lesson 1.3, Rates and Ratios with Complex Fractions, the Student Lesson describes the concept using mathematical terminology and highlights new vocabulary by providing examples. “A complex fraction is a fraction that contains a fraction expression in its numerator, denominator or both. The following are examples of complex fractions.”