7th Grade - Gateway 2

The materials reviewed for EdGems Math (2024) Grade 7 meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific content standards of cluster headings.

Materials develop conceptual understanding throughout the grade level, providing opportunities for students to independently demonstrate their understanding through various program components. Each lesson incorporates an Explore! activity for students to discover new concepts using diverse methods, a Lesson Presentation with slides that supports reasoning and sense-making through examples and communication breaks, a Student Lesson featuring mathematical representations, and a selection of Teacher Gems designed to target conceptual understanding through engaging activities such as Always, Sometimes, Never, Categories, Four Corners, and Climb the Ladder. Examples include: 

• Unit 1, Lesson 1.2, Leveled Practice T, Exercises 1-7, students develop conceptual understanding of unit rates with ratios written in fraction form as they write unit rate ratios using division to find equivalent ratios (7.RP.1). “Find each unit rate. 1. \frac{100 miles}{4 gallons} 2. \frac{30 words}{2 minutes}. \frac{200 miles}{8 hours} 4. \frac{65 dollars}{5 books} ” As shown below each problem, a model is provided with the original ratio, followed by a division symbol for each part, an equal sign, and a new ratio with '1' labeled at the bottom. "5. \frac{210 kilometers}{7 hours} 6. \frac{24 dollars}{4 people} 7. A store sells 2 dozen cookies for $12.00. Find the price per cookie. a. 2 dozen cookies = _____ cookies. b. Write a rate of total dollars to total cookies. c. Find the unit rate (price per cookie).”

• Unit 6, Lesson 6.1, Teacher Gems, Always, Sometimes, Never, Statement 4, students work in pairs or partners to develop a conceptual understanding of evaluating numerical expressions using integer operations (7.EE.1). '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 #4: The value of a numerical expression is positive if it contains an even power.'"

• Unit 10, Lesson 10.1, Student Lesson, Example 1, students develop a conceptual understanding of probability by describing the likelihood of a real-world scenario, ranging from impossible to certain (7.SP.5). An example provided is as follows: “Determine whether each event is impossible, unlikely, equally likely, likely, or certain: a. A playing card thrown on the floor lands face up. b. You roll a 2 on a regular number cube. c. You are older now than when you were born. Solutions: a. Equally likely. It can land face up or face down – neither is more likely than the other. b. Unlikely. There are 6 numbers on a number cube, and 2 is only one of them. c. Certain. You are older now than when you were born. Extra Example 1: Determine whether each event is impossible, unlikely, equally likely, likely, or certain: a. A card picked from a full deck is a king. b. A coin lands on its side when it is flipped on a flat surface. c. You roll a 3, 4, or 5 on a regular number cube.”

The materials provide students with opportunities to engage independently with concrete and semi-concrete representations while developing conceptual understanding. Examples include:

• Unit 4, Lesson 4.2, Student Lesson, Exercise 12, students demonstrate conceptual understanding of adding rational numbers when given a real-world scenario (7.NS.1). An example provided is as follows: “Jonah went scuba diving. On his first dive he descended -9\frac{2}{3}feet, paused and then descended an additional -12\frac{5}{6} feet. a. Explain why -9\frac{2}{3} + -12\frac{5}{6} represents his total change in depth. b. Find his total change in depth.”

• Unit 5, Lesson 5.1, Student Lesson, Exercises 15, students demonstrate conceptual understanding of multiplication of integers by modeling integer multiplication on a number line (7.NS.2.C). An example provided is as follows: “Model 3(−2) on one number line and −2(3) on a different number line. How are these the same? How are they different?” Two number lines are provided, each labeled with integers from -6 to 6.

• Unit 7, Lesson 7.2, Explore!, students demonstrate conceptual understanding of solving equations by using an equation mat and algebra tiles to solve an equation (7.EE.2). An example provided is as follows: “Step 2: On your equation mat, place two positive variable tiles on one side with one positive unit tile. On the other side of the mat, place seven negative unit tiles. What two-step equation does this represent? Step 3: The first step in solving an equation is to isolate the variables. The integer tiles that are with the variable tiles must be canceled out. Use zero pairs to cancel out the positive unit tile on the left side of the mat. Remember that whatever you add to one side must be added to the other side of the mat. Draw a picture of what your mat looks like now. Step 4: Divide the unit tiles on the mat equally between the two variable tiles. How many unit tiles are equal to one variable tile? This is what x equals.”

The materials reviewed for EdGems Math (2024) Grade 7 meet expectations for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency.

There are opportunities for students to develop procedural skills and fluency in each lesson. The materials support the development of these skills and fluencies through Starter Choice Boards, Student Gems, Lesson Examples, Student Exercises, and Teacher Gems. Examples include:

• Unit 2, Lesson 2.1, Starter Choice Board, Target Skill 2, students develop procedural skill and fluency by adding and subtracting decimals (7.NS.1). Five expressions are provided: “3.2 + 5.5, 9.8 - 3.2, 5.5 - 2.1, 6.2 + 5.9, 5.1 - 4.6.”

• Unit 5, Lesson 5.3, Teacher Gem: Ticket Time, students develop procedural skills and fluency in evaluating expressions using multiplication and division with rational numbers (7.NS.2). This activity is completed with peer partners and teacher support. Examples include: “Ticket 3: Find the value of 6\div(-\frac{2}{3}). Ticket 4: Find the value of -\frac{2}{3}\div1.5. Ticket 6: Find the value of -3.84\div2.4.

• Unit 8, Lesson 8.2, Teacher Gems, Partner Math, students develop procedural skills and fluency by working with a partner to solve for unknown angles using relationships of vertical, complementary, and supplementary angles (7.G.5). B. Examples include: “The measure of \angle DEF is 34^{\circ}. The measure of its complement is (3x-1)^{\circ}. What is the value of x? C. \angle TEN and \angle KEV are vertical angles. The measure of \angle TEN is 125^{\circ}. What is the measure of \angle KEV?”

There are opportunities for students to develop procedural skill and fluency independently throughout the grade level. Examples include:

• Unit 2, Lesson 2.2, Leveled Practice P, Exercises 6-8, students independently demonstrate procedural skill and fluency as they use proportional relationships to solve multistep ratio problems (7.RP.3). Examples include: “6. A truck driver travels 93 miles in 1 hour and 30 minutes. At this rate, how far will he travel in 4 hours? 7. Mark walked 21,129 feet in one hour. At that speed, about how many miles will he walk in two hours? 8. A 12-ounce soda costs $1.25 in the vending machine. At that rate, how much would a 32-ounce soda cost?”

• Unit 6, Lesson 6.4, Student Lesson, Exercises 6-8, students independently demonstrate procedural skill and fluency as they use properties of operations to simplify linear expressions (7.EE.1). An example provided is as follows; “Simplify each algebraic expression. 6. 7-3(x-2). 7. (2x+1)-(6x-7). 8. -5(2f-7)+10f-30.”

• Unit 7, Lesson 7.4, Explore! students independently demonstrate procedural skill and fluency as they use variables in mathematical problems of inequalities to solve problems by reasoning about their quantities (7.EE.4). An example provided is as follows: “Step 1: Add 4 to both sides of each of the true inequalities below. Are the resulting inequalities true statements? a. 5 < 8 b. 1 > −6 c. −10 < −2. Step 2: Add −3 to both sides of each of the true inequalities below. Are the resulting inequalities true statements? a. 5 < 8 b. 1 >−6 c. −10 < −2. Step 3: Based on your trials above, do you think inequalities can be balanced using addition in the same way an equation can be balanced using the Addition Property of Equality? Explain.”

The materials reviewed for EdGems Math (2024) Grade 7 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics.

There are opportunities for students to develop routine and non-routine applications of mathematics in each lesson. The materials develop application through the Student Lesson Exercises in the Apply to the World Around Me section, Teacher Gems, a Storyboard Launch/Finale, and Performance Tasks.

Materials include multiple routine and non-routine applications of mathematics throughout the grade level. Examples include:

• Unit 3, Lesson 3.4, Explore!, Step 6, solve a non-routine problem by identifying an error in a multistep percent problem and then providing a correct solution (7.RP.3). An example provided is as follows: “Mikayla lived in a town that had an 8% tax rate on food served in restaurants. She planned to purchase food and drinks that cost a total of $100 and give a 15% tip on food and drink before tax. She determined her total cost was $124.20 after tax and tip. Her calculations are shown below. Explain what she did wrong and calculate the correct total. $100\times0.08=$8 tax; $100 + $8 = $108; $108”

• Unit 5, Lesson 5.1, Student Lesson, Exercise 19, students solve a routine real-world problem involving operations with rational numbers (7.NS.3). Example 19 states, “Maggie borrowed $4 from one friend and $8 from another friend. She did this five days in a row. What integer represents the amount of money she owes?”

• Unit 7, Lesson 7.2, Teacher Gems, Masterpiece, students solve a non-routine problem by constructing simple equations to solve a problem (7.EE.4). Masterpiece states, “The Erickson family is planning a family vacation and are looking for vacation rentals in the area. They have found three comparable rentals, each with different costs and fees which will accommodate their four person family. Option A: $150 cleaning cost plus $115 per night Option B: $20 cleaning cost per night plus $159 per night Option C: $190 per night with no cleaning cost 1. Write an expression to model the cost, 𝐶, of each rental option for 𝑛 nights. 2. The Ericksons budget will allow them to spend up to $750 on lodging prior to additional fees and taxes. Which option should the Ericksons choose if they want the longest vacation possible?” 

Materials provide opportunities for students to independently demonstrate multiple routine and non-routine applications of mathematics throughout the grade level. Examples include:

• Unit 1, Materials, Tic-Tac-Toe Board, Home Floor Plan, students independently solve a non-routine problem by creating a scale drawing using a given scale (7.G.1) The Home Floor Plan states, “Use the scale 1 in : 2.5 ft to create a scale drawing of a home. The floor plan must include a minimum of: 1 family room. 1 kitchen, 1 bathroom, 2 bedrooms Step 1: Sketch the design of the floor plan. Step 2: Draw the floor plan using a ruler. Carefully measure each room. Step 3: Next to each wall in the home, label the length of the measurement in the drawing. Step 4: Next to each wall in the home, label the length of the actual measurement in the home. Step 5: List the area of each room in real-life on the drawing.”

• Unit 2, Lesson 2.1 Student Lesson, Exercise 13, students independently solve routine problems by determining whether ratios in a real-world context form a proportion (7.RP.2). Exercise 13 states, “Larry paid $15 for 6 bags of pretzels. Nancy bought 4 bags of the same pretzels at another store. She paid $11. a. Find the ratio of the cost to the number of bags of pretzels for Larry’s purchase. b. Find the ratio of the cost to the number of bags of pretzels for Nancy’s purchase. c. Do the ratios form a proportion? Explain how you know your answer is correct.” 

• Unit 6, Lesson 6.4, Student Lesson, Exercise 17, students independently solve a routine multi-step real-life problem involving algebraic expressions (7.EE.3). Exercise 17 states, “The length of a rectangular table runner is one less than three times the width. a. Write a simplified expression to represent the perimeter of the table runner. b. If the width of the table runner is 2.5 feet, what is the area of the table runner?”

The materials reviewed for EdGems Math (2024) Grade 7 meet expectations 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 level. 

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

• Unit 2, Lesson 2.2, Student Lesson, Exercise 7, students apply their understanding of ratios in lengths and areas to solve a real-world problem (7.RP.1). “Carrie made a design that was 8.5 inches by 11 inches. She enlarged the design so the new design had a scale factor to the old design of 3 : 1. a. What is the perimeter of the larger design? b. What is the area of the larger design? c. Carrie’s friend said the new design has an area three times as large as the area of the original design. Carrie disagrees. She says the new area is nine times as large as the area of the original design. Who is correct? Use mathematics to justify your answer.” 

• Unit 6, Lesson 6.3, Student Lesson, Exercise 8, students deepen their conceptual understanding as they use the distributive property to generate equivalent expressions (7.EE.1). Exercise 8 states, “Use the Distributive Property to rewrite each expression without parentheses. -3(m+2h-5)” 

• Unit 8, Lesson 8.2, Student Lesson, Exercise 1, students demonstrate procedural skill by applying facts about vertical angles, adjacent angles, and linear pairs to solve problems (7.G.5). Exercise 1 states, “1. m\angle1=50^{\circ}, find the following. a. m\angle2= __. b. m\angle3= __ c. m\angle4= __” A pair of vertical angles is provided. 

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

• Unit 4 Lesson 4.2 Leveled Practice T, students use all three aspects of rigor, conceptual understanding, procedural fluency, and application as they add positive and negative fractions and decimals (7.NS.1). Leveled Practice T states, “Find each sum. 11. -4.6 + 2.34. 12. 3.6 + (-5.8). 13. Mr. Peters bought a jar of jellybeans for his clients. He gave out 3.6 pounds of jellybeans over the month. Mr. Peters then refilled the jar with 1.2 pounds of jellybeans. a. One of the decimals should be written as a negative number. Which one is it and why? b. Find the sum of the two numbers to determine the total change weight of the jellybeans.”

• Unit 5, Materials, Unit Review, Exercise 15, students use all three aspects of rigor, conceptual understanding, procedural fluency, and application to solve real-world problems involving fractions, including division of mixed numbers, and apply their knowledge to determine practical solutions (7.NS.2). Exercise 15 states, “Nai’s dog eats about 1\frac{3}{4} cups of dog food per day. If Nai has 12\frac{1}{3} cups of dog food left, about how many days can Nai feed his dog before having to buy more?”

• Unit 10, Lesson 10.1, Student Lesson, Exercise 9, students use conceptual understanding and apply their understanding by calculating theoretical probabilities, interpreting expected outcomes, and explaining the reasoning behind the likelihood of events in real-world contexts (7.SP.5). Exercise 9 states, “Rupees are a type of coin used as currency in India. A rupee was used for an experiment where the side with the number was tails and the side with the lion was heads. a. What is the theoretical probability of a flipped rupee landing tails? b. You flip a rupee 40 times. How many times would you expect it to land tails based on theoretical probability? c. Are you guaranteed to get tails 20 times? Explain your reasoning.”

The materials reviewed for EdGems Math (2024) 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. Students have opportunities to engage with the Math Practices throughout the year, with each lesson’s Unit Overview and Teacher Guide clearly outlining the teacher and student actions for implementing these practices.

MP1 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students make sense of problems and persevere in solving them as they work with support of the teacher and independently throughout the units. Examples include:

• Unit 2, Planning & Assessment, Performance Assessment, Problem 3, students make sense of problems and persevere in solving and representing a real-world proportional relationship. Problem 3 states, “The next week, the store sells every video game at an additional 10% discount off the first sale price. a. The owner graphed the first sale price, 𝑥, with the new sales price, 𝑦. What is the constant of proportionality of the graph? Show work that supports your answer. b. The manager graphed the original price of each video game, 𝑥, with its new sale price, 𝑦. She said the constant of proportionality was 0.45. Explain why this is the constant of proportionality and explain what this constant of proportionality means in terms of the situation. c. Rhane has $20. He wants to purchase a video game that was originally priced at $36. If the sales tax is 8%, can Rhane afford the video game that he wants? Show work that supports your answer.”

• Unit 6, Lesson 6.4, Student Lesson, Exercise 14, students make sense of problems as they generate equivalent expressions using properties of operations. Exercise 14 states, “Explain two different ways to simplify 3(4x-10+3.8x).” 

• Unit 10, Lesson 10.4, Lesson Presentation, Example 1 and 2, students make sense of problems and then persevere in solving with teacher guidance as they make sense of random sampling and what comprises a true representative sample. Example 1 states, “Margo wanted to know if people in her city would vote for a dog park. She could not ask everyone in the city so she decided she would call 50 random people from the list of people who take their animals to the vet. Explain whether or not the sample of people surveyed will most likely give an accurate prediction of how people will vote in the election.” The teacher provides the answer after students attempt the problem. “This sample would most likely NOT give an accurate prediction. People who go to the vet own animals and may be more likely to support a vote for a dog park. The sample needs to include a better representation of the entire city, both people who own animals and people who do not. It is a biased sample.” Example 2 states, “A company is making a new video game. There is a possibility some of the games have a defect and do not work. The owner of the company wants to make sure the games work before he sells them. He has made 10,000 games and packed them into 100 boxes that each holds 100 games. He is trying to decide if he should test 100 games from a single box or if he should test 100 games by testing one game in each of the 100 different boxes. Which sample do you think would give a more accurate prediction about the games and the possible defect?” The teacher provides the answer after students attempt the problem. “Choosing a single game from each of 100 different boxes would most likely give a more accurate prediction of whether or not the games have defects. Games from the same box are usually manufactured and packed at the same time. Games from different boxes were most likely made at different times and will better represent the entire population of games.”

MP2 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students reason abstractly and quantitatively as they work with support of the teacher and independently throughout the units. Examples include:

• Unit 2, Lesson 2.2, Student Lesson, Exercise 6, students reason abstractly and quantitatively as they create a real-world situation that could be represented with a given proportion. Exercise 6 states, “Write a real-world scenario with a missing value that could be modeled by the proportion \frac{30}{4}=\frac{x}{7}.”

• Unit 7, Lesson 7.2, Student Lesson, Exercise 14, students reason abstractly and quantitatively as they write and solve a two-step equation representing a real-world situation. Exercise 14 states, “Each month the fire department hosts a pancake feed. This month 75 people attended the pancake feed. This was 13 less than twice as many people as were at the pancake feed last month. How many people attended last month’s pancake feed? Write and use an equation to find the answer.” 

• Unit 9, Lesson 9.2, Lesson Presentation, Example 1, students reason abstractly and quantitatively as they calculate the surface area of prisms. Example 1 states, “A shipping container used by an overseas shipping company is a rectangular prism with a length of 4.5 feet, a width of 2 feet and a height of 3 feet. What is the surface area of the shipping container?” Teachers guide students on how to solve through directions and illustrations. “Draw the net of the solid. Label the dimensions. Find the area of each rectangle.” 

The materials reviewed for EdGems Math 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.

Students have opportunities to engage with the Math Practice throughout the year, with each lesson’s Unit Overview and Teacher Guide clearly outlining the teacher and student actions for implementing these practices.

Students construct viable arguments, in connection to grade-level content, as they work with support of the teacher and independently throughout the units. Examples include:

• Unit 3, Planning & Assessment, Performance Assessment, Exercise 1d, students construct a viable argument as they use proportional relationships to solve multistep ratio and percent problems. Exercise 1d states, “Employees in an electronics store earn commission on items that customers purchase. That means they earn a percentage of what the item costs in addition to their salary. Jacob earns \frac{3}{100} commission on items he sells. Peggy earns 5% commission on items she sells. Lamar earns 2% commission if the total cost of items he sells is less than $100. If he sells more than $100, he earns 10% commission on the entire sale. Chen earned $50 from her commission after selling $400 worth of electronics. Jacob and Peggy each earned $200 in commission from sales on Tuesday. Peggy says that means they each sold the same amount of items in terms of dollars. Is she correct? Justify your answer using words and mathematics.”

• Unit 6, Lesson 6.1, Student Lesson, Exercise 14, construct a viable argument as they explain their reasoning when solving mathematical problems involving exponents. Exercise 14 states, “Explain in words the difference between (-5)^{2} and -5^{2} .”

• Unit 8, Lesson 8.3, Student Lesson, Exercises 9-11, students construct viable arguments when determining if a triangle can be constructed given three side lengths. The materials state, “In problems 9-11, determine if each statement is always, sometimes or never true. Explain your reasoning. 9. The sum of the two shortest sides of a triangle must be less than the longest side. 10. If three sides of one triangle are equal in length to three sides of another triangle, the triangles are exactly the same size and shape. 11. If two figures are the same shape, then they are also the same size.”

Students critique the reasoning of others, in connection to grade-level content, as they work with support of the teacher and independently throughout the units. Examples include:

• Unit 1, Planning & Assessment, Unit 1 Assessment Form A, Problem 8, students critique the reasoning of others as they compute unit rates associated with ratios of fractions. Problem 8 states, “It took 56 minute to fill 10 equal-sized glasses with water. Assume the water filled the cups at the same rate. What was the rate of water in minutes per glass? Ethan answered this question but made a mistake. Explain Ethan’s mistake. Then complete the work correctly to answer the question.” Ethan’s work in solving the problem is provided.

• Unit 2, Lesson 2.2, Student Lesson, Exercise 7, students critique the reasoning of others as they evaluate the clarity and accuracy of arguments, identify strengths and weaknesses, and suggest improvements based on valid mathematical principles. Exercise 7 states, “Carrie made a design that was 8.5 inches by 11 inches. She enlarged the design so the new design had a scale factor to the old design of 3 : 1. a. What is the perimeter of the larger design? b. What is the area of the larger design? c. Carrie’s friend said the new design has an area three times as large as the area of the original design. Carrie disagrees. She says the new area is nine times as large as the area of the original design. Who is correct? Use mathematics to justify your answer.”

• Unit 10, Lesson 10.2, Student Lesson, Exercise 6, students critique the reasoning of others as they use models to find probabilities of events. Exercise 6 states, “Tisa heard that about 5 out of every 18 people that apply for a summer job end up with their first choice job. She knew that 80 of her classmates were hoping to get a job so she set up the following proportion to find out how many will get their first choice. \frac{5}{18}=\frac{80}{x}. When she solved the proportion, she got an answer of 288 classmates. This answer did not make sense. Explain to Tisa what she did wrong and then find the correct prediction.”

The materials reviewed for EdGems Math (2024) 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. Students have opportunities to engage with the Math Practices throughout the year, with each lesson’s Unit Overview and Teacher Guide clearly outlining the teacher and student actions for implementing these practices.

MP4 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students have many opportunities to solve real-world problems, model situations with appropriate representations, and describe what they do with the model and how it relates to the problem. Students model with mathematics as they work with support of the teacher and independently throughout the units. Examples include:

• Unit 1, Planning and Assessment, Performance Assessment, Exercise 2, students model with mathematics as they solve real-world situations involving ratios and rates. Exercise 2 states, “Tamara made a scale drawing of her new house. She used a scale of 3 inches to 8 feet in her drawing. The family room and office are shown below.” An image of a square labeled 'Office,' adjacent to a rectangle labeled 'Family Room,' is provided. “The walls in Tamara’s home will be 8 feet tall. One of the two longer walls in the family room has no windows or doors, and Tamara wants to paint it red. How many square feet of paint will Tamara need for the one long family room wall? Justify your answer using words and mathematics.”

• Unit 4, Lesson 4,1, Lesson Presentation, Explore!, with the support of the teacher, students model with mathematics as they add and subtract integers. The materials state, “Integer chips or tiles are helpful for modeling integer operations. Each blue chip will represent the integer +1. Each red chip will represent the integer −1. When a positive integer chip is combined with a negative integer chip, the result is zero. This pair of integer chips is called a zero pair. Step 1: Explain in your own words why a positive integer chip and a negative integer chip form a zero pair. Step 2 Model 4+(−2) with integer chips. Step 3 Group as many zero pairs as possible. Step 4 Because zero pairs are worth zero, remove all zero pairs. What is the result? Step 5 Write an expression for the following model. Create the same model using your own integer chips.” A model that shows three blue plus five red chips. “Step 6 Group as many zero pairs as possible and remove them. What is the result of your expression? Step 7 Model −5 + (−3) with integer chips. Step 8 Are there any zero pairs? If so, remove them. What is the result of −5 + (−3)? Step 9 Write an integer addition expression. Model it with integer chips to find the sum. Step 10 Use integer chips to help you determine what type of answer (positive, negative or zero) you think you will get for each situation. Explain your reasoning. a. the sum of two positive numbers. b. the sum of two negative numbers. c. the sum of a number and its opposite. d. the sum of a negative and positive integer.”

• Unit 9, Materials, Performance Task, students model with mathematics as they identify quantities, explain their meanings, determine how quantities relate to each other and the situation, and then connect their solutions back to the context. The Performance Task states, “PART 1: Brighton has a candle-making business and has a new design for a candle in the shape of a square pyramid. 1. What are some questions related to the situation above that could be solved using mathematics? 2. What additional information do you need to answer the questions you created? PART 2: Brighton designed the candle to have a base length of 3 inches, a height 4 inches and a slant height of 5 inches. Brighton wants to determine the total cost of materials to make each candle. 3. If the wax costs $0.10 per cubic inch and the wicks cost $0.02, how much will it cost to make one candle? 4. To protect the candle, the Brighton also covers the whole candle in a thin cardboard material that costs $0.05 per square inch. What is the cost to cover one candle? PART 3: Brighton packages each individual candle in a rectangular box that has the same width and height as each candle. To ensure that the candles do not move within the box, the rest of the box is filled with padding material. 5. If the box costs $0.08 per square inch, how much does it cost to make one candle box? 6. If the padding material costs $0.03 per cubic inch, how much does it cost to package the candle, including the cost of the box? 7. If Brighton sells each candle for $35, what is the profit per candle?”

MP5 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students have multiple opportunities to identify and use a variety of tools or strategies, working with the support of the teacher and independently, throughout the units to support their understanding of grade-level math. Examples include:

• Unit 2, Lesson 3.4, Student Lesson, Exercise 7, students choose appropriate tools as they calculate multi-step percentages, verify the clearance price, and explain their reasoning. Exercise 7 states, “Bradley’s uncle wants to buy him a basketball that was originally priced $36. It was marked down 40% and then was put on clearance an additional 10% off the sale price. Bradley’s uncle told him the basketball was half price. Is he correct? Explain your reasoning then determine the clearance price of the basketball.”

• Unit 4, Lesson 4,1, Student Lesson, Exercise 11, students choose appropriate tools as they solve problems involving integers. Exercise 11 states, “Maria is in a two-day golf tournament. She scored −3 on the first day. On the second day, her score was −5. In golf, the lowest score wins. a. On which day did Maria have a better golf score? Explain. b. What is her overall score for the entire tournament?” Teacher Guide, Math Practices: Teacher and Student Moves, Teacher Moves state “Provide students with integer chips, horizontal number lines, vertical number lines, blank number lines, grid paper and other manipulatives from which to choose. Before students begin solving a problem, instruct them to choose a tool and explain why they chose the tool for the problem. After solving the problem, encourage students to reflect on the effectiveness of the tool.” Student Moves, “For Student Lesson Exercises #11-13, choose a tool, explain why you chose the tool, solve the problem, then reflect on how useful the tool was. With a partner, discuss anything that you noticed because of the use of the tool.”

• Unit 7, Lesson 7.1, Student Lesson, Exercise 4, with the support of the teacher, students use tools strategically as they solve algebraic equations. Exercise 4 states, “Solve each equation. Check each solution. −6d = −24.” Teacher Guide, Focus Math Practice, Math Practices: Teacher and Student Moves, Teacher Moves, states, “Equation mats and algebra tiles are used in this and future lessons as a physical model to help students see the process of solving equations. Provide students with these and other manipulatives from which to choose. Instruct students to explain how each tool they choose is helpful for visualizing and solving equations.”

The materials reviewed for EdGems Math (2024) 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 have many opportunities to attend to precision and the specialized language of math, in connection to grade-level content, as they work with support of the teacher and independently throughout the units. Examples include:

• Unit 2, Lesson 2.3, Student Lesson, Exercise 7, students attend to the specialized language of mathematics as they identify proportional relationships in a table and state the constant of proportionality. Exercise 7 states, “The table at right could represent a proportional relationship if one pair of numbers was removed. Which (x, y) pair of numbers should be removed and what is the constant of proportionality once it is removed?” A table is provided with x values 6, 9, 12, 15 and corresponding y values 4, 6, 9, 10.

• Unit 3, Lesson 3.3, Student Lesson, Exercise 8, students attend to the specialized language by using precise mathematical terms such as 'percent decrease,' 'original amount,' and '100% decrease' to explain and justify their reasoning clearly. Exercise 8 states, “Hailey claimed that the sale price of an item was more than a 100% decrease compared to the original price. Do you agree or disagree? Explain your reasoning.”

• Unit 4, Lesson 4.2, Student Lesson, Exercise 7, students attend to precision as they add rational numbers given in various forms. Exercise 7 states, “Find the sum of −2.38, 5.6, 4\frac{1}{4} and -1\frac{2}{5}” Teacher Guide, Math Practices, Teacher Moves “As students perform operations with rational numbers, they must deal with a variety of components (e.g., the sign of a solution, common denominators, simplest form, etc.). Have students revisit the sign of the solution after completing the operation based on the instructions they learned in Lesson 4.1.” 

• Unit 7, Lesson 7.4, Student Lesson, Exercise 12, students attend to precision as they create number lines for real-world situations represented by inequalities. Exercise 12 states, “Callie and BreShay went to the mall. They each spent the exact same amount during the day. Callie spent less than or equal to $45. BreShay spent more than $40. Create a number line that shows all the possible amounts that Callie or BreShay could have spent during the day.”

The materials reviewed for EdGems Math (2024) 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 have opportunities to engage with the Math Practices throughout the year, with each lesson’s Unit Overview and Teacher Guide clearly outlining the teacher and student actions for implementing these practices.

MP7 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students have many opportunities to look for, describe, and make use of patterns within problem-solving as they work with support of the teacher and independently throughout the units. Examples include:

• Unit 4, Lesson 4.1, Student Lesson, Exercise 17, students make use of structure as they add positive and negative numbers. Exercise 17 states, “Esther opened a checking account by depositing $60. Her next four transactions are shown in the table at right. Complete the “Integer” column of the table by writing an integer to represent each transaction. Find her balance at the end of this transaction period.” A table is shown with two columns: the first column lists transactions ('Deposited $60,' 'Withdrew $22,' 'Withdrew $6,' 'Deposited $35,' 'Withdrew $20,' and 'Balance'), and the second column, titled 'Integer,' starts with +60. Students use the information from the transactions in the first column to complete the corresponding rows in the 'Integer' column.

• Unit 7, Lesson 7.4, Lesson Presentation, Explore!, students, with the support of the teacher, make use of structure in their development of generalizations based on the similarities and differences found between solving equations and solving inequalities. The materials state, “An inequality is a statement that compares one expression to another expression. In the last three lessons, you have solved equations by isolating the variable. You performed operations on both sides of the equation and the equation remained true. In this activity, you will examine if that process works for inequalities as well. Step 1: Add 4 to both sides of each of the true inequalities below. Are the resulting inequalities true statements? a. 5 < 8 b. 1 > −6 c. −10 < −2 Step 2: Add −3 to both sides of each of the true inequalities below. Are the resulting inequalities true statements? a. 5 < 8 b. 1 > −6 c. −10 < −2 Step 3: Based on your trials above, do you think inequalities can be balanced using addition in the same way an equation can be balanced using the Addition Property of Equality? Explain. Step 4: Multiply each number by 3 to create new inequalities below. Are the resulting inequalities true statements? a. 5 < 8 b. 1 > −6 c. −10 < −2 Step 5: Multiply each number by −2 to create new inequalities below. Are the resulting inequalities true statements? a. 5 < 8 b. 1 > −6 c. −10 < −2 Step 6: Based on your trials from Steps 4 - 5, do you think inequalities can be balanced using multiplication (or division) in the same way an equation can be balanced using the Multiplication Property of Equality? Explain. Step 7: What did you notice when you multiplied each number in an inequality by a negative number in Step 5? Step 8: What could you do to each of the new inequalities in Step 5 to make a true statement (without changing the values of the numbers)? Step 9: Complete the following statement: When you multiply or divide both sides of an inequality by a _________________ number, you must _____________ the inequality sign.”

• Unit 8, Lesson 8.1, Student Lesson, Exercise 13, students make use of structure as they use facts to describe adjacent angles. Exercise 13 states, “\angle PRM and. \angle MRT are adjacent angles. Will \angle PRT always, sometimes or never be greater than \angle PRM? Explain how you know your answer is correct.” Teacher Guide, Math Practices, Teacher Moves, “Facilitate Exercise 13 using the Always Sometimes Never Teacher Gem structure. Extend the activity by listing the properties of operations on the board and asking students to identify which property could be used to support their claim.” 

MP8 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students have multiple opportunities to use repeated reasoning in order to make generalizations and build a deeper understanding of grade-level math concepts as they work with support of the teacher and independently throughout the units. Examples include:

• Unit 1, Lesson 1.3, Student Lesson, Exercises 1-3, students, with the support of the teacher, use repeated reasoning to compute unit rates associated with ratios of fractions. “Find each unit rate. 1. \frac{\frac{4}{3}inches}{\frac{4}{9}minute} 2. \frac{\frac{5}{7}feet}{\frac{15}{14}seconds}3. \frac{1\frac{1}{2}miles}{\frac{1}{4}hour}” Teacher Guide, Math Practices, Teacher Moves, “Reinforce the concept of a fraction bar representing division. With repeated reasoning, students should draw connections between division and complex fractions.”

• Unit 2, Lesson 2.4, Student Lesson, Exercise 13, students look for regularity and use repeated reasoning to represent proportional relationships as a table and an equation. “ Exercise 13 states, Juanita opened her own movie theater. She plans to charge $5.00 per person and hopes to fill her 50-seat theater once in the late afternoon and once in the evening. a. Copy and complete this table to show how much money Juanita will get for selling the given numbers of tickets to a show. b. Is this a proportional relationship? Explain your reasoning. c. Write an equation for the amount of money collected (y) based on the number of tickets sold (x). d. If Juanita sells out both shows in the afternoon and evening, how much money will she collect?” A table containing 'Number of tickets sold (x): 0, 10, 15, 20, 40, 50' and 'Money collected ($, y)' is provided.

• Unit 10, Lesson 10.5, Student Lesson, Exercise 15, students look for regularity and use repeated reasoning to find multiple sets of dimensions that satisfy the volume formula while keeping the volume constant. Exercise 15 states, “The volume of a rectangular prism is 36 in^{3}. Give two different sets of possible measurements for its length, width and height.”