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. $ 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.”

The materials reviewed for EdGems Math (2024) Grade 7 meet expectations for providing teacher guidance with useful annotations and suggestions for how to enact the student materials and ancillary materials, with specific attention to engaging students to guide their mathematical development.

Examples of where and how the materials provide comprehensive guidance that will assist teachers in presenting the student and ancillary materials include:

• Key instructional support through resources designed to enhance teacher effectiveness. The Unit Planning & Assessment pages offer access to both general course and unit-specific instructional information, ensuring teachers have the necessary materials for lesson execution. The PD Library includes written and video-based professional development on implementing Teacher Gems, Communication Breaks, Fluency Boards & Routines, and the 5E Instructional Model, equipping teachers with techniques for effective instruction. Additionally, the ELL Supports Guide provides strategies for ELL Proficiency Levels, Instructional Design, Mathematical Language Routines (MLRs), and Scaffolding Techniques. This guide includes resources such as a Word Problem Graphic Organizer, Target Trackers, Math Practice Trackers, a Math Self-Assessment Rubric, and a Vocabulary Journal Format, ensuring multilingual learners receive appropriate language supports.

• Lesson planning guidance is structured through unit resources that outline daily instructional expectations. The Unit Launch Guide provides a two-day lesson plan for introducing each unit, detailing required and optional components with class time allocations and facilitation instructions. These components include the Target Tracker Launch, Storyboard Launch, Fluency Board Launch, Readiness Check, and Unit Launch Teacher Gem, all designed to establish foundational knowledge. The Unit Finale Guide supports teachers in unit review, differentiation, and assessment through a three-day lesson plan incorporating the Unit Review, Unit Finale Teacher Gem, Fluency Board Finale, Storyboard Finale, and Assessments, along with explanations of assessment options.

• Lesson implementation support is embedded within the Teacher Guides, which contain detailed two-day lesson plans with structured guidance on instruction and differentiation. The At a Glance section provides a one-page lesson summary covering Standards, Materials, Starter Choice Board, Lesson Planning Overview, and Learning Outcomes. The Deep Dive section offers explicit lesson planning guidance, outlining both required and optional components with recommended class time. Day 1 lessons include the Starter Choice Board, Explore! Activity, Lesson Presentation, and Independent Practice, while Day 2 includes the Starter Choice Board, Teacher Gem options, Exit Card & Target Tracker, and additional Independent Practice. The Deep Dive also incorporates formative assessments, Focus Math Practices, Math Practices: Teacher and Student Moves, and Supports for Students with Learning and Language Differences, ensuring teachers have clear implementation strategies for diverse learners.

Materials include sufficient and useful annotations and suggestions that are embedded within specific learning objectives to support effective lesson implementation. Preparation materials, lesson narratives, and instructional supports provide teachers with structured lesson planning guidance, differentiation strategies, formative assessment recommendations, and opportunities for student engagement. These supports are found in resources such as the Unit Launch Guides, Unit Finale Guides, Lesson Planning Guidance, Teacher Guides, Deep Dive sections, Starter Choice Boards, and Small Group Instruction recommendations.

• Unit 3, Planning & Assessment, Unit Launch Guide, Lesson Planning Guidance: Day 1, “Fluency Board Launch (20-25 minutes) The Fluency Boards provide students with opportunities to build number sense through discourse and practice. This unit’s Fluency Boards focus on two skills: (1) multiplying decimals and (2) solving proportions. Each Fluency Board also includes a ‘Mix It Up’ skill. To launch the Fluency Board for this unit, complete the following: (1) Fluency Board Pre-Assessment: Have students fold their pre-assessment in half. For this Fluency Pre-Assessment, students should be given two minutes to complete Target Skill 1 and two minutes to complete Target Skill 2. (2) Fluency Board Cover Sheet: Once completed, have students self-correct their pre-assessment. They should record the number of items completed and number correct for each skill on the cover page and then self-assess their current level of understanding. As a class, come up with key understandings for each skill that are needed for procedural fluency and record those in the corresponding box on the cover sheet. (3) If time allows, you may choose to have students start on the Fluency Board in the first lesson of this unit.“

• Unit 3, Planning & Assessment, Unit Finale Guide, Lesson Planning Guidance: Day 1, “Unit Review (20-25 minutes) Two Unit Review options are available. The style of Unit Assessment you choose for Day 3 of the Unit Finale may guide your choice of Unit Review. Unit Review: The Unit Review is a print-ready resource which addresses all of the Focus Standards covered in the unit. This resource offers constructed response items that are similar to the items on the Unit Assessment without moving beyond Depth of Knowledge (DOK) Level 2 (Basic Skills + Concepts) so as to maintain authenticity when assessing students at a DOK Level 3 (Strategic Thinking) during the actual assessment. The Unit Review resource has a built-in reflection component on groups of items that align to the Pathways Teacher Gem activity that can be used after the Unit Review is completed. Online Unit Review: Like the Unit Review, the Online Unit Review addresses all of the Focus Standards covered in the unit. Unlike the Unit Review, which consists of constructed response items, the Online Unit Review consists of selected response items, such as multiple choice, select all that apply, true/false and yes/no. Teachers have the option to provide a print-version of the Online Unit Review, found in the Editable Resources spreadsheet, to encourage students to show their work before submitting their answers digitally. The Online Unit Review provides teachers and students with instant data for each item. The correlation of the Online Unit Review items to the Pathways Teacher Gem are below. Unit Review Skills Review Skill 1: I can convert between fractions, decimals and percents. (Items 1, 2) Review Skill 2: I can use percents to find a missing number using proportions and equations. (Items 3, 4, 5) Review Skill 3: I can find the percentage of increase or decrease or the percent error when given a real-world scenario. (Items 6, 7) Review Skill 4: I can solve problems involving mark-ups, discounts, tips and taxes. (Items 8, 9)”

• Unit 6, Lesson 6.2, Teacher Guide, Deep Dive, Lesson Planning Guidance: Day 1, “Explore! Activity: ‘Formula Frenzy’ (10-15 minutes) In this activity, students will evaluate geometric formulas by substituting values for the variables in the formulas. This activity connects to previous work students have done finding area, surface area and volume as well as vocabulary terms learned in previous grades (variable, evaluate and expression). The activity concludes by asking students to reflect on what it means to “substitute” values in an expression or equation. Implementation Option #1: Have students use Step 1 for an individual brainstorming session before joining with a group of three to four students to discuss and then complete Steps 2-4. As a class, discuss Step 5. Implementation Option #2: Discuss Step 1 as a class and then ask students to complete Steps 2-3 individually. Check answers together and then have students join with a partner to complete Steps 4-5.”

The materials reviewed for EdGems Math (2024) Grade 7 meet expectations for containing adult-level explanations and examples of the more complex grade/course-level concepts and concepts beyond the current course so that teachers can improve their own knowledge of the subject.

Each Unit’s Planning & Assessment page includes a PD Library that provides teachers with access to Achieve the Core open-source publications from Student Achievement Partners. These documents offer adult-level explanations of mathematical content, organized by vertical progression within each domain. Additionally, the Planning & Assessment page contains a Unit Overview with the following information:

The Content Analysis section explains the major mathematical concepts taught in the unit, providing examples and explanations to enhance teachers’ understanding of both the content and its vertical progression within the standards. It also illustrates the types of tasks and procedures students will encounter. For example: 

• Unit 6, Planning & Assessment, Unit Overview, Content Analysis states, “This unit builds directly upon students’ work in Grade 6 with expressions. The key differences in Grade 7 are the inclusion of negative rational numbers and more complex expressions, including the use of absolute value bars as grouping symbols. Students will spend time in this unit making sense of the inclusion of negative rational numbers, especially as part of the base of a power and as coefficients when distributed to a term inside parentheses. These concepts will be explored in both numerical and algebraic expressions.” Visual student examples of Numerical Expressions and Algebraic Expressions are provided for teachers to review. “Through the practice of generating equivalent expressions, students will apply and develop their understanding of properties of operations, particularly the Associative, Commutative and Distributive Properties, as they work toward fluency in the norms associated with mathematical annotation. For example, given the expression 3 − 2(5x − 6), a student might mistakenly rewrite the expression as 3 − 10x − 12, in which the negative within the parentheses was not taken into account. Alternatively, a student might aim to simplify the expression within the grouping symbols, recalling their work with the order of operations, although the terms within the parentheses are not like terms. Students may choose to organize their work in different manners, such as applying the Commutative Property to group like terms in an expression, or carrying out the Associative Property to multiply the whole numbers in the expression 0.5(−6)(−4) before finding half of the resulting product. The exploration of the properties of operations in this unit will lay the foundation for continued work with algebraic reasoning in this course and beyond.”

The Learning Progression section explains and provides specific examples of the vertical progression of standards within the unit’s targeted domains. These examples include diagrams, models, numerical or algebraic representations, sample problems, and solution pathways. The Learning Progression is structured under the headings: ‘Previously, students have…, ‘In this unit, students will…,’ and ‘In the future, students will…’ with corresponding standards identified. For example:

• Unit 5, Planning & Assessment, Unit Overview, Content Analysis, Readiness Check & Learning Progression states, “In this unit, students will… Understand that multiplication is extended from fractions to rational numbers through the properties of operations. 7.NS.A.2a Understand integers can be divided as long as the divisor is not zero. 7.NS.A.2b Solve real-world problems involving multiplication and division with rational numbers 7.NS.A.2c, 7.NS.A.3”

• Unit 7, Planning & Assessment, Unit Overview, Content Analysis, Readiness Check & Learning Progression states,  “In the future, students will… Use the formulas for area and circumference of a circle to solve problems. 7.G.B.4 Analyze and solve pairs of simultaneous linear equations. 8.EE.C.8 Create equations and inequalities in one variable and use them to solve problems. HS.A.CED.A.1” 

Each lesson’s Teacher Guide includes a Common Misconceptions section, which identifies common errors and provides explanations and recommendations to help students develop a stronger understanding. For example: 

• Unit 8, Lesson 8.6, Teacher Guide, Supports for Students with Learning and Language Differences, Common Misconceptions states, “The formulas for the area of a circle and the circumference of a circle are often confused by students. Teaching students to memorize these formulas without any understanding of how they relate to a circle increases the chance for confusion. Build the understanding before presenting the formulas. Also, make the connection to area representing square units and the area formula having a value (the radius) squared. Guide a derivation of the relationship between the circumference and area of a circle. Use a circle as a model. Cut the circle into as many equal-sized pie pieces as possible. Lay the pie pieces to form a shape similar to a parallelogram. Have students write an expression for the area of the parallelogram related to the radius (note: the length of the base of the parallelogram is half the circumference, or $\pi r$, and the height is r, resulting in an area of $\pi r^{2}$, which is the area of the circle). This derivation is shown at the beginning of the student lesson. For both circumference and area, students may struggle with the idea of the solution being an approximation versus an exact solution. This goes back to their introductory level of understanding of pi being an irrational number. Help students understand this by going back to the idea that they are most likely using an estimate of pi (3.14) or rounding at the end (when using pi on their calculator) so both of these methods lead to approximations rather than exact answers.”

The materials reviewed for EdGems Math (2024) Grade 7 meet expectations for including standards correlation information that explains the role of the standards in the context of the overall series.

Standards correlation information is included to support teachers in making connections from grade-level content to prior and future content. Standards can be found in multiple places throughout the course, including the Course Level, Unit Level, and Lesson Level of the program. Examples include:

• Each Unit’s Planning and Assessment section includes a Pacing Guide & Correlations, where the EdGems Math Course 2 Content Standards Alignment lists all grade-level standards along with the specific lessons where they are addressed. The program provides a structured approach to standards alignment through its Focus and Connecting Standards framework. A correlation chart is included, organizing standards into columns that indicate where each standard is taught as a Focus Standard in specific lessons and as a Connecting Standard across different units. This structure helps ensure that concepts are reinforced and revisited throughout the course.

  • “EdGems Math supports students’ proficiency in the Common Core State Standards through a program design which supports the interconnectivity of mathematical ideas while providing clear learning objectives. This is achieved by designating Focus Standards in each lesson and Connecting Standards in each unit. The qualifiers of Focus and Connecting Standards were developed by the EdGems Math authoring team to design a scope and sequence in which mathematical ideas build upon each other and are revisited throughout the course. Each EdGems Math lesson identifies one or more standards as a Focus Standard to provide a focal point for the lesson objectives. The unit then provides opportunities for further connections to other standards across clusters and domains. These Connecting Standards offer opportunities for students to draw up and apply many mathematical ideas throughout the unit. The following chart shows when each standard is aligned as a Focus Standard or Connecting Standard throughout the course. Further explanations of the Focus and Connecting Standards are available within each Unit Overview.”

• Unit 3, Planning and Assessment, Unit Overview, Standards Correlation, Focus Content Standards states, “The Focus Content Standards for this unit conclude the concentration on the Ratios and Proportional Relationships domain within the Grade 7 and the K-12 standards collectively, though these standards will appear as Connecting Standards throughout the rest of the course. In future grades, students will continue to work with proportional relationships, distinguishing these from non-proportional linear relationships within the Functions domain. The standards in this unit are formatively assessed throughout the unit and summatively assessed in the unit’s Test Prep, Performance Assessment and Unit Assessments.”

• Each Unit's Planning & Assessment section includes a PD Library with resources from Achieve the Core to support professional learning and instructional planning. These resources offer in-depth explanations of mathematical progressions aligned with the Common Core State Standards.

  • “CCSS Math Learning Progressions: Student Achievement Partners, a nonprofit organization, developed Achieve The Core to provide free professional learning and planning resources to teachers and districts across the country. The narrative documents below provide adult-level descriptions of the progression of mathematical ideas within domains or topics within the Common Core State standards for Mathematics.”

The Planning & Assessment sections within each unit provide coherence by summarizing content connections across grades. These sections highlight how mathematical concepts build upon prior knowledge and prepare students for future learning. Examples of where explanations of the role of specific grade-level mathematics appear in the context of the series include:

• Unit 4, Planning and Assessment, Unit Overview, Readiness Check & Learning Progression includes a structured progression of learning, outlining prior knowledge, current instructional goals, and future learning connections to reinforce coherence across grades. It states, "Previously, students have… Added and subtracted fractions with unlike denominators (5.NF.A.1), understood that the opposite of a positive or negative number is the same distance from zero on the opposite side of zero (6.NS.C.5), and ordered rational numbers on a number line (6.NS.C.6-7). In this unit, students will… Describe situations in which opposite quantities combine to make zero and understand absolute value as a distance from zero (7.NS.A.1a, 7.NS.A.1b), understand subtraction of rational numbers as adding the additive inverse (7.NS.A.1c), and solve mathematical problems involving addition and subtraction of rational numbers (7.NS.A.3). In the future, students will… Solve multi-step problems involving positive and negative rational numbers in various forms (7.EE.B.3), use rational approximations of irrational numbers to compare, order, and estimate the value of irrational numbers (8.NS.A.2), and rewrite expressions involving radicals and rational exponents using the properties of exponents (HS.N-RN.2)."

• Unit 5, Lesson 5.3, Teacher Guide, Standards Overview states,“Focus Content Standard(s): 7.NS.A.2b,c (Major), 7.NS.A.3 (Major)” Starter Choice Board Overview “Storyboard: Multiply rational numbers (7.NS.A.2, 7.NS.A.3) Building Blocks: Divide positive decimal numbers and fractions (6.NS.A.1, 6.NS.B.3) Blast from the Past: Solve a percent problem (6.RP.A.3) Fluency Board Skills: Add and subtract integers, add and subtract decimals, simplify using the distributive property.” 

• Unit 9, Planning & Assessment, Unit Overview, Connecting Content Standards states, “In this unit, students will use rational number operations (7.NS.A.1-3) and work with expressions and equations (7.EE.A.1, 7.EE.B.3-4) to solve problems involving surface area and volume of prisms and pyramids. Prism nets and cross sections will offer students more opportunities to practice the two-dimensional geometry skills from the previous unit. Students will also apply proportional reasoning as they consider how areas of figures compare after increasing or decreasing the dimensions by applying a scale factor (7.RP.A.3, 7.G.A.1).” 

The materials reviewed for EdGems Math (2024) Grade 7 meet expectations for providing explanations of the instructional approaches of the program and identification of the research-based strategies.

Instructional approaches of the program and identification of the research-based strategies can be found throughout the materials, but particularly within each unit’s Planning & Assessment, About EdGems Math, Research Guide. 

Research Guide states, “Middle school is a critical stage for math instruction. Students form conclusions about their mathematical abilities, interests, and motivation.^1-10 Middle school students in the United States are falling behind compared to other countries in their math performance.² Studies have shown that struggles with math are particularly acute in middle school grades. The transition from elementary to middle school can lead to students falling behind with accumulated learning gaps.^3-5 Research shows that the mathematical achievement of middle schoolers has a direct impact on the likelihood that they will persist through the challenging material in pathways that can prepare them for the broadest range of options in high school and beyond.^6-7 Within this crucial time frame, a principal goal for middle school math teachers is to create a learning environment in which students are encouraged to see themselves as capable thinkers and doers of mathematics. Research demonstrates that to do this successfully, instructional materials must provide teachers with opportunities to 1) build upon and expand students’ cultural knowledge bases, identities, and experiences, 2) actively support students’ conceptual understanding, engagement, and motivation, 3) provide relevant, problem-oriented tasks that enables them to combine explicit instruction about key ideas with well-designed inquiry opportunities, and 4) spark student peer-to-peer discussion, perseverance and curiosity as they think and reason mathematically to solve problems in mathematical and real-world contexts.⁸ EdGems Math has been intentionally designed to support the diverse mathematical journeys of middle school students as they grow in their learning, critical thinking, and reasoning abilities. To reach the goal of higher order thinking for all, the EdGems Math curriculum connects each grade’s foundational math concepts to authentic, real-world contexts taught in multi-dimensional ways that meet a variety of learning needs. EdGems Math empowers teachers to adjust the content and instructional strategy and tailor outcomes of how learning is assessed.^9-10 EdGems Math curriculum is comprehensive, rigorous, and focused. It draws on decades of research exploring the best methods for teaching and learning math.”

The Unit Planning & Assessment, About EdGems Math, Research Guide incorporates multiple research-based strategies to support student learning:

• “Explore! Activities: The lesson-based Explore! Activities engage students in scaffolded tasks, guiding students as they begin grappling with the big ideas of the lesson and discovering new concepts (Small, M. and Lin, A.). The steps of the Explore! Activities move students through ‘Comprehension Checkpoints’ (National Council of Teachers of Mathematics) to guide information processing, ensure prior knowledge is activated, and discover patterns, big ideas, and relationships. Utilizing a student-centered approach, Explore! Activities engage students in the Standards of Mathematical Practice, which allows teachers to better facilitate learning using effective mathematical teaching practices (McCullum, W.). Every Explore! Activity provides students with ways to connect to key concepts through investigative, discovery-based tasks, culminating in an opportunity to generalize or transfer learning and move toward procedural and strategic proficiencies (California Department of Education).”

• “Lesson Presentation Communication Breaks: Communication Breaks are integrated into each Lesson Presentation as an opportunity for students to make sense of their learning. Each Lesson Presentation features two of seven structures to support students in the communication of their ideas or questions directly with their peers. The use of sentence stems in each Communication Break increases accessibility, enabling students to develop both social and academic language as they reflect on their learning (Smith et al.). The structures of Communication Breaks allow teachers to elicit student thinking, provide multiple entry points, focus students’ attention on structure, and facilitate student discourse (Chapin, S.H., O’Connor, C., Anderson, N.). As a result, students engage in the Standards of Mathematical Practice and gradually become more secure in their understanding and abilities to develop their knowledge (Bay-Williams, J.M., & Livers, S.).”

• “Mathematical Language Routines: Within every Teacher Lesson Guide are instructional supports and practices called Mathematical Language Routines to help teachers recognize and support students’ language development in the context of mathematical sense-making when planning and delivering lessons (Aguirre, J. M. & Bunch, G. C.). While Mathematical Language Routines can support all students when reading, writing, listening, conversing, and representing in math, they are particularly well-suited to meet the needs of linguistically and culturally diverse students. When students use language in ways that are purposeful and meaningful for themselves, they are motivated to attend to ways in which language can be both clarified and clarifying (Mondada, L. & Pekarek Doehler, S.). These routines help teachers ‘amplify, assess, and develop students’ language in math class’ (Zwiers, J. et al).”

• “Lesson Presentation: The editable Lesson Presentation enables teachers to shape lessons of balanced instruction on mathematical content and practice as they guide students through productive perseverance, small group instruction, and growth mindset activities. Components of the Lesson Presentation include Fluency Routines to develop number sense, vocabulary terms with definitions, examples with solution pathways, extra examples, the Explore! activity, Communication Breaks, and a formative assessment Exit Card. Teaching tips provide guidance on independent, group, and whole-class instruction. Studies identify explicit attention to concepts and students’ opportunity to struggle (as during the Explore! activity and Communication Breaks) as key teaching features that foster conceptual understanding (Hiebert, J., & Grouws, D. A.).”

The materials reviewed for EdGems Math (2024) Grade 7 meet expectations for providing a comprehensive list of supplies needed to support instructional activities. 

Comprehensive lists of materials needed for instruction can be found in the PD Library link on the Unit Planning & Assessments page. The Required & Recommended Materials document provides a lesson-by-lesson breakdown of necessary resources for the course. Additionally, the Teacher Guide for each lesson includes a list of required materials. Examples include:

• Unit 3, Lesson 3.4, Teacher Guide, Materials states, “Required: Take-out restaurant menu (for the Explore! activity) Optional: Calculators”

• Unit 4, Lesson 4.1, Teacher Guide, Materials states, “Required: Integer chips or tiles (for the Explore! activity) Optional: Blank number lines, horizontal number lines, vertical number lines, grid paper”

• Unit 8, Lesson 8.5, Teacher Guide, Materials states, “Required: Collection of circular objects (such as lids, cups, cans, Frisbee, etc.), tape measures (for the Explore! activity)”

The materials reviewed for EdGems Math (2024) Grade 7 meet expectations for having assessment information included in the materials to indicate which standards are assessed.

Each Unit Overview outlines standards alignment for formal assessments, including the Readiness Check, Performance Assessment, and Fluency Pre- and Post-Assessments. The EdGems website provides grade-level standards alignment for all formal assessments, including Assessments, Tiered Assessments, Performance Assessment, Unit Review (Print and Online), and Review & Assessments, accessible via the information (“i”) button at the bottom right corner of the icon. Each question's assessed standard(s) is listed for teachers, while only the Performance Assessment and Performance Task include practice standards. Examples include:

• Unit 2, Planning & Assessment, Tiered Assessments, Exercise 4 states, “Jim ran 10 laps in 5 minutes. How many laps will Jim run in 12 minutes?” The assessment information identifies the standard alignment as 7.RP.A.3.

• Unit 7, Planning & Assessment, Unit Overview, Performance Task states, “In ‘Cargo Elevator,’ students solve equations and inequalities to determine how many boxes can be loaded onto an elevator and in how many trips. Students will make use of structure (SMP7) as they explore changing quantities using one- and two-step equations and inequalities (6.EE.B.7, 7.EE.A.3-4) to predict which method of moving boxes will take less time.”

• Unit 7, Planning & Assessment, Unit Overview, Performance Assessment states, “In this Performance Assessment, students will write and solve equations and inequalities involving the dimensions, area and perimeter of various rectangles. Students will reason abstractly and quantitatively (SMP2) as they discover shape and space using one- and two-step equations and inequalities (6.EE.B.7, 7.EE.A.3-4) to understand how changing dimensions affect the area and perimeter of a rectangle.”

• Unit 10, Materials, Online Review & Assessments, Online Unit Review, Item 3 states, “Scott polled 30 seventh graders. Twelve students said they would vote for Pedro in the upcoming election. There are 250 students who will vote in the election. How many votes do you predict Pedro will get?” The assessment information identifies the standard alignment as 7.SP.C.6/7.SP.C.7/7.RP.A.3.

The materials reviewed for EdGems Math (2024) Grade 7 meet expectations for including an assessment system that provides multiple opportunities throughout the grade, course, and/or series to determine students’ learning and sufficient guidance to teachers for interpreting student performance and suggestions for follow-up. 

Each Lesson At A Glance in the Teacher Guide outlines specific learning goals, ensuring that teachers know the math standards that students should be able to demonstrate by the end of the lesson. Exit Cards serve as formative assessments, providing a real-time snapshot of student understanding, while a related Student Lesson exercise offers another checkpoint to identify students needing additional support. The EdGems website provides rubrics for scoring Exit Cards, ensuring consistency in evaluation.

The Online Class Results page generates automatic proficiency ratings based on student performance in Online Practice, Online Challenge, Test Prep, and Online Unit Reviews. These ratings align with state assessment benchmarks, helping teachers interpret mastery levels. Assessment Scoring Guides for Unit Assessments and Tiered Assessments follow the same ranking system, allowing teachers to track progress across multiple assessment formats.

Teachers receive real-time student performance insights through Teacher Gem activities, which embed informal assessment opportunities. For example, in Relay, teachers track how often a team revises an answer before moving forward, and in Ticket Time, class dot plots allow teachers to identify common errors. The PD Library provides written and video-based facilitation guides to support teachers in implementing these strategies effectively.

The Lesson Guide Deep Dive helps teachers analyze assessment data and adjust instruction accordingly. Exit Card results inform assignments for Leveled Practice and Differentiation Days, while the Differentiation Day guides provide self-assessments and targeted rotations to meet diverse learning needs. The Deep Dive section also identifies common misconceptions, equipping teachers with strategies to proactively address misunderstandings.

Performance assessments include rubrics for teacher grading and student self-reflection prompts, reinforcing the Standards for Mathematical Practice (SMP). The Unit Overview and Lesson Guide At A Glance ensure that all assessments and activities align with content and practice standards, with detailed mapping to Readiness Check skills, Storyboards, Performance Tasks, Fluency Boards, and Tiered Assessments.

To further support follow-up instruction, the Online Class Results tool provides recommended activities based on student proficiency levels, allowing teachers to tailor instructional strategies. By incorporating structured assessments, clear proficiency guidance, real-time monitoring tools, and differentiation strategies, EdGems Math ensures teachers have the necessary resources to assess student learning, interpret performance data, and provide targeted follow-up instruction.

Examples include: 

• Unit 2, Planning & Assessment, Performance Assessment, Performance Assessment Rubric ~ Student Reflection states, “Describe at least two ways you demonstrated the Focus Math practice below while completing this performance assessment. SMP4 I can represent everyday situations using models and other representations. I represented an everyday situation using a picture or model. I represented an everyday situation using algebraic expressions or equations. I represented an everyday situation using data sets and displays.” The Performance Assessment Rubric ~ Teacher Grading Rubric consists of four categories rated on a scale of 4 to 1 and a space for Comments. The four categories are: “Making Sense of the Problem: Interpret the concepts of the assessment and translate them into mathematics. Representing and Solving the Problem: Select an effective strategy that uses models, pictures, diagrams, and/or symbols to represent and solve problems. Communicating Reasoning: Effectively communicate mathematical reasoning and clearly use mathematical language. Accuracy: Solutions are correct and supported.”

• Unit 9, Planning & Assessment, includes Form A and Form B of the Unit 9 Assessment, along with an Answer Key and Assessment Scoring Guidelines for each question. The materials state, “3-point Items: #1, 2, 3, 11, 12 Items that are each worth three points consist primarily of Depth of Knowledge Level 3 items considered “Strategic Thinking.” Students may earn partial credit on items when showing progress on a solution pathway that connects to the concept being assessed with one or more errors. Students can earn either 0, 1, 2 or 3 points for items in this category. 0 points: An incorrect solution is given with no work or with work that does not show understanding of the concept. 1 point: Progress is made towards a correct solution, but multiple errors have been made. OR A correct solution is given with no supporting work or explanation. 2 points: Progress is made towards a correct solution, but one small error is made. OR A correct solution is given with partial supporting work provided. 3 points: The correct solution is given and is supported by necessary work or explanation. Total Points Possible: 27 Not Yet Met 0-16, Nearly Meets 17-18, Meets 19-24, Exceeds 25-27” Similar guidance is provided for 1 and 2 point assessment items.

The materials reviewed for EdGems Math (2024) Grade 7 meet expectations for providing assessments that include opportunities for students to demonstrate the full intent of the course-level standards and practices across the series.

Assessments align with grade-level content and practice standards through various item types, including multiple-choice, short answer, extended response prompts, graphing, mistake analysis, and constructed response items. They are available as downloadable PDFs for in-class printing and administration or can be completed through the online platform. Examples include:

• Unit 2, Materials, Online Review & Assessments, Unit Assessment, Form A, Item 8 demonstrates the full intent of 7.RP.2c. Item 8 states, “Which equations represent proportional relationships? Select all that apply. a. $y=\frac{x}{7}$ b. $y=5x$ c. $y=0.2x$ d. $y=\frac{9}{x}$ e. $y=4x^{2}$ f. $y=\frac{3}{2}x$”

• Unit 6, Planning & Assessment, Assessments, Form B, includes three items that demonstrate the full intent of 7.EE.1, Exercise 15. The materials state, "$2(3x-7)+10$" Exercise 17 states, “Factor each expression using the greatest common factor. a. 5𝑚 + 20 b. 9𝑥 − 21.” Exercise 20 states, “Explain two different ways to simplify $2(3.5x-10+1.6x)$. Show that both ways lead to the same simplified expression. Method #1 Method #2”

• Unit 7, Planning & Assessment, Performance Assessment, includes four items that demonstrate the full intent of 7.EE. 4 and SMP2. The materials state, “Ricardo and Emilia are solving problems about a rectangle. The area of the rectangle is twenty-three more than one-half its perimeter. The area of the rectangle is 50 square inches. 1. What is the perimeter of the rectangle? Show all work necessary to justify your answer. 2. What are the dimensions of the rectangle? Show all work necessary to justify your answer. 3. The length and width of the rectangle were cut in half. Karly stated, “The perimeter and area of the rectangle are now half as big as they were originally.” Is she correct? Justify your answer using words and mathematics. 4. Jaime has a rectangle that he is comparing to Ricardo’s and Emilia’s rectangle. Half of Jaime’s rectangle perimeter is at least 10 inches more than three times the perimeter of Ricardo and Emilia’s rectangle. a. Write an inequality to represent the situation. b. Solve the inequality you wrote for part a. Show the work that leads to your answer. c. Explain your solution to part b in the context of the problem. Justify your answer using words and mathematics. d. Give an example of dimensions that could describe Jaime’s rectangle. Show work that supports your answer. e. Give an example of dimensions that could not describe Jaime’s rectangle. Show work that supports your answer.”

The materials reviewed for EdGems Math (2024) Grade 7 meet expectations for providing strategies and supports for students in special populations to support their regular and active participation in learning grade-level mathematics.

For each lesson, teacher guidance is provided alongside the Teacher Gem activity, which includes strategies to support instruction and student engagement. Printable PDF resources, such as the Student Lesson Textbook and Interactive consumable, include tools like graphic organizers, sentence stems, number lines, and coordinate planes. Lessons are also available as e-books with features including adjustable font sizes, text highlighting, text-to-speech, and note-taking tools.

Resources that support students in special populations to actively participate in learning grade level mathematics include:

• Differentiation Days: Differentiation Days are designed to provide teachers with structured opportunities to work with small groups based on specific learning targets. During these sessions, other students participate in mixed-ability group rotations, including the Teacher Small Group Rotation, Additional Practice Rotation, Application Rotation, and Tech Rotation. 

• Leveled Practice: The program includes three levels of leveled practice to address varying student needs. “Leveled Practice-T” is structured for students with learning and language differences, offering shorter problem sets, additional workspace, and simplified terminology and numbers to align with accessibility needs while maintaining grade-level alignment.

• ELL Supports: ELL supports are provided in the Planning and Assessment menu of each unit. These include explanations of Mathematical Language Routines (MLRs) and specific directions for incorporating these routines into lessons and activities. 

Each lesson includes Spanish translations of the student lesson, Explore! activities, leveled practice, and Exit tickets. Accompanying videos are included to guide students through the lesson content. These resources are structured to support student learning and accessibility.

Examples of the materials providing strategies and support for students in special populations include:

• Unit 4, Lesson 4.2, Lesson Presentation, Slide 21, Communication Break, Think, Ink, Pair, Square states, “How would you add two rational numbers if one was written as a decimal and the other as a fraction? Think by yourself. Write down an idea. Share with a partner. Join with another partner set. We think… Do you agree or disagree? I respectfully agree/disagree because…” Teacher Guide, Lesson Presentation states, “Have students utilize their interactive textbooks or composition notebooks to participate in guided note taking using the Lesson Presentation. Have students attempt Extra Examples with partners, in small groups or independently. Use ‘Communication Break’ slides as opportunities for meaningful discourse. Communication Break–Think, Ink, Pair, Square: Use the prompt ‘How would you add two rational numbers if one was written as a decimal and the other as a fraction?’ Have students think and write independently before joining with a partner to share. Then have two partner sets join together. Ask one group to start and the other group respond using the sentence stems provided.”

• Unit 7, Lesson 7.2, Teacher Gem, Climb the Ladder, Climb the Ladder Instructions states, “NOTE: It is helpful to use four different colors of paper for the four different ladders. One way this is useful is in assessing the progress of each partner set. During the activity, you can quickly scan the room and be able to tell which sets of students are falling behind or are almost finished. Also, you can use colored paper for the expert tents to match the colors of the ladders so students are able to find the appropriate experts quickly. Another way the colored papers are helpful is to allow students who are struggling to be called to a huddle. This is where the teacher can call all of the partner sets with a certain color of paper to huddle with experts around the room. The expert’s job is to stop what they are doing and help these students complete this ladder. The teacher may choose to allow the students who have huddled with an expert to take cuts in line at the scoring table. At the end of the activity, students who finish early can be asked to join groups as a third person. The teacher can direct them to partner sets with a certain color of ladder that is the earliest in the progression.”

The materials reviewed for EdGems Math (2024) Grade 7 meet expectations for providing extensions and/or opportunities for students to engage with grade-level/course-level mathematics at higher levels of complexity.

Each unit includes multiple opportunities for students to engage with grade-level mathematics at increasing levels of complexity. These opportunities are embedded within lessons and available to all students, supporting a range of learners in exploring mathematical concepts at higher levels of complexity.

Several resources and features within the program provide opportunities for extended mathematical exploration.

• Performance Task: “The Performance Task provides applications of most or all of the standards addressed in the unit. This task contains Depth of Knowledge Level 3 and 4 strategic and extended thinking questions where students apply multiple standards in a non-routine manner to solve. These tasks provide entry points for all levels of learners and encourage students to explain their thought processes or critique the reasoning of others.”

• Performance Assessment: These non-routine problems require students to engage in higher-level thinking while applying their knowledge of the standards.

• Leveled Practice: The “C” in each lesson is designed for students who have already demonstrated proficiency. It extends their learning by making connections to future standards and incorporating Depth of Knowledge (DOK) Level 3 or 4 exercises.

•  Tic-Tac-Toe Boards: According to the EdGems Math Program Components state, “Each Tic-Tac-Toe Board includes nine activities that extend or look at the content of the unit in different ways. The Tic-Tac-Toe Boards include activities that make use of a variety of multiple intelligences.”

• Teacher Gems: Teacher Gems include problem sets at multiple levels of complexity, allowing for differentiated problem-solving experiences. For example, activities such as Four Corners, Relay, and Stations include multiple levels of complexity within the tasks.

• Online Practice & Exit Card Resource: This resource offers five options for each lesson: Two Online Practice sets (A and B), each containing six items at the proficient level. Two Online Challenge sets (A and B), each containing four challenge questions. Attempt A provides immediate feedback on correctness, while Attempt B includes worked-out solution pathways to help students identify errors in their work.

The materials include structured activities that provide opportunities for students to engage with mathematical concepts at increasing levels of complexity.

• Unit 3, Lesson 3.3, Teacher Gem, Relay, Directions state, “Print the two sets of relay cards. The first set, numbered 1 through 8, provides students practice in the standard at a proficiency level. A challenge set of cards, A through H, provide opportunities for students to extend and apply their thinking around the standard. The questions in each set of relay cards increase in difficulty throughout the activity.” Relay 2, “Identify the percent of change as an increase or a decrease. Find the percent of change. Round to the nearest whole percent if necessary. 40 to 44.” Relay B, “Office Supply Depot buys notebooks at a bulk price of $0.25 each. They sell them to students for $0.75 each. What is the percent of increase?”

• Unit 7, Lesson 7.3, Online Practice and Exit Card, Challenge A, Item 4 states, “Nolan went to the same store three days this week to buy groceries. This store charges the same rate for milk per gallon, even when bought in containers smaller than one gallon in size. The first day he bought two gallons of milk. On the second day he bought a loaf of bread for $2.50 and $\frac{1}{2}$ a gallon of milk. The third day he bought just one gallon of milk. Nolan spent a total of $17.20 on groceries this week. How much did Nolan spend on his first trip to the grocery store?”

The materials reviewed for EdGems Math (2024) Grade 7 meet expectations for providing strategies and supports for students who read, write, and/or speak in a language other than English to regularly participate in learning grade-level mathematics.

Guidance is consistently provided for teachers to support students who read, write, and/or speak in a language other than English, providing scaffolds for them to meet or exceed grade-level standards. According to the ELL Supports Guide, “We view the background knowledge, experiences, and insights that English Learners bring to the classroom as strengths to be leveraged, and we are committed to ensuring that they receive academic success with rigorous grade-level curriculum. In recognition of the unique needs of learners, including those with diverse levels of mathematical proficiency, our curriculum includes research-based guidance for differentiated English Language Learner (ELL) instruction."

• The ELL Supports Guide outlines strategies for students who read, write, and/or speak in a language other than English to engage with grade-level mathematics. Key areas of focus include scaffolding tasks, fostering mathematical discourse, and incorporating instructional strategies informed by research. Tasks include scaffolds and language supports designed to facilitate mathematical understanding. The instructional design integrates opportunities for students to express their mathematical thinking both orally and in writing.

• The ELL Supports Guide contain recommendations related to student assessments. Additional resources in the materials include Target Trackers and Math Practice Trackers, which align with structured conferencing planned three times per unit. A Math Self-Assessment Rubric is included to support student reflection, along with a Sample Vocabulary Journal Format that provides space for root words, home-language translations, definitions, images, and sentence frames.

• Each lesson’s Teacher Guide includes three lesson-specific Mathematical Language Routines (MLRs), with two MLRs suggested for implementation per lesson. Strategies described in the materials include language modeling through think-alouds, the use of visual aids featuring key vocabulary, and a multilingual glossary with online vocabulary available in ten languages. Videos within the ELL Supports Guide provide examples of teachers breaking down tasks, using cognates, and prompting students to explain their thinking. Language functions are also included to structure discussions.

Examples where the materials provide strategies and supports for students who read, write, and/or speak in a language other than English include:

• Unit 4, Lesson 4.1, Teacher Guide, Lesson Presentation, states, “Have students utilize their interactive textbooks or composition notebooks to participate in guided note taking using the Lesson Presentation. Have students attempt Extra Examples with partners, in small groups or independently. Use ‘Communication Break’ slides as opportunities for meaningful discourse. Communication Break – Vocabulary Bank: After the vocabulary is introduced, use the Vocabulary Bank slide to give students a set of vocabulary words and ask them to create their own sentence(s) that connect the words. Select specific students in each group to share their sentence(s) and have others respond using the sentence stems.” Lesson Presentation states, “Communication Break – Vocabulary Bank Integer, Opposites, Absolute Value Write one or two sentences using all the works in a context that makes sense mathematically. Compare your sentence(s) with another group and give them feedback. Your sentence helped me understand. I think the word was used correctly/incorrectly because…”

• Unit 7, Lesson 7.2, Teacher Guide, Supports for Students with Learning and Language Differences, Mathematical Language Routines states, “MLR 1 – Stronger & Clearer Each Time: After groups complete Exercise #5 from the Teacher Gem activity Masterpiece, but before they check with the teacher, groups exchange their work with 2 -3 other groups to get feedback on how to improve their work and their explanations. When students receive their work back with the feedback, they revise their work and explanations. While groups are providing feedback, display the following questions to help them think about what types of feedback will be helpful: Did the group answer the questions? Did the group follow all of the constraints of the problem? Are the group’s calculations accurate? Did the group justify their reasoning, referencing their work? How can the group make their work and explanations stronger?”

• Unit 9, Lesson 9.1, Teacher Guide, Supports for Students with Learning and Language Differences, Common Misconceptions states, “Students may not count attributes of a three-dimensional figure that are not visible in a two-dimensional drawing. Students may also count certain attributes twice within an image of a three-dimensional figure. Provide hands on models (i.e., plastic nets) where possible to support students counting attributes of three-dimensional figures. Teachers will find that it helps to keep showing students shapes in different ways and have them describe what they see. Students have been exposed to many of the vocabulary terms in this unit in previous grades but the amount of terminology introduced in one lesson may cause many students to mix up definitions. Have students create an organizer that shows each term, its definition and/or diagram. It may also be helpful for them to think of real-world objects that connect to each solid. Many students struggle with visualizing slices of solids. It may be helpful to have clay and plastic knives available for students to have hands-on manipulatives to support their understanding.”

The materials reviewed for EdGems Grade 7 meet expectations for providing manipulatives, physical but not virtual, that are accurate representations of the mathematical objects they represent and, when appropriate, are connected to written methods.

Materials consistently include suggestions and links to manipulatives to support grade-level math concepts. The Teacher and Student Moves for Math Practice 5 and Explore! Activities incorporate physical manipulatives when appropriate, with required materials listed in the full course materials list under the Teacher Guide At A Glance section. Student Gems in each lesson provide virtual manipulatives, such as Desmos and Geogebra, to help students make sense of concepts and procedures. Examples include:

• Unit 2, Lesson 2.1, Student Gems, Desmos, Resource Info states, “This activity will help your students understand the definition of a proportional relationship. They'll create a giant and then make sure all of his features are proportional. They'll see the representation of his proportions on a graph and manipulate the graph to see the giant change dynamically.”

• Unit 4, Lesson 4.1, Teacher Guide, Explore! Activity, Integer Chips states, “In ‘Integer Chips,’ students use integer chips or tiles to model the addition of integers. The activity begins by introducing vocabulary terms and the concept of a zero pair. Students then model various expressions using integer chips and learn how to form zero pairs to determine the sum of each expression. The activity culminates in an opportunity for students to generalize their learning and develop their own rules for when a sum will be positive, negative or zero. Implementation Option #1: Facilitate the activity as a whole class without the activity sheet, using the steps from the activity sheets as verbal prompts to guide the activity. Ensure each student has their own set of integer chips to model each expression on their table. Implementation Option #2: Assign partner sets and give each set of partners a bag of integer chips. Work through Steps 1-4 together as a class and then have students complete Steps 5- 10 with their partner. Discuss Step 10 together as a class, allowing a variety of students to share their ideas.”

• Unit 7, Lesson 7.2, Teacher Guide, Math Practices: Teacher and Student Moves, SMP5 Teacher Moves states, “Provide students with unifix cubes, base ten blocks, colored pencils, graph paper, equation mats, balances, algebra tiles and other manipulatives from which to choose. Instruct students to choose one or more tools to help represent and visualize solving two step equations, and explain their choice. After solving the equations, ask students to reflect on their choice of tools and determine if a different tool would be a better choice.”