7th Grade - Gateway 1

The materials reviewed for Eureka Math² Grade 7 meet expectations for assessing grade-level content and, if applicable, content from earlier grades.

Assessment System includes lesson-embedded Exit Tickets, Topic Quizzes, and Module Assessments. According to the Implementation Guide, “Exit Tickets are not graded. They are paper based so that you can quickly review and sort them. Typical Topic Quizzes consist of 4-6 items that assess proficiency with the major concepts from the topic. You may find it useful to grade Topic Quizzes. Typical Module Assessments consist of 6-10 items that assess proficiency with the major concepts, skills, and applications taught in the module. Module Assessments represent the most important content taught in the module. These assessments use a variety of question types, such as constructed response, multiple select, multiple choice, single answer, and multi-part. There are two analogous versions of each Module Assessment available digitally. Analogous versions target the same material at the same level of cognitive complexity.” Examples of summative Module Assessments items that assess grade-level standards include:

• Module 1, Module Assessment 1, Item 6, “Lily uses a scale factor of 2.5 to create a scale drawing of rectangle A. She claims that the area of her scale drawing is 2.5 times the area of rectangle A. (rectangle shown has dimensions 8.7 cm by 2 cm). Part A: Do you agree or disagree with Lily? Explain your answer. Part B: What is the area of Lily’s scale drawing?” (7.G.1)

• Module 2, Module Assessment 1, Item 8, “Enter the expression -3-8 as an equivalent addition expression.” (7.NS.1c)

• Module 3, Module Assessment 2, Item 9, “What is the solution set of the inequality -2>7-\frac{3}{8}x?” Answers provided, “x<-24, x>24, x<-\frac{40}{3}, x>\frac{40}{3}.” (7.EE.4b)

• Module 5, Module Assessment 1, Item 7, “A store owner purchases milk for $2.72 per gallon. She marks up the price by 33%. What is the selling price per gallon of milk? ___ per gallon ____.” (7.RP.3)

• Module 6, Module Assessment 2, Item 1, “Students are assigned art supplies by randomly picking a piece of paper from a hat. Each piece of paper has one of three art supplies written on it. The hat holds the following: 15 pieces of paper that say pastels, 19 pieces of paper that say paint, and 17 pieces of paper that say charcoal. What is the theoretical probability that paint is the first art supply assigned?” Answers provided, “\frac{19}{32}, \frac{19}{51}, \frac{32}{19}, and \frac{51}{19}.” (7.SP.7a)

The materials reviewed for Eureka Math² Grade 7 meet expectations for giving all students extensive work with grade-level problems to meet the full intent of grade-level standards.

Each lesson consists of four sections (Fluency, Launch, Learn, and Land) that provide extensive work with grade-level problems and to meet the full intent of grade-level standards. The Fluency section provides opportunities for students to practice previously learned content and activates students’ prior knowledge to prepare for new learning. Launch activities build context for learning goals. Learn activities present new learning through a series of learning segments. During the Land section, teachers facilitate a discussion to address key questions related to the learning goal. Practice pages can be assigned to students for additional practice with problems that range from simple to complex.

Instructional materials engage all students in extensive work with grade-level problems. Examples include:

• Module 2, Topic A, Lesson 3: Adding Integers Efficiently, Fluency, Problem 2, students add numbers efficiently to prepare for adding integers. Launch, students make true number sentences using additive inverses, “We have determined that 12 and -12 are additive inverses. Now what do you notice and wonder about the expression 12+(-13)?” Learn, Problem 7, students find a strategy to solve integer addition problems containing large numbers, “You have $54 in your bank account. Then you buy a game for $30. a. What integer represents the change in the balance of your bank account? b. Write an addition expression that represents the balance of your bank account after you buy the game. c. What is the balance of your bank account after you buy the game?” Land, students describe the sum of an integer and its opposite are zero, “What strategies can we use to add integers? How do we add more than two integers?” Exit Ticket, students write a numerical expression to represent points lost and earned in the context of a quiz show, “Maya competes in a quiz show. Her answer to the first question is incorrect, and she loses 425 points. Her answer to the second question is correct, and she earns 300 points. a. After the second question, is Maya’s score positive or negative? Explain your thinking. b. Write a numerical expression to explain this situation. c. What is Maya’s correct score after answering the second question? Show your work.” Practice, Problem 3, “For problems 3-5, represent the numerical expression on the number line, 14+(-14).” Students engage in extensive work with grade-level problems of 7.NS.1, ”Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.”

• Module 3, Topic A, Lesson 5: Factoring Expressions, Fluency, Problem 4, students identify GCF to prepare for factoring expressions, “6x and 27x.” Launch, Problem 1, students compare equivalent expressions, “Compare the expressions 3(5x-4) and 15x−12. Identify similarities and differences between them.” Learn, Problem 4, students factor expressions using a tabular model, “Factor the expression. Organize your work by using the tabular model.” Students are shown an area model of 27x and -81. Land, students use the distributive property to generate equivalent expressions, “Give me an example of equivalent expressions from today’s lesson and explain how you know they’re equivalent.” Exit Ticket, Problem 1, “Factor the expression. 6m+8n+4.” Practice, Problem 4, “For problems 3-7, factor the expression. Organize your work using the tabular model.” Students are shown the area model 8x and -28. Students engage in extensive work with grade-level problems of 7.EE.1 (Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients) and 7.EE.2 (Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related).

• Module 4, Topic C, Lesson 10: The Outside of a Circle, Fluency, Problem 3, students multiply decimals to prepare for calculating circumference, “2.3(4).” Launch, students compare strategies to measure the distance around a circle, “How does using a string and a ruler compare to using only a ruler to measure the distance around a circle?” Learn, students measure circular bases and determine that there is a constant number that relates the distance around a circle to its diameter, “Introduce problem 3 and divide students into groups of three. Have each group use a string and a ruler to measure around the circular base and find the diameter of four of the objects that have been placed throughout the classroom. The group’s members should work together to hold the string in place. Encourage students to measure a variety of different-size circular-based objects.” Students record their data in a table and answer, “b. Write the ratio of the distance around the circle to the diameter for each object. c. Find the value of each ratio. If needed, round to the nearest hundredth.” Land, students describe the relationship between the circumference and diameter of any circle as a proportional relationship, “What is the relationship between the circumference and the diameter? Why does our equation give us an approximation of the circumference instead of the exact circumference? How can we be even more precise when determining the circumference of a circle?” Exit ticket, “Find the approximate circumference of the circle. Use 3.1 as the constant number.” Students are shown a circle with a radius of 6 cm. Practice, Problem 6, “For problems 2-9, find the approximate circumference of the circle. Use 3.1 as the constant number. A circle that has a diameter of 8 inches.” Students engage in extensive work with grade-level problems of 7.G.4 (Know the formulas for the area and circumference of a circle and use them to solve problems; give an informational derivation of the relationship between the circumference and area of a circle).

Instructional materials provide opportunities for all students to engage with the full intent of grade-level standards. Examples include: 

• Module 3, Topic A, Lesson 4: Adding and Subtracting Expressions, Fluency, Problem 5, students combine like terms to prepare for adding expressions, (3x+4)+(2x-7)+5x. Launch, Problem 1, students explore common errors when subtracting expressions, “Shawn earns the same amount of money each day. Noor earns 4 dollars more than Shawn each day. Write an expression that represents the number of dollars Noor earns if she and Shawn each work for 5 days. Let w represent the amount Shawn earns in dollars each day. The expression 5(w+4) represents the number of dollars Noor earns in total for 5 days of work. The distributive property can be used to write that expression as 5w+20. Noor typically earns the same amount of money each week, 5w+20. Unfortunately, she is sick and unable to work one day this week. That means her pay decreases by the amount she earns in one day. Write an expression to model the amount Noor earns after one day’s pay is taken away.” Learn, Problem 6, students learn expressions can be written in different ways and still be equivalent, “Three students got the same equivalent expression, but in different ways. Analyze the work shown and then complete parts a-d. 8(x+y)-5(x+y). Student 1: 8(x+y)-5(x+y)=8(x+y)+(-5(x+y))=(8x+8y)+(-5x)+(-5y)=(8+(-5))x+(8(-5)y)=3x+3y. Student 2: 8(x+y)-5(x+y)=8(x)+8(y)-5(x)-5(y)=8x+8y-5x-5y=8x-5x+8y-5y=(8-5)x+(8-5)y=3x+3y. Student 3: 8(x+y)-5(x+y)=(8-5)(x+y)=3(x+y)=3x+3y. a. How did student 1 write an equivalent expression? b. How did student 2 write an equivalent expression? c. How did student 3 write an equivalent expression? d. Which method do you prefer? Why?” Land, generate equivalent expressions using properties of operations to add and subtract expressions, “In what ways can we apply the distributive property to expressions? How can we use the structure of an expression to help us write the expression in an equivalent form? How can we add and subtract expressions?” Exit Ticket, Problem 1, “For problems 1-3, use the distributive property to write an equivalent expression, 2(4a-3)+6a.” Practice, Problem 5, “For problems 1-11, use the distributive property to write the expression in an equivalent form. 2.4x+3.2(3+x).” The materials meet the full intent of 7.EE.2 (Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related).

• Module 5, Topic D, Lesson 17: Simple Interest and Proportionality, Fluency, Problem 3, students determine the unit rate from a table to prepare for finding the constant of proportionality in simple interest contexts, “Find the unit rate.” Students are shown a table with x values: 0.5, 3, 5 and y values: 5.25, 31.5, 52.5. Launch, students consider two options for saving money, “You have decided it is time to start saving money for college. Would you rather save your money in a jar at home or save it in a bank account? Explain.” Learn, students understand simple interest and generate a formula for calculating simple interest. Students review two graphs (simple interest and compound interest) to make a prediction regarding the type of interest problems they will study, “There are two types of interest: simple interest and compound interest. This year we will only study one of these types of interest. Which do you think we will study? Why?” Classwork, Problem 1, students complete a table calculating simple interest and reflect on the amounts, “Liam deposits $1,200 into a bank account that earns an annual simple interest rate of 2%. Complete the table. What do you notice about the completed table in problem 1? Is this a proportional relationship? If so, what is the constant of proportionality? Write an equation to find interest earned, I, in any number of years, t.” The table shown has time in years: 1, 2, 3, 4 and total interest earned in dollars. Students derive the simple interest formula and use it to solve simple interest problems. Classwork, Problem 3, “For problems 3–6, use the simple interest formula to write and solve an equation. Mrs. Kondo takes out a $10,500 loan to buy a used car. The bank charges an annual simple interest rate of 5%. How much will Mrs. Kondo pay in interest if she pays off the loan in 5 years?” Land, students identify how simple interest is an example of a proportional relationship. “How is simple interest an example of a proportional relationship? What value represents the constant of proportionality?” Exit Ticket, “Sara’s parents give her $500 for her high school graduation. She deposits the money into a bank account with an annual simple interest rate of 0.5%. How much interest will she earn in 3 years?” Practice, Problem 3, “Dylan’s brother takes out a $628 loan to buy an electric scooter. The bank charges an annual simple interest rate of 11%. How much will Dylan’s brother pay in interest if he pays off the loan in 3 years? a. Identify the principal amount, the interest rate, and the time of the loan. b. Use the simple interest formula to write and solve an equation.” The materials meet the full intent of 7.RP.3 (Use proportional relationships to solve multistep ratio and percent problems).

• Module 6, Topic B, Lesson 7: The Law of Large Numbers, Fluency, Problem 2, students find the theoretical probabilities of events when rolling a fair 6-sided die to prepare for using empirical probability to estimate theoretical probability, “Find the theoretical probability that the event occurs when rolling a 6-sided die with the numbers 1–6 on its sides. Rolling a 4.” Launch, students use theoretical probability to predict the outcome of a chance experiment, “If you flip your coin two times, how many times would you expect to flip heads? How many times would you expect to flip tails? How would you describe this result by using relative frequencies? Flipping your coin two times is a chance experiment. What is the sample space of this chance experiment? When a coin is flipped two times, what is the theoretical probability of flipping exactly 1 heads? How do you know?” Learn, Problem 1, students examine the effect of repeated trials on empirical probabilities, “Flip a coin 10 times and record your results in the following table. Write relative frequencies as fractions and as decimals rounded to the nearest hundredth.” Problem 3, “Summarize your results from problems 1 and 2 in the following table. Write the relative frequency of heads as a fraction and as a decimal rounded to the nearest hundredth.” Problem 4, “Graph the number of flips and the corresponding relative frequencies from the table in problem 3 as points. Connect consecutive points with line segments.” Land, students discuss using empirical probability to estimate theoretical probability and compare probabilities from a theoretical model to observed relative frequencies, “How can you restate the law of large numbers in terms of the coin flipping experiment? How can we estimate the theoretical probability of an event when the outcomes are not equally likely? How can we use the results of a chance experiment to judge whether a theoretical probability model seems reasonable?” Exit Ticket, Problem 1, “After flipping a penny 30 times, Dylan reports that the relative frequency of heads is about 0.47. a. How many times did the coin land heads up? Explain. b. Dylan flips the coin one more time. Is it possible for the new relative frequency of heads to be 0.55? Explain.” Practice, Problem 3, “Shawn rolls a 6-sided die 36 times and records the results in the table. a. Complete the relative frequency column of the table. Write each relative frequency as a fraction. b. Determine the empirical probability of Shawn rolling a 2. c. Determine the empirical probability of Shawn rolling a 3. d. Determine the empirical probability of Shawn rolling a 6. e. Do you think Shawn’s die is fair? Explain.” The materials meet the full intent of 7.SP.7 (Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy).

The materials reviewed for Eureka Math² Grade 7 meet expectations that, when implemented as designed, the majority of the materials address the major work of each grade.

• There are 6 instructional modules, of which 4 modules address major work of the grade or supporting work connected to major work of the grade, approximately 67%.

• There are 137 instructional lessons, of which 88.5 lessons address major work of the grade or supporting work connected to major work of the grade, approximately 65%.

• There are 169 instructional days, of which 107.5 address major work of the grade or supporting work connected to the major work of the grade, approximately 64%. Instructional days include 137 instructional lessons, 26 topic assessments, and 6 module assessments.

A lesson-level analysis is most representative of the instructional materials as the lessons include major work and supporting work connected to major work. As a result, approximately 65% of the instructional materials focus on major work of the grade.

The materials reviewed for Eureka Math² Grade 7 meet expectations that supporting content enhances focus and coherence simultaneously by engaging students in the major work of the grade.

Each lesson contains Achievement Descriptors that provide descriptions and about what the students should be able to do after completing the lesson and lists standards. Materials do not provide information about connections between standards in lessons.

Materials connect learning of supporting and major work to enhance focus on major work. Examples include:

• Module 1, Topic C, Lesson 18: Relating Areas of Scaled Drawings, Launch, Problem 1, students construct a scale drawing, compute the area, and determine the scale factor, “Create a scale drawing of the triangle by using the scale factor of 3.” Students are shown a triangle with a base of 4 units and a height of 2 units. Learn, Problem 15, students prove that the scale factor used to determine length and width can also be used to determine area, “Consider the triangles from problem 1. Are the areas of the scaled triangles related in the same way as the areas of the scaled squares and rectangles? Explain. Suppose these two triangles had areas of 5 square units and 80 square units. Determine the scale factor that would enlarge the smaller triangle to the larger triangle.” This connects the supporting work of 7.G.1 (Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale) to the major work of 7.RP.2 (Recognize and represent proportional relationships between quantities).

• Module 3, Topic B, Lesson 7: Angle Relationships and Unknown Angle Measures, Learn, Problem 3, students find unknown angle measures, “In the diagram, \angle D⁢A⁢B  is a straight angle. a. Describe the relationship between \angle D⁢A⁢C and \angle C⁢A⁢B. b. Write an equation for the angle relationship and solve for x. c. Check your solution to the equation.” This connects the supporting work of 7.G.5 (Use facts about supplementary, complementary, vertical and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure) to the major work of 7.EE.4 (Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities). 

• Module 6, Topic A, Lesson 4: Theoretical Probability, Learn, Problem 3, students identify a sample space and use it to find theoretical probabilities, “Consider a chance experiment of rolling a fair 6-sided die. a. How many outcomes are in the sample space for this chance experiment? b. Is each outcome equally likely? c. Complete the table by listing the possible outcomes for this chance experiment and the theoretical probability for each outcome. d. What is the theoretical probability of rolling an odd number? Explain. e. What is the theoretical probability of rolling a number less than 5? Explain. f. What is the theoretical probability of rolling a number greater than 7? Explain.” This connects the supporting work of 7.SP.6 (Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability) to the major work of 7.NS.3 (Solve real-world and mathematical problems involving the four operations with rational numbers).

The materials reviewed for Eureka Math² Grade 7 meet expectations for including problems and activities that serve to connect two or more clusters in a domain or two or more domains in a grade.

Each lesson contains Achievement Descriptors that provide descriptions and about what the students should be able to do after completing the lesson and lists standards. Materials do not provide information about connections between standards in lessons.

Materials provide connections from major work to major work throughout the grade-level when appropriate. Examples include.

• Module 1, Topic B, Lesson 11: Constant Rates, Learn, Problem 6, students explore rate language and contexts, notice the structure of rate equations, and use equations to find unknown values, “You count 20 heartbeats in 15 seconds. a. What is the unit rate and what does it mean in this situation? b. Write an equation to represent this situation. Define the variable in your equation. c. At this rate, how many times does your heart beat in a minute?” This connects the major work of 7.RP.A (Analyze proportional relationships and use them to solve real-world and mathematical problems) to the major work of 7.EE.B (Solve real-life and mathematical problems using numerical and algebraic expressions and equations).

• Module 2, Topic E, Lesson 25: Writing and Evaluating Expressions with Rational Numbers, Part I, Exit Ticket, students write, evaluate, and interpret numerical expressions given mathematical and real-world contexts, “Logan has $36.52 in her bank account. She puts all but $40 of her $314.87 paycheck into her bank account 1. Write an expression to represent the new balance of Logan's bank account. 2. Evaluate the expression to find the new balance.” This connects the major work of 7.EE.B (Solve real-life and mathematical problems using numerical and algebraic expressions and equations) to the major work of 7.NS.A (Apply and extend previous understandings of operations with fractions to add, subtract, multiply and divide rational numbers).

• Module 3, Topic B, Lesson 9: Solving Equations to Determine Unknown Angle Measures, Practice, Problem 3, students write and solve multi-step equations, “Two angles are vertical angles. One angle has a measure of (5x+12)\degree, and the other angle has a measure of 136\degree. a. Write an equation for the angle relationship described and solve for x. b. Check your solution to the equation.” This connects the major work of 7.EE.A (Use properties of operations to generate equivalent expressions) to the major work of 7.EE.B (Solve real-life and mathematical problems using numerical and algebraic expressions and equations). 

Materials provide connections from supporting work to supporting work throughout the grade-level when appropriate. Examples include:

• Module 4, Topic A, Lesson 1: Sketching, Drawing, and Constructing Geometric Figures, Learn, Figure D, students sketch vertical angles freehand, construct them using tools, and find the missing angle measures, “Sketch vertical angles that measure about 80\degree. Construct vertical angles that measure 80\degree. Label each angle with its measure.” This connects the supporting work of 7.G.A (Draw, construct, and describe geometrical figures and describe the relationships between them) to the supporting work of 7.G.B (Solve real-life and mathematical problems involving angle measure, area, surface area, and volume).

• Module 4, Topic D, Lesson 16: Solving Area Problems by Composition and Decomposition, Launch, Problem 1, students consider methods for finding the area of a composite figure, “The diagram on the right models the front of the iron bridge in Slovenia that is shown next to it. Explain how you can estimate the area of the figure on the right. Identify which lengths you need to know.” This problem connects the supporting work of 7.G.A (Draw, construct and describe geometrical figures and describe the relationships between them) to the supporting work of 7.G.B (Solve real-life and mathematical problems involving angle measurement, area, surface area, and volume).

• Module 6, Topic D, Lesson 17: Comparing Sample Means, Launch, Problem 1, students select random samples from different populations and calculate the sample means, “Organize students in groups of 3 and distribute one set of prepared bags to each group. Within each group, assign each student bag A, bag B, or bag C. Share with students that all bags with a given label have identical contents. Randomly select a sample of ten slips of paper from your bag. Record the label on your bag and then record the number from the slips of paper you selected. a. Create a dot plot of your sample. b. Find the sample mean and record the means calculated by each person in your group. Are all the sample means that your group calculated equal? Do you think the population means of your bag are equal or unequal? Why?” This connects the supporting work of 7.SP.A (Use random sampling to draw inferences about a population) to the supporting work of 7.SP.B (Draw informal comparative inferences about two populations).

The materials reviewed for Eureka Math² Grade 7 meet expectations that content from future grades is identified and related to grade-level work, and materials relate grade-level concepts explicitly to prior knowledge from earlier grades.

Each Module Overview contains Before This Module and After This Module looking forward and back respectively, to reveal coherence across modules and grade levels. The Topic Overview includes information about how learning connects to previous or future content. Some Teacher Notes within lessons enhance mathematical reasoning by providing connections/explanations to prior and future concepts.

Content from future grades is identified and related to grade-level work. Examples include:

• Module 2: Operations with Rational Numbers, Module Overview, After This Module, Grade 7 Modules 3 and 4, “Students use rational number operations throughout grade 7. They use rational numbers and properties of operations when working with equations and inequalities in module 3 and when finding area, surface area, and volume in module 4.” Grade 8, “Students’ knowledge of extending operations to rational numbers supports their work in grade 8 as they solve equations, evaluate functions, and determine the distance between two points in a coordinate plane.”

• Module 4, Topic A, Lesson 4: Angles of a Triangle, Learn, Teacher Note, “All mentions of angles in this section refer to interior angles of a triangle. Grade 8 introduces exterior angles of a triangle, so the interior and exterior angles of a triangle are distinguished in grade 8.”

• Module 6, Topic D: Comparing Populations, Topic Overview, “An understanding of random sampling and informal statistical inference is the foundation for understanding bivariate data and linear regression in grade 8. It is important in this topic to reinforce the difference between sample statistics and population characteristics, along with the reasonableness of using sample statistics to make inferences about population characteristics.”

Materials relate grade-level concepts explicitly to prior knowledge from earlier grades. Examples include:

• Module 1: Ratios and Proportionality, Module Overview, Before This Module, Grade 6 Module 1, “Students apply knowledge of multiplicative comparisons to understand ratio relationships. They represent the two values in a ratio as a quotient—known as the value of the ratio—and then use that value to determine rates and unit rates of ratio relationships. Throughout the coursework of grade 6, students apply ratio reasoning to work with percents, equations, graphs, geometry, and statistics. Grade 7 module 1 elevates the work of grade 6 by introducing the terms proportional relationships and scale factor.”

• Module 4, Topic C: Circumference and Area of Circles, Topic Overview, “In topic B, students construct triangles given conditions. That sets up constructing circles and analyzing their characteristics in this topic. Students build on what they know about composite area from grade 6 to find the areas of composite figures that include circular regions.”

• Module 5, Topic A, Lesson 3: Percent as Rate per 100, Launch, Teacher Note, “In grade 6, students solve percent problems by using a tape diagram, a double number line, and mental math. Encourage students to model the problems in this lesson by using these strategies in any way that makes the most sense to them.”