7th Grade - Gateway 2

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

The materials include problems and questions that develop conceptual understanding throughout the grade level. For example:

• In Module 2, Lesson 2, students develop conceptual understanding of rational numbers by representing integers on a number line with arrows and use the length of the arrows to understand absolute value. The number line is used to model addition and subtraction with integers to help develop the concept visually. (7.NS.1)
• In Module 3, Lesson 4, students develop conceptual understanding by using area models to learn how to write expressions. The area model shows how products are written as sums which the students will use to solve real-life situations. (7.EE.2)
• Module 3, Lesson 17, Example 2, students develop conceptual understanding as they use the formula for the area of the circle to solve, "A sprinkler rotates in a circular pattern and sprays water over a distance of 12 feet, What is the area of the circular region covered by the sprinkler? Express your answer to the nearest foot. Draw a diagram to assist you in solving the problem. What does the distance of 12 feet reporesent in this problem?"(7.G.4)

The materials provide opportunities for students to demonstrate conceptual understanding independently throughout the grade level. For example:

• In Module 2, Lesson 2, students independently demonstrate a conceptual understanding of integer addition as they complete the Exit Ticket. The students determine if the arrows on a number line correctly represent an expression involving the addition of three integers, draw a correct number line model and write a real-world situation that would represent the sum. (7.NS.1)
• In Module 3, Lesson 6, students independently demonstrate a conceptual understanding of equivalent expressions as they write two different expressions that represent the new cost of a shirt and a pair of pants, and explain the different information each one shows. Example Problem 2 states, “At a store, a shirt was marked down in price by $10.00. A pair of pants doubled in price. Following these changes, the price of every item in the store was cut in half. Write two different expressions that represent the new cost of the items, using s for the cost of each shirt and p for the cost of a pair of pants. Explain the different information each one shows.” (7.EE.2)

The instructional materials for Eureka Grade 7 meet expectations that they attend to those standards that set an expectation of procedural skill.

The instructional materials develop procedural skill throughout the grade level. For example:

• In Module 2, Lessons 1-11, students develop procedural skill with integer operations. Students develop procedural skill of integer operations through a card game, number-line models, and real-world contexts. (7.NS.1-2)
• In Module 2, Lessons 22-23, students develop procedural skill with expressions and equations when translating word problems into algebraic equations. Students solve equations of the form p + q = r and p(x+q)=r, where p, q, and r are specific rational numbers. (7.EE.1, 7.EE.4a)
• In Module 3, Lessons 8-9, students develop procedural skill with writing sequences of equations underneath each other, linked together by if-then moves and/or properties of operations. (7.EE.1, 7.EE.4a).

The instructional materials provide opportunities to demonstrate procedural skill independentlythroughout the grade level. For example:

• In Module 2, Lessons 15-16, students independently demonstrate procedural skill when determining the product of integers within a “Sprint” activity. Problem 31 states, “11 x -33” (7.NS)
• In Module 3, Lesson 2, students independently demonstrate procedural skill when writing equations in standard form within a “Sprint” activity. (7.EE.1)

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

The instructional materials include multiple opportunities for students to engage in routine and non-routine application of mathematical skills and knowledge of the grade level. For example:

• In Module 1, Lesson 14, students engage in grade-level mathematics by solving multi-step problems including fractional markdowns, markups, commissions, and fees. Classwork Example 2 states, “A used car salesperson receives a commision of 1/12 of the sales price of the car for each car he sells. What would the sales commission be on a car that sold for $21,999?” (7.RP.3)
• In Module 3, Lesson 9, students engage in grade-level mathematics by using addition, subtraction, multiplication, and substitution properties of equality to solve word problems.. Exit Ticket states, “Brand A scooter has a top speed that goes 2 miles per hour faster than Brand B. If after 3 hours, Brand A scooter traveled 24 miles at its top speed, at what rate did Brand B scooter travel at its top speed if it traveled the same distance? Write an equation to determine the solution. Identify the if-then moves used in your solution.” (7.EE.B)
• In Module 4, Lesson 12 students engage in grade-level mathematics by creating scale drawings of given drawings. Problem Set Question 3 states, “The accompanying diagram shows that the length of a pencil from its eraser to its tip is 7 units and that the eraser is 1.5 units wide. The picture was placed on a photocopy machine and reduced to 66 2/3%. Find the new size of the pencil, and sketch a drawing. Write numerical equations to find the new dimensions..” (7.G.1, 7.RP.A)

The instructional materials provide opportunities for students to demonstrate independently the use of mathematics flexibly in a variety of contexts. For example:

• In Module 1, Lesson 17, students independently demonstrate the use of mathematics by using a table or an equation to create a scale drawing of a rectangular swimming pool. The Exit Ticket states, “A rectangular pool in your friend’s yard is 150 ft. × 400 ft. Create a scale drawing with a scale factor of 1/600. Use a table or an equation to show how you computed the scale drawing lengths.” (7.G.1)
• In Module 3, Lesson 13, students independently demonstrate the use of mathematics by using inequalities to solve word problems. The Exit Ticket states, “Shaggy earned $7.55 per hour plus an additional$100 in tips waiting tables on Saturday. He earned at least $160 in all. Write an inequality and find the minimum number of hours, to the nearest hour, that Shaggy worked on Saturday.” (7.EE.4)
• In Module 4, Lesson 16, students independently demonstrate the use of mathematics by writing and using algebraic expressions and equations to solve percent word problems. Problem Set Question 3 states, “During lunch hour at a local restaurant, 90% of customers order a meat entrée and 10% order a vegetarian entrée. Of the customers who order a meat entrée, 80% order a drink. Of the customers who order a vegetarian entrée, 40% order a drink. What is the percent of customers who order a drink with their entrée?” (7.RP.3)

The instructional materials for Eureka Grade 7 meet expectations that the three aspects of rigor are not always treated together and are not always treated separately.

Conceptual understanding is addressed in Classwork. During this time, the teacher guides students through a new concept or an extension of the previous day’s learning. Students practice solving procedural problems in problem sets. The materials provide engaging applications of grade-level concepts throughout each lesson. The program balances all three aspects of rigor in every lesson.

All three aspects of rigor are present independently throughout the program materials. For example:

• In Module 1, Lesson 18, students engage in the application of mathematics when finding actual lengths and areas from scale drawings. Problem Set 4 Question states, “A model of a skyscraper is made so that 1 inch represents 75 feet. What is the height of the actual building if the height of the model is 18 ⅗ inches?” (7.G.1)
• In Module 4, Lesson 1, students practice the procedural skill of converting from one form of a number to another (decimal, percent, fraction) as they play the “I have, who has?” game. In Exercise 1 cards, one of the cards states, “I have the equivalent value, 0.11. Who has the card equivalent to 350 percent?” (7.EE.3)
• In Module 6, Lesson 5, students develop conceptual understanding of the structure of a triangle, noticing the conditions that determine a unique triangle, more than one triangle, or no triangle. Classwork Example 2 states, “Two identical triangles are shown below. Give a triangle correspondence that matches equal sides and equal angles.” (7.G.2)

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

• In Module 4, Lesson 3, students practice procedural skill when solving word problems involving percent more and percent less than a quantity. Exit Ticket Question 1 states, “Solve each problem below using at least two different approaches. Jenny’s great-grandmother is 90 years old. Jenny is 12 years old. What percent of Jenny’s great-grandmother’s age is Jenny’s age?” (7.RP.3)
• In Module 6, Lesson 21, students develop conceptual understanding of area and practice procedural skill when using the distributive property on expressions. Class work Example 2 states, “Bobby draws a square that is 10 units by 10 units. He increases the length by x units and the width by 2 units. Draw a diagram that models this scenario. Assume the area of the large rectangle is 156 units squared. Find the value of x.” (7.G.6)

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

All of the eight MPs are identified within the grade-level materials. The Standards for Mathematical Practice are identified at the beginning of each module under the Module Standards. The tab named “Highlighted Standards for Mathematical Practice” lists all of the MPs that are focused on in the module. Each MP is linked to the definition of the practice as well as in which lessons throughout the series that practice can be found.

Each Module Overview contains a section titled, “Focus Standard for Mathematical Practice." Every practice that is identified in the module has a written explanation with specific examples of how each practice is being used to enrich the content of the module. For example:

• In Module 5, the explanation for MP 4 states, “Model with mathematics. Students use probability models to describe outcomes of chance experiments. They evaluate probability models by calculating the theoretical probabilities of chance events and by comparing these probabilities to observed relative frequencies.”

Each lesson specifically identifies where MPs are located, usually within the margins of the teacher edition.

The instructional materials reviewed for Eureka Grade 7 meet expectations for prompting students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics.

Student materials consistently prompt students to construct viable arguments and analyze the arguments of others.

• In Module 3, Lesson 1, students construct viable arguments and analyze the arguments of others when determining if a fictional student’s statement and justification are correct. Example 3f states, “Alexander says that 3x + 4y is equivalent to (3)(4) + xy because of any order, any grouping. Is he correct? Why or why not?”
• In Module 5, Lesson 12, students construct viable arguments and analyze the arguments of others when estimating probabilities. Students determine if their results agree with a fictional students results and explain their decision. Exercise 3 states, “Collect data for Sylvia. Carry out the experiment of shaking a cup that contains four balls, two black and two white, observing, and recording whether the pattern is opposite or adjacent. Repeat this process 20 times. Then, combine the data with those collected by your classmates. Do your results agree with Philippe’s equally likely model, or do they indicate that Sylvia had the right idea? Explain.”
• In Module 5, Lesson 14, students construct viable arguments and analyze the arguments of others when creating random sequencing. Exercise 2 states, “Working with a partner, toss a coin 20 times, and write down the sequence of heads and tails you get. Compare your results with your classmates. How are your results from actually tossing the coin different from the sequences you and your classmates wrote down? Toni claimed she could make up a set of numbers that would be random. What would you say to her?”
• In Module 6, Lesson 25, students construct viable arguments and analyze the arguments of others when finding the volume of rectangular prisms. Exercise 2 states, “Two aquariums are shaped like right rectangular prisms. The ratio of the dimensions of the larger aquarium to the dimension of the smaller aquarium is 3:2. Addie says the larger aquarium holds 50% more water than the smaller aquarium. Berry says that the larger aquarium holds 150% more water. Cathy says that that larger aquarium hold over 200% more water. Are any of the girls correct? Explain your reasoning.”

The instructional materials reviewed for Eureka Grade 7 meet expectations for explicitly attending to the specialized language of mathematics.

In each module, the instructional materials provide new or recently introduced mathematical terms that will be used throughout the module. The mathematical terms that are the focus of the module are highlighted for students throughout the lessons and are reiterated at the end of most lessons.

Each mathematical term that is introduced has an explanation, and some terms are supported with an example. The terminology that is used in the modules is consistent with the terms in the standards.

The materials provide explicit instruction in how to communicate mathematical thinking using words, diagrams and symbols. For example:

• In Module 1, Lesson 3, the materials prompt students to respond to the questions, “Is the pay proportional to the hours worked? How do you know?” A sample response is included and states, “Yes, the pay is proportional to the hours worked because every ratio of the amount of pay to the number of hours worked is the same. The ratio is 8:1, and every measure of hours worked multiplied by 8 will result in the corresponding measure of pay.”
• In Module 4, Lesson 2, the materials guide teachers through the teaching process of finding a part when given the percent of the whole. Example 1 states, “Is 30 the whole unit or part of the whole? What percentage of Ty’s class does the quantity 30 students represent? Solve the problem first using a tape diagram.”

The materials use precise and accurate terminology and definitions when describing mathematics and support students in using them. For example:

• In the Module 1 Overview, a description of proportional relationships is provided: “A proportional relationships is a correspondence between two types of quantities such that the measures of quantities of the first type are proportional to the measures of the quantities of the second type. Note that proportional relationships and ratio relationships describe the same set of ordered pairs but in two different ways.”
• In Module 1, Lesson 2, the mathematical term proportional is introduced. The Student Outcomes section provides a definition for the constant of proportionality and a description for proportional. “Students understand that two quantities are proportional to each other when there exists a constant (number) such that each measure in the first quantity multiplied by this constant gives the corresponding measure in the second quantity.”
• In Module 3, Lesson 1, the mathematical term expression in standard form is introduced. The teacher edition calls attention to an important note regarding the term standard form: “Important: An expression in standard form is the equivalent of what is traditionally referred to as a simplified expression. This curriculum does not utilize the term simplify when writing equivalent expressions, but rather asks students to put an expression in standard form or expand the expression and combine like terms. However, students must know that the term simplify will be seen outside of this curriculum and that the term is directing them to write an expression in standard form.”