7th Grade - Gateway 2

The instructional materials for Big Ideas Math: Modeling Real Life Grade 7 partially meet expectations that the materials develop conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings. The instructional materials do not always provide students opportunities to independently demonstrate conceptual understanding throughout the grade-level. 

Each lesson begins with an Exploration section where students develop conceptual understanding of key mathematical concepts through teacher-led activities. For example:

• In Chapter 1, Section 2, Exploration 1 (7.NS.1.d), students are taught to add integers with chips and using number lines. “Write an addition expression represented by the number line. Then find the sum.” After these examples, students are asked to use conceptual strategies (number line or chips). 
• In Chapter 3, Lesson 2, Exploration 1, students use algebra tiles to model a sum of terms equal to zero and simplify expressions. In the Concepts, Skills and Problem Solving section, students have two additional problems where they use algebra tiles to simplify expressions. (7.EE.1)
• Chapter 1, Section 4, “Subtracting Integers,” Exploration 1 asks students to work with partners and use integer counters to find the differences and sums of several problems with two different representations. For example, “4 - 2” and “4 + (-2)”; “-3 - 1” and “-3 + (-1)” and “13 - 1”.  Student pairs are asked to generate a rule for subtracting integers. Students who can’t generate a rule are prompted to use a number line. After working independently students share their rule with a partner and discuss any discrepancies. (7.NS.1)
• Chapter 4, Section 1 “Solving Equations Using Addition or Subtraction” Exploration 1, students are asked, “Write the four equations modeled by the algebra tiles. Explain how you can use algebra tiles to solve each equation.” (7.EE.3)

The instructional materials do not always provide students opportunities to independently demonstrate conceptual understanding throughout the grade-level. The shift from conceptual understanding, most prevalent in the Exploration Section, to procedural understanding occurs within the lesson.  The Examples and “Concepts, Skills, and Problem Solving” sections have a focus that is primarily procedural with limited opportunities to demonstrate conceptual understanding. For example: 

• In Chapter 3, Section 2, only Problems 8 and 9 ask students to demonstrate conceptual understanding. For example, Problems 10-17 ask students to “Find the Sum.”  Problem 10: “(n+8) + (n-12)”; Problem 16: “(6-2.7h) + (-1.3j-4).” Problems 19-26 ask students to “Find the difference.” Problem 19: “(-2g+7) - (g+11)”; Problem 26: “(1-5q) - (2.5s+8) - (O.5q+6)”. (7.EE.1)
• In Chapter 2, Section 2, Concepts, Skills & Problem Solving, the majority of the questions require procedural knowledge and do not ask students to demonstrate conceptual understanding. For example, Problems 13-28 ask students to “Find the quotient, if possible”, such as Problem 16: “-18 ÷ (-3)"; and Problem 22: “-49 ÷ (-7)”. (7.NS.1)

The instructional materials for Big Ideas Math: Modeling Real Life Grade 7 partially meet expectations that the materials are designed so that teachers and students spend sufficient time working with engaging applications of mathematics. 

The instructional materials present opportunities for students to engage in application of grade-level mathematics; however, the problems are scaffolded through teacher-led questions and procedural explanation. The last example of each lesson is titled, “Modeling Real Life,” which provides a real-life problem involving the key standards addressed for each lesson. This section provides a step-by-step solution for the problem; therefore, students do not fully engage in application. For example:

• Chapter 5, Lesson 1, Example 3, Modeling Real Life, “You mix 1/2 cup of yellow paint for every 3/4 cup of blue paint to make 15 cups of green paint. How much yellow paint do you use?” Students are given two methods to solve the questions with both methods being explained and answered. For example, “Method 1: The ratio of yellow paint to blue paint is 1/2 to 3/4. Use a ratio table to find an equivalent ratio in which the total amount of yellow paint and blue paint is 15 cups.” [A completed ratio table with annotated description as to how it was filled out is included.] “Method 2: You can use the ratio of yellow paint to blue paint to find the fraction of the green paint that is made from yellow paint. You use 1/2 cup of yellow paint for every ¾ cup of blue paint, so the fraction of the green paint that is made from yellow paint is 2/5 [included equation and solution]. So, you use  2/5 ⋅ 15 = 6 cups of yellow paint.” (7.RP.1)
• Chapter 1, Lesson 1, Example 3, Modeling Real Life, “A moon has an ocean underneath its icy surface. Scientists run tests above and below the surface. [Table Provided] The table shows the elevations of each test. Which test is deepest? Which test is closest to the surface?” The explanation from this point provides students with step-by-step directions on how to solve the problem. “To determine which test is deepest, find the least elevation. Graph the elevations on a vertical number line. [Vertical line provided.] The number line shows that the salinity test is deepest. The number line also shows that the atmosphere test and the ice test are closest to the surface. To determine which is closer to the surface, identify which elevation has a lesser absolute value. Atmosphere:  ∣0.3∣  = 0.3  Ice:  ∣−0.25∣  = 0.25 So, the salinity test is deepest and the ice test is closest to the surface.” (7.NS.1)

Throughout the series, there are examples of routine application problems that require both single and multi-step processes; however, there are limited opportunities to engage in non-routine problems. For example:

• Chapter 2, Lesson 1, Problem 17, “On a mountain, the temperature decreases by 18°F for each 5000-foot increase in elevation. At 7000 feet, the temperature is 41°F. What is the temperature at 22,000 feet? Justify your answer.” (7.NS.3, multi-step, routine)
• Chapter 3, Lesson 4, Problem 41, Dig Deeper, “A square fire pit with a side length of s feet is bordered by 1-foot square stones as shown. [Diagram provided] a. How many stones does it take to border the fire pit with two rows of stones? Use a diagram to justify your answer.” (routine) "b. You border the fire pit with n rows of stones. How many stones are in the nth row? Explain your reasoning.” (non-routine) (7.EE.3)
• Chapter 6, Lesson 3, Problem 32, Dig Deeper, “At a restaurant, the amount of your bill before taxes and tip is $19.83. A 6% sales tax is applied to your bill, and you leave a tip equal to 19% of the original amount. Use mental math to estimate the total amount of money you pay. Explain your reasoning. (Hint: Use 10% of the original amount.)” (7.RP.3, routine)

The instructional materials for Big Ideas Math: Modeling Real Life Grade 7 partially meet expectations that the three aspects of rigor are not always treated together and are not always treated separately. 

The instructional materials present opportunities in most lessons for students to engage in each aspect of rigor, however, these are often treated together. There is an over-emphasis on procedural skill and fluency. For example:

• In Chapter 4, Lesson 3, Solving Two-Step Equations, students begin with an Exploration example that uses algebra tiles to show the steps for solving an equation and the relationship to the properties of equality. These examples show the conceptual solving of an equation through models. The lesson shifts to a procedural steps of solving two step equations with Examples 1: “-3x + 5 = 2” and Example 2: “x/8- 1/2 = -7/2”. Example 3 is a procedural example of solving two step equations by combining like terms “3y - 8y = 25”. The lesson progresses to independent application of the skill in Concepts, Skills, and Problem Solving. Students solve equations procedurally.
• Chapter 6, Lesson 1, Fractions, Decimals and Percents, students begin the lesson with an Exploration activity where they compare numbers in different forms based on a variety of strategies. Example 1, presents a conceptual model of a decimal using a hundredth grid, and how to convert a decimal to a percent. Example 2, shows the students how to procedurally build on what they have learned to convert a fraction to a decimal to a percent using division. The lesson then moves to independent practice in Concepts, Skills, and Problem Solving where students procedurally convert between decimals, percents, and fractions.
• Chapter 7, Lesson 2, Experimental and Theoretical Probability, students’ learning begins with an Exploration activity in which students conduct two experiments to find relative frequencies (Flip a Quarter and Toss and Thumbtack) to understand the concept behind probability. The lesson moves on to Example 1, Finding an Experimental Probability by utilizing a formula. “$$P(event) =\frac {number of times the event occurs}{total number of trials}$$”, and Example 2, Finding a Theoretical Probability, by utilizing the  formula “$$P (event)= \frac{number of favorable outcomes}{number of possible outcomes}$$”. Example 3, shows the steps for applying each formula to compare probabilities. The bar growth shows the results of rolling a number cube 300 times. How does the experimental probability of rolling an odd number compare with the theoretical probability?” The independent practice in Concepts, Skills, and Problem Solving has the students finding an experimental probability and theoretical probability based on an event. 
• Chapter 9, Lesson 1, Circles and Circumference, begins with Exploration 1, where students use a compass to draw circles and conceptually see the length of the diameter and circumference. Exploration 2, continues to explore diameter and circumference through hands on modeling. The lesson continues with three examples showing the steps of applying the formula for finding radius, circumference, and perimeter of a circle. The independent work of the students is within the Concepts, Skills, and Problem Solving in which students are asked to procedurally solve for the radius, diameter, circumference and perimeter.

The instructional materials reviewed for Big Ideas Math: Modeling Real Life Grade 7 partially meet expectations that the instructional materials prompt students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics. 

“You be the Teacher”, found in many lessons, presents opportunities for students to critique the reasoning of others, and construct arguments. Examples of where students engage in the full intent of MP3 include the following:

• Chapter 4, Lesson 2, Problem 28, You Be the Teacher, “Your friend solves the equation -4.2x=21. Is your friend correct? Explain your reasoning.” The student work is provided to examine.
• Chapter 6, Lesson 1, Problem 20, You Be the Teacher, “Your friend uses the percent proportion to answer the question below. Is your friend correct? Explain your reasoning. ‘40% of what number is 34?’” The student work is provided to examine.

The Student Edition labels MP3 as “MP Construct Arguments,” however, these activities do not always require students to construct arguments. In the Student Edition, “Construct Arguments” was labeled only once for students and “Build Arguments” was labeled once for students. For example:

• Chapter 2,  Lesson 1, Construct Arguments, students construct viable arguments by writing general rules for multiplying (i) two integers with the same sign and (ii) two integers with different signs. Students are prompted to “Construct an argument that you can use to convince a friend of the rules you wrote in Exploration 1(c).”  
• Chapter 8, Lesson 4, Exploration 1, Build Arguments is identified in the Math Practice blue box with the following question, “How does taking multiple random samples allow you to make conclusions about two populations?”

The instructional materials reviewed for Big Ideas Math: Modeling Real Life Grade 7 partially meet expectations that the instructional materials assist teachers in engaging students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics. 

There are some missed opportunities where the materials could assist teachers in engaging students in both constructing viable arguments and analyzing the arguments of others. For example:

• In Chapter 1, Lesson 4, Subtracting Integers, students are shown an example of subtracting integers. In Laurie’s notes, teachers are prompted, “Ask students if it is possible to determine when the difference of two negative numbers will be positive and when the difference of two negative numbers will be negative.” 
• In Chapter 5, Lesson 2, Example 1, students find a unit rate based on given information. In Laurie’s notes, teachers are prompted, “There are several ways in which students may explain their reasoning. Take time to hear a variety of approaches.” This is labeled as MP3, but there is no support for teachers to assist students in constructing a viable argument or critiquing the thoughts of others. 
• Chapter 1, Lesson 2, Example 2, The Teacher’s Guide is noted with MP3 with the following directions, “‘When you add two integers with different signs, how do you know if the sum is positive or negative?’ Students answered a similar question in Example 1, but now they should be using the concept of absolute value, even if they don’t use the precise language. You want to hear something about the size of the number, meaning its absolute value.” There is no reference to MP3 in the Student Edition in this Lesson.