7th Grade - Gateway 2

The instructional materials for Into Math Florida 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 designed to develop conceptual understanding and provide opportunities for students to independently demonstrate conceptual understanding throughout the grade. Build Understanding and Step it Out introduce mathematical concepts, and students independently demonstrate their understanding of the concepts in Check Understanding and On My Own problems at the end of each lesson.

• In Lessons 3.1 and 3.2, students develop conceptual understanding of integer addition and subtraction by representing the operations on number lines. For example, in Build Understanding, part A states, “Latrell spins a wheel to find out how many points he adds to his score. The wheel stops on “-5 points.” Use the number line to add -5 points to Latrell’s score. Then complete the equation.” (7.NS.1.1)
• In Module 5, students build understanding of multiplying and dividing rational numbers. For example, students represent multiplication of positive and negative numbers with repeated addition on a number line. In Lesson 5.2, students investigate signs of products as the number of negative factors change. (7.NS.1.2)
• In Lesson 7.1, Build Understanding, Question 2, students rewrite the discount expression p+p+0.60p as 2.60p and explain why the expressions are equivalent. In Lesson 7.2, On My Own, Question 6, students solve, “A pentagon has side lengths: (12 + 4x), (10 + 8x), (15 + 3x), (9 + 2x), and (14 + 3x). Write a simplified expression that represents the perimeter of the pentagon. Explain.” (7.EE.1.2)

The instructional materials for Into Math Florida Grade 7 meet expectations for attending to those standards that set an expectation of procedural skills.

The materials include problems and questions designed to develop procedural skills and provide opportunities for students to independently demonstrate procedural skills throughout the grade. The materials develop procedural skills in On Your Own, and students demonstrate procedural skills in More Practice/Homework.

• Throughout Modules 3 and 4, students evaluate addition and subtraction problems with various rational numbers. For example, in Lesson 3.3, On My Own, “Questions 10-13, 102.8 − 98.6; 99.4 − 101.7; -98.6 + (-1.4); -19 + 11; Question 16, $$\frac{1}{2}$$ − (-$$\frac{1}{2}$$).” In Lesson 4.1 More Practice/Homework: “Question 10, 21 − 57; Question 12, -23 − (-5); Question 15, 24 − (-37).” (7.NS.1.1)
• Throughout Modules 5 and 6, students evaluate multiplication and division problems with various rational numbers. For example, in Lesson 5.2, On My Own, Question 7, “Find the product: (0.5)(-0.5)(-4)(-4)(-1)”; More Practice/Homework, Question 7, “Find each quotient: -27/5”. In Lesson 6.1, More Practice/Homework, Question 3, 5 - 44 $$\times$$ (0.75) - 18/ $$\frac{2}{3}$$ $$\times$$ 0.8+ (-$$\frac{4}{5}$$). (7.NS.1.2)
• In Lesson 7.2, students write linear expressions and apply properties of operations to generate equivalent expressions. For example, “students factor expressions using the GCF: 3x-24; students simplify expressions using properties of operations: (-r-5)-(-2r-4); students also expand expressions using the Distributive Property and then simplify the expressions: (4x-7.2)+(-5.3x-8).” (7.EE.1.1)
• In Lesson 7.4, students solve two-step equations involving rational numbers and equations written in the forms px + q = r and p(x+q) = r. For example, On My Own, Questions 8-15 and 19-27, students solve equations in this form, “-9d-1=17; 2+1/6a=-4; 2x-5=15”, etc.). (7.EE.2.4a)

The instructional materials for Into Math Florida Grade 7 meet expectations for teachers and students spending 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 concepts of the grade-level, and students independently demonstrate the use of mathematics flexibly in a variety of contexts. During Independent Practice and On My Own, students often engage with problems including real-world contexts and present opportunities for application. More Practice and Homework contains additional application problems.

• In Module 2, students solve multi-step ratio and percent problems. For example, in Lesson 2.1, Check Understanding, Question 1 states, “Peggy earned $20 for each lawn she mowed last summer. This summer, she raised her price to $23 per lawn. What is the percent of Peggy’s change?” In Lesson 2.2, On My Own, Question 8 states, “Shelly’s boutique had a labor day sale featuring 25% off any item. Tammy wanted to buy a blouse that originally sold for $21.99. To the nearest cent, how much will it cost her before tax to buy the blouse during the sale?” (7.RP.1.3)
• In Modules 4 and 5, students solve real-world problems involving the four operations with rational numbers. For example, in Lesson 4.4, Check Understanding, Question 1 states, “Soojin is adding some lengths of wood she used for a project. The lengths of wood are 1 $$\frac{2}{3}$$, 3 $$\frac{3}{4}$$, and 2 $$\frac{1}{4}$$ feet. How much wood did she use in all?” In Lesson 5.4, On My Own, Question 4 states, “A butterfly is flying 8 $$\frac{3}{4}$$ feet above the ground. It descends at a steady rate to a spot 6 $$\frac{1}{4}$$ feet above the ground in 1 $$\frac{2}{3}$$ minutes. What is the butterfly’s change in elevation per minute?” (7.NS.1.3)
• In Lesson 6.3, Check Understanding, Question 1 states, “A block of clay contains 20 4-ounce portions of clay. A ceramics teacher wants to use the block to make as many spheres of clay as possible, each weighing $$\frac{2}{5}$$ pound. How many spheres can she make?” In Lesson 6.2, students use estimation to check reasonableness, “The dimensions of a room are shown. Ivan wants to have the four walls painted. The painter says the area is 542.2 square feet. Estimate the total area that needs to be painted. Was the painter’s answer reasonable. Explain.” (7.EE.2.3)
• In Lesson 8.3, students write and solve two-step inequalities given a real-world scenario, “Ariana started a saving account with $240. She deposits $30 into her account each month. she wants to know how many months it will take for her account to have a balance greater than $500. Write and solve an inequality that represents this situation. How many months will it take for her account to have a balance greater than $500? Explain.” (7.EE.2.4b)

The instructional materials for Into Math Florida Grade 7 meet expectations for the three aspects of rigor not always being treated together and not always being treated separately. In general, two or all three, of the aspects are interwoven throughout each module. The Module planning pages include a diagram showing the first few lessons addressing understanding and connecting concepts and skills and the last lessons addressing applications and practice.

All three aspects of rigor are present independently throughout the program materials, and examples include: 

• In Lesson 12.1, the materials address conceptual understanding of statistics by having students determine where to locate a fast-food restaurant. Students discuss the terms population, sample, representative, and bias and identify them in different contexts. Students participate in class discussion and Turn and Talk, making generalizations and inferences, before working independently. In Build Understanding, students answer, “Which of the representative samples of the population is more representative of the population? Explain.”
• In Module 8, students develop procedural skill by writing, solving, and graphing the solutions to one- and two-step inequalities. The inequalities incorporate rational numbers in various forms, i.e. integers, fractions, and decimals. An example is, “...solve the inequality. Graph the solution. -$$\frac{3}{4}$$m + $$\frac{1}{4}$$ $$\ge$$ 4 $$\frac{3}{4}$$”.
• In Lesson 15.4, students conduct simulations and find experimental probabilities, “Allie is a softball player. She has a batting average of 0.600. This means Allie gets a hit 60% of the time. Design a simulation using slips of paper and a box to predict the probability of Allie getting a hit in at least 2 of her next 5 at bats. Perform the simulation and use your results to predict the probability of Allie getting a hit in at least 2 of her next 5 at bats.”

Multiple aspects of rigor are engaged simultaneously to develop students’ mathematical understanding of a single topic/unit of study throughout the materials, examples include: 

• In Lesson 5.3, students develop conceptual understanding of converting fractions to decimals using multiple representations including a double number line with one side as decimals and one side as fractions and long division. Students develop procedural skill by converting fractions to decimals in numerous problems such as, “Hayley is buying herbs. She wants to buy ⅚ ounce of basil. The scale she is using to weigh the basil displays the weight as a decimal. How will she know when the display on the scale is correct to the tenths’ place? Explain your reasoning.”
• In Lesson 10.1, students develop conceptual understanding of the formula for circumference by examining the relationship between circumference (C), diameter (d), and pi. Students complete a table finding the ratio, C/d, for various circular objects, and they rewrite pi = C/d to reveal the formula for the circumference of a circle. Throughout the remainder of the lesson, students use the formula to solve real-world problems, for example, “Toni rides the Ferris wheel shown for 15 revolutions. A. How far does Toni travel in one revolution? How far does Toni travel for the entire ride?”

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

An often-used strategy in these materials is Turn and Talk with a partner about the related task. Often, Turn and Talks require students to construct viable arguments and analyze the arguments of others. In addition, students are often asked to justify their reasoning in practice problems.

• In Lesson 3.2, Question 7 states, “Leah said that when you add two negative integers the result must be negative. Do you agree or disagree? Use a number line to help explain your answer.”
• In Lesson 3.3, Question 2 states, “Jamin evaluated the expression 10-(-3) and says it is equal to 7. Is he right? If not, what was his mistake?”
• In Lesson 4.2, Spark Your Learning, Turn and Talk prompts, “Share your solution with your partner. If you didn’t already, come up with an addition expression for solving the problem. Discuss with your partner: Does order matter for this problem? Why or why not?”
• In Lesson 9.4, “Travis draws a triangle with three 60 degree angles and side lengths of 5 inches each.  A. Can Margarita draw a triangle with the same angle measures as Travis’s triangle and different side lengths? Why or why not?  B. Can she draw a triangle with the same side lengths and different angle measures? Explain.”
• In Lesson 10.2, Question 7 states, “A classmate states that if the radius of a circle is doubled, then the area is doubled. Do you agree or disagree, how much larger do you think the area will be?”
• In Lesson 13.1, students analyze data on wait times for two food trucks using dot plots and tables and they answer, “Based on the data, which food truck would have more predictable wait times? Explain.”
• In Lesson 13.2, On My Own states, “After looking at the box plots, Tomas expresses surprise that most award-winning actresses are under the age of 40. Martha disagrees, pointing out that the right whisker is the longest part of the Best Actress box plot. Therefore, she argues, there are more winners between the age of 41 and 62 than in the other intervals. Determine which friend is correct, and explain why.”
• In Lesson 14.4, Question 13 states, “Construct Arguments. Jiang was concerned about a particularly busy intersection because it did not have a stop sign. She took a survey of 100 people who used that intersection. 75 people she spoke to support putting in a stop sign. The town’s population is 4,480. Write a percent equation and also use proportional reasoning to predict how many people in the town support the stop sign. Then make an argument as to why Jiang’s survey results may be unreasonable.”

The instructional materials reviewed for Into Math Florida Grade 7 meet expectations for assisting teachers in engaging students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics.

Many of the lesson tasks are designed for students to collaborate. Teacher prompts promote explaining their reasoning to each other during collaborative lesson tasks. Independent problems provided throughout the lessons also have teacher guidance to assist teachers in engaging students.

• The Teacher Edition provides Guided Student Discussion with questions to encourage students to explain their thinking. For example, in Lesson 5.3, “When you divide two nonzero integers, how do you know if the quotient is positive or negative?” In Lesson 3.1, “When you start with a negative temperature and move down on the number line, is the temperature increasing or decreasing? Why?”
• Turn and Talks are provided multiple times per lesson. For example, in Lesson 10.3, Task 2 Turn and Talk states, “What do you notice about the width of a vertical cross section through the centers of the bases and the diameter of the horizontal cross section? Is this true for all cylinders? Explain.” Teachers are given a possible answer, as well as, additional guidance to assist students in constructing arguments, for example, “If some students are having trouble understanding that the width of a vertical cross section through the center of a cylinder is the same length as the diameter of a horizontal cross section from the same cylinder, ask other students who understand to explain their thinking.”
• The Teacher Edition includes Let’s Talk in margin notes to prompt student engagement. For example, in Lesson 4.1, “Select students who used various strategies and have them share how they solved the problem with the class. Encourage students to ask questions of their classmates. Look for students who used integer sums in steps to determine the number of hours it would take for the submarine to be above water.”
• The Teacher Edition also provides Cultivate Conversation prompts in the lessons. For example, in Lesson 4.2, “Stronger and Clearer. Ask students to describe in their own words what it means that addition and subtraction are inverse operations. Ask how inverse operations are like opposite numbers. Ask them to describe how the addition of a negative number is like subtraction and the subtraction of a negative number is like addition.”
• In lesson planning pages, sometimes Professional Learning provides a rationale for a lesson labeled with a Mathematical Practice. For example, in Lesson 5.2 which is labeled MP.3.1, “In this lesson, students derive the rules of multiplying three or more signed numbers by experimentation, reducing their observations to a general rule. They approach the products of multiple factors strategically and justify their choices by reference to the properties of operations. They then share their strategies with other class members and compare their ideas about solving the problems in the lesson.”

The instructional materials reviewed for Into Math Florida Grade 7 meet expectations for explicitly attending to the specialized language of mathematics. The materials provide explicit instruction on communicating mathematical thinking with words, diagrams, and symbols. The materials use precise, accurate terminology and definitions when describing mathematics and support students in using them. Examples are found throughout the materials.

• Key Academic Vocabulary is listed at the beginning of the module in a table including any prior vocabulary relevant to the lesson and new vocabulary.
• Each lesson includes a Language Objective emphasizing mathematical terminology. For example, in Lesson 5.3, “Use mathematical terminology to explain how to express quotients in different forms.”
• In the Module planning pages, a Linguistic Note located on the Language Development page provides teachers with possible misconceptions relating to academic language. For example, in Module 2, “Speak with students about words which can have multiple meanings. The word change can have multiple meanings in mathematics. When working with money, change can indicate an amount of money returned from a transaction or it can indicate coins. In this lesson, percent change, describes the percent of increase or decrease in an amount compared to its original amount.”
• In Sharpen Skills located in the lesson planning pages, some lessons include Vocabulary Review activities. For example, in Lesson 3.2, students use a graphic organizer to make sense of the term absolute value by using examples and visual models.
• Guided Student Discussion often provides prompts related to understanding vocabulary, such as in Module 13, “Students should be familiar with the terms mean, median, mode, range, interquartile range, center of data, and spread of data. Ask student to explain what they mean if they use those terms.”
• Student pages include vocabulary boxes defining content vocabulary.
• Vocabulary is highlighted and italicized within each lesson in the materials.
• Vocabulary review is located at the end of each Module where students match new vocabulary terms with their meaning and/or examples provided, fill-in-the-blank with definitions or examples, or create a graphic organizer to help make sense of terms.
• The Teacher Edition sometimes suggests creating an Anchor Chart to “connect math ideas, reasoning, and language” where students define terms with words and pictures, trying to make connections among concepts. For example, Lesson 10.2 shows a sample anchor chart including vocabulary related to circumference, area, and cross sections.”
• The Interactive Glossary at the end of the text provides the definition and a visual (diagrams, symbols, etc.) is provided for each vocabulary word. In the student book, the instructions read, “As you learn about each new term, add notes, drawings, or sentences in the space next to the definition. Doing so will help you remember what each term means.”