6th Grade - Gateway 2

The instructional materials for Into Math Florida Grade 6 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 Lesson 5.1, students use ratio language to describe a relationship between two quantities such as birds visiting birdhouses, orange and black kittens, and colored fabrics in a quilt. For example, students “write a part-to-whole or whole-to-part comparison about the quilt using symbols and using ratio language, such as “for each”, “for every,” or “per”.” (6.RP.1.1, 6.RP.1.2)
• In Lesson 7.1, students use ratio tables to represent equivalent fractions leading to a percent of 100. Other visual representations include double number lines and place value blocks. In Lesson 7.2, students use a variety of strategies to find the percent of a whole, such as estimation, tape diagrams, and equivalent ratios. (6.RP.1)
• In Lesson 8.5, Step It Out, students substitute a value into two algebraic expressions to determine if the expressions are equivalent. This provides the understanding of different expressions can be equal. In the next task, students substitute missing values to rewrite an algebraic expression and identify the mathematical property used to rewrite the expression. Independent problems include similar exercises and provide students opportunities to build their own understanding. (6.EE.1.3)
• In Lesson 10.1, Build Understanding, students are presented with a situation represented in words, with a table, and with an equation. Students analyze the equation to show their understanding of the dependent and independent variables. In On My Own, students demonstrate understanding of the variables and the multiple representations showing the relationship between them by identifying the dependent and independent variable and completing a table and a graph representing the situation. (6.EE.3.9)

The instructional materials for Into Math Florida Grade 6 meet expectations for attending to those standards that set an expectation of procedural skill and fluency.

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

• In Lesson 3.4, students practice division of fractions using the standard algorithm. For example, in Lesson 3.4, Question 9: “To paint her bedroom, Jade estimates she will need to buy 3 $$\frac{1}{4}$$ gallons of paint. How many $$\frac{1}{2}$$ gallon cans of paint should she buy? Explain.”; Question 5: “$$\frac{5}{4}$$ ÷ $$\frac{1}{10}$$”. (6.NS.1.1)
• In Module 4, students practice adding, subtracting, multiplying and dividing with multi-digit decimals using multiple strategies, including the standard algorithm. Expressions are presented both vertically and horizontally (6.NS.2.2 and 6.NS.2.3). Examples of problems for addition and subtraction include: 0.807-0.408; 0.13 + 0.58; “sum of 0.26 pound of red grapes and 0.34 pound of green grapes”; “difference between a cardinal weighing 1.5 ounces and a bluebird weighing 1.09 ounces.” Examples of problems for multiplication and division include: 10.05 × 5.6; 5104/116; 5.44 divided by 3.4; “the number of months it will take someone that pays $18 a month for a phone that costs a total of $432”; “how many shipping boxes are needed to ship 4,630 shirts when 130 shirts will fit in each box.”
• In Lesson 8.2, students evaluate exponents in numerical expressions and use order of operations, such as “$$72\div(15-6)+3\times2^2$$” and “$$15+32-6+(5+2)^2$$”, and in Lesson 8.4, students evaluate expressions by substituting values in for variables such as “Evaluate x − 12 when x = 18.6.” (6.EE.1.2c)

The instructional materials for Into Math Florida Grade 6 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 Lesson 5.3, students answer, “Two farms grow lettuce and tomatoes as shown in the labels to the right. Write ratios to compare acres of lettuce to acres of tomatoes at each farm. Which farm has a greater ratio of acres of lettuce to acres of tomatoes? How do you know?” In Lesson 5.5, Check for Understanding, Question 1 states, “A garden center is running a special on houseplants. A selection of 2 plants costs $7. If a designer buys 22 plants for new homes, how much does the designer spend on plants?” (6.RP.1.3)
• In Lesson 7.1, students solve real-world problems involving percent, such as, “Ryan got 36 out of 40 questions right on a test. Tessa got 92% on the same test. Who got a better score? Explain.” (6.RP.1.3)
• In Module 3, students solve word problems involving division of fractions by fractions such as, “Patrick has $$\frac{7}{10}$$ pound of flour. A batch of biscuits requires ⅛ pound of flour. How many whole batches of biscuits can Patrick make? Explain.” Also, “Daryl has $$\frac{2}{3}$$ of a bag of dog food. His dog eats $$\frac{4}{9}$$ of a bag per week. How many weeks will the dog food last?” (6.NS.1.1)
• In Module 9, students write and solve one-step equations for a variety of contexts, including differences in measurements and money. Examples respectively include, “Annie is 152.5 centimeters tall. She is 49 centimeters taller than her brother. Write and solve an equation to find her brother’s height in centimeters.” Also, “One ride on a city bus costs $1.50. Martina has $18 on her bus pass. Write and solve an equation to find how many rides she can take without loading more money on her bus pass.” (6.EE.2.7)

The instructional materials for Into Math Florida Grade 6 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.

• Lesson 3.1 attends to conceptual understanding. Students use models to represent fraction division. For example, Part A) “Draw the fraction bar that you could use to begin to find the solution.” Part B) “Will the fraction bar you drew in Part A help you make groups of $$\frac{3}{8}$$? If not what other fraction bar could help? Explain why?” Then in part C, students use the new fraction bar to find out “How many groups of $$\frac{3}{8}$$ are in $$\frac{3}{4}$$? Explain.” The problems throughout the lesson ask students to use models to solve division of fractions or to write and solve problems based on the models presented.
• Lesson 15.2 develops procedural skill. After measures of center are defined, students calculate the mean, median, and mode in a variety of contexts including temperature, 40-yard dash times, and cat food consumption. For example, “The 40-yard dash time (in seconds) for 8 runners is shown. A) What is the mean of the data?  B) What is the median of the data?  C) What is the mode of the data?  D) Which of the measures of center has more than one possible value?”
• Lesson 2.4 emphasizes application of unit rates. Students find and interpret unit rates and apply the concept in a variety of contexts including: recipes, better buys, pool drainage, swimmers, painting, dog walking, reading a book, a model truck, entertainment, and more. An example problem includes, “Greg drove 300 miles at a constant speed in 4 hours. The speed limit was 70 miles per hour. Was he speeding?”

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

• Lesson 4.5 attends to procedural skills and application related to computing with multi-digit numbers and finding common factors and multiples. Problems in the lesson require application of the four operations with multi-digit decimals. For example, “Anwar waters his front lawn with a sprinkler that sprays 0.7 gallons of water per minute. If it takes 45 minutes to water the front lawn, how many gallons of water does Anwar use?” The lesson also requires students to demonstrate procedural fluency. For example, Questions 19-24 present students with decimal operation problems and they are instructed to “add, subtract, multiply, or divide.”
• Lesson 12.1 attends to all three aspects of rigor as students find the area of polygons. To build conceptual understanding, students derive the formula for the area of a parallelogram using visual representations and connecting to the area of a rectangle. The lesson includes multiple problems requiring students use the formula to find area of parallelograms using diagrams and given dimensions, such as “The height of a parallelogram is 4 times its base. The base measures 2 $$\frac{1}{2}$$ feet. Find the area of the parallelogram. Show your work.” Another example, “A window is in the shape of a rhombus, with each side being 20 in. long. The height of the window is 16 in. What is the area in square inches of the glass needed for the window?”

The instructional materials reviewed for Into Math Florida Grade 6 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. Regularly, 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 2.4, Question 4 states, “D’Marc lists the sizes, in inches, of a set of screws: $$\frac{9}{64}$$, $$\frac{5}{32}$$, $$\frac{1}{16}$$, 18. He reasons that because the denominators are in order from greatest to least, the list is in order from least to greatest. Is D’Marc correct? Why or why not?”
• In Lesson 3.3, Question 14 states, “Dan says that 24 $$\frac{1}{2}$$ divided by 12 $$\frac{1}{2}$$ = 2, because $$\frac{24}{12}$$ = 2. Sam disagrees and thinks there will be fewer than 2 groups of 12 $$\frac{1}{2}$$ in 24 $$\frac{1}{2}$$. Who is is correct and why? What is 24 $$\frac{1}{2}$$ divided by 12 $$\frac{1}{2}$$?”
• In Lesson 4.4, Step It Out states, “Yan borrowed his parents’ car for a weekend camping trip. He drove the car 276.3 miles and used a total 10.230 gallons of gas. Kierra and Shawna both calculate how many miles Yan’s car drove per gallon. Whose solution is correct? Explain the error in the incorrect solution. How could you check the result?”
• In Lesson 5.1, Spark Your Learning, the Turn and Talk prompts, “Share your solutions. Did you use the same method? If not, explain your reasoning to make sure both methods are correct.”
• In Lesson 8.4, “Bill and Tia are trying to evaluate the expression $$5x^2$$ when x = 3. They both agree that 3 should be substituted for x. Tia says they should multiply 3 by 5, and then square the result. Bill says they should square 3 and then multiply by 5. Who is correct and why? What is the value of the expression?”
• In lesson 9.3, a Turn and Talk states, “Why can’t you divide both sides of an equation by zero? Explain.”
• In Lesson 12.1, Question 14 states, “For the two quadrilaterals below, Dan says that the one on the left has a larger area than the one on the right because it is longer. Bob says that both quadrilaterals have the same area. Who is correct? Why?”
• In Lesson 16.1, Question 3 states, “Construct Arguments. The dot plot shows the number of hours that 40 students studied each week. Make a statement that summarizes the data in the plot. Support your statement by describing clusters, gaps and/or peaks.”

The instructional materials reviewed for Into Math Florida Grade 6 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 4.1, “How is adding decimals similar to adding whole numbers? How is it different? How close to 10 kilometers is the length of your route?”, and in Lesson 14.3: “The movie theater recorded sales data for 14 days. How is this number represented in the histogram? Can you analyze the histogram to determine the total number of pretzels that the theater sold over the 14 days? Explain.”
• Turn and Talks are provided multiple times per lesson. For example, in Lesson 10.1, Task 3 Turn and Talk states, “Do the equations and the tables describe the same relationships? Explain how you know.” 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 equations and tables show the same relationship, have students substitute the values for each row of each table into the related equation.”
• The Teacher Edition includes Let’s Talk in margin notes to prompt student engagement. For example, in Lesson 3.1, “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. Discuss how the division problem can be modeled with fraction strips, or how division can be calculated by multiplying by the reciprocal.”
• The Teacher Edition also provides Cultivate Conversation prompts in the lessons. For example, Lesson 2.4 includes, “Stronger and Clearer. Have students share how they solved the problem. Remind students to ask each other questions of each other that focus on how they approached the problem. Then have the students refine their answers.”
• In the margin notes for practice questions identified as a mathematical practice, an explanation about why that practice is labeled. For example, in Lesson 16.1, Question 3 is labeled Construct Arguments. In the Teacher Edition, the notes explain the problem “gives students an opportunity to demonstrate an understanding of clusters, gaps, and peaks to describe a data distribution in the context of a real-world situation.”
• In Lesson 6.2, Connect Math Ideas, Reasoning, and Language states, “Before beginning the task, have students describe and give examples on their own words where they might convert measurements, such as one foot to inches or 1 yard to feet. Have partners share their work and discuss how their descriptions compare and connect.”

The instructional materials reviewed for Into Math Florida Grade 6 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 10.1, “Use the terms dependent and independent to describe variables represented in equations, tables, and graphs.”
• In Module planning pages, a Linguistic Note on the Language Development page provides teachers with possible misconceptions relating to academic language. For example, in Module 1, “Listen for students who do not understand the meanings of the terms positive, negative, and opposite as they refer to numbers. Students may already know the words positive and negative in phrases such as positive attitude or negative thinker. Ensure that students understand that, in mathematics, positive and negative numbers don’t have a meaning of “good” or “bad.” Model the correct language for students.”
• In Sharpen Skills in the lesson planning pages, some lessons include Vocabulary Review activities. For example, in Lesson 3.5, students use a Frayer Model to define and explain the terms: least common multiple, greatest common factor, and common denominator. Students explain to each other how the terms are related.
• Guided Student Discussion often provides prompts related to understanding vocabulary such as, “Listen for students who correctly use review vocabulary as part of their discourse. Students should be familiar with the terms origin, x-axis, y-axis, and ordered pair. Ask students 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.
• The vocabulary review at the end of each Module require 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 13.1 shows a sample anchor chart including vocabulary related to nets, surface area, and volume.
• 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.”