6th Grade - Gateway 2

The instructional materials for Kendall Hunt’s Illustrative Mathematics 6-8 Math, 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 that develop conceptual understanding throughout the grade level, and multiple opportunities exist for students to access concepts from different perspectives and independently demonstrate conceptual understanding throughout the grade.

Units 2 and 3 address 6.RP.A by exploring a variety of real-world applications using multiple mathematical representations, and multiple opportunities exist for students to work with ratios through the use of visual representations, interactive examples, and different strategies, for example, in Unit 2:

• In Lessons 1 and 2, students use physical objects to develop ratio language to describe a relationship between two quantities (6.RP.1). Students sort and categorize concrete objects such as different color binder clips and analyze a picture of snap cubes to write a sentence to describe the ratio shown in their diagram.
• In Lessons 3 and 4, students develop a conceptual understanding of equivalent ratios (6.RP.3). Lesson 3 extends the concept of ratios as described in the lesson introduction: “Students see that scaling a recipe up (or down) requires multiplying the amount of each ingredient by the same factor, e.g., doubling a recipe means doubling the amount of each ingredient (MP7). They also gain more experience using a discrete diagram as a tool to represent a situation.
• Lesson 6 introduces double number lines for students to use and interpret, alongside the more familiar discrete diagrams, in the context of recipes.
• Lesson 8 introduces students to the concept of unit price. They continue their work on ratios involving one unit “of something” in a real-world context (6.RP.2). For example, “Eight avocados cost $4. How much do 16 avocados cost? How much do 20 avocados cost? How much do 9 avocados cost?” Students also choose whether to draw double number lines or use other representations to support their reasoning.
• In Lesson 10, a short video shows a person walking at a constant speed on a treadmill for a few seconds. Students compare the length of time it takes two different people to run three miles and explain their reasoning.
• In Lessons 11 through 15, students continue to use recipes as they explore tables, tape diagrams, and double number lines to solve problems including equivalent ratio problems and part-part-whole ratios. 
• In Lesson 16, students use all the methods learned throughout Unit 2 “to solve ratio problems that involve the sum of the quantities in the ratio.”

Unit 4 develops conceptual understanding of division of fractions, 6.NS.A, using a variety of applets, for example:

• In Lesson 1 Activities, students use hands-on activities to explore the size of quotients, based on divisors and dividends. The second Activity, which is optional, includes an applet for students to model a variety of division problems and interpret the quotients. Students examine the divisor and dividend (they do not perform the operation) to order the fractions from least to greatest and group them as close to 0, close to 1, or much greater than 1. In the third Activity, students interpret division situations. In the Cool-Down, students determine proximity to 1, based on the given division problems in order to demonstrate an understanding of the concepts within the lesson.
• In Lessons 4 and 5, students manipulate pattern blocks to determine how many groups can be formed. Students use pattern blocks to find how many times a fraction goes into a number starting with whole numbers, then mixed numbers and fractions. The Lesson 4 Cool-Down states: “Answer the following questions. If you get stuck, use pattern blocks. a) How many 1/2 are in 3 1/2? b) How many 1/3 are in 2 2/3?” c) How many 1/6 are in 2/3?” Students examine division of fractions from a multiplication perspective and use a diagram to understand the connection between multiplication and division.

Unit 6 presents opportunities for students to develop their conceptual understanding of applying and extending previous understandings of arithmetic to algebraic expressions and developing reasoning to solve one-variable equations and inequalities, 6.EE, for example: 

• Lesson 1 introduces students to tape diagrams to represent equations with and without variables, and students match the equation with the related diagram then use the diagrams as needed to solve equations (6.EE.6).
• Lesson 3 introduces students to “hanger diagrams” (to represent balance scales), and students reason about concrete representations of equations. They identify what is true and/or false about the diagrams, as well as reason about how balanced hangers with two shapes are related when the shapes are not equally represented on each side, connecting the “hanger diagrams” to equations.
• In Lesson 8, students draw diagrams of two separate expressions to show that they are equivalent for given values (6.EE.4).
• In Lesson 10, the first Activity, students calculate the area of partitioned rectangles, as both a product of length and width and as the sum of the area of two smaller rectangles, and write expressions to represent both calculations. In comparing their expressions, students recognize equivalence through the distributive property. (6.EE.3, 4).

The instructional materials for Kendall Hunt’s Illustrative Mathematics 6-8 Math, Grade 6 meet expectations for attending to those standards that set an expectation of procedural skill and fluency. Materials attend to the Grade 6 expected fluencies, particularly fluency with multi-digit decimals and computing with them in expressions and equations.

Procedural skills and fluencies develop with conceptual understanding and are built upon work students have accomplished with operations and equations from prior grades. Students practice developed procedures throughout practice problem sets that follow the units, and students use emerging fluencies in the context of solving problems. According to the How to Use the Materials, Design Principles, “As the unit progresses, students are systematically introduced to representations, contexts, concepts, language, and notation. As their learning progresses, they make connections between different representations and strategies, consolidating their conceptual understanding, and see and understand more efficient methods of solving problems, supporting the shift towards procedural fluency. The distributed practice problems give students ongoing practice, which also supports developing procedural proficiency.” Number Talks included in many Warm-Ups often revisit fluencies developed in earlier grades and specifically relate to the Activities found in the lessons.

Unit 5 addresses 6.NS.2, 3, developing fluency in adding, subtracting, multiplying, and dividing, multi-digit decimals using the standard algorithm, and specific examples include:

• In Lesson 1, students review decimal work and utilize the four operations to solve problems in real-world contexts, such as money or planning a party (6.NS.3), using strategies such as mental math to estimate with decimals.
• In Lesson 3, students add and subtract decimals. Students encounter decimals beyond thousandths, find missing addends, and work with decimals in contexts. Students “evaluate mentally: 1.009 + 0.391.”
• In Lesson 5, Items 5 and 6, students solve problems using multiplication. 
• In Lesson 11, students use the standard algorithm in the second Activity (6.NS.2).
• In Lesson 12, practice problems explicitly state to use “long division.”

The instructional materials for Kendall Hunt’s Illustrative Mathematics 6-8 Math, Grade 6 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics. Engaging applications include single and multi-step problems, both routine and non-routine, presented in a context in which the mathematics is applied.

Applications occur throughout the materials and are used throughout the curriculum to build conceptual understanding. The Grade 6 Course Guide states: “Students have opportunities to make connections to real-world contexts throughout the materials. Frequently, carefully-chosen anchor contexts are used to motivate new mathematical concepts, and students have many opportunities to make connections between contexts and the concepts they are learning. In some cases, students spend more time developing mathematical concepts before tackling more complex application problems, and the focus is on the mathematical contexts. The first unit on geometry is an example of this.”

In Unit 2, students use ratio and rate reasoning to solve real-world and mathematical problems (6.RP.3).

• Lesson 14 optional activity directions read, “Lin read the first 54 pages from a 270-page book in the last three days. Diego read the first 100 pages from a 320-page book in the last four days. Elena read the first 160 pages from a 480-page book in the last five days. If they continue to read every day at these rates, who will finish first, second, and third? Explain or show your reasoning.”
• Students encounter non-routine word problems as they apply ratio and rate reasoning to problems with multiple solutions. In Lesson 15, students “Invent another ratio problem that can be solved with a tape diagram and solve it. If you get stuck, consider looking back at the problems you solved in the earlier activity. Create a visual display that includes: The new problem that you wrote, without the solution, and enough work space for someone to show a solution. Trade your display with another group and solve each other’s problem. Include a tape diagram as part of your solution. Be prepared to share the solution with the class. When the solution to the problem you invented is being shared by another group, check their answer for accuracy.” In Lesson 16, a multiple-solution problem from openmiddle.com is included: “Use the digits 1 through 9 to create three equivalent ratios. Use each digit only one time. ____ : ____ is equivalent to ____ : ____ and ____ : ____.”

In Unit 4, students solve word problems involving division of fractions by fractions (6.NS.1).

• In Lesson 3, students “analyze a division context and tell if it represents a ‘how many groups?’ question, or a ‘how many in each group?’ question.” Students use unit fractions, non-unit fractions with whole-number dividends, and mixed-number dividends with non-unit fraction divisors.
• In Lesson 11, the following division problem is included: “If 4/3 liters of water are enough to water 2/5 of the plants in the house, how much water is necessary to water all the plants in the house? Write a multiplication equation and a division equation for the situation, then answer the question. Show your reasoning.”
• The end of the unit includes opportunities to use division in word problems involving multiplicative comparison and problems involving length and area.

The instructional materials for Kendall Hunt’s Illustrative Mathematics 6-8 Math, Grade 6 meet expectations that the three aspects of rigor are not always treated together and are not always treated separately. The materials address aspects of rigor independently, and there are instances when multiple aspects of rigor are used to support student learning and mastery of the standards. There are multiple lessons where one aspect of rigor is emphasized.

Examples of conceptual understanding include:

• In Unit 1, concepts are built from the use of physical models and visual representations as students develop understanding of formulas. In Lessons 4 through 6, students work with parallelograms, and in Lessons 7 through 10 they work with triangles.
• In Unit 6, Lesson 1, conceptual understanding of the connections between multiplication and addition (6.EE.6) are reinforced. In the Warm-Up, visual models using tape diagrams are revisited. Students “draw a diagram that represents each equation, “4 + 3 = 7 and 4 ⋅ 3 = 12” in the first activity. Students then “use what they know about relationships between operations to identify multiple equations that match a given diagram.”

Examples of procedural skills and fluency include:

• In Unit 5, Lessons 11 through 13, students divide decimals (6.NS.2) using the standard algorithm. First, students mentally solve four division problems using structure and patterns in the Warm-Up. In the second Activity, students evaluate the division algorithm as performed by a given student by answering questions such as, “Lin subtracted 5 groups of 4 from 20. What value does the 4 in the quotient represent?” Fluency is further developed over additional practice problems found in Lessons 11 through 13.

Examples of application include:

• In Unit 1, Lesson 19, students interpret a tent design problem and create a tent design that meets certain specifications. They calculate surface area and estimate the amount of fabric they will need (6.G.4). In the second part of the lesson, students present and justify their design to a peer and reflect on similarities and differences in the different designs of their group.
• In Unit 2, Lesson 17, students solve Fermi problems (e.g., “How many times does your heart beat in a year?”), clarify and narrow a problem, and apply what they’ve learned about rates and ratios to estimate a solution (6.RP.3). Students also develop and create an estimated solution to their own Fermi problem.

Examples of lessons where two or three aspects of rigor are connected include:

• In Unit 3, Lesson 1 provides students with facts about the Burj Khalifa (world’s tallest building) and this information: “A window-washing crew can finish 15 windows in 18 minutes.” Students determine how long it would take the crew to wash all the windows of the Burj Khalifa. This task is designed to develop students' understanding of the unit rate in solving problems within this context. This lesson also extends students’ work in the last lesson of the previous unit (Unit 2 Lesson 17) with using rate and ratio reasoning to solve Fermi problems.
• Typically, there are six problems included in the practice problems. Procedural practice, visual representations, contexts, and/or standard methods of solving said problems are present. For example, in Unit 6, Lesson 1 Practice Problems, students solve several problems involving tape diagrams to develop conceptual understanding and procedural skill with one-step equations. The last few problems anticipate Unit 3 material and require students to apply previous knowledge.

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics 6-8 Math, Grade 6 meet expectations that the Standards for Mathematical Practice are identified and used to enrich mathematics content within and throughout the grade level. All eight MPs are clearly identified throughout the materials. The MPs are initially identified in the narrative for each unit described within the course information, for example:

• In Unit 1, the Unit Narrative states, “Students learn strategies for finding areas of parallelograms and triangles and use regularity in repeated reasoning (MP8) to develop formulas for these areas, using geometric properties to justify the correctness of these formulas.”
• In Unit 4, an excerpt from the Unit Narrative states, “The second section of the unit focuses on equal groups and comparison situations. It begins with partitive and quotitive situations that involve whole numbers, represented by tape diagrams and equations. Students interpret the numbers in the two situations (MP2) and consider analogous situations that involve one or more fractions, again accompanied by tape diagrams and equations.”

The MPs are identified in the Lesson under the Preparation tab. Lesson narratives often highlight when an MP is particularly important for a concept or when a task may exemplify the identified practice, for example:

• In Unit 2, Lesson 1, the Lesson Narrative introduces ratios and ratio language, “Expressing associations of quantities in a context - as students will be doing in this lesson - requires students to use ratio language with care (MP6).”
• In Unit 2, Lesson 8, the Warm-Up narrative states, “Students choose whether to draw double number lines or other representations to support their reasoning. They continue to use precision in stating the units that go with the numbers in a ratio in both verbal statements and diagrams (MP6)."
• In Unit 6, Lesson 13, the narrative for the first Activity states, “The purpose of this task is to give students experience working with exponential expressions and to promote making use of structure (MP7) to compare exponential expressions. To this end, encourage students to rewrite expressions in a different form rather than evaluate them to a single number.”

The MPs are used to enrich the mathematical content and are not treated separately from the content in stand-alone lessons. MPs are highlighted and discussed throughout the lesson narratives to support a teacher’s understanding of the MP itself as the teacher is provided direction regarding how the content is connected to the MP, for example:

• In Unit 2, Lesson 2 Narrative, an explanation is provided for ratio language and its connection to MP6, “Students used physical objects to learn about ratios in the previous lesson. Here they use diagrams to represent situations involving ratios and continue to develop ratio language. The use of diagrams to represent ratios involves some care so that students can make strategic choices about the tools they use to solve problems. Both the visual and verbal descriptions of ratios demand careful interpretation and use of language (MP6).”
• In Unit 7, Lesson 1, the first Activity, understanding of positive and negative integers is enriched as “students reason abstractly and quantitatively when they represent the change in temperature on a number line (MP2).”

The MPs are not identified in the student materials; however, there are questions posed with activities that engage students with MPs. For example, in Unit 7, Lesson 1, the first Activity poses the following question in relation to MP2: “Do numbers below 0 make sense outside of the context of temperature? If you think so, give some examples to show how they make sense. If you don’t think so, give some examples to show otherwise.”

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics 6-8 Math, Grade 6 meet expectations for carefully attending to the full meaning of each practice standard. The materials attend to the full meaning of each of the 8 MPs. The MPs are discussed in both the unit and lesson narratives, as appropriate, when they relate to the overall work. They are also explained within individual activities, when necessary. Over the course of the year, students have ample opportunity to engage with the full meaning of every MP. Examples include:

MP1 - Make sense of problems and persevere in solving them.

• In Unit 2, Lesson 1, the first part of MP1 is captured as students make sense of ratios. Students sort shapes and various objects into categories with similar characteristics and then use their like traits to establish ratio relationships.
• In Unit 9, Lesson 2, the first Activity, students are given the following problem, “There are 7.4 billion people in the world. If the whole world were represented by a 30-person class: 14 people would eat rice as their main food, 12 people would be under the age of 20, 5 people would be from Africa. 1. How many people in the class would not eat rice as their main food? 2. What percentage of the people in the class would be under the age of 20? 3. Based on the number of people in the class representing people from Africa, how many people live in Africa?” In solving this problem, students have to look for entry points to the solution; analyze given information, constraints, relationships, and goals; and finally, make conjectures about the form and meaning of the solution and plan a solution pathway.

MP2 - Reason abstractly and quantitatively.

• In Unit 5, Lesson 1, students think about solving problems in the context of money. For example, “Clare went to a concession stand that sells pretzels for $3.25, drinks for $1.85, and bags of popcorn for $0.99 each. She bought at least one of each item and spent no more than $10. Could Clare have purchased 2 pretzels, 2 drinks, and 2 bags of popcorn? Explain your reasoning.”

MP4 - Model with mathematics.

• Throughout the Activities in Unit 7, Lesson 1, students model with mathematics using number lines to represent thermometers and scenarios involving weather. The second Activity introduction states, “The purpose of this task is to present a second, natural context for negative numbers and to start comparing positive and negative numbers in preparation for ordering them.” Students again model a context using vertical number lines, but this time it is with elevation using a digital applet.
• In Unit 9, Lesson 1, students answer, “How long would it take an ant to run from New York City to Los Angeles?” The Fermi problem requires students to make a rough estimate for quantities that are difficult or impossible to measure directly. Often, they use rates and require several calculations with fractions and decimals, making them well-aligned to Grade 6 work. Fermi problems are examples of mathematical modeling because one must make simplifying assumptions, estimates, research, and decisions about which quantities are important and what mathematics to use.

MP5 - Use appropriate tools strategically.

• Each lesson in Unit 1 lists a geometry toolkit containing tracing paper, graph paper, colored pencils, scissors, and an index card to use as a straightedge or to mark right angles as Required Materials. For Unit 1, the narrative explains, “Providing students with these toolkits gives opportunities for students to develop abilities to select appropriate tools and use them strategically to solve problems.” In addition, many lessons of Unit 1 include activities in which students use digital applets which allow for making simulations and exploring compositions and decompositions of figures. The unit narrative also explains, “Apps and simulations should be considered additions to their toolkits, not replacements for physical tools.”

MP7 - Look for and make use of structure.

• In Unit 1, Lesson 7, the second Activity, students are given several quadrilaterals and directed to draw a line that would decompose them into two identical triangles. In order to make generalizations about quadrilaterals that can be decomposed into identical triangles, students first need to analyze the features of the given shapes and look for structure.
• In Unit 5, Lesson 4, students notice and use structure in the second Activity. The Lesson Synthesis states, “In this lesson, students practiced adding and subtracting numbers with many decimal places, both in and outside of the context of situations. They noticed the benefits of vertical calculations and used its structure to solve problems.”

MP8 - Look for and express regularity in repeated reasoning.

• In Unit 1, Lesson 18, the second Activity, students are told that a cube has an edge length of s. These prompts follow: “1) Draw a net for the cube. 2) Write an expression for the area of each face. Label each face with its area. 3) Write an expression for the surface area. 4) Write an expression for the volume.” In doing this, students express regularity in repeated reasoning to write the formula for the surface area of a cube.
• In Unit 5, Lesson 8, Optional activity, students solve problems with decimals and look at patterns in solving problems with decimals. First, students “write the following expressions as decimals (1−0.1, 1−0.1+10−0.01, 1−0.1+10−0.01+100−0.001). Describe the decimal that results as this process continues. What would happen to the decimal if all of the positive and negative signs became multiplication symbols? Explain your reasoning.”

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics 6-8 Math, Grade 6 meet expectations for prompting students to construct viable arguments and/or analyze the arguments of others concerning key grade-level mathematics. The student materials consistently prompt students to both construct viable arguments and analyze the arguments of others. Students explain their reasoning and compare their strategies for solving in small group and whole class settings, and examples include:

• In Unit 3, Lesson 6, the second Activity, students explore two unit rates related to a given ratio. They decide which unit rate is correct and extend the unit rate into a related problem. In this scenario, both unit rates are correct, and students could use either unit rate to solve the related problems.
• In Unit 7, Lesson 6, Lesson synthesis, students answer, “What do you notice about the order of numbers after taking absolute value? Explain why this happens.” Questions such as these are present throughout the lessons, providing students the opportunity to construct viable arguments in both verbal and written form.
• In Unit 6, Lesson 16, Warm-Up, students find the unit price to determine which price option is a better deal. Students engage in constructing arguments and critiquing the reasoning of their classmates. Students are asked: “Which one would you choose? Be prepared to explain your reasoning. A 5-pound jug of honey for $15.35 [or] three 1.5-pound jars of honey for $13.05?”
• In Unit 7, Lesson 1, the Cool-Down includes the following prompts with which students must agree or disagree and explain their reasoning: “A temperature of 35 degrees Fahrenheit is as cold as a temperature of -35 degrees Fahrenheit. A city that has an elevation of 15 meters is closer to sea level than a city that has an elevation of -10 meters. A city that has an elevation of -17 meters is closer to sea level than a city that has an elevation of -40 meters.”

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics 6-8 Math, 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. The teacher materials assist teachers in engaging students in both constructing viable arguments and analyzing the arguments of others throughout the program.

• In Unit 1, Lesson 9, Warm-Up students study examples and non-examples of bases and heights in a triangle. They select all the statements that are true about bases and heights in a triangle. The teacher is given the following direction: “As students discuss with their partners, listen for how they justify their decisions or how they know which statements are true.”
• The Unit 2, Lesson 4 Warm-Up provides guiding questions in the Activity Synthesis to engage students in MP3, such as: “Who can restate ___’s reasoning in a different way? Did anyone solve the problem the same way but would explain it differently? Did anyone solve the problem in a different way? Does anyone want to add on to ____’s strategy? Do you agree or disagree? Why?” This strategy is used repeatedly throughout the series.
• In Unit 4, Lesson 5, the first Activity provides guidance for the teacher as they observe student groups using pattern blocks to solve a task: “As students discuss in groups, listen for their explanations for the question ‘How many rhombuses are in a trapezoid?’ Select a couple of students to share later - one person to elaborate on Diego's argument, and another to support Jada's argument.”
• The Unit 7, Lesson 1 Warm-Up states: “The purpose of this task is to introduce students to temperatures measured in degrees Celsius.” This prompt assists teachers in engaging students in constructing viable arguments, precisely the types of questions teachers can ask to aid in the discussion and includes possible student responses.

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics 6-8 Math, Grade 6 meet expectations for attending to the specialized language of mathematics. The materials provide explicit instruction on communicating mathematical thinking using words, diagrams, and symbols. The materials use precise and accurate terminology and definitions when describing mathematics and support students in using them.

• In the teacher materials, the Grade 6 Glossary is located in the Course Guide. Lesson-specific vocabulary can be found in bold within the lesson, and is listed and defined at the end of the student lesson. In the student materials, the entire Glossary is accessible by a tab on the student home page. 
• Both the unit and lesson narratives contain specific guidance for the teacher on methods to support students to communicate mathematically. Within the lesson narratives, new terms are in bold print and explained as related to the context of the material.
• In Unit 2, Lesson 1 introduces ratios and ratio language. Within the Warm-Up and the first Activity, students categorize items and verbally compare the sorted groups. The definition of ratio is developed and applied to the sorted groups using correct language. For example, “The ratio of purple to orange dinosaurs is 4 to 2.” or “There are 4 purple dinosaurs for every 2 orange dinosaurs.” Within the second Activity, students write ratio sentences comparing two categories. The Lesson Synthesis provides further practice and discussion questions for the teacher on the concept of a ratio. “Consider posing some more general questions, such as: 'What things must you pay attention to when writing a ratio? What are some words and phrases that are used to write a ratio?'”
• In Unit 7, students interpret signed numbers in contexts (e.g., temperature above or below zero, elevation above or below sea level). Students use the context to build proper mathematical vocabulary. In Lesson 1, students explore the idea of a temperature that is less than zero. This activity is used to introduce the term negative as a way to represent a quantity less than zero.

No examples of incorrect use of vocabulary, symbols, or numbers were found within the materials.