7th Grade - Gateway 2

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

In Unit 6, students develop conceptual understanding of equivalence through the manipulation of expressions, using the properties of operations in order to identify and generate equivalent expressions (7.EE.A).

• In Lessons 2 through 5, students develop an understanding of equations as they manipulate equal parts, different parts, and total parts. Students also label terms in a model to represent a given context, write their own contexts, and develop the corresponding equation.
• In Lesson 7, students use “hanger diagrams” to model and maintain equivalency in equations.
• In Lessons 20 through 22, students use multiple strategies including expanded form with variables, partitioned rectangle areas, and properties of operations to simplify expressions and identify equivalent expressions (7.EE.1). In Lesson 20 Activity 2, students “Replace each ? with an expression that will make the left side of the equation equivalent to the right side: “Set A: 6x + ? = 10x, 6x + ? = 2x, 6x + ? = -10x, 6x + ? = 0, 6x + ? = 10.” Students respond to questions: “Why didn't you combine x terms and numbers?” “How did you decide on the components of the missing term?” “Did you use the commutative property?” “Did you use the distributive property?” “What are some ways we can tell that 7x+2 is not equivalent to 9x?” “Someone is doubtful that 3b−8b is equivalent to -5b, but they do understand the distributive property. How could you convince them that these expressions are equivalent?” “What are some ways we could rearrange the terms in the expression -2x + 6y − 6x + 15y and create an equivalent expression?” In this activity, students develop and articulate their understanding of equivalent expressions.

Unit 5 addresses applying and extending previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers, 7.NS.A:

• In Lesson 2, students add and subtract rational numbers on the number line (7.NS.1). In the first Activity, students draw number lines, write equations, and verbally explain number line movement. A thermometer applet, which students can manipulate to represent the given situation, is included in the second Activity. Arrows are applied within the applet, and the thermometer context serves as a vertical number line representation on which students add integers.
• In Lesson 3, conceptual understanding is extended as students apply the information from given situations to visually model elevation changes (7.NS.1).
• The Lesson 4 Introduction explains that students will use money as a way “to practice performing operations on signed values, but the emphasis is really on noticing that money can be represented with positive and negative values.” The narrative further describes conceptual understanding with: “Any situation in which we use a negative number to represent a debt (for example), we could equally well just use a positive number and distinguish it by calling it a debt. The reason we use signed numbers in this context is that it allows us to represent a whole class of problems with the same expression. For example, if a person has $50 in the bank and writes a $20 check, we can represent the balance as 50−20. If they had written an $80 check, we can still write the balance as 50−80, as long as we have adopted the convention that negative numbers represent what the person owes the bank (and assuming the bank allows overdrafts).”
• Lessons 8 through 12 address multiplication and division (7.NS.2). In Lesson 8, students explore multiplication of a negative and a positive value in terms of speed in a certain direction over a number of seconds. Students use a number line and a table to establish a pattern in 8.2. For example, the second row in the table states: “starting at zero” “left” “4 units per second” “6 seconds.” Students use a number line to come up with the equation. In Lesson 9, students determine the pattern for multiplying two negative values (7.NS.2).

The instructional materials for Kendall Hunt’s Illustrative Mathematics 6-8 Math, Grade 7 meet expectations for attending to those standards that set an expectation of procedural skill and fluency. Materials attend to the Grade 7 expected procedural skills, particularly those related to rational numbers and solving 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 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. Additionally, students demonstrate procedural skills throughout the year in a variety of practice problems. Examples of practice problems include:

• In Unit 6, Lesson 10, Practice Problem 5, students solve equations such as “7(x + 2) = 91” (7.EE.4a).
• In Unit 5, students perform computations with rational numbers in Lessons 2 through 12 and rational numbers within expressions and equations in Lessons 13 through 17 (7.NS.A):
• Lesson 5 introduces subtracting integers with number lines, the coordinate plane, tables, and the relationship between addition and subtraction of integers, and students solve expressions with rational numbers.
• In Lesson 7, students apply their knowledge of operations with integers to real-world problems. Multiplying integers is introduced in Lesson 8 with additional opportunities to build procedural skill in Lessons 9 through 11.
• In Lesson 11, Items 1-4 provide practice with multiplication and division with rational numbers.
• The Lesson 13 Warm-Up includes true/false reasoning with signed numbers. The first Activity, a Card Sort, states: “In this activity students continue to build fluency operating with signed numbers as they match different expressions that have the same value.” The cards include both integers and fractions.
• In Unit 6, procedural fluency with grade-level operations are revisited in the Lesson 18 and 19 Warm-Ups in preparation for working with equations including rational numbers. In Lessons 18 through 22, students work with rational numbers and negative terms (7.EE.1) in ways that continue to build fluency when computing with signed numbers.

In Unit 6, students use properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients (7.EE.1).

• In Lesson 19, the first activity, students factor and expand expressions with signed numbers in the given table. The following Cool-Down Activity includes two more opportunities to expand and factor equations with rational values.
• In Lesson 20, the first Activity, students use properties of operations to justify equivalence of expressions. In the second Activity , students fill in missing terms to balance equations and are provided questions such as, “How did you decide on the missing components of the term?” to develop the procedures of combining like terms.

The instructional materials for Kendall Hunt’s Illustrative Mathematics 6-8 Math, Grade 7 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 7 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. Additionally, most units include a real-world application lesson at the end. In some cases, students spend more time developing mathematical concepts before tackling more complex application problems, and the focus is on mathematical contexts.”

In several Units, students work with speed and water filling/draining contexts in increasingly complex ways and also use proportional reasoning skills (7.RP.A) in more complex and non-routine ways alongside work with surface area/volume and later with probability. Examples include:

• In Unit 2, Lesson 5, students are given a rate of water filling an object, find multiple equations that model the situation, and identify and interpret the rates in context. The lesson introduction states: “Students are expected to use methods developed earlier: organize data in a table, write and solve an equation to determine the constant of proportionality, and generalize an equation from repeated calculations.”
• In Unit 2, Lesson 6, students determine the constant of proportionality in the context of concert ticket sales, recipes, and recycling. In Lesson 7, students make adjustments to quantities used in recipes, compare costs for various group sizes to enter a state park, and determine if several runners are moving at a constant pace. In the last context, students reason about making determinations about proportionality in a discrete context (7.RP.2a) and their inability to establish proportionality given a continuous relationship when considering the limitations of intervals in the table.
• In Unit 4, Lesson 10, students apply percentage to taxes and tips. The first Activity provides students with two tables giving the sales tax charges on the same items in two different cities. Students complete the table, find the tax rate in each city, and compare them. The second Activity adds another city to the problem, and students realize through their calculations that the tax amount must be rounded.
• In Unit 4, Lesson 16, students sort actual newspaper clippings, decide if they are percents of increase or decrease, choose a clipping that is interesting to them, and “create a visual display that includes: a title that describes the situation, the news clipping, your diagram of the situation, the two questions you asked about the situation, the answers to each of your questions, [and] an explanation of how you calculated each answer.”
• In Unit 5, Lesson 8 Warm-Up, three problem scenarios are given in which students apply understanding of proportional relationships in a constant speed context. Students first find the total distance a plane travels; then the speed of a train; and finally the time a car traveled. This work prepares students for thinking about contexts involving negative numbers in context of velocity, time, and position.
• In Unit 7, Lesson 16, students apply understanding of proportional relationships to find surface area and volume and the total cost of constructing a ramp in a non-routine problem context. The second Activity presents another non-routine problem involving surface area and volume using proportional relationships: “The daycare has two sandboxes that are both prisms with regular hexagons as their bases. The smaller sandbox has a base area of 1,146 square inches and is filled 10 inches deep with sand.” The following prompts accompany the given real-world scenario: “It took 14 bags of sand to fill the small sandbox to this depth. What volume of sand comes in one bag? (Round to the nearest whole cubic inch.); The daycare manager wants to add three more inches to the depth of the sand in the small sandbox. How many bags of sand will they need to buy?; The daycare manager also wants to add three more inches to the depth of the sand in the large sandbox. The base of the large sandbox is a scaled copy of the base of the small sandbox, with a scale factor of 1.5. How many bags of sand will they need to buy for the large sandbox?; A lawn and garden store is selling six bags of sand for $19.50. How much will they spend to buy all the new sand for both sandboxes?”

In Unit 6, students solve multi-step real-life and mathematical problems with rational numbers in various forms (7.EE.3). Examples include:

• In Lesson 2, students encounter many routine problems that require algebraic thinking to solve: “Noah’s family bought some fruit bars to put in the gift bags. They bought one box each of four flavors: apple, strawberry, blueberry, and peach. The boxes all had the same number of bars. Noah wanted to taste the flavors and ate one bar from each box. There were 28 bars left for the gift bags.” Students use tape diagrams to develop equations in the forms p(x+q)=r and px+q=r.
• In Lesson 12, the first Activity, students use the given tape diagram and sample student responses to make connections between the context such as change in temperature over three days and equations that model percent increase. In the second Activity, students solve four multi-step real-world problems. In the Cool-Down, students write and solve an equation for this problem: “The track team is trying to reduce their time for a relay race. First, they reduce their time by 2.1 minutes. Then, they are able to reduce that time by 1/10. If their final time is 3.96 minutes, what was their beginning time? Show or explain your reasoning.”

The instructional materials for Kendall Hunt’s Illustrative Mathematics 6-8 Math, Grade 7 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 6, Lessons 7 and 8 intentionally isolate the conceptual understanding related to solving equations in the form of px + q = r and p(x+q) = r by using hanger diagram models to visualize balance between the sides of an equality (7.EE.4a).
• In Unit 6, Lesson 18, Activity 1, students use area models to develop conceptual understanding of how the distributive property works with both positive and negative terms. In the Lesson Synthesis, students develop a conceptual understanding on how the commutative property works with addition but not subtraction.

Examples of procedural skills and fluency include:

• In Unit 6, Lesson 10, students develop procedural skills in solving equations, 7.EE.4, by sharing their solution strategies. The Warm-Up involves an Algebra Talk where students have 30 seconds to think about methods of solving and solutions to five different problems in p(x+q) = r form. In the first Activity, students perform an error analysis of given students’ methods. In the second Activity, students solve five problems using two different methods and compare them. This task also prompts students to evaluate more inefficient methods and explain why they are not effective.
• In Unit 6, Lesson 22 Overview states that “students have an opportunity to demonstrate fluency in combining like terms and look for and make use of structure (MP7) to apply the distributive property in more sophisticated ways.” In the first Activity, students match equivalent expressions which include negative terms with a partner.

Examples of application include:

• In Unit 2, Lesson 15, students answer whether baths or showers use more water (7.RP.2). Students determine what information they would need to solve the problem, gather the data/information they need, and make assumptions using what they find.
• In Unit 5, Lesson 17, students apply their understanding of percent increase and decrease and work with rational numbers to calculate changes in stock value and total value of investment portfolios in non-routine stock market situations (7.RP.3).

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

• In Unit 2, Lesson 14, students apply conceptual understanding of proportional relationships. Students use lists of items (creatures, length units, time units, volume units, body parts, area units, etc.) to create a situation that shows a proportional relationship between two different things in the list, describe a situation where two things are not proportional, and create a table, graph, and equation that represent the proportional/non-proportional relationship.
• In Unit 5, Lessons 1 and 2 build upon the conceptual development of negative numbers in Grade 6, including placing them on the number line, comparing and ordering them, and interpreting them in the contexts of temperature and elevation. Procedural skill is integrated with conceptual understanding and application when students draw diagrams to represent temperature changes, write equations to represent the context, and solve by adding and/or subtracting integers. In Lesson 3, students represent quantities on number lines, identify opposites, and add integers. In Lesson 4, students apply their knowledge of integers to real-world scenarios with banking. Students “understand that when representing a debt with a negative number, the additive inverse tells how much money is needed to pay off the debt.”

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics 6-8 Math, 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 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 4, the Unit Narrative states, “In small groups, students identify important quantities in a situation described in a news item, use diagrams to map the relationship of of the quantities, and reason mathematically to draw conclusions (MP4).”
• In Unit 7, an excerpt from the Unit Narrative states, “[Students] understand and use the formula for the volume of a right rectangular prism, and solve problems involving area, surface area, and volume (MP1, MP4). Students should have access to their geometry toolkits so that they have an opportunity to select and use appropriate tools strategically (MP5).”

Within a lesson, the MPs are identified within the Lesson Narratives under the Preparation tab or before each of the activities. 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 4, the Lesson Narrative states, “In this lesson, students build on their work with tables and represent proportional relationships using equations of the form y = kx. The activities revisit contexts from the previous two lessons, presenting values in tables and focusing on the idea that for each table, there is a number k, so that all values in the table satisfy the equation y = kx. By expressing the regularity of repeated calculations of values in the table with the equations, students are engaging in MP8.”
• In Unit 3, Lesson 4, the Lesson Narrative connects the application of the circumference formula to two math practices, “Students think strategically about how to decompose and recompose complex shapes (MP7) and need to choose an appropriate level of precision for π and for their final calculations (MP6).”
• In Unit 8, Lesson 5, the narrative for the first Activity states, “Students have a chance to construct arguments (MP3) about why probability estimates based on carrying out the experiment many times might differ from the expected probability.”

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 5, Lesson 11, the narrative for the first Activity states, “The purpose of this activity is to understand that the division facts for rational numbers are simply a consequence of the multiplication done previously. Students work several numerical examples relating multiplication to division and then articulate a rule for the sign of a quotient based on the signs of the dividend and divisor (MP8).”
• In Unit 2, Lesson 7, the first Activity, students use a table to explore the cost of parking and admission per person. “These diagrams may be helpful in illustrating to them that their resulting prices are including more than one vehicle. This gives them an opportunity to make sense of problems and persevere in solving them (MP1).”

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 2, Lesson 7, the first Activity, the prompt includes, “How might you determine the entrance cost for a bus with 50 people? Is the relationship between the number of people and the total entrance cost a proportional relationship? Explain how you know.” (MP1)

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics 6-8 Math, Grade 7 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 2, students build on their understanding of scale and try to find the scale factor. The teaching narrative for second activity states, “Students grapple with finding missing values for ratios of whole numbers presented in a table where identifying a usable scale factor.” Students must make sense of new terms and apply them to their learning specifically the constant of proportionality and proportional relationship.

MP2 - Reason abstractly and quantitatively.

• In Unit 2, Lesson 3, the narrative for the second Activity states: “Students need opportunities to make the connection to proportional relationships; students who successfully make this connection are reasoning abstractly about contexts with constant speed.” Students make connections between constant speed and proportional relationships, with special attention to the constant of proportionality. 
• In Unit 4, Lesson 2, students are given the following optional problem: “In real life, the Mona Lisa measures 2 1/2 feet by 1 3/4 feet. A company that makes office supplies wants to print a scaled copy of the Mona Lisa on the cover of a notebook that measures 11 inches by 9 inches. Students are asked, “1. What size should they use for the scaled copy of the Mona Lisa on the notebook cover? 2. What is the scale factor from the real painting to its copy on the notebook cover? 3. Discuss your thinking with your partner. Did you use the same scale factor? If not, is one more reasonable than the other?” An applet is provided for students to experiment and understand the problem by changing the size of the Mona Lisa (abstract). The applet will display the new dimensions of the picture, but the information given for the picture cannot be scaled to the exact size of the notebook. Further investigation with equivalent ratios will lead students to one of the many possible dimensions and scale factors appropriate for the picture (quantitative).

MP4 - Model with mathematics.

• In Unit 1, Lesson 9, students apply the mathematics they know to solve problems arising in everyday life. They create their own scale drawing of a floor plan, two different scale drawings of the state of Utah, noticing how the scale impacts the drawing, and make a scale drawing of a swimming pool.
• In Unit 4, Lesson 16, students work in groups to collect news clippings that mention percentages and sort them according to whether they are about percent increase or percent decrease, formulate questions about them, and then share their questions with other groups in a gallery walk. The purpose is for students to apply percentages in a real-world context.

MP5 - Use appropriate tools strategically.

• In Unit 1, Lesson 3, the optional Activity, students use a digital platform to create scaled copies of an original figure by increasing or shrinking the dimensions with the use of a fractional scale factor. In the Activity, “Students continue to work with scaled copies of simple geometric figures, this time on a grid. When trying to scale non-horizontal and non-vertical segments, students may think of using tracing paper or a ruler to measure lengths and a protractor to measure angles.”
• In Units 1 and 3, the lesson preparation suggests that each student have access to a geometry toolkit. These contain tracing paper, graph paper, colored pencils, scissors, centimeter ruler, protractor, and an index card to use as a straightedge or to mark right angles. “Providing students with these toolkits gives opportunities for students to select appropriate tools and use them strategically to solve problems.”

MP7 - Look for and make use of structure.

• In Unit 1, Lesson 2, Warm-Up Number Talk, students review multiplication strategies and develop the idea that multiplying by a unit fraction is the same as dividing by its whole number reciprocal. Students find 7.2 x 1/9 mentally. Students are guided to think about 72 * 1/9 (72 $$\div$$ 9) and then consider what happens to the decimal. Students are encouraged to use the structure of base ten numbers and the properties of operations.
• In Unit 4, Lesson 4, students work with the distributive property. It is noted in the lesson that when students look for opportunities to use the distributive property to write equations in a simpler way, they are looking for and making use of structure.

MP8 - Look for and express regularity in repeated reasoning.

• In Unit 1, Lesson 2, students continue their work with corresponding parts to develop the understanding and use of scale factors. The narrative for the second Activity states, “Students see that there is a single factor that relates each length in the original triangle to its corresponding length in a copy (MP8). They learn that this number is called a scale factor.”

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics 6-8 Math, Grade 7 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 5, Lesson 3, the third Activity, which is optional, students work with a partner to analyze a number line using variable expressions in order to compare two expressions with an inequality or equal sign. They also give an explanation to support their answer. For example, “-a ___ -b” based on their given locations on the number line.
• In Unit 5, Lesson 5, the first Activity, students find the length of the missing arrow on a number line and analyze two equations (both are correct) written to represent the situation. Students write equations in the same form as the original two and explain their solutions.
• In Unit 4, Lesson 16, students work in groups to collect news clippings that mention percentages and sort them according to whether they reflect percent increase or percent decrease. They formulate questions about them and share their questions with other groups in a gallery walk. During the gallery walk, students use sticky notes to ask questions about the information presented on each poster.

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

• The Unit 2, Lesson 1 Warm-Up provides teacher guidance as students complete a double number line and invent a situation. Teachers are instructed, “As students discuss their answers with their partner, select students to share their answers during the whole-class discussion...Invite selected students to explain how they reasoned about possible labels for each of the number lines and the units of each. After each student shares, invite others to agree, disagree, or question the reasonableness of the number line descriptions.”
• The Unit 5, Lesson 6 Warm-Up Activity Synthesis provides teachers with questions to encourage constructing viable arguments. This strategy is used repeatedly throughout the series. “To involve more students in the conversation, use some of the following questions: Who can restate ___’s reasoning in a different way? Did anyone find the value of n the same way, but would explain it differently? Did anyone find the value of n in a different way? Does anyone want to add on to ____’s strategy? Do you agree or disagree? Why?”
• In Unit 2, Lesson 4 Warm-Up states, “This Number Talk encourages students to think about the numbers in division problems and how they can use the result of one division problem to find the answer to a similar problem with a different, but related, divisor...Each problem is chosen to elicit a slightly different reasoning, so, as students explain their strategies, ask how the factors impacted their product.”

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics 6-8 Math, Grade 7 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 7 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 4, Lesson 11 introduces students to a variety of percentage contexts: markups, markdowns, tax, tip, and commission. In the Optional Activity provided in this lesson, students are paired and given the task of completing a “card sort.” Taking turns, the students match the term with the percentage scenario and explain their reasoning for the match. If the other student disagrees, they must explain why, and the pair must work to an agreement. The focus is on explaining the definition of these terms and relating them to specific scenarios.
• Unit 3 builds on students' understanding of a circle. In Lesson 2, the formal definition of a circle (the set of points that are equally distant from the center, enclosing a circular region) is developed. Also in this lesson, students develop the idea that the size of a circle can be measured by its diameter, radius, circumference, or the enclosed area, depending on the context.

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