7th Grade - Gateway 2

The instructional materials reviewed for The Utah Middle School Math Project Grade 7 meet the expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings.

Each chapter starts with an Anchor problem which poses a mathematical situation that students will learn to solve. Many of these are conceptual in nature. For example, Chapter 3 Anchor Problem 3.0 engages students in the understanding that there are different ways to write equivalent expressions and that the different ways shed light on ways of thinking about the problem. The Teacher’s Notes for that problem emphasize developing understanding.

Many Class Activity problems involve hands-on activities or models. In Chapter 3, students use the properties of operations to generate equivalent expressions. This chapter gives students practice with algebra tiles to build a conceptual understanding of equivalent expressions. In Chapter 3 Class Activity 3.1d, students learn how to use algebra tiles to build a representation of factoring. Later in Class Activity 3.1h, students are shown two ways to factor. Method 1 encourages the use of a model, and in Method 2 students use the greatest common factor.

The teacher notes for each lesson describe the purpose of the lesson and how to guide students to develop their conceptual understanding. The notes include prompts and questions during instruction that lead to conceptual understanding.

Chapters 1 and 2 address 7.NS.A.

• The Chapter 1 Section 1.2 Overview states: “The concept of equivalent fractions naturally leads students to the issues of ordering and estimation. Students will represent order of fractions on the real number line.” Students understand where rational numbers are placed on a number line and use models to solve multi-step problems.
• The Chapter 2 Section 2.1 Overview summarizes the use of hands-on manipulatives and number lines so that students can eventually “reason through addition and subtraction of integers without a model.” Students develop a conceptual understanding of negative numbers and additive inverse by adding integers on a number line, using chips to model addition problems, and using the number line to model subtraction problems.

Chapters 3 and 6 address 7.EE.A.

• In Chapter 3 Class Activity 3.1a and Homework, students determine if two expressions are equivalent and justify their conclusions, consolidating their understanding of the properties of operations.
• In Chapter 3 Class Activity 3.1c and Homework, students use algebra tiles to rewrite algebraic expressions.
• Students demonstrate conceptual understanding to solve Chapter 6 Class Activity 6.2c Problem 1: “Matt, Rosa, and Kathy are cousins. If you combine their ages, they would be 40 yrs. old. Matt is one-third of Rosa's age. Kathy is five years older than Rosa. How old are they? Show several ways to solve the problem. Be able to explain how you came to your answer.”

The instructional materials reviewed for The Utah Middle School Math Project Grade 7 meet the expectations for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency. Overall, when the intention is that procedural skill and fluency be developed, the materials offer opportunities for their development.

There are examples and repetition in practice in each lesson and homework. Spiral Reviews are found in each chapter that set an expectation of procedural skill and fluency. For example, the Chapter 2 Spiral Review (page 7WB3 - 24) addresses a number of computational standards from previous grades as well as 7.NS.A. Question 5 asks students to solve 5 × (−9).

The following standards are addressed within the course:

• 7.NS.A: Section 2.1 and 2.3 give students practice adding and subtracting rational numbers. Students begin to describe situations in which opposite quantities combine to make 0, and as teachers introduce the properties of arithmetic, students use these properties to add and subtract fluently. Students practice multiplying and dividing rational numbers. Extra Practice sections are also provided.
• 7.EE.1: In Chapter 3, students begin the concept of generating equivalent expressions through the use of concrete models. Students use the commutative property as well as the distributive property to generate an equivalent expression. Students continue procedural practice with solving equations. By the end of Chapter 3, students are expected to be fluent with the properties of operations.
• 7.EE.4: Students move from translating contexts to numeric expressions in Chapter 3 Class Activity 3.1b and Homework to translating contexts to algebraic expressions in Class Activity 3.1c Homework and Additional Practice. In Section 3.2 students build procedural skill and fluency by modeling two-step equations and by using their knowledge of properties. Class Activities 3.2a-c give students practice using models (algebra tiles) to solve two-step equations with and without rational numbers. Class Activities 3.2d and 3.2.e provide opportunities to continue working on this skill to gain fluency.

The instructional materials reviewed for The Utah Middle School Math Project Grade 7 meet the expectation that teachers and students spend sufficient time working with engaging applications of mathematics, without losing focus on the major work of the grade. Overall, the materials have opportunities for students to apply mathematical knowledge and/or skills in a real world context.

Throughout the materials students engage in application problems in Class Activity and Anchor Problems. These problems are contextual, and some include multiple representations and steps. Students are asked to present their solutions in ways that demonstrate their understanding of the mathematics in the context.

• Chapter 3 Class Activity 3.3a (7WB3 - 119) includes contextual problems. For example, “Today is Rosa’s 12th birthday. She has a savings account with $515 in it, but her goal is to save $10,000 by the time she turns 18. How much money should she add to her savings account each month to reach her goal of $10,000 between now and her 18th birthday?” Students draw a model, write an equation that represents the model, solve the equation, and answer the question in a full sentence.
• Chapter 4’s Anchor Problem, “Tasting Lemonade,” is a multi-step, real-world, contextual problem that develops analysis of proportional relationships (7.RP.A). It emphasizes solving the problem using a variety of strategies. Students are presented with the context that “you want to sell lemonade in a park” and have five different recipes to choose from, consisting of different concentrate and water ratios. The following problems develop students’ understanding on how the different ratios would affect the flavor of the lemonade, and the Teacher Notes that follow provide a variety of solution strategies to share with students to help them develop flexibility in their application of mathematics.
  • “Which one would be the most 'lemony'?"
  • “Which would use 10 cups of water?”
  • “How much would you need to make 50 cups of each recipe?”
• Chapter 4 Class Activity 4.3c has a variety of multi-step and contextual problems. For example, Question 1 reads as follows: “Ginger and her brother Cal have red and green planting buckets in the ratio of 3:1. a. If there are 5 green buckets, how many red buckets are there? b. Ginger and Cal bought more buckets because they have more to plant. They purchased the buckets in the same red:green ratio of 3:1. If they now have 28 buckets total, how many red and green buckets do they have? c. How are the problems different?”

In Grade 7, some specific standards that include application are 7.NS.3 and 7.EE.3. Examples of problems that address these standards include:

• On page 7WB1 – 55, students are asked to solve problems involving investment rates, target heart rates, and the cost of dinner with tax and tip. (7.NS.3) “Rico's resting heart rate is 50 beats per minute. His target exercise rate is 350% of his resting rate. What is his target rate?” Students use a model and write a number sentence to solve the multi-step problem.
• In Chapter 2 Lesson 2.3a Problem 25 students use a number line to model situations, answer questions using their knowledge of the number line, write an addition equation, and explain their thinking. (7WB2 - 20) (7.EE.3)
• In Chapter 6 Section 6.2 students work in two “different directions.” In some sections, students are given a context and asked to find relationships and solutions while in other sections students are given relationships and asked to write contexts.
  • “Write a context that models the following equation. 2L + 2(3L) = 990.” (7WB6 - 66) (7.EE.3)
  • “Martha divides $94 amongst her four friends. Leon gets twice as much money as Kokyangwuti. Jill gets five more dollars than Leon. Isaac gets ten less dollars than Kokyangwuti. How much money does each friend get?” (7WB6 - 67) (7.EE.3)

The instructional materials reviewed for The Utah Middle School Math Project Grade 7 meet the expectation that the materials balance all three aspects of rigor with the three aspects not always combined together nor are they always separated. Every chapter includes all three aspects of rigor. In some lessons the aspects of rigor are addressed separately, and in some lessons multiple aspects of rigor are addressed. Overall, the three aspects of rigor are balanced in this program.

There are lessons where the aspects of rigor are not combined.

• In Homework 4.3b students practice their procedural skill in solving proportions.
• Spiral Reviews throughout the materials provide opportunities for students to reinforce their procedural skills and fluencies from previous standards and lessons.

There are multiple lessons where two or all three of the aspects are interwoven.

• Class Activity 2.1a (page 7WB2 - 7) begins with exploring additive inverses in contexts. For example, “A hydrogen atom has one proton and one electron.” Students demonstrate their understanding by creating a model/picture, writing the net result in words, and answering how many zero pairs exist in the context.
• In Class Activity 2.2c students make connections between multiplication and division of integers. Students solve division problems and also solve contextual problems. At the end of this lesson/homework there is an extra practice section for students to gain fluency with integer operations as well as more contextual problems with integers.

The instructional materials reviewed for The Utah Middle School Math Project Grade 7 meet expectations that the Standards for Mathematical Practice are identified and used to enrich mathematics content within and throughout the grade.

The Standards for Mathematical Practices are identified in both the Teacher and Student Workbooks in most lessons. The MPs are explained in the beginning of the chapter and are identified using an icon within the lessons where they occur.

Overall, the materials clearly identify the MPs and incorporate them into the lessons. All of the MPs are represented and attended to multiple times throughout the year, and MPs are used to enrich the content and are not taught as a separate lesson.

• Chapter 1 Class Activity 1.1c Question 5 asks students to "look for and express regularity in repeated reasoning" as students determine patterns emerging in the previous examples of probability (MP8).
• The Chapter 2 Anchor Problem presents a number line with 0, 1, and variables (a) and (b). Students are asked: “Which of the following numbers is negative? Choose all that apply. Explain your reasoning.” Students reason abstractly and quantitatively (MP2) as well as construct viable arguments (MP3).
• Chapter 4 Class Activity 4.1f asks students to "attend to precision" as they find the unit rate in word problems and compare two quantities (MP6). For example, “Frosted Flakes has 11 grams sugar per ounce and Raisin Bran 13 grams per 1.4 ounces. Which cereal has more sugar per ounce?

The instructional materials reviewed for The Utah Middle School Math Project Grade 7 meet the expectation for prompting students to construct viable arguments concerning grade-level mathematics detailed in the content standards.

In many cases, students are asked to construct arguments and justify their thinking.

• Throughout the materials students are asked to justify their thinking. For example, Chapter 4 Homework 4.2b Question 3b asks, “Which solution is saltier, Solution A or Solution B? Justify your answer with at least two pieces of evidence.”
• There are instances where students are asked to make conjectures. For example, in Chapter 1 Class Activity 1.1a Question 7 students are asked to “make a conjecture about how many GREEN tiles are in your bag if the bag contains 12 total tiles.”
• Students are asked to engage in Error Analysis in some of the lessons. For example, in Chapter 4 Class Activity 4.1b Question 6 students must identify the error in the table of values that is represented. In the given solution the error was in adding 2 to each value in column A to get Column B rather than multiplying by 3/2 in Column A.

The instructional materials reviewed for The Utah Middle School Math Project Grade 7 meet the expectation of assisting teachers in engaging students in constructing viable arguments and analyzing the arguments of others concerning key grade-level mathematics detailed in the content standards. Many of the directions for MP3 are the same as those found written in the Student Workbook. Guidance is given on how to assist students in expressing arguments.

A few examples of guidance provided for teachers include:

• In Chapter 6 Homework 6.1a Questions 11-15 the Teacher Notes state: “Note: constructing an argument to disprove a statement only requires one counterexample, while constructing an argument to 'prove' something is more involved. In other words, one affirmative example does not prove a statement. In 7th grade attention to precision in making statements is an important first step towards building arguments. So, for #14, press students to explain why the statement is true; look for statements that build on understanding of supplementary angles and transitivity.”
• In Chapter 4 Class Activity 4.1b the students are given: “The values in the table below represent the lengths of corresponding sides of two similar figures. The side lengths are proportional to one another. Darcy filled in the remaining values in the table and has made a mistake. Find her mistake and fix it by filling in the correct values in the table on the right. Then provide an explanation as to what she did wrong.” The Teacher Note says: ”This problem allows students to critique Darcy’s reasoning and then make their own conjectures about the proportional constant.”
• There are some prompts for the teachers in the form of questions to ask or problems to present. For example, in Chapter 1 Class Activity 1.1b the students roll dice to simulate a horse race. Students determine a specific answer about which horse won most often and why. The Teacher Notes clarify the question and prompt the teacher to ask follow up questions: “Have students justify their arguments. Ask them for evidence to support their claims. Do you think that this game is fair? Why or why not?”

The instructional materials reviewed for The Utah Middle School Math Project Grade 8 meet the expectation for attending to the specialized language of mathematics. Overall, the materials provide explicit instruction on how to communicate mathematical thinking using words, diagrams, and symbols. When students are introduced to new mathematical vocabulary, it is explained, and teachers are encouraged to tell students to use the new terms.

• Each chapter in the workbook begins with a vocabulary list of words used in the chapter that includes words from previous learning as well as new terms.
• Throughout the chapter, these terms are used in context during Class Activities, Homework, and Self-Assessments.
• Vocabulary is bold in the context of the lesson.
• Vocabulary is presented throughout the Textbook: Mathematical Foundations along with accurate definitions. For example on 7MF2 - 17, “A golden rectangle is a rectangle that is not a square, but has this property: if we remove the square of whose side is the length of the smaller side of the rectangle, the remaining rectangle is a smaller version of the original.”
• Students are encouraged to use vocabulary appropriately. For example, Class Activity 1.1c Question 2f asks: “Have you been computing theoretical or experimental probability? Explain.” Class Activity 3.2a, between questions #10 and #11 asks: “What do the terms evaluate and solve mean? What is the difference between an equation and an expression?”
• At times the Teacher Notes give suggestions for using vocabulary in a lesson. For example, in Chapter 1 Class Activity 1.1a, students are learning about experimental probability, and the Teacher notes recommend, “Discuss again as a group. Compare their thinking now with their thinking before the experiment. Formalize the definition.”.
• The terminology that is used in the course is consistent with the terms in the standards.

Although it is not included in the CCSSM, the word simplify is used throughout the instructional materials. For example, in Chapter 3 Class Activity 3.1e, between Questions 8 and 9: “Your friend is struggling to understand what it means when the directions say, 'simplify the expression.' What can you tell your friend to help him? Teacher Note: Answers will vary. Discuss 'simplify' vs. 'evaluate' vs. 'solve' and 'expression' vs. 'equation.' Also discuss why we simplify—when does it help and when is it easier to not simplify? You might refer back to Activity 2 above."