3.1b Homework Translating Contexts To Numerical Expressions Worksheet
The instructional materials reviewed for The Utah Middle School Math Project Grade 7 meet the expectations for alignment. The materials spend the majority of the time on the major work of the grade, and the assessments are focused on grade-level standards. Content is aligned to the standards and progresses coherently across the grades and within each grade. The lessons include conceptual understanding, fluency and procedures, and application. There is a balance of these aspects for rigor. The Standards for Mathematical Practice (MPs) are used to enrich the learning.
Focus & Coherence
The instructional materials reviewed for The Utah Middle School Math Project Grade 7 meet the expectation for being focused on and coherent with the Common Core State Standards in Mathematics. The Unit Assessments do not assess above grade-level topics, and the instructional materials devote over 65 percent of class time to major work. Supporting work is connected to the major work of the grade, and the amount of content for one grade level is viable for one school year and will foster coherence between the grades. The materials explicitly relate grade-level concepts to prior knowledge from earlier grades, and the materials foster coherence through connections at a single grade, where appropriate and required by the standards.
The instructional materials reviewed for The Utah Middle School Math Project Grade 7 meet the expectations for focus within the grade. The Unit Assessments address grade-level material, and the instructional materials meet the expectations for focus within major clusters. Approximately 82 percent of the instructional days are on major work of the grade, including days in which work addressing supporting clusters directly reinforces major work of the grade. Overall, students and teachers using the materials as designed devote the majority of class time to the major work of the grade.
Materials do not assess topics before the grade level in which the topic should be introduced.
The instructional materials reviewed for The Utah Middle School Math Project Grade 7 meet the expectation for not assessing topics before the grade-level in which the topic should be introduced. The materials did not include any assessment questions that were above grade-level.
The instructional material assesses the grade-level content and, if applicable, content from earlier grades. Content from future grades may be introduced but students should not be held accountable on assessments for future expectations.
The instructional materials reviewed for The Utah Middle School Math Project Grade 7 meet the expectations for focus within assessment. Overall, the instructional material does not assess content from future grades within the assessment sections of each unit.
There are multiple Self-Assessments within each unit. Each assessment includes a scoring rubric that helps students articulate their understanding of key concepts being assessed. All assessments have answer keys provided in the Teacher Workbook.
On grade-level examples include:
- Chapter 2 Section 2.3- Students demonstrate their knowledge of 7.NS.2 by applying and extending previous understandings of multiplication and division of fractions to multiply and divide rational numbers. Question 2c on the Self-Assessment states: “Estimate each product or quotient. Then find the actual product or quotient of -89(0.5).”
- Chapter 6 Section 6.3- Students solve word problems leading to linear inequalities demonstrating their knowledge of 7.EE.4b. Question 3b on the Self-Assessment states: “Write an inequality to represent each of the following word problems. Solve each problem. Explain your solution in context. 'Jeremy is two years older than Rachel. The sum of the ages of Jeremy and Rachel is less than 46. How old could Jeremy be?'”
Students and teachers using the materials as designed devote the large majority of class time in each grade K-8 to the major work of the grade.
The instructional materials reviewed for The Utah Middle School Math Project Grade 7 meet the expectations for having students and teachers using the materials as designed, devoting the large majority of class time to the major work of the grade. Overall, the materials devote approximately 82 percent of class time to major work.
Instructional material spends the majority of class time on the major cluster of each grade.
The instructional materials reviewed for Grade 7 meet expectations for focus by spending a majority of class time on the major clusters of the grade including all clusters in 7.RP, 7.NS, and 7.EE. To determine this, three perspectives were evaluated: 1) the number of chapters devoted to major work, 2) the number of lessons devoted to major work, and 3) the number of weeks devoted to major work. Of the three perspectives, the number of lessons is most representative and was used to determine the score for this indicator.
Overall, the materials spend approximately 82 percent of instructional time on the major clusters of the grade. The Grade 7 materials have 8 chapters that contain 139 lessons, which accounts for a total of 31 weeks of class time including Anchor Problems and Self-Assessments.
- Grade 7 instruction is divided into eight chapters. More than half of Chapter 1 addresses 7.NS. Chapter 2 addresses 7.NS. Chapter 3 addresses 7.EE. Chapter 4 addresses 7.RP. More than half of Chapter 6 addresses 7.EE. Therefore, approximately 4.5 out of 8 chapters (56 percent) focus exclusively on the major work of the grade.
- Grade 7 instruction consists of 139 lessons. Approximately 114 lessons out of 139 (82 percent) focus on the major work of the grade level, which includes supporting work that connects to the major work of the grade.
- Grade 7 instruction is divided into 31 weeks. Approximately nineteen out of 31 weeks (61 percent) focus exclusively on the major work of the grade.
The instructional materials reviewed for The Utah Middle School Math Project Grade 7 meet the expectations for being coherent and consistent with the standards. Supporting work is connected to the major work of the grade, and the amount of content for one grade level is viable for one school year and fosters coherence between the grades. Content from prior or future grades is clearly identified, and the materials explicitly relate grade-level concepts to prior knowledge from earlier grades. The objectives for the materials are shaped by the CCSSM cluster headings, and they also incorporate natural connections that will prepare a student for upcoming grades.
Coherence: Each grade's instructional materials are coherent and consistent with the Standards.
Supporting content enhances focus and coherence simultaneously by engaging students in the major work of the grade.
The Instructional materials reviewed for The Utah Middle School Math Project Grade 7 meet the expectation for the supporting content-enhancing focus and coherence simultaneously by engaging students in the major work of the grade. Overall, the lessons that focus on supporting content also engage students in major work where natural and appropriate.
The following examples demonstrate where the supporting work enhances understanding of the major work of Grade 7.
- Chapter 5: Sections 5.2b, 5.2c, and 5.4b work with 7.G.1, and 7.G.6 supports 7.RP.2 by having students create and solve ratios and proportions to find similar figures. For example, in the overview teachers are told that the central idea of Section 5.2 is scale and its relationship to ratio and proportion. The standard for ratio and proportion are not listed. Ratio language is used in the Activities and Homework problems.
- Chapter 6: Section 6.1 supports 7.EE.4 and 7.NS.1 by having students find angle pairs which involves working with rational numbers and creating/solving equations.
- Chapter 7: Activities 7.1a and 7.1b support 7.NS.1 by having students compare populations which involves working with rational numbers.
- Chapter 7: Sections 7.1 and 7.2 include problems that are related to ratio and proportions (7.RP.2) while working with statistics. For example, Chapter 7, Class Activity 7.2b, Teacher Workbook, page 7WB7 – 43, students use the ratio of colors of jellybeans for a statistical experiment.
- Chapter 8: Activity 8.1c supports 7.EE.4 by having students create and solve equations in real-life mathematical problems based on composite area.
The amount of content designated for one grade level is viable for one school year in order to foster coherence between grades.
The instructional materials reviewed for Grade 7 meet the expectations for the amount of content designated for one grade level being viable for one school year in order to foster coherence between grades. The instructional materials are designed to take approximately 155 days. According to the publisher, completing the work would take a total of 31 weeks. That includes days for Anchor Problems, Class Activities, Homework, and Spiral Review. According to the Preface, “Each lesson covers classroom activity and homework for a 50-minute class. Sometimes the demands of the material exceed this limitation; when we recognize this, we say so; but some teachers may see different time constraints, and we defer to the teacher to decide how much time to devote to a lesson, how much of it is essential to the demands of the relevant standard. What is important are the proportions dedicated to the various divisions, so that it all fits into a year’s work. Within a lesson, the activities for the students are graduated, so that, in working the problems, students can arrive at an understanding of a concept or procedure. In most cases there is an abundance of problems, providing the teacher with an opportunity to adapt to specific needs.” The number of weeks was converted to days for this review. Each chapter has built-in days for Self Assessments. Overall, the amount of content that is designated for this grade level is viable for one school year.
Materials are consistent with the progressions in the Standards i. Materials develop according to the grade-by-grade progressions in the Standards. If there is content from prior or future grades, that content is clearly identified and related to grade-level work ii. Materials give all students extensive work with grade-level problems iii. Materials relate grade level concepts explicitly to prior knowledge from earlier grades.
The materials reviewed for The Utah Middle School Math Project Grade 7 meet expectations for consistency with the progressions in the Standards. In general, materials develop according to the grade-by-grade progressions in the Standards and provide extensive work with grade-level problems. Materials consistently relate grade-level concepts explicitly to prior knowledge from earlier grades.
Content from prior and future grade levels is identified in Connections to Content at the beginning of each student and teacher workbook chapter. Chapter overviews/summaries, as well as section overviews, include a written explanations of what students will be doing throughout the chapter. Summaries explain what students will learn and how they will use this knowledge in future learning.
- Chapter 1 explains that “throughout the chapter, students are provided with opportunities to review fractions, decimals and percents.” (page 7WB1 - 2)
- Chapter 2: “The development of rational numbers in 7th grade is a progression in the development of the real number system that continues through 8th grade. In high school students will move to extending their understanding of number into the complex number system.” (pages 7WB2 – 3)
- Chapter 4: “The chapter begins by reviewing ideas from 6th grade as well as 7th grade chapters 1-3 and transitioning students to algebraic representations. Student will rely on knowledge developed in previous chapters and grades in finding unit rates, proportional constants, comparing rates and situations in multiple forms, writing expressions and equations, and analyzing tables and graphs.“ (page 7WB4 – 2)
- Chapter 6: “Work on inequalities in this chapter builds on Grade 6 understandings where students were introduced to inequalities represented on a number line. The goal in Grade 7 is to move to solving simple one-step inequalities, representing ideas symbolically rather than with models.” (page 7WB6 - 2)
- Chapter 8: “In 8th grade, students will continue working with volume, formalizing algorithms for volume of cylinders and adding methods for finding the volume of cones and spheres.” (page 7WB8 – 3)
Materials consistently relate grade-level concepts explicitly to prior knowledge from earlier grades. Connections between concepts are addressed in the Connections to Content, chapter overviews/summaries, and Math Textbook. Examples of these explicit connections include:
- Chapter 4, Class Activity 4.1a: “Equivalent Ratios, Fractions, and Percents (Review from 6th grade): They should know a ratio expresses a numerical relation between two quantities. Students studied ratios extensively in 6th grade.” (p. 7WB4 – 14)
- Chapter 8, Connections to Content: “Towards the end of this section students review the use of nets (a concept from 6th grade) to find surface area of prisms and cylinders and then to differentiate this measure from volume, which they will also find.” (page 7WB8 – 2)
Materials foster coherence through connections at a single grade, where appropriate and required by the Standards i. Materials include learning objectives that are visibly shaped by CCSSM cluster headings. ii. Materials include problems and activities that serve to connect two or more clusters in a domain, or two or more domains in a grade, in cases where these connections are natural and important.
The instructional materials reviewed for The Utah Middle School Math Project Grade 7 meet the expectations for fostering coherence through connections at a single grade, where appropriate and required by the Standards.
In the teacher's workbook, the CCSSM are identified on the introduction page of each chapter. Each chapter correlates to a Grade 7 domain, with sections within the chapter focusing on standards within the domain. There is a section titled, “Concepts and Skills to Master," which identifies specific learning objectives for each section in the teacher, parent, and student workbooks.
- “Investigate chance processes, develop/use probability models, as well as the work within the section,” a learning objective from Chapter 1 Section 1.1, reflects Cluster 7.SP.C (Investigate chance processes and develop, use, and evaluate probability models).
- In Chapter 3, students are engaged in activities aligned to Cluster 7.EE.A (Use properties of operations to generate equivalent expressions). In Section 3.1a (page 7WB3-7 - 12), the first Activity, “Naming Properties of Arithmetic,” has an objective to “recognize properties of arithmetic and use them in justifying work when manipulating expressions.” Students are engaged in using the identified properties and identifying pairs of equivalent expressions. In Anchor Problem 3.0 (page 7WB3 - 6), a teacher’s note reflects the cluster heading: “A big idea you’re after right now is that one can write equivalent expressions in a number of ways and that different ways shed light on different thinking.”
The materials include problems and activities that serve to connect two or more clusters in a domain where connections are natural and important.
- The Chapter 3 Overview connects 6.EE.1 and 6.EE.2 through this statement: “By the end of this section (3.1) students should be proficient at simplifying expressions and justifying their work with properties of arithmetic. Section 3.2 uses the skills developed in the previous section to solve equations…Section 3.3 ends the chapter with application contexts.” (pages 7WB3 - 2 and 7WB3 - 3)
The materials include problems and activities that serve to connect two more domains in a grade where connections are natural and important.
- Chapter 3 Section 3.1 and 3.2 connects 7.NS.A and 7.EE.A as students transfer integer properties to algebraic expressions. Students use the Distributive Property of Multiplication and Division over Addition and Subtraction to write equivalent algebraic expressions and to develop an understanding of combining coefficients of like terms and calculating the product of two numbers. (pages 7WB3 – 6 and 7WB3 - 93 through 7WB3 - 109)
- In Chapter 4, Class Activities 4.3d and 4.3e 7.RP.A, 7.NS.A and 7.EE.B are connected as students write equations and compute to solve percent problems. (pages 7WB4 - 173 through 7WB4 - 181)
Rigor And Mathematical Practices
The instructional materials reviewed for The Utah Middle School Math Project Grade 7 meet the expectations for rigor and mathematical practices. The materials meet the expectations for rigor as they balance and help students develop conceptual understanding and procedural skill and fluency. The materials meet the expectations for mathematical practices as they identify and use each of the MPs and support the Standards' emphasis on mathematical reasoning.
Rigor and Balance
The instructional materials reviewed for The Utah Middle School Math Project Grade 7 meet the expectations for rigor and balance. The materials meet the expectations for rigor as they help students develop conceptual understanding, procedural skill and fluency, and application with a balance in all three aspects of rigor.
Rigor and Balance: Each grade's instructional materials reflect the balances in the Standards and help students meet the Standards' rigorous expectations, by helping students develop conceptual understanding, procedural skill and fluency, and application.
The instructional materials reviewed for The Utah Middle School Math Project Grade 7 meet the expectations for rigor and balance. The materials meet the expectations for rigor as they help students develop conceptual understanding, procedural skill and fluency, and application with a balance in all three aspects of rigor.
Attention to conceptual understanding: Materials develop conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings.
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.”
Attention to Procedural Skill and Fluency: Materials give attention throughout the year to individual standards that set an expectation of procedural skill and fluency.
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.
Attention to Applications: Materials are designed so that teachers and students spend sufficient time working with engaging applications of the mathematics, without losing focus on the major work of each grade
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)
Balance: The three aspects of rigor are not always treated together and are not always treated separately. There is a balance of the 3 aspects of rigor within the grade.
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.
Mathematical Practice-Content Connections
The instructional materials reviewed for The Utah Middle School Math Project Grade 7 meet the expectations for practice–content connections. The materials show strengths in identifying and using the MPs to enrich the content along with attending to the specialized language of mathematics. The instructional materials also support the Standards' emphasis on mathematical reasoning.
Practice-Content Connections: Materials meaningfully connect the Standards for Mathematical Content and the Standards for Mathematical Practice
The Standards for Mathematical Practice are identified and used to enrich mathematics content within and throughout each applicable grade.
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?
Materials carefully attend to the full meaning of each practice standard
The instructional materials reviewed for The Utah Middle School Math Project Grade 7 partially meet the expectations for attending to the full meaning of each Mathematical Practice Standard. The MPs are most frequently identified in Teacher Notes where they are aligned to a particular practice activity or question. Many times the note is guidance on what the teacher does or says rather than engaging students in the practice.
The intent of the MPs is often not met since teachers engage in the MPs as they demonstrate to students how to solve the problems.
- Many problems marked MP1 do not ensure that students have to make sense of problems and persevere in solving them. For example, Chapter 2 Class Activity 2.1 directs students to “use the idea of 'zero pairs' to complete the worksheet.” Students are not making sense of problems but answering problems based on how the teacher models the problem.
- MP4 is identified throughout the program; however, it is rarely identified in situations where students are modeling a mathematical problem and making choices about that process. In many situations, it is labeled when directions are provided for how the teacher models. For example, in Chapter 3 Class Activity 3.1i students are given a number line as the model.
- Where MP5 is labeled, the materials suggest a specific tool for students to use which does not lead students to develop the full intent of MP.5. For example, in Chapter 4 Class Activity 4.2b students are told to use the graph and table to model the context.
Emphasis on Mathematical Reasoning: Materials support the Standards' emphasis on mathematical reasoning by:
Materials prompt students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics detailed in the content standards.
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.
Materials assist teachers in engaging students in constructing viable arguments and analyzing the arguments of others concerning key grade-level mathematics detailed in the content standards.
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?”
Materials explicitly attend to the specialized language of mathematics.
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."
PARTIALLY MEETS EXPECTATIONS
The instructional materials for The Utah Middle School Math Project Grade 7 partially meet the expectations for usability. In reviews for use and design, the problems and exercises are developed sequentially, and each activity has a mathematical purpose. Manipulatives and models are used to enhance learning, and the purpose of each is explained well. All materials include support for teachers in using questions to guide mathematical development, and the teacher editions have many annotations and examples on how to present the content and an explanation of the math of each unit and the program as a whole. Although materials include opportunities for ongoing review and practice, there are no assessments that purposely identify prior knowledge within and across grade levels.The teacher materials identify common misconceptions and errors, but there are no specific strategies to address these when they arise. Tasks provide students with multiple entry points that can be solved with a variety of solutions and representations, and there are suggestions for students to monitor their own progress. Activities provide few ELL strategies, support strategies for special populations, or strategies for advanced students to investigate mathematics content at greater depth.
Use and design facilitate student learning: Materials are well designed and take into account effective lesson structure and pacing.
The instructional materials for The Utah Middle School Math Project Grade 7 meet the expectations for use and design. Materials are well-designed, and lessons are intentionally sequenced. Students are presented with an Anchor Problem at the beginning of each chapter to introduce new concepts. Anchor Problems are sometimes referenced throughout the chapter. Students produce a variety of types of answers including both verbal and written answers. Manipulatives are used in the instructional materials as mathematical representations and to build conceptual understanding.
The underlying design of the materials distinguishes between problems and exercises. In essence, the difference is that in solving problems, students learn new mathematics, whereas in working exercises, students apply what they have already learned to build mastery. Each problem or exercise has a purpose.
The instructional materials for Grade 7 meet the expectation that the underlying design of the materials distinguishes between problems and exercises.
The chapters begin with a non-routine problem that introduces new concepts and is labeled as an Anchor Problem. The chapters are subsequently sectioned into Class Activities, Homework, Spiral Reviews, and Assessments.
Generally, each Class Activity has problems to solve together as a class with instructor guidance. Occasionally, they are intended to review previous grades' concepts in order to connect them to seventh grade concepts. Most often, the Class Activities are for the students to apply what they have already learned.
The mathematics taught in each Class Activity is reinforced by an accompanying Homework component.
Design of assignments is not haphazard: exercises are given in intentional sequences.
The instructional materials for Grade 7 meet the expectation that the design of assignments is not haphazard; exercises are given in intentional sequences.
Students are presented with an Anchor Problem at the beginning of each chapter to introduce new concepts. Anchor Problems are sometimes referenced throughout the chapter.
Within each chapter, concept development is sequential. During Class Activities, the teacher introduces new concepts or builds upon prior knowledge. Students work individually or as a whole class when engaged in the Class Activities. The Homework component reinforces the mathematical concepts taught during the previous Class Activity. Spiral Reviews are used to provide continued practice of newly learned mathematical concepts throughout the year.
The progression of lessons taught is intentional and assists students in building their mathematical understanding and skill. Students begin with activities to build conceptual understanding and procedural skill, and progress to applying the mathematics with more complex problems and procedures.
There is variety in what students are asked to produce. For example, students are asked to produce answers and solutions, but also, in a grade-appropriate way, arguments and explanations, diagrams, mathematical models, etc.
The instructional materials for Grade 7 meet the expectation for the variety in what students are asked to produce.
Throughout the Class Activities, students are asked to produce answers and solutions, discuss ideas, make conjectures, explain solutions and justify reasoning, make sketches and diagrams, and use appropriate models. These aspects are found individually within problems as well as in combination with others, such as provide an explanation of a solution and include a diagram.
- Chapter 3 Class Activity 3.1b: Models are used to represent the quantities and relationships stated in contextual problems. Students examine the models and write numeric expressions that represent the quantities and relationships represented by the models as well as explain why the various correct representations are equivalent. In subsequent tasks students determine and explain which expressions are equivalent, write contexts for expressions, and explain how they determined if various expressions adequately represent given contexts. The final problems provide opportunity for application.
- Chapter 3 Class Activity 3.1c: Students transition from writing numeric expressions to algebraic expressions using the same types of tasks and problem formats as those presented in 3.1b. The materials provide additional practice for students to draw models, define variables, and write expressions that model given situations.
Manipulatives are faithful representations of the mathematical objects they represent and when appropriate are connected to written methods.
The instructional materials for Grade 7 meet the expectation that manipulatives are faithful representations of the mathematical objects they represent and, when appropriate, are connected to written models.
Colored tiles are used when students work with probability. In Class Activity 1.1a, students are learning the difference between experimental and theoretical probability with the activity, “How Many Green Tiles Are In Your Bag?” Students draw several tiles out of a bag and record the color each time. By using the tiles, students are able to make conjectures, as well as compare theoretical and experimental probability.
The Anchor Problem in chapter 7, “The Teacher Always Wins,” uses teacher-created colored number cubes to create data through a game between the students and the teacher. Students use the data collected from the game to analyze the probability of winning when using different colored cubes.
The visual design (whether in print or online) is not distracting or chaotic, but supports students in engaging thoughtfully with the subject.
The instructional materials for Grade 7 meet the expectation that the visual design is not distracting or chaotic and supports students in engaging thoughtfully with the subject.
- The student materials are clear and consistent between activities within a grade level as well as across grade levels.
- Each Class Activity and Homework is clearly labeled and provides consistent numbering for each investigation and problem set with both a lesson number and page number.
- The examples shown in the Textbook: Mathematical Foundation are consistently labeled and numbered within each section.
Teacher Planning and Learning for Success with CCSS: Materials support teacher learning and understanding of the Standards.
The instructional materials for The Utah Middle School Math Project Grade 7 meet the expectations for supporting teachers’ learning and understanding of the standards. The instructional materials provide questions that support teachers in delivering quality instruction. The teacher’s edition is easy to use and consistently organized and annotated. The teacher’s edition explains the mathematics in each unit as well as the role of the grade-level mathematics within the program as a whole. The instructional materials are all aligned to the standards, and the instructional approaches and philosophy of the program are clearly explained.
Materials support teachers in planning and providing effective learning experiences by providing quality questions to help guide students' mathematical development.
The instructional materials for Grade 7 meet the expectation for supporting teachers in planning and providing effective learning experiences by providing quality questions to help guide students' mathematical development.
Let's take a look at what's expected in the middle grades. W.3.3, W.4.3, and W.5.3 ask students to "write narratives to develop real or imagined experiences and or events using effective technique, descriptive details, and clear event sequences."
This is further spelled out in five ways:
- Write a Strong Beginning (W.3.3a/W.4.3a/W.5.3a) - Third, fourth, and fifth grade students are expected to "establish a situation and introduce a narrator and/or characters" and "organize an event sequence that unfolds naturally."
- Use Effective Technique (W.3.3b/W.4.3b/W.5.3b) - Techniques include dialogue and descriptions to develop the plot or "show the response of characters to situations." Fifth graders are also required to use pacing.
- Add Words to Establish Order and Create Smooth Transitions (W.3.3c/W.4.4c/W.5.4c) - The Common Core lists temporal words and phrases for third graders, transitional words and phrases for fourth, and adds transitional clauses for fifth.
- Choose Exact Words (W4.3d/W.5.d) - Students in Grades 4 and 5 begin to "use concrete words and phrases and sensory details to convey experiences and events precisely."
- Write a Strong Conclusion (W.3.3 d/W.4.3e/W.5.3e) - Students need to provide closure to their stories.
- Start simple.
- Use model texts.
- Build writing skills as the year goes on.
1. Start Simple - One of the simplest forms of narrative writing is the fable. These short pieces allow students to practice simple beginnings, simple dialogue, simple descriptions, simple transitions, simple word choice, and simple conclusions. (Did I mention that they're simple?)
I'm a big believer in "read first, write second." Start off by reading some fables as models. You can access several versions of Aesop's Fables free of charge at Project Gutenberg. Because the copyrights of these texts have expired, you may use them freely in your classroom. The Aesop for Children, illustrated by Milo Winter, includes over 100 illustrated fables (such as the one featured below) that have been adapted especially for children.
Discuss how the author employed each of the tactics listed in the Common Core: beginning, dialogue, description, transition words, concrete words (specific nouns and active verbs), and conclusion. Work together to determine what makes a story a fable (brief narrative with a moral, uses animals with human characteristics, etc.)
This is the perfect place to teach (or review) how to write dialogue. Students only need to use a few lines of dialogue in a fable.
To plan a fable, students choose a moral/lesson, think of brief plot that will teach the lesson, and select animals that portray appropriate human traits. Easy! As your students write these brief, age-appropriate narratives, they will take their first steps toward mastering CCRA.W.3.
You have everything you need to get started! If you want the complete lesson plans, you can purchase Writing Fables for $5.00 in my Teachers pay Teachers store.
2. Use model texts - I've already talked a bit about this with Aesop, but there's more. The first two "P's" we'll talk about today are parody and parroting. Parody allows students to adapt familiar story lines to create original narratives. Parroting challenges students to copy effective writing techniques in their own stories. Both of these will again use "read first, write second."
Parody - So much of what we read (or watch on television or at the movie theater) is parody! Authors are constantly recycling what worked before. Kids can do this too . . . with great success.
As we discuss parody, think about texts that you already use in your classroom (or personal favorites). Consider their strengths and how your students might twist their story lines into something new. I'm going to explain how I use Cinderella stories in my classroom, but you might use something totally different in yours.
Read First - I begin by reading a few Cinderella (folklore) picture books aloud. We discuss similarities and differences then list elements common to all Cinderella stories. Each time we read a book, we analyze how the Cinderella elements are portrayed on a table that lists elements across the top and book titles down the side. Next, I introduce parody, and we read and analyze a bunch of Cinderella parodies. In my class, the culminating book is Ella Enchanted, a full-length novel, which is also a Cinderella parody. All of this (and more!) can be found in Comparing Cinderella Stories with Ella Enchanted on Teachers pay Teachers.
Write Second - This is really fun! Kids can choose whatever twist on the Cinderella story that they want. I've had everything from "Treearella" to "Mozzarella." This year, one of the Cinderella characters was even a famous football player whose family didn't like his choice for a wife. The sky is the limit! Students plug their ideas into the Cinderella elements list and start writing. In my class, we type, illustrate, and bind the stories into our own series of picture books.
Parroting - Certain authors have distinctive voice. How do they do it? Parroting involves reading one or more texts by a certain author, analyzing his or her distinctive style, then trying it on for size (again, "read first, write second"). My class worked with Just So Stories by Rudyard Kipling this year and the results were astonishing. You can read about it on my May 19th blog, "Just So Writing." Next year I want to try using a different author each month!
3. Build writing skills as the year goes on - Students' writing should steadily improve as time wears on. How can we ensure that this happens? First, point out strategies used in each text you read, whether simple or complex. (For example, "Did you notice how the author used the words 'first,' 'second,' and 'third' to establish time order?" or "Wow, the author just hinted at something that might happen in the future. That's called foreshadowing. How did he do that?") Second, ask students to employ specific strategies in their own writing. Finally, let them practice with simpler assignments before you ask them to pull stories out of their hats.
This brings me to the third "P": personal narrative. Many English textbooks use personal narrative as the keystone for narrative writing in fourth grade. To me, this "P" needs to come after structured practice with parodies and parroting. Personal narrative requires students to determine the entire event sequence on their own, and many students this age just aren't ready.
For example, when I ask my students to write about a favorite vacation, inevitably they'll write a full page elaborating on waking up, getting dressed, eating breakfast, and driving to, let's say, an amusement park. Then, in one or two sentences, they tell which roller coasters they rode. Before you know it, they're back in the car heading home. Wait? What just happened? They worked hard at the beginning to do everything the teacher asked . . . and then they ran out of steam and quickly wrapped up the story.
In the intermediate grades, I've found that the most important part of instruction for writing personal narratives is focusing on one compelling part of the story and cutting the rest. I tell them to pretend they're movie directors with hours and hours of film that they need to cut down to a fifteen-minute segment. It's brutal. They don't want to cut anything! But I insist.
There's so much more work that goes into a personal narrative, as well. By nature, the students want to tell instead of show. This sets a student on a much more complex path of "show, not tell." They begin to see the differences between describing and narrating, between passive verbs and active verbs, etc.
You might be thinking, "She didn't even mention writing original stories in today's blog." While that's the final goal, I wanted to emphasize the importance of the "3 P's" - - - parody, parroting, and personal narrative - - - in that order. You see, this isn't about letting kids write stories. It's about teaching students to write effective, powerful narratives. Some students are ready to generate original stories in Grades 3-5; others need to be inched along through modeling and targeted techniques.
I'm sure not all of you out there agree with me, but this is what's worked for me. Feel free to comment! Much of what I've learned in my teaching career has come from the great ideas of my fellow educators!