5th Grade - Gateway 2
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Rigor & Mathematical Practices
Gateway 2 - Partially Meets Expectations | 77% |
|---|---|
Criterion 2.1: Rigor | 7 / 8 |
Criterion 2.2: Math Practices | 7 / 10 |
The instructional materials reviewed for Grade 5 partially meet the expectations for rigor and the mathematical practices. The materials meet the expectations for rigor as they help students develop conceptual understanding, procedural skill and fluency, and applications. However, the materials partially meet the expectations for mathematical practices as they do not attend to the full meaning for each of the MPs and rarely prompt, or have the teachers prompt, students to analyze the arguments of others.
Criterion 2.1: 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 Grade 5 partially meet the expectations for rigor and the mathematical practices. The materials meet the expectations for rigor as they help students develop conceptual understanding, procedural skill and fluency, and applications. However, the materials partially meet the expectations for mathematical practices as they do not attend to the full meaning for each of the MPs and rarely prompt, or have the teachers prompt, students to analyze the arguments of others.
Indicator 2a
Attention to conceptual understanding: Materials develop conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings.
The materials reviewed for Grade 5 meet the expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings. Overall, the instructional materials often call for visual representations, verbal explanations, and written equations.
- In Unit 6 Session 2.2 students use representations to add tenths, hundredths, and thousandths through reasoning about place value, equivalents, and representations by creating posters that show their solution to a problem with adding decimals. Their posters show, “How did you add the decimals? Make sure you explain your method carefully on your posters. If [someone] walked in, would that person be able to understand clearly what problem you were solving and how you solved it from your poster?” (5.NBT.A)
- In Unit 7 Session 1.1 students use representations to multiply a fraction and a whole number by first creating a representation of a multiplication problem making sure to use a few different representations such as number lines, shapes divided into fractional parts, and equations. Students also complete Student Activity Book pages 421-424 which require them to make a representation, write a multiplication equation, and solve the problem (5.NF.B).
- In Unit 7 Session 1.5 students represent a fractional part of a fractional quantity by using fraction bars and tables to talk through what each of the numbers refers to in the representations given (5.NF.B). Students also complete Student Activity Book pages 443-447 which require them to use fraction bars and explain their thinking and representations.
- In Unit 7 Session 1.7 students use arrays to represent multiplication of fractions and also understand the relationship between the denominator and numerator of factors and relationship between the denominator and numerator of products. Students solve problems by using unmarked arrays to express the fractions making sure to keep track of what the whole is and how each vertical and horizontal line should be placed and what needs to be shaded to represent the fraction problem given (5.NF.B). Students also complete Student Activity Book pages 453-456 which require them to not only solve a story problem using an array, but then also write an equation based off of what is represented.
Indicator 2b
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 materials reviewed for Grade 5 meet the expectations for giving attention throughout the year to individual standards that set an expectation for procedural skill and fluency. The materials include opportunities to practice and review in order to build procedural skill and fluency. Students are provided Daily Practice in every session and Homework in many sessions. The instructional materials provide direct instruction regarding the standard algorithm, 5.NBT.5, and additional practice is provided.
Standard 5.NBT.5 requires students to fluently multiply multi-digit numbers using the standard algorithm.
- The Ten-Minute Math activities indirectly work on the requirement for fluency of the grade. The Ten-Minute Math activities addressing 5.NBT.5 begin in Unit 3 before the algorithm is taught in Unit 4. In Unit 3, “Today’s Number,” students create expressions that equal (n), use multiplication, and use no more than 2 factors.
- In Unit 5 Session 1.1 students solve multi-digit multiplication problems in the “Solve Two Ways” activity in the Student Activity Book page 276.
- Unit 4- all of Investigation 1; Sessions 2.4, 2.5, 2.7; and all of Investigation 3- provide opportunities to practice procedural skill and fluency of the grade. During Session 1.1 students review multiplication strategies by breaking numbers apart, changing one factor and adjusting, and creating an equivalent problem. During Session 1.2 students estimate products by rounding numbers. During Session 1.3 students are shown solutions to the same problem solved using two different algorithms followed by the teacher working through a problem step-by-step using the US standard algorithm for multi-digit multiplication.
Indicator 2c
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 materials reviewed for Grade 5 meet the expectations for teachers and students spending sufficient time working with engaging applications of the mathematics, without losing focus on the major work of each grade. Though at times the problems are scaffolded in such a way the students are guided through the question, they are engaged consistently in problem solving during the year.
Practice with application of 5.NF.6 is found in two Units, Units 7 and 8. Unit 7 provides practice with real-world application of 5.NF.6. Sessions 1.1, 1.2, 1.3, 1.4, 1.7, and 1.8 require students to solve real-world problems such as “Janet is using a recipe for muffins that calls for ¾ cup of of milk. She is going to make 3 times the recipe. How many cups of milk does she need?” Unit 8 Session 2.4 provides practice with finding the area of different size rectangles for a class garden.
Work for standard 5.NF.7c, solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem is found in Unit 7. In Sessions 1.9, 1.10, and 1.11 students are guided through problems that either divide a whole number by a fraction or divide a fraction by a whole number. Students practice these real world problems in the Student Activity Book Daily Practice. Most of the Daily Practice problems follow the same format such as “Felix has 2 yards of ribbon. He needs ¼ yard to make 1 bow. How many bows can Felix make?” or “How much popcorn would each person get if 2 people shared ½ of a bag of popcorn equally?”
Indicator 2d
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 Grade 5 partially meet the expectations for balance of the three aspects of rigor within a grade. Although the instructional materials meet expectations for each aspect of rigor, these aspects of rigor are often addressed in separate parts of the Sessions. Materials targeting application are often scaffolded, detracting from the balance of rigor. Overall, the three aspects of rigor are most commonly treated separately.
In general, conceptual understanding, procedural skill and fluency, and application are addressed in the Sessions; however, for the most part they are addressed in separate sections of the instructional materials. Conceptual understanding is typically addressed in the Discussion and Math Workshop portions of the Sessions. Procedural skill and fluency is typically introduced in separate Sessions and then practiced in the Daily Practice portion of sessions. Application consists of routine word problems in the instructional materials. As a result, all aspects of rigor are almost always treated separately within the curriculum including within and during Sessions, Practice, and Homework.
Criterion 2.2: Math Practices
Practice-Content Connections: Materials meaningfully connect the Standards for Mathematical Content and the Standards for Mathematical Practice
The instructional materials reviewed for Grade 5 partially meet the expectations for practice-content connections. Overall, the materials show strengths in identifying and using the MPs to enrich the content along with attending to the specialize language of mathematics. However, the materials do not always attend to the full meaning of each MP, and there are few opportunities for students to analyze the arguments of others either through prompts from the materials or from their teachers.
Indicator 2e
The Standards for Mathematical Practice are identified and used to enrich mathematics content within and throughout each applicable grade.
The instructional materials for Grade 5 meet the expectations for identifying the Standards for Mathematical Practice (MPs) and using them to enrich the mathematical content. The MPs are clearly identified in Implementing Investigations on page 44 and can also be found in each Unit. The instructional materials highlight two MPs in every unit. During the sessions, Math Practice Notes dialogue boxes are given to provide tips to the teacher on how to engage students in the MPs. Additionally, Math Practice Notes are provided for the MPs that are not highlighted so that students continue to work on the practices all year.
The Introduction and Overview of each unit includes a “Mathematical Practices in this Unit” section. This section of each unit highlights the two MPs that are the focus of the unit. The MPs are described and examples from the Unit are provided. A chart showing where Mathematical Practice Notes occur and when the MP is assessed is also included in this section.
- The Unit 2 “Mathematical Practices in this Unit” is found on pages 8-11. This unit focuses on MP4 and MP5. An example of MP4 from Session 2.3 is included.
- The Unit 7 “Mathematical Practices in this Unit” is found on pages 10-13. This unit focuses on MP1 and MP8. An example of MP1 from Session 1.4 is included.
Math Practice Notes are provided in sessions alongside content. Math Practice notes are provided for the MPs highlighted within the unit and MPs that are not the highlighted practices for the unit.
- Unit 2 Session 1.1 includes a Math Practice Note for MP1 and MP4. MP4 is a practice highlighted in the unit. Students are developing approaches for determining the volume of a rectangular solid which will lead to formulas for volume.
- Unit 4 Session 3.5 includes a Math Practice Note for MP1, a practice not highlighted in the unit. The note describes how students compare approaches for multiplying and dividing large numbers with known strategies.
- Unit 5 Session 2.5 includes a Math Practice Note for MP4 and MP5, practices that are highlighted in the unit. The note discusses how choosing from various tools allows students to model with mathematics in different ways.
- Unit 7 Session 1.8 includes a Math Practice Note for MP2 and MP8. MP8 is a practice highlighted in the unit. Students are discussing problems related to the multiplication of fractions.
Indicator 2f
Materials carefully attend to the full meaning of each practice standard
The instructional materials reviewed for Grade 5 partially meet expectations that materials carefully attend to the full meaning of each practice standard (MP). Although the instructional materials attend to the full meaning of some of the MPs, there are some MPs for which the full meaning is not developed.
At times, the instructional materials only attend superficially to MPs. The following are examples:
- The Unit 2 Session 2.1 Math Practice Note lists MP5 and has students determining the volume in cubic centimeters of a small prism. The Math Practice Note states that “students might use centimeter cubes or a centimeter ruler to find the dimensions.” In this activity, students are told which tools to use.
- The Unit 7 Session 1.1 Math Practice Note lists MP1. The teacher states, “Earlier in the year you worked on adding and subtracting fractions, and in the last unit you worked on adding and subtracting decimals. In this unit you will be multiplying and dividing fractions and decimals. We are going to begin with multiplying fractions.” The Math Practice Note discusses that part of making sense of problems is being able to connect unfamiliar work with prior learning. However, the statements to be made by the teacher explicitly guide students to where connections could be made, and the session itself does not assist the teachers or students to make or discuss these connections.
- The Unit 7 Session 2.1 Math Practice Note lists MP1 and has students making sense of the problem by recognizing that different pathways to a solution may yield answers that look different even if they are equivalent. However, they are not persevering in solving the problems given.
- The Unit 7 Session 2.2 Math Practice Note lists MP5 and has students identifying decimal and fraction equivalence through the use of a calculator. The Math Practice Note states that in this activity, students explore how to use a calculator to determine the records of several basketball players. This activity tells students which tool to use.
At times, the instructional materials fully attend to a specific MP. The following are examples:
- The Unit 1 Session 2.6 Math Practice Note lists MP5 and has students reminded of the various tools, such as story contexts and arrays, that they can use to represent not only the problem but also the steps of their solutions to solve multiplication problems. They are never specifically told which tool to use and are able to choose the tool that is most appropriate for them and the lesson.
- The Unit 4 Session 1.1 Math Practice Note lists MP6 and has students describing and comparing strategies used to solve multi-digit multiplication problems. The Math Practice Note specifically calls out that in the discussions students need to explain the steps of their solution, name the strategy used, and question classmates making sure to develop clear explanations and concise notation for their work.
- The Unit 7 Session 1.5 Math Practice Note lists MP6 and asks students to make a conjecture about what happens when they multiply a number by a fraction smaller than 1. The Math Practice Note discusses that articulating conjectures requires that students attend carefully to the language they use and to state all components of their reasoning.
Indicator 2g
Emphasis on Mathematical Reasoning: Materials support the Standards' emphasis on mathematical reasoning by:
Indicator 2g.i
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 Grade 5 partially meet the expectations for prompting students to construct viable arguments and analyze the arguments of others concerning key grade level mathematics.
When MP3 is referenced, students are often asked to solve and share solutions. The independent work of the student is most often about finding the solution to a problem without creating a viable argument. Students often listen to peer solutions without being asked to critique the reasoning of the other student. Much of the student engagement in the class discussion is teacher prompted without giving students the opportunity to create their own authentic inquiry in the thinking of others.
- In Unit 6 Session 1.2 students neither construct an argument nor critique the arguments of others when the materials reference MP3 and direct students to “make sure that they think about how many parts out of 1,000 are shaded.”
- In Unit 6 Session 1.5 MP3 is referenced when students are asked to “share their responses and justifications with the whole class.” The students are not required to analyze the arguments of others.
- In Unit 7 Session 1.1 MP3 is referenced when students are asked to share their solutions and equations for Problem 1 on Student Activity Book page 421.
At times, the materials prompt students to construct viable arguments and analyze the arguments of others.
- In Unit 1 Session 1.2 students are told “[Hana] and [Yumiko] say that all these numbers are multiples of 12 and that any solution to this number puzzle will be a multiple of 12. Do you agree? Why or why not?”
- In Unit 6 Session 1.7 the following guided exchange supports students in critiquing the arguments of others: “[Stuart] says that we can’t play the card because, in the bottom row, we’ve already played 825 thousandths and 975 thousandths, and this card goes in between them. Are there any comments? [Shandra] disagrees. she says that we could play it in the last spot in the middle row because it’s greater than 6 tenths and goes between 45 hundredths and 975 thousandths in the last column. Are there any comments? [Shandra] is right; we can play the card there. Remember the rules: the decimals have to be in order from left to right and from top to bottom, but it’s all right if the last card in one row is bigger than a card in the first one or two spaces of the next row.”
Indicator 2g.ii
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 Grade 5 partially meet the expectations for 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.
Most of the time when MP3 is referenced, teachers are asked to have students share or explain their solutions. Teachers are also directed to have students ask questions but are not supported in focusing those questions toward critiquing the arguments of others.
- In Unit 3 Session 3.3 students are playing a game, and the teacher says, “a player wins a round if the sum of his or her two cards is greater than the sum of the other player’s cards. Who has the greater sum? How do you know?” These questions would prompt students to construct their own argument, but there are no other questions or prompts to help students who are not able to construct an argument. There are also no questions or prompts for students to analyze the arguments of others.
- In Unit 5 Session 1.4 there are a series of scaffolded questions for teachers that would assist them in having students construct arguments from graphs about the height and age of two people. There are no questions or prompts for teachers to assist them in having students analyze others’ arguments when different interpretations might arise.
- In Unit 8 Session 2.3 students are asked to explain their thinking about how the perimeter and area of rectangles change when each dimension is doubled. There are question that assist the teacher with engaging students in constructing their own arguments, but there are no questions or prompts to assist teachers with having students analyze the arguments of others.
The materials assist teachers, at times, in engaging students in constructing viable and analyzing the argument of others.
- In Unit 3 Session 1.1 teachers are prompted to provide the students with questions such as, “How did you know how many marbles to circle in Problem 5a?....Deon says that since there are 12 marbles, ⅓ is four marbles. Deon , how do you know that? Who has another way to think abou this problem? I noticed in Problem 7 that almost everyone agrees ⅔ is greater than 2/6, but who can explain why this is true?” The Math Practice Note tells teachers to ask students to paraphrase each other’s explanations or to ask one another questions to encourage students’ working to make sense of each other’s approaches.
Indicator 2g.iii
Materials explicitly attend to the specialized language of mathematics.
The instructional materials reviewed for Grade 5 meet the expectations for explicitly attending to the specialized language of mathematics.
The instructional materials provide opportunities for teachers to say mathematical terms to students during the whole group portion of the lessons. The materials use precise and accurate terminology when describing mathematics. New terminology is introduced on the summary page of the TE at the beginning of the session where it will first be used. The mathematical terminology is highlighted in italics throughout the sessions within the TE. There is also an index at the end of each unit manual in which math terms are listed for the unit.
- In Unit 1 Session 1.4 students are introduced to the order of operations. The materials prompt the teacher to state, “Mathematicians realized that when you have more than one operation in an expression like this, people could have different interpretations of how to solve it. We don’t want to have two different answers for the same computation. So mathematicians agreed on rules for what you do first. This is called the order of operations.”
- In Unit 3 Session 3.4 students are adding and subtracting mixed numbers. The materials prompt the teacher to state, “You used many of the same strategies for adding and subtracting mixed numbers that you have used for adding and subtracting whole numbers. It seems as though the main difference is sometimes you have to find ways to deal with adding or subtracting fractions that have different denominators.”
- In Unit 7 Session 2.1 students are dividing fractions. The materials prompt the teacher to state, “You know that ⅜ is a fraction, and it can also be used to represent a division problem, just like the other notations. In the next few sessions, we’re going to continue to think about fractions as division, and we’re also going to look at their decimal equivalents.”