6th Grade - Gateway 1
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Focus & Coherence
Gateway 1 - Partially Meets Expectations | 71% |
|---|---|
Criterion 1.1: Focus | 0 / 2 |
Criterion 1.2: Coherence | 4 / 4 |
Criterion 1.3: Coherence | 6 / 8 |
The instructional materials reviewed for Dimensions Math Grade 6 partially meet expectations for focus and coherence in Gateway 1. For focus, the instructional materials do not meet the expectations for assessing grade-level standards, but the amount of time devoted to the major work of the grade is at least 65 percent. For coherence, the instructional materials are partially coherent and consistent with the Standards. The instructional materials contain supporting work that enhances focus and coherence simultaneously by engaging students in the major work of the grade and foster coherence through connections at a single grade.
Criterion 1.1: Focus
The instructional materials reviewed for Dimensions Math Grade 6 do not meet expectations for not assessing topics before the grade level in which the topic should be introduced. The instructional materials include assessment items that align to standards above this grade level.
Indicator 1a
The instructional materials reviewed for Dimensions Math Grade 6 do not meet expectations for assessing grade-level content. The FAQ page on the website for Singapore Math states, “There are currently no tests, but the workbook could be used as a test bank.” In Dimensions Math workbooks 6A and 6B, above grade-level items are present and could not be modified or omitted without a significant impact on the underlying structure of the instructional materials. For example:
- Students solve multi-step percent problems involving discount, tax, and percent increase/decrease (7.RP.3). For example, in Lesson 7.2 pages 150-151, problem 6 states, “A pair of pants is discounted 25 percent. They now cost $104.25. What was the cost before the discount?”
- In Lesson 10.3, pages 58-63, students write equations in the form px + q = r (7.EE.4a). For example, problem 6 states, “The internet service at the airport costs $11 to sign on and an additional $2.50 per half hour. Let h represent the amount of time Frances used the internet, and t represent the total cost in dollars. Write an equation that represents this scenario.”
Criterion 1.2: Coherence
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 Dimensions Math Grade 6 meet expectations for devoting the large majority of class time to the major work of the grade. The instructional materials spend at least 65% of instructional time on the major work of the grade.
Indicator 1b
Instructional material spends the majority of class time on the major cluster of each grade.
The instructional materials reviewed for Dimensions Math Grade 6 meet expectations for spending a majority of instructional time on major work of the grade.
- The approximate number of chapters devoted to major work of the grade (including assessments and supporting work connected to the major work) is 11 out of 13, which is approximately 86 percent.
- The number of lessons devoted to major work of the grade (including assessments and supporting work connected to the major work) is 28.5 out of 34, which is approximately 84 percent.
- The number of days devoted to major work of the grade (including assessments and supporting work connected to the major work) is 107 out of 134, which is approximately 80 percent.
A lesson-level analysis (which includes lessons and sublessons) is most representative of the instructional materials because it addresses the amount of class time students are engaged in major work throughout the school year. As a result, approximately 84 percent of the instructional materials focus on major work of the grade.
Criterion 1.3: Coherence
Coherence: Each grade's instructional materials are coherent and consistent with the Standards.
The instructional materials reviewed for Dimensions Math Grade 6 partially meet expectations for being coherent and consistent with the Standards. The instructional materials contain supporting work that enhances focus and coherence simultaneously by engaging students in the major work of the grade and foster coherence through connections at a single grade. The instructional materials include an amount of content that is partially viable for one year, do not attend to the full intent of some standards, and do not give all students extensive work with grade-level problems.
Indicator 1c
Supporting content enhances focus and coherence simultaneously by engaging students in the major work of the grade.
The instructional materials reviewed for Dimensions Math Grade 6 meet expectations that supporting work enhances focus and coherence simultaneously by engaging students in the major work of the grade.
Supporting standards/clusters are connected to the major standards/clusters of the grade with two exceptions. No connections are explicitly stated. Examples of supporting work that engage students in the major work of the grade include:
- In Lesson 6.1, students calculate average weight, average height, and average distance (supporting standard 6.SP.5c), and these are connected to unit rates (major standard 6.RP.2).
- In Lesson 10.1, students graph points to draw a polygon on the coordinate plane (supporting standard 6.G.3), and this is connected to graphing points in all four quadrants (major standard 6.NS.8).
- In Lesson 1.4, students use division of multi-digit numbers (supporting standard 6.NS.2) when writing equivalent expressions and solving equations (major clusters 6.EE.A,B).
- In Chapter 3, students evaluate expressions (major cluster 6.EE.A) that include multi-digit decimals (supporting standard 6.NS.3).
- In Chapters 11 and 12, students evaluate expressions arising from area and volume formulas (supporting standards 6.G.1,2), and this connects to writing and solving equations for unknown lengths (major standards 6.EE.2,7).
- In Lesson 13.1B, students evaluate expressions (major standard 6.EE.1) to find the mean of a data set (supporting standard 6.SP.5c).
- In Lesson 13.2, students calculate percentages (major standard 6.RP.3) from analyzing histograms (supporting cluster 6.SP.B).
- In Lesson 6.2, students solve problems involving unit rates (major standard 6.RP.3) by dividing multi-digit numbers (supporting standard 6.NS.2).
- In Lesson 12.1, students use the ratio of length to width to height of a right rectangular prism (major standard 6.RP.3) to find the volume of the prism (supporting standard 6.G.2).
Examples of missed connections between supporting and major work include:
- These is no connection between finding factors (6.NS.4) and generating equivalent expressions (6.EE.3).
- In Lesson 11.1, problem 13 involves division of fractions (major standard 6.NS.1) within the context of area, surface area, and volume (supporting cluster 6.G.A). There are no other opportunities to connect 6.G.A and 6.NS.1.
Indicator 1d
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 for Dimensions Math Grade 6 partially meet expectations that the amount of content designated for one grade level is viable for one year.
As designed, the instructional materials can be completed in 134 days. The total days were computed in the following manner:
- Each lesson was counted as one day of instruction.
- A “lesson” with subsections (i.e., 1a, 1b, 1c) counted as three lessons or three days.
- A practice day was added for each chapter.
The total days were computed based on a pacing chart provided in the teacher guide. The suggested time frame for the materials and/or the expectations for teachers and students are not viable. Some significant modifications would be necessary for materials to be viable for one school year.
Indicator 1e
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 instructional materials for Dimensions Math Grade 6 partially meet expectations for being consistent with the progressions in the standards. In general, materials follow the progression of grade-level standards, though they don’t always meet the full intent of the standards. In addition, lessons utilize standards from prior grade levels, though these are not always explicitly identified in the materials.
Examples where standards from prior grades are utilized but not identified include:
- In Lesson 1.1, students write numeric expressions for statements (5.OA.2). This material is not identified as content from a prior grade.
- In Lesson 2.1, the materials reference Multiplication of a Proper Fraction by a Whole Number as learning from a previous grade, but the materials do not identify Multiplication of a Proper Fraction by a Fraction as previous learning. Instead, the materials treat this topic and Division of a Whole Number by a Fraction and Division of a Fraction by a Whole Number (Lesson 2.2) as grade-level topics, though they are prior learning (5.NF.4).
- In Lesson 1.4, students examine division as sharing and division as grouping (3.OA.2 & 3.OA.6), but the material do not reference this as prior learning.
- Lesson 3.4 is not identified as work from a prior grade (5.MD.1).
The materials typically develop according to the grade-by-grade progressions in the standards, but one missed opportunity is in the unit on The Number System (Chapters 1-3). The standards include students representing numbers on a number line, but students are not given that opportunity. The model most commonly used in this unit is the bar model. There is some emphasis on a number line with the introduction of integers to help students compare values.
The “Notes on Teaching” in Teaching Notes and Solutions provide some direction for teachers to explicitly relate the content to prior learning:
- In Lesson 2.1, the lesson states, “In Grade 5, we interpreted a fraction as a division of the numerator by the denominator…let’s recap the multiplication of a fraction by a whole number.”
- In Lesson 3.1B, adding and subtracting decimals are connected to prior knowledge: “Since both whole numbers and decimals are written using the base-ten number system of numbers, we can use the same method for adding and subtracting whole numbers to add and subtract decimals.”
- In Lesson 11.1, the materials state, “In earlier grades, we learned the area of a rectangle can be found by multiplying the side lengths.”
Examples where the student workbooks reference content learned in earlier grades include:
- In Workbook 6B, page 109 states, “In earlier grades, we learned that the area of a rectangle can be found by multiplying the side lengths. That is, area of a rectangle = length x width.”
- In Workbook 6B, page 154 states, “In earlier grades, you have already come across shapes like rectangular prisms and cubes. To review, a rectangular prism is a three-dimensional solid shape.”
The instructional materials do not attend to the full intent of some standards. Examples include:
- In Chapter 1, students find the GCF of two numbers but do not have an opportunity to “use the distributive property to express a sum of two whole numbers 1-100 with a common factor as a multiple of a sum of two whole numbers with no common factor,” as stated in 6.NS.4.
- In Lesson 3.4, students convert measurements to different units, but ratio reasoning is not used for these conversions (6.RP.3d).
- In Chapter 5, rate language is not used to develop the concept of ratios (6.RP.2).
- In Chapters 5 and 6, ratios are not represented with tables (6.RP.3a).
- Unit rate is defined in Chapter 6 on page 172, but there is no opportunity for students to “understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0” (6.RP.2).
- In Chapter 8, students write algebraic expressions for given statements but do not have an opportunity to “identify parts of an expression using mathematical terms.” (6.EE.2b)
- The materials do not provide an opportunity for students to “view one or more parts of an expression as a single entity.” (6.EE.2b)
- In Chapter 10, students use coordinates and absolute value to find distances between points on a coordinate plane (6.NS.8) but do not apply this understanding to real-world problems.
The materials do not give all students extensive work with grade-level problems for some standards. Examples include:
- In Chapter 2, students divide whole numbers, and in Chapter 3, students divide decimals by decimals using the standard algorithm. The materials do not provide opportunities for extensive work dividing multi-digit numbers using the standard algorithm (6.NS.3).
- On pages 175 and 189, students examine the net of a triangular prism (6.G.4), but they do not “represent three-dimensional figures using nets made up of rectangles and triangles, and use nets to find the surface area of these figures,” as stated in the standard.
Indicator 1f
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 for Dimensions Math Grade 6 meet expectations for fostering coherence through connections at a single grade, where appropriate and required by the standards.
The materials include learning objectives that are visibly shaped by CCSSM cluster headings, and there are correlations between Dimensions Math Grade 6 learning objectives and CCSSM cluster headings. Examples include:
- In Chapter 5, learning objectives are shaped by 6.RP.A, “Understand ratio concepts and use ratio reasoning to solve problems.”
- Examples of learning objectives shaped by 6.RP.A are: “Compare quantities using ratios and use ratio language to describe a ratio relationship between two quantities; simplify ratios to obtain equivalent ratios; and relate ratios and fraction and apply ratio relationships to solve real-world problems.” In Chapter 4, some learning objectives are: “Recognize the use of positive and negative numbers in the real-world context; compare and order positive and negative numbers and plot them on a number diagram; and interpret the absolute values of a positive and negative quantity in real-world situations.” These objectives are shaped by 6.NS.C, “Apply and extend previous understandings of numbers to the system of rational numbers.”
- In Chapter 2, some learning objectives are: “multiply fractions by whole numbers; multiply fractions by fractions; divide whole numbers by fractions and fractions by whole numbers; divide fractions by fractions; and solve word problems involving multiplication and division of fractions.” These objectives are shaped by 6.NS.A, “Apply and extend previous understandings of multiplication and division to divide fractions by fractions.”
The materials include problems and activities that serve to connect two or more clusters in a domain or two or more domains in a grade. Examples include:
- In Chapters 11 and 12, students use formulas to calculate area, volume, and surface area (6.G.A) involving measures that are given in both decimal and fraction forms (6.NS.B).
- In Chapter 13, students investigate appropriate use of measures of center in different contexts (6.SP.A) and make comparisons among the three measures (6.SP.B).
- In Lesson 8.1, students evaluate expressions (6.EE.A) using multiplication and division of fractions (6.NS.A).
- In Lesson 9.1, students solve equations (6.EE.B) with fractional coefficients (6.NS.A).
- In Lesson 1.3, students model the distributive property (6.NS.B) to solve area problems (6.G.A).
- In Lesson 10.3A, students write and solve equations (6.EE.B) to describe relationships between dependent and independent variables (6.EE.C).
Students do not compare rates of two or more quantities using graphs of quantities, missing a connection between 6.NS.C and 6.RP.A.