6th Grade - Gateway 1

The instructional materials reviewed for Achievement First Mathematics Grade 6 meet expectations that they assess grade-level content. Assessment questions are aligned to grade-level standards. No examples of above grade-level assessment items were noted. Each unit contains a Post-Assessment which is a summative assessment based on the standards designated in that unit. The assessments contain grammar and/or printing errors which could, at times, interfere with the ability to make sense of the materials. Examples of assessment items aligned to grade-level standards include: 

• In Unit 1 Assessment, Question 11 states, “Write an equivalent expression to $$40 + 32$$ using the distributive property. The sum of two whole numbers within the parentheses should have no common factor greater than 1.” (6.NS.4)
• In Unit 5 Assessment, Question 6 states, “Manufacturer A requires 70 buttons for the manufacture of 10 shirts. Manufacturer B requires 68 buttons for the manufacture of 17 shirts. How many buttons will each require to manufacture 15 shirts?” (6.RP.3b)
• In Unit 7 Assessment, Question 6 states, “Given the set {4.2, 4.7, 5}, determine which of the values are solutions to the inequality, $$4.3 + x < 9$$. Defend your answer by explaining why or why not for each of the values.” (6.EE.5)
• In Unit 9 Assessment,  Question 7 states, “Julia is repainting her jewelry box with a length of 8 inches, a height of 5.5 inches, and a width of $$3\frac{1}{2}$$ inches. She is painting all 4 sides blue, the top purple, and she is not painting the bottom. How many square inches of each paint color does Julia need?” (6.G.4) 
• In Unit 10 Assessment, Question 1 states, “For each of the following, identify whether or not it would be a valid statistical question you could ask about people at your school. Explain for each why it is, or is not, a statistical question. a) What was the mean number of hours of television watched by students at your school last night? (Yes, this is a valid statistical question because there is more than one answer and you can collect information from multiple sources.) b) What is the school principal’s favorite television program? c) Do most students at your school tend to watch at least one hour of television on the weekend?” (6.SP.1)

The instructional materials reviewed for Achievement First Mathematics Grade 6 meet expectations that supporting work enhances focus and coherence simultaneously by engaging students in the major work of the grade. Although connections are rarely explicitly stated, problems clearly connect supporting and major work throughout the curriculum. Examples where supporting work enhances major work include:

• In Unit 1, Lesson 16, supporting standard 6.NS.4 enhances the major work of 6.EE.3. Students use the distributive property to create an expression that expresses the sum of two whole numbers between 1-100 and explains how to apply the concept of factoring to the distributive property. For example, Independent Practice Question 7 (Master level) states, “What values of a and b make the two expressions below equivalent? $$36 + 54 = a(2 + b)$$.” 
• In Unit 3, Lesson 12, supporting standard 6.G.3 enhances major work standards 6.NS.6 and 6.NS.8. Students solve real-world and mathematical problems that involve points, lines, and polygons on the coordinate plane. In Independent Practice Question 4 (PhD level), students are given a coordinate graph and instructed, “Edwina is creating a diagram of her bedroom so that she can plan how to rearrange her furniture before moving anything. Use the following information below to help Edwina rearrange her room. Her room is in the shape of a rectangle. The area of her room is 86 square feet. Her bed covers 48 square feet of the floor. Her dresser covers 3 square feet of the floor. Her two night stands each have the dimensions $$\frac{3}{4}$$ ft. by $$1\frac{1}{4}$$ ft. Use the information provided above and draw a plan for arranging Edwina’s furniture. Start by drawing the floor of her room. Be sure to label all vertices with coordinate pairs and label the dimensions of each figure you create.”
• In Unit 4, Lesson 11, supporting standard 6.NS.3 enhances major work standards 6.RP.2 and 6.RP.3. Given a unit rate, students find ratios associated with the unit rate and recognize that all ratios associated to a given unit rate are equivalent. For example, Independent Practice #3 (Master level) states, “Aubrey has to type a 5-page article but only has 18 minutes until she reaches the deadline. If Aubrey is able to type at a constant rate of 0.25 page every 1 minute, will she meet her deadline? Show your work to defend your answer.”
• In Unit 9, Lesson 4, supporting standard 6.G.2. enhances the major standard 6.EE.7. Students apply the formulas V = lwh and V = bh to find the volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems. For example, Independent Practice Question 5 (Master level) states, “A rectangular box is going to be filled with sand. The length of the box is 412 feet. The width $$2\frac{1}{4}$$ feet and the height is 9 feet. If sand is sold in bags of 12 cubic feet, how many bags of sand will be needed to fill the box?”

Instructional materials for Achievement First Mathematics Grade 6 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 140 days. For example:

• There are 10 units with 130 lessons total; each lesson is 1 day. 
• There are 10 days for summative Post-Assessments.
• There is an Optional Cumulative Project at the end of Unit 10 on Statistics. The amount of time is not designated. Since it is optional, it is not included in the total count.

According to The Guide to Implementing Achievement First Mathematics Grade 6, each lesson is completed in one day, which is designed for 90 minutes. 

• Each day includes a Math Lesson (55 minutes) and Cumulative Review (35 minutes). 
• The Implementation Guide states, “If a school has less than 90 minutes of math, then the fluency work and/or mixed practice can be used as homework or otherwise reduced or extended.”

The instructional materials for Achievement First Mathematics Grade 6 meet expectations for the materials being consistent with the progressions in the Standards. 

The materials clearly identify content from prior and future grade levels and use it to support the progressions of the grade-level standards. These connections are made throughout the materials including the Implementation Guide, the Unit Overviews, and the lessons. For example:

• The Unit Overview includes “Previous Grade Level Standards and Previously Taught and Related Standards” which describes in detail the progression of the standards within each unit. For example, Unit 3 states, “Prior to 6th grade, students learned to represent positive rational numbers on a number line (3.NF.2 and 4.NF.6) and to plot points in the first quadrant of the coordinate grid (5.G.A). Students have also compared and ordered positive rational numbers, including decimal fractions (5.NBT.3b) and fractions (4.NF.2). This prior knowledge of location on a number line, coordinate geometry, and ordering positive numbers is fundamental for students being introduced to negative rational numbers for the first time in this unit.” The end of this section adds, “Mastery of this unit is essential to 6th grade and in future grades. In later units in 6th grade, students apply coordinate geometry when working with areas of a variety of polygons. In the 7th grade, students rely heavily on the number line to make sense of and form generalizations about rational number operations. They apply rational numbers to represent and solve real world and mathematical problems as well as to evaluate expressions and solve equations and inequalities. Additionally, students graph proportional relationships on the coordinate plane. In 8th grade, students work heavily on the coordinate plane as they learn about transformations, functions, linear equations, systems of equations, and bivariate data. Students also work with rational number operations when solving equations and systems of equations. Rational numbers and coordinate geometry continue to be integral throughout High School mathematics as well.”
• Throughout the narrative for the teacher in the Unit Overview, there are descriptions of how the lessons will be used as the grade level work progresses. For example, Unit 8 Overview states, “In lesson 6, students use what they learned about calculating the area of a triangle to derive the formula for the area of trapezoids.”
• The last paragraph of each narrative for the teacher in the Unit Overview describes the importance of the unit in the progressions. For example, Unit 7 states, “This unit is essential for 6th grade and future grades. Students will work with equations and inequalities throughout all future math courses. It is imperative in 6th grade that the conceptual foundations for equations and inequalities are deeply understood in order to set students up for more abstract manipulation and application of equations and inequalities in future grades. Students learn to represent problems with and solve multi-step equations and inequalities using inverse operations and number properties in 7th grade and continue to apply their understanding of and skills with equations and inequalities throughout the remainder of their math career.”
• For units that correlate with the progressions document, the materials attach the relevant text so that connections are made. For example, in Unit 7, Appendix A: Teacher Background Knowledge (after the assessment), the “6-8 Expression and Equations” progression document is included with the footnote, “‘Common Core Expressions and Equations Progressions 6-8’ by Common Core Tools. Achievement First does not own the copyright in ‘CC Expressions and Equations Progression’ and claims no copyright in this material.”
• Each lesson includes a “Connection to Learning and Conceptual Understanding” section that describes the progression of the standards within the unit. In Unit 5, Lesson 11 states, “In the previous four lessons, students learned to use a DNL” (double number line) “to find an unknown percent, part, and total given two other pieces of known information. In this lesson, students make a connection between using an equation and a DNL in order to make sense of and use equations to solve ‘percent of’ problems. This is the first day of two days working with equations to solve percent problems.”
• In the Scope and Sequence Detail from the Implementation Guide, there are additional progression connections made. The Cumulative Review column for each unit provides a list of lesson components and the standards addressed. Prior (Remedial) standards are referenced with an “R” and grade level standards are referenced with an “O.” In Unit 2, The Number System- Dividing Fractions states, “Skill Fluency (4 days a week): 6.NS.3 (O)* Division, 6.NS.2 (O), 6.NS.4 (O)* GCF,LCM, Distributive Prop. Mixed Practice (3 days a week): 5.NBT.3 (R), 5.NBT.4 (R), 6.NS.3 (O), 6.NS.1 (O).”

The materials attend to the full intent of the grade-level standards by giving all students extensive work with grade-level problems. Each lesson provides State Test Alignment practice, Exit Tickets, Think About It, Test the Conjecture or Exercise Based problems, Error Analysis, Partner Practice, and Independent Practice, which all include grade-level practice for all students. The Partner and Independent Practice provide practice at different levels: Bachelor, Masters and PhD. Each unit also provides Mixed Practice, Problem of the Day, and Skill Fluency practice. By the end of the year, the materials address the full intent of the grade-level standards. Examples include:

• In Unit 1, Mixed Practice 1.3, Day 1, Question 4, students fluently subtract and multiply multi-digit decimals using the standard algorithm. The materials state, “Rashad drove at an average speed of 50.55 miles per hour for 1.75 hours. He stopped at a rest stop and then drove at an average speed of 45.2 miles per hour for 2.25 hours. Did Rashad drive more miles before or after the rest stop? How many more miles?” (6.NS.3)
• In Unit 2 Lesson 7, Independent Practice Question 3 (Masters level), students use ratio and rate reasoning to solve percent problems. The materials state, “Steven orders 8 pizzas for his birthday party. He expects that each person will eat 25% of a pizza. How many people can attend the party based on his prediction? Use a model to prove your answer.” (6.RP.3c)
• In Unit 6, Lesson 7, Independent Practice Question 3 (Master level), students understand that a variable can represent an unknown number. The materials state, “Jeff planned to run a few miles over a certain number of weeks. He planned to run half a mile each weeknight and 4 miles on Saturday. Which expression(s) shows how many miles Jeff planned to run over a certain number of weeks. Circle all that apply. a) $$\frac{5}{2}+4$$; b) $$w(\frac{5}{2}+4)$$; c) $$\frac{1}{2}w+4$$; d) $$\frac{5}{2}w+10$$; e) $$\frac{5}{2}+4w$$; f) $$5w+(2)(4)$$.” (6. EE.6)
• In Unit 7, Lesson 3, Independent Practice, Question 9 (PhD level), students write and solve real-world problems. The materials state, “Michael bought 8 Granny Smith apples, 7 Macintosh apples, and p Red Delicious apples. She bought a total of 27 apples. Write and solve an equation that represents this problem.” (6.EE.7)

The instructional materials relate grade-level concepts explicitly to prior knowledge from earlier grades. This can be found in the progressions descriptions listed above, but also often focuses explicitly on connecting prior understanding. For example:

• Each Unit Overview provides a narrative for the teacher that introduces the student learning of the unit and the background students should have. In Unit 10, Overview, Statistics and Probability – Representing and Analyzing Data states, “Prior to this unit, students have had little exposure to statistics. Throughout their elementary schooling, however, they do talk about data analysis in each grade. While their study of measuring, representing, and interpreting data starts in $$1^{st}$$ grade, I will quickly review the previous three years. In $$3^{rd}$$ grade (3MDB), students generate, represent, and interpret data in bar graphs. In $$4^{th}$$ (4MD4) and $$5^{th}$$ (5MD2) grades, students represent and interpret data using line/dot. Unit 10 is the first time students learn about statistical questions and measures of center and variability. They also learn about new graphical representations – the box plot, frequency table, and histogram. On account of this, the unit focuses the majority of the time on these topics in order to develop students’ understanding of statistical representations and analysis.”
• The narrative for the teacher in the Unit Overview makes connections to current work. Unit 8 states, “Additionally, in $$5^{th}$$ grade, students used area models as a way to understand multiplication and division of whole numbers as well as multiplication of fractions. All of this work with area provides students with (a) strong base understanding of the concept from which to build as students dive deeper into their study of area in Unit 8.”
• Each lesson includes a “Connection to Learning and Conceptual Understanding” section that relates to prior knowledge. In Unit 1, Lesson 11 Connection to Learning explains, “More directly, students learn to find factor pairs of numbers between 1 and 100, and identify numbers as prime or composite in 4th grade (4.OA.4). Ss (students) build directly off of this foundational knowledge in this lesson. FENCEPOST #1: Factors are numbers that you multiply together to get a product. FENCEPOST #2: Common factors are factors shared by two numbers.”
• In the Scope and Sequence Detail from the Implementation Guide, the Notes + Resources column for some lessons includes a lesson explanation that makes connections to prior learning. Unit 6 states, “Students have been taught the order of operations in 5th grade. The inclusion of exponents into the process is new.”