6th Grade - Gateway 2

The instructional materials for Achievement First Mathematics Grade 6 partially meet expectations that the materials are designed so that teachers and students spend sufficient time working with engaging applications of the mathematics. Students are given multiple opportunities to engage in real world applications especially within exercise based lessons as well as the problem of the day in each cumulative review. However, students do not have consistent opportunities to explore non-routine problems.

The instructional materials include multiple opportunities for students to engage in routine application of mathematical skills and knowledge of the grade level. Students are rarely presented with problems that involve a context that they have not already practiced. Examples include:

• In Unit 2, Mixed Practice 2.3, Day 3, Question 3, students apply skills related to fraction division. The materials state, “Ellie ran a $$1\frac{3}{4}$$ mile special race course. Every $$\frac{1}{8}$$ mile, there was an obstacle, with a final obstacle at the finish line. How many obstacles did Ellie encounter in the race? Use a model and/or words to explain your thinking.” (6.NS.1)
• In Unit 3, Lesson 12, Independent Practice, Question 3 (Master), students solve real life problems such as graphing points on the coordinate plane and using them to find area. The materials state, “Mason Rice Elementary School is creating a new playground in the park behind the school. The playground’s perimeter is rectangular and is 60 feet long with a width of $$15\frac{1}{2}$$ feet. The planning committee is drafting the design of the new playground on a coordinate grid. They started by placing one corner of the park at (-8, 8). Each unit on the coordinate plane represents 1 foot. a) Plot the other three corners of the playground, label the coordinates of each corner, and connect the corners to create a rectangle. b) The committee is planning on splitting the playground diagonally in order to make two separate spaces for younger kids and older kids. Draw a line that divides the playground diagonally. How many square feet of space is the committee allocating for each part of the playground?” (6.NS.6c, 6.NS.8, 6.G.1)
• In Unit 5, Problem of the Day 5.2, Day 2, students apply skills related to solving problems using ratio reasoning. The materials state, “Gylissa and Alicia are developing a business of making and selling slime. The table below shows corresponding amounts of all ingredients they use to make their slime. Part A: If Gylissa and Alicia always use the same recipe when making slime, what are the values of $$x$$ and $$y$$? Part B: Gylissa and Alicia receive a huge order of 6 cups of slime for each student in their class of 24 students. How many cups of each ingredient will they need to fill the order? Part C: Jadine and Tiarah also decide to make a slime business, but their recipe uses 6 cups of water, 8 cups of glue, and 3 cups of borax. Whose recipe will make a stickier slime? Show your work below to prove your answer.” (6.RP.3a)
• In Unit 7, Lesson 7, Independent Practice, Question 7 (PhD level), students apply skills related to writing and solving one-step equations. The materials state, “Nadia bought five food tickets that each cost $$x$$ dollars and three drink tickets that each cost $2 to attend a spaghetti fundraiser at her school. She spent a total of $33.50. Write an equation that represents the cost of each food ticket.” (6.EE.7)
• In Unit 7, Lesson 12, Partner Practice, Question 2 (Master Level), students represent and analyze quantitative relationships between dependent and independent variables. The materials state, “Sam drove his car at a constant speed for t minutes and traveled a total of $$m$$ miles. This relationship is represented in the table below. (3 data points provided leading to $$1.5t = m$$) If Sam drove 14.25 miles in all, how many minutes had he been traveling?” (6.EE.9)
• In Unit 9, Problem of the Day 9.2, Day 3, Question 1, students apply skills related to solving rate problems using decimals. The materials state, “Alan and his family are on their way to visit a national park that is 780 miles from their house. They choose a route so that it will take them 3 days to get to their destination. The first day they drive for 4 hours at an average speed of 60 miles per hour. The second day they drive for 6 hours at an average speed of 65 miles per hour. If the average speed on the third day is 60 miles per hour, how many more hours will it take them to reach the national park? Show your work.” (6.RP.3b, 6.NS.3)

The instructional materials reviewed for Achievement First Mathematics Grade 6 partially meet expectations that the Standards for Mathematical Practice are identified and used to enrich mathematics content within and throughout the grade level. The Mathematical Practices are listed in the Unit Overviews as well at the beginning of each lesson, however there is little direction provided about how the MPs enrich the content or make connections to enhance student learning.

All eight MPs are clearly identified throughout the materials, with few or no exceptions, though they are not always accurate. For example:

• In the Guide to Implementing AF Math, “Math Lesson Types” explains how different types of lessons engage students with the Mathematical Practice Standards. For example, Conjecture Based Lessons states, “Purpose: Through the use of investigation and guided inquiry, students develop conceptual understanding of math topics and strategies. They persevere by developing and proving mathematical conjectures. Structurally based off of the Standards for Mathematical Practice 3, these lessons push students to make viable arguments and critique the thinking of others to generate a conjecture that will then be tested. They must make connections to previously learned content, apply sound mathematical practices, and think flexibly.”
• The MPs are listed at the beginning of each lesson in the Standards section. 
• All MPs are represented throughout the materials, though lacking balance. For example, MP5 is emphasized in two units, while MP6 is emphasized in ten.
• The Mathematical Practices are not always identified accurately. For example:
• At the unit level for Unit 2, MP3 is not identified as an emphasized practice. However, at the lesson level, 11 out of the 12 lessons identify it as connected. MP6 and MP7 are bold (indicating emphasis) at the unit level, but aren't connected in any of the lessons.
• The Unit 5 Overview bolds MPs 1, 5, 6 and 7 as the emphasis of the unit. However, eight of the lessons bold MP2, and 10 lessons bold MP4, while MP7 is only bolded in Lesson 6. 
• In Unit 7, MPs 2, 4, 6, 7, and 8 are bold; however, MP7 does not appear in any of the lessons in this unit. 
• There is no stated connection to the MPs within the skill fluency, mixed review, problem of the day, or assessments. In the Guide to Implementation in Problem of the Day Overview it explains, “The problem of the day provides students with practice applying mathematical practices and multiple standards to a rigorous problem.” While the learning standards are listed for these problems, the relevant MPs are not identified.

There are a few instances where the MPs are addressed, but are not clear in the content. For example:

• It is generally left to the teacher to determine where and how to connect the emphasized mathematical practices within each lesson.
• There are connections to the content described in the Overview, though not specifically linked to an MP. If a teacher was not familiar with the MPs, the connection may be overlooked. Examples include:
• Unit 4 Overview states, “In lesson 2, students use tape diagrams to model ratios and solve problems given a part-to-part ratio and one quantity from the second ratio. Students understand that a tape diagram is used to model relationships that involve ‘like units’ because the size of each part is the same." 
• Unit 5 Overview states, ”In lessons 6-12, students are introduced to percentages for the first time and apply their understanding of ratios and rates to make sense of and solve problems involving percentages.”
• Unit 7 Overview states, “In lesson 6, students understand that mathematical situations and structures can be translated and represented abstractly using one-step equations that are made up of variables, symbols, and numbers.”

Unit 1 is the exception in the materials. The Unit 1 Overview makes connections to help teachers understand how to emphasize the content to incorporate the MPs. None of the other units make these connections. For example:

• MP1 - Unit 1 Overview states, “In lesson 10, students solve problems that incorporate all four operations with decimals and multiple steps. The lesson is intended to push students to make decisions regarding the correct operation to utilize given known and unknown information as well as to help them make sense of multi-step problems and apply their learning flexibly (SMP.1; 6.NS.3).” 
• MP2 - Unit 1 Overview states, “By the end of the lesson, students are fluent in creating and recognizing equivalent division expressions to find quotients when divisors are decimal numbers. Again, throughout the division lessons, teachers consistently reinforce the use of estimation and reasoning about the magnitude of a quotient relative to the dividend to gauge reasonableness (SMP.2).”
• MP7 - Unit 1 Overview states,  “...Once they have realized that, they can factor out the GCF from the addends and rewrite the expression as the product of the GCF and the sum of the two other factors (SMP.7).”
• MP8 - Unit 1 Overview states, “Given a pair of addends made up of whole numbers from 1-100, students recognize that they can find factors of the addends as a strategy to apply the distributive property to rewrite the expression. Students recognize that both addends are multiples with common factors (SMP.8).”

The instructional materials reviewed for Achievement First Mathematics Grade 6 partially meet expectations that the instructional materials carefully attend to the full meaning of each practice standard. 

The materials do not attend to the full meaning of two MPs. For example:

• MP4: Model with mathematics - Students have limited opportunities to develop their own solution pathways that would best support mathematical tasks and are often directed to represent the problem in a certain way. 
• In Unit 3, Lesson 1, Think About It!, students find the absolute value of numbers. The materials state, “Use the number line below to answer the following prompts. a) Labeling the number line has been started for you. Label all the hash marks to the left of 0; b) What is the relationship between 2 and -2?” Students complete a number line without developing any strategies or applying any real-world context.
• In Unit 5, Lesson 6, Independent Practice, Question 1, (Bachelor Level), students use a five-column chart to represent a value as a percentage - decimal - fraction - ratio - model (given a 10x10 grid). In this example, MP4 is a visual representation but students do not choose a strategy, nor is there any real-world connection.
• In Unit 7, Lesson 5, Independent Practice, Question 5, (Master Level) states, “Directions: Draw a model to solve each equation and check your answer using substitution. $$\frac{3.15}{k}=0.5$$.” Again, students create a visual representation without any real-world context. 

• MP5: Use appropriate tools strategically - Students have limited opportunities to choose tools that would best support mathematical tasks and are often provided with only one type of tool to solve a problem. MP8 is only identified in Unit 8 of the materials. 
• In Unit 8, Lesson 8, students calculate the area of figures given a coordinate plane. There is no opportunity to choose a tool to help solve the problems, all problems include a coordinate grid that is pre-numbered and labeled. 
• In Unit 8, Lesson 9, Partner Practice, Question 1, (Bachelor Level) states, “Determine the area of the figure drawn on the grid below.” The grid provided is pre-numbered and labeled; students do not have a chance to choose tools. 
• In Unit 8, Lesson 11, students decompose compound figures and calculate area. Students are not given a choice of tools, as calculators are simply listed in the materials.

The following practices are connected to grade-level content and are developed to their full intent over the course of the materials. 

• MP1: Make sense of problems and persevere in solving them  
• In Unit 2, Lesson 10, Independent Practice Question 6 (PhD level), students find an entry point and persevere to solve a multi-step problem involving rational numbers. The materials state, “Amare paid $16.50 to buy a book. The cost of the book is $$2\frac{1}{2}$$ the cost of a magazine. He bought 3 books and 2 magazines with a $50 bill. How much change should he receive?”
• In Unit 7, Lesson 5, Independent Practice, Question 8 (PhD Level), students make sense of an unusual situation in order to find a solution. The materials state, “A toy-maker is attempting to make a stackable set of discs to help recreate the famous Tower of Hanoi problem. To do this, the toy-maker wants to create discs whose diameters decrease by 1 inch for each new level of the stacked discs. The toy-maker wants to eventually have 5 discs stacked on top of each other and the largest disc will be 10 inches in diameter. What will be the diameter of the smallest disc?”
• In Unit 8, Lesson 7, Interaction with New Material, students are provided with an unusual shape and the additional challenge of determined cost. The materials state, “Bob the builder is constructing a lot with the shape and dimensions below. (Given a trapezoid with all 4 sides and the height labeled.) Part A: Before starting to build, he ropes off the entire lot to ensure no one walks on it during construction. How many feet of rope does he need? Part B: After roping off the lot, he covers the entire lot with a layer of cement. Cement costs $4 per bag and each bag covers 20 sq. ft. How much does he need to spend on cement?”

• MP2: Reason abstractly and quantitatively 
• In Unit 2, Lesson 3, Independent Practice Question 2 (Master Level), students have to use ratio reasoning with fractions and then put their answer back into context in order to answer the question. The materials state, “Melanie is planning a hiking trip. She knows that she estimates that she will finish a liter of water every $$\frac{3}{4}$$ mile that she hikes. Using the table below (given 3 trails and their distance), answer each of the following questions. a) How many liters of water will Melanie drink if she does the Lily Pond loop? b) How many liters of water will Melanie drink if she hikes to Starlight point and back?”
• In Unit 5, Lesson 9, Independent Practice, Question8 (PhD level), students use percent reasoning to quantitatively derive a final solution. The materials state, “Peter and his friends are all driving from Boston to New York. Peter is driving 120 miles per hour. Lauren is driving 80% as fast as Peter. Gabe is driving 75% as fast as Lauren. And, Deshawn is driving 25% as fast as Lauren. What percent of Peter’s speed is Deshawn driving?” 
• In Unit 7, Lesson 14, Exit Ticket, students reason abstractly and quantitatively as they solve and graph the solution for one-step inequalities. The materials state, “For each inequality below, solve, graph the solution, and check your answer. a) $$20≥m+7$$; b) $$\frac{1}{3}<4$$”

• MP6: Attend to precision  

• In Unit 3, Lesson 11, Independent Practice, Question 1, (Bachelor Level), students must accurately create an absolute value expression to calculate the distance of line segments. The materials state, “Find the lengths of the line segments whose endpoints are given below. Write an expression using absolute value to represent the length of each line segment. a) (-3, 4) and (-3, 9); b) (2, -2) and (-8, -2); c) (0, -11) and (0, 8).”
• In Unit 6, Lesson 3, Exit Ticket Question 3, students must clearly communicate using accurate vocabulary. The materials state, “When you take a taxi, the driver charges an initial fee when you start a ride and then an additional charge for every mile driven in the cab. If the total fare for a cab ride is $$3.5m + 2.5$$ after riding for m miles, what does the 2.5 represent? Explain (use appropriate math vocabulary in your explanation).”
• In Unit 7, Lesson 6, Independent Practice, Question 4, (Master Level), students must understand mathematical symbols in order to demonstrate understanding of a description. The materials state, “Which equation(s) represent the following description? Select all that apply. One-fourth of a number is $$3\frac{1}{4}$$. a) $$\frac{1}{4}x=3\frac{1}{4}$$; b) $$x-\frac{1}{4}=3\frac{1}{4}$$; c) $$x\div\frac{1}{4}=3\frac{1}{4}$$; d) $$x\div4=3\frac{1}{4}$$; e) $$(4)(3\frac{1}{4})=x$$; f) $$x=3\frac{1}{4}+3\frac{1}{4}+3\frac{1}{4}+3\frac{1}{4}$$.”

• MP7: Look for and make use of structure
• In Unit 8, Lesson 1, Opening, THINK ABOUT IT!, students understand that rectangles with the same area can have different dimensions and perimeters. The materials state, “Mr. Boyer is creating a rectangular garden outside of his house that has an area of 18 square units. Using the grid below, draw all the possible gardens that he can create with whole number dimensions. Fill out the table below for each of the figures you drew. Mr. Boyer believes that no matter which garden he chooses, the perimeter will be the same. Do you agree or disagree? Explain.”
• In Unit 6, Lesson 12, Think About It! states, “You can represent a number next to parentheses as repeated addition.” During instruction, the teacher states, “The expression $$2(a + 4)$$ means, ‘$$2 × (a + 4)$$.’ Using this understanding, write two equivalent expressions to $$2(a + 4)$$ and explain how you came up with the two expressions.” Students create various examples, and provided with the conjecture, “Applying the distributive property creates equivalent expressions.” Then students are guided through identifying a “quicker route” to use the distributive property. Students are guided through two expressions in Test the Conjecture, followed by independent practice. 
• In Unit 6, Lesson 2, Think About It!, students analyze the structure of expressions with order of operations. The materials state, “Antoine, Rosa, and Michelle are having a debate about how to evaluate an expression. Analyze their work and settle the debate by explaining: 1) Who solved correctly. 2) How you know that they solved correctly.”

• MP8: Look for and express regularity in repeated reasoning 

• In Unit 1, Lesson 8, THINK ABOUT IT!, Debrief, students have the opportunity to generalize understanding about multiplying by form of 1 using powers of 10. The teacher prompts, “Let’s zoom in on A and C. Aside from multiplying each expression by a form of one to create an equivalent expression, what else do you notice is similar about how equivalent expressions were created? Why do you think that similarly is important?  Discuss. SMS: I notice that in both cases the first expression is being multiplied by a form of one made up of two powers of ten $$-\frac{10}{10}$$ in the first and $$\frac{100}{100}$$ in the second. This is necessary because multiplying by a power of ten shifts the digits and in both cases we are now dividing by a whole number divisor. We don’t know how to divide by a decimal divisor, so this step helps us create an expression that we can evaluate. BPQ: What is the impact of multiplying by $$\frac{10}{10}$$ on the dividend and divisor in part A? What about the impact of multiplying by $$\frac{100}{100}$$ in part C? BPQ: What is different about the divisors in the equivalent expressions from the original expressions? How do you think that would be helpful for division? BPQ: Why did the S multiply by a $$\frac{10}{10}$$ and not another form of 1, like $$\frac{6}{6}$$?”
• In Unit 4, Lesson 11, Independent Practice, Question 6 (Master Level), students develop an understanding of repeated addition to create equivalent rates and ratios, and determine that all ratios with the same unit rate are equivalent. The materials state, “At Fun Burger, the burger master can make 25 hamburgers every 5 minutes. The master’s apprentice can make 35 burgers every 7 minutes. The master says that the apprentice still has much to learn because it takes the apprentice more time to make burgers. The apprentice says that they make the same number of burgers in the same amount of time. With whom do you agree? Explain why.”
• In Unit 6, Lesson 1, Think About It, students have the opportunity to make sense of repeated reasoning as they evaluate exponents. The materials state, “Complete the table below. Explain how you came up with the exponential expressions.” The table includes the multiplication expression, product, and exponential expression.