7th Grade - Gateway 2

The instructional materials for Achievement First Mathematics Grade 7 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, Lesson 10, Exit Ticket, Question 2, students apply skills related to operations with rational numbers. The materials state, “An hourglass loses $$8\frac{1}{4}$$ oz of sand every five minutes. How much sand will be in the hourglass after a half hour if it starts with 50 oz and 20.4 oz are added at the end of the 30 minutes?” (7.NS.3)
• In Unit 3, Lesson 7, Exit Ticket, Question 1, students solve real life problems such as football yardage using positive and negative numbers. The materials state, “Nquan is playing in a football game and rushes for 7.4 yards on the first play. For the next three plays, he loses y yards on each play. The last play he gains 1 yard. Write an algebraic expression to represent this situation and solve it if $$y=3\frac{1}{2}$$ yards.” (7.EE.3)
• In Unit 4, Lesson 8, Independent Practice, Question 9 (PhD Level), students apply skills related to reasoning about quantities by constructing simple equations. The materials state, “Ben and Jerry saved up their pennies to buy a present for their dad’s birthday. By the end of the first week, Ben had saved $15 dollars and Jerry had saved d dollars. By the end of the second week, they had tripled their savings, and had $66 in total. How much did Jerry save in the first week? Show two methods for solving this problem.” (7.EE.4)
• In Unit 4, Lesson 19, Independent Practice, Question 6 (Master Level), students solve real life problems such as finding the area of a circle. The materials state, “Brian’s dad wants to put a circular pool in their pool (yard). He can choose between pools with diameters of 15 ft, 17 ft, or 22 ft. Step A: Determine how much more space the pool with a diameter of 22 feet would take up compared to the 15 foot diameter pool. Step B: Determine how much more space the 15ft and 17ft pools combined would take up compared to the 22ft pool.” (7.G.4)
• In Unit 6, Lesson 13, Independent Practice, Question 1 (Bachelor Level), students solve real life problems such as finding final costs using percent problems. The materials state, “A snowboard originally costs $260. The sports store is having a sale of 10% off of items less than $100 and 15% off of items above $100. The sales tax is 12%. What is the final price for the snowboard, including tax?” (7.RP.A)
• In Unit 9, Problem of the Day 9.1, Day 1, Question 1, students apply skills related to using proportional relationships to solve percent problems. The materials state, “Last year, a property manager bought five identical snow shovels and six identical bags of salt. The total cost of the snow shovels was $172.50, before tax, and each bag of salt cost $6.20, before tax. This year, the property manager bought two identical snow shovels and four identical bags of salt. The total cost of the snow shovels was $70.38, before tax, and the total cost of the bags of salt was $26.04, before tax. Determine the item with the greatest percent increase in the price from last year to this year. Be sure to include the percent increase of this item to the nearest percent.” (7.RP.3)

The instructional materials reviewed for Achievement First Mathematics Grade 7 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, MP8 is emphasized in two units, while MP2 is emphasized in 10.
• The Mathematical Practices are not always identified accurately. For example:
• At the unit level for Unit 3, MP6 is not identified as an emphasized practice. However, at the lesson level, 5 lessons identify it as connected. MP8 is bold (indicating emphasis) at the unit level, but isn’t connected in any of the lessons.
• In Unit 6, MPs 1, 2, 5, and 7 are bold; however, MP5 does not appear in any of the lessons in this unit. 
• The Unit 8 Overview does not bold MP1 as an emphasis, yet 5 out of 9 of the lessons include this practice. Whereas MP4 is an emphasis, but is not bolded in any lessons.

• 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 2 Overview states, “Students can reason from here that the sign of a must be negative because the product is negative meaning the signs of the factors must be different and 5 is positive.”
• Unit 5 Overview states, “Students also work backwards and reason that if they are given a dependent value, then the value divided by the CoP will produce the corresponding independent value.”
• Unit 6 Overview states, “In both lessons, students will still be expected to use double number lines to make sense of the problem.”
• Unit 7 Overview states, “Simulation uses tools (coin, number cube, etc.) to generate outcomes that represent real outcomes that otherwise might have been difficult to do in real time.”
• In Unit 3, Lesson 7 prompts the teacher to ask if student answers are reasonable and shows a key strategy as “Check for reasonableness.” But there is no specific connection to MPs. 

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:

• MP2 - Unit 1 Overview states, “To develop a conceptual understanding of addition and subtraction with integers, students return to the number line to model the operations (SMP2, SMP4). In the first four lessons, students move from using number lines to understand and perform addition of integers to applying abstract rules to add integers fluently.” Although SMP4 isn’t one of the emphasized practices in this unit.
• MP6 - Unit 1 Overview states, “For subtracting integers using rules (SMP6), students may utilize one of two paths. They may visualize subtraction on a number line. Or, they can understand subtraction as adding the additive inverse and apply the rules for adding integers.”
• MP7 - Unit 1 Overview states, “...Therefore, regardless of the order of the minuend and subtrahend, if one takes the absolute value of the difference, they will find the distance between the two values (SMP7).”

The instructional materials reviewed for Achievement First Mathematics Grade 7 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 1, Lesson 5, Independent Practice, Question 3 (Master Level) states, “Evaluate and model both expressions by drawing your own number line. a) $$1-(-7)$$; b) $$-9-(-6)$$.” In this example, MP4 is a visual representation but students do not choose a strategy, nor is there any real-world connection.
• In Unit 4, Lesson 1, Exit Ticket, Question 2 states, “Use a balance model to represent and solve the equation $$3x+3=24$$. Explain your answer and how you kept the equation balanced.” Students do not choose a strategy, nor is there any real-world connection.
• In Unit 4, Lesson 10, Think About It states, “Ted is planning a vacation this summer. The resort he wants to stay at charges $125.25 per day plus tax. Ted books the trip for three days and it costs him $422.25. How much did Ted pay in taxes per day of his vacation? See below for a potential model.” A model is provided to the students, supplying a pathway for students so they do not choose strategies. 

• 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. 
• In Unit 1, Lesson 1, Independent Practice Question 3 (Bachelor Level) states, “Jamie’s bank account currently has a negative balance of -2 dollars. How much money will be in her account if she deposits 8 dollars?” Instead of being able to choose a tool that could be best used to solve this problem, a number line has been given, suggesting that this is the correct tool to use. 
• In Unit 4, Lesson 16, THINK ABOUT IT! states, “Record the diameter and circumference of the given circle using the tools provided.” Students are given rulers and string to calculate circumference without an opportunity to choose tools. 
• In Unit 10, Lesson 4, AIM states, “Construct line segments and angles given measurements using a ruler and protractor.” In the problems, students are given rulers and protractors specifically by the teacher and do not self-select tools.

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

• MP1: Make sense of problems and persevere in solving them. 
• In Unit 1, Lesson 11, THINK ABOUT IT!, students use any estimation or an integer operation strategy to solve this problem and determine if their solution makes sense. The materials state, “Maggie said that she could determine the answer to the problem below by just estimating the answer and comparing it to the given answer choices. Do you agree with Maggie? If so, explain and use estimation to prove she is correct. If not, explain and solve the problem to determine the actual answer. Dominic jumped from a height of 14.3 feet above the surface of a pool.  He traveled 18.7 feet straight down into the water. From there he traveled up 25.55 feet to the top of the biggest water slide at the pool. What is the height of the tallest waterslide? a) -55.55; b) -29.55; c) 21.15; d) 55.55.”
• In Unit 4, Lesson 19, Independent Practice, Question 7 (PhD Level), students solve an unrehearsed and unfamiliar problem by decoding information to work backwards. The materials state, “Explain how you would be able to determine the area of a circle if you were given the circumference. Draw a diagram and provide an example in your explanation. Explain why it would be more difference to determine the circumference of a circle given the area.”
• In Unit 6, Lesson 4, Independent Practice, Question 6 (Master level), students already know the answer to the question, but must figure out how they got there. The materials state, “Tom Brady threw 52 completions in the Super Bowl XLL loss to the New York Giants. His completion rate of completed passes to total passes was 65%. How many total passes did Brady attempt?”

• MP2: Reason abstractly and quantitatively.
• In Unit 5, Lesson 3, Partner Practice, Question 2 (Bachelor Level), students determine what numbers and quantities mean in a relationship. The materials state, “The table below shows the relationship between the cost of renting a movie (in dollars) to the number of days the movie is rented. Read each statement below the table and determine if it is true or false. a) Dollars represents the independent variable; b) The relationship between the cost and the number of days is proportional because $$6\div2=3$$ and $$9\div3=3$$; c) The relationship between the cost and the number of days is not proportional because the values do not increase in order; d) The relationship between the cost and the number of days is proportional because there is a CoP of $$\frac{1}{3}$$.” 
• In Unit 6, Lesson 16, Independent Practice, Question 3 (Master Level), students understand the proportional relationship between two figures and represent the math problem symbolically. The materials state, “If the area of the shaded region in the larger figure is approximately 21.5 square inches, write an equation that relates the areas using scale factor and explain what each quantity represents. Determine the area of the shaded region in the smaller scale drawing. Equation: ___ Area shaded region (smaller square): ___.” The shaded area is square and the unshaded area is an oval. 
• In Unit 8, Lesson 4, Independent Practice, Question 1 (Bachelor Level), students analyze a dot plot, then find a solution using the data to determine if the quantity is reasonable. The materials state, “Consider the distribution below. Part A: Describe the distribution of the data Part B: Describe the variability of the data. Part C: What would you expect the distribution of a random sample of size 10 from this population to look like?”

• MP6: Attend to precision.
• In Unit 1, Lesson 2, students “represent and add integers (-100 to 100) using a horizontal or vertical number line.” The Anticipated Misconceptions and Errors section identifies several points to help students attend to the precision of the mathematics and set up equations correctly. The materials state, “Anticipated Misconceptions and Errors: Students might not draw arrows proportionally to each other resulting in incorrect signs; Students might move in the wrong direction when drawing their arrows; Students might start their q-value arrow at zero instead of at the p-value; Students might add or subtract incorrectly (carrying or borrowing) when given larger numbers.” 
• In Unit 8, Lesson 8, Interaction with New Material, students must understand vocabulary in order to communicate an accurate explanation. The materials state, “Ex. 1) The distances that two different track teams run every day for practice are compiled. The team that runs fewer miles has a mean distance of 6.5 miles. The distance between the means between the two teams is 4 miles. Use the number lines below to visualize the problem based on Step A and B. Step A: Would you expect there to be any visual overlap of the data if both teams had a MAD of 2 miles? Step B: Could you still explain Part B if you didn’t know the mean of the slower team? Explain.”
• In Unit 9, Lesson 11, Interaction with New Material, students must understand the language of the problem in order to find surface area and volume, and calculate accurately in order to compare the two shapes. The materials state, “An airfreight company uses a box in the shape of a triangular prism to pack blueprints, posters, and other items that can be rolled up to fit inside the box. Each base is an equilateral triangle. The dimensions of the box are shown below. a) The freight company needs to first wrap the entire box in packing tape. How many square inches will be covered with tape? b) How many cubic inches of material can be packed within the prism? c) Another airfreight company uses a box shaped like a rectangular prism for the same purposes. The rectangular prism is also 38 inches long, and each of its square bases has a length of 3 inches.  Which box takes up more space?”  

• MP7: Look for and make use of structure.  
• In Unit 2, Lesson 6, the narrative describes the structure of fraction division which teachers guide students through before they have the opportunity to practice independently. The materials state, “Students understand that any division equation can be rewritten as a multiplication equation where the dividend of the division equation is equal to the product of the multiplication equation and the divisor and quotient are the factors of the multiplication equation. Students understand that because all division equations can be rewritten with multiplication that the rules for multiplying integers extends to dividing integers.”
• In Unit 3, Lesson 1, Test the Conjecture, Question 1, students use repeated addition and the commutative property within expressions to understand combining like terms. The materials state, “Write two expressions that are equivalent to the expression $$3x + 5 + 4x + 2$$ and indicate the expression that is in simplest form.” Teacher prompts include, “In order to help us combine like terms, we are going to expand each term in the expression that contains a variable. How could we expand this expression? How can we group the like terms so they are with each other?” 
• In Unit 5, Lesson 4, Partner Practice, Question 3 (Master Level), students use structure by creating a table to discern the pattern of repeated addition. The materials state, “Mary is filling out a table to keep track of how much money is in her account. On the first day of the month, she has $50. On the third day she has $150. On the 4th day, she has $200. If her account continues the same way, write an expression to determine how much money she will have on the 9th day and how long it will take her to have $n in her account.”

• MP8: Look for and express regularity in repeated reasoning. 
• Unit 1, Lesson 3, Exit Ticket, Question 1, students use repeated reasoning to add integers. The materials state, “Evaluate the expression: $$(-22) + 15 + (-9)$$ and explain how you used the generalized rules for adding integers.”
• Unit 4, Lesson 17, THINK ABOUT IT!, “For the circle below, Chandler says that there isn’t enough information to determine the circumference without measuring. Joey disagrees and says that he can write an equation to solve for the circumference. Who do you agree with and why?” Then the teacher states, “The circumference of a circle is equal to Pi multiplied by the diameter. What will we be able to do if our conjecture is true? We will be able to write an equation for the circumference of a circle and substitute to determine either the circumference or diameter.” Students use repeated reasoning about the relationship between circumference, diameter, and Pi. 
• Unit 6, Lesson 3, AIM, students “develop the formula part = $$\frac{p}{100}×$$ total using a double number line diagram.” In THINK ABOUT IT!, the teacher prompts, “The number line below shows a general percent problem with the percent, part and whole. Write an equation and solve for the part. Using your equation, describe how you can find the percent of a number.” Then the teacher states,  “The percent of a number is equal to the percent (as a decimal) multiplied by the total.” Students use repeated reasoning to determine the percent of a number with a double number line.