7th Grade - Gateway 2

The instructional materials for Achievement First Mathematics Grade 7 meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings.

The instructional materials develop conceptual understanding throughout the grade level. Materials include problems and questions that promote conceptual learning. Examples include:

• Unit 1, Lesson 1, THINK ABOUT IT!, students develop conceptual understanding of addition with integers by modeling problems using number lines. “Model the expressions -2 + 9 and -2 + (-4) by accurately labeling using the number lines below.” (7.NS.1b)

• Unit 3, Lesson 1, Partner Practice, Question 2 (Bachelor level), students develop conceptual understanding of equivalence of equations by expanding expressions to combine like terms. “Expand the following expressions and then combine like-terms: a) 4x + 6 + 2x + 3; b) 2r + 3y + 4 + 5y; c) 4n + 3f + 5 + 5f + 2n + 1.” (7.EE.A) 

• Unit 10, Lesson 2, Independent Practice, Question 4 (Master), students develop conceptual understanding of angle congruence to find unknown angles. "Why must vertical angles always be congruent? Draw a diagram to help explain your answer." (7.G.5) 

Materials provide opportunities for students to independently demonstrate conceptual understanding throughout the grade. Examples include:

• Unit 2, Lesson 6, Independent Practice, Question 3 (Master Level), students demonstrate conceptual understanding of multiplying and dividing rational numbers by using a number line. “Use multiplication to prove that the quotient of -12 ÷ 4 is negative.” Question 4, “Use your answer to question 3 for the following two steps: Step A: Model the expression on the number line provided below. (number line from -15 to 15 provided). Step B: Explain how your number line in Step A could also represent multiplication.” (7.NS.2)

• Unit 4, Lesson 1, Independent Practice, Question 5 (Master Level), students demonstrate conceptual understanding of reasoning about quantities in a simple equation by using a balance model. “Model the equation 9n+31=66 using a balance model and apply your model to solve for the variable arithmetically.” (7.EE.3, 7.EE.4)

• Unit 6, Lesson 2, Independent Practice, Question 8 (Master Level), students demonstrate conceptual understanding of using proportional relationships to solve percent problems by using a number line. “Set up a double number line to write and solve an equation for the given problem. a) 40 is 80% of what number? b) 18 is what percent of 72?” (7.RP.3)

The materials for Achievement First Mathematics Grade 7 meet expectations for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency. Although there are not many examples to practice within a lesson, students are provided opportunities to practice fluency both with a partner and individual practice, especially within exercise based lessons and the skill fluency of the cumulative review section. 

The materials develop procedural skill and fluency throughout the grade level. Examples include:

• Unit 1, Lesson 3, Exit Ticket Question 2, students develop procedural skill and fluency by using operations with rational numbers. “Which of the following expressions with have a positive sum? Select all that apply: a) -14 + (-42); b) 34 + (-24); c) -7 + 10; d) -50 + 45; e) 8 + 88; f) -6 + 7.” (7.NS.A)

• Unit 3, Lesson 3, Independent Practice, Question 6 (Masters level), students develop procedural skill and fluency by rewriting equivalent expressions. “Write at least four different expressions that are equivalent to -18 + 6n.” (7.EE.2)

• Unit 4, Lesson 8, Independent Practice, Question 2 (Bachelor level), students develop procedural skill and fluency by solving word problems that lead to 2-step equations. “A dog is starting a diet to get in better shape. The dog starts at 89.5 points and loses 0.5 points each week for a certain number of weeks. Halfway through the diet, the dog weighs 80 pounds. How many weeks has the dog been dieting for?” (7.EE.4a)

The materials provide opportunities to independently demonstrate procedural skill and fluency throughout the grade-level. Examples include:

• Unit 1, Lesson 7, Independent Practice, Question 5 (Master level), students demonstrate procedural skill and fluency by using operations with rational numbers. “Evaluate the following expression: -42 - (-23) + (-37 - 5).” (7.NS.A)

• Unit 4, Lesson 2, Independent Practice Question 5 (Master level), students demonstrate procedural skill and fluency by solving word problems that lead to 2-step equations. “Mari is twice as old as Harry. Jacob is three times older than Harry plus two years. Their combined age is 50. How old is each person?” (7.EE.4a) 

• Unit 9, Skill Fluency, 9.2, Day 1, Question 3, students demonstrate procedural skill and fluency related by rewriting equivalent expressions. “Which expression is equivalent to (4x - 5) - (3x - 2)? a) 7x - 7; b) 7x - 3; c). x - 7; d) x - 3.” (7.EE.2)

The materials reviewed for Achievement First Mathematics Grade 7 meet expectations for being 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. 

Materials include multiple routine and non-routine applications of the mathematics throughout the grade level. Examples include:

• Unit 5 Lesson 1, Mixed Practice, Day 2, Question 5, students apply skills related to using understanding of circumference in a non-routine problem. “Kate bent some wire around a rectangle to make a picture frame. The rectangle is 8 inches by 10 inches. a. Find the perimeter of the wire picture frame. Explain or show your reasoning. b. If the wire picture frame were stretched out to make one complete circle, what would its radius be?” (7.G.4)

• Unit 7, Lesson 3, Problem of the Day, Day 1, Question 1, students apply skills related to routine real-world problems using rational numbers. “Emily leaves her house at exactly 8:25 am to bike to her school, which is 3.42 miles away. When she  passes the post office, which is \frac{3}{4} mile away from her home, she looks at her watch and sees that it is 30 seconds past 8:29 am. If Emily’s school starts at 8:50 am, can Emily make it to school on time without increasing her rate of speed? Show and explain the work necessary to support your answer.” (7.NS.3)

• Unit 8, Lesson 1, Day 1, Mixed Practice, Question 5, students apply skills related to using random sampling to make predictions in a routine problem. “A carnival game has 160 rubber ducks floating in a pool. The person playing the game takes out one duck and looks at it. If there’s a red mark on the bottom of the duck, the person wins a small prize. If there’s a blue mark on the bottom of the duck, the person wins a large prize. Many ducks do not have a mark. After 50 people have played the game, only 3 of them have won a small prize, and none of them have won a large prize. Estimate the number of the 160 ducks that you think have red marks on the bottom. Then estimate the number of ducks you think have blue marks. Explain your reasoning.” (7.SP.1)

• Unit 9, Problem of the Day 9.1, Day 1, Question 1, students apply skills related to using proportional relationships to solve percent problems in a non-routine format. “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)

Materials provide opportunities for students to independently demonstrate routine and non-routine applications of the mathematics throughout the grade level. Examples include:

• Unit 2, Lesson 10, Exit Ticket, Question 2, students apply skills related to routine operations with rational numbers. “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)

• Unit 4, Lesson 8, Independent Practice, Question 9 (PhD Level), students apply skills related to reasoning about quantities by constructing simple equations in a non-routine format. "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)

• Unit 4, Lesson 19, Independent Practice, Question 6 (Master Level), students solve routine real life problems such as finding the area of a circle. "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 15 ft and 17 ft pools combined would take up compared to the 22 ft pool.” (7.G.4)

• Unit 6, Lesson 13, Independent Practice, Question 1 (Bachelor Level), students solve routine real life problems such as finding final costs using percent problems. "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)

The materials reviewed for Achievement First Mathematics Grade 7 meet expectations in that the three aspects of rigor are not always treated together and are not always treated separately. There is a balance of the three aspects of rigor within the grade. Overall, there is an emphasis on the application aspect with the conceptual component of rigor being slightly less represented; however, each aspect of rigor is demonstrated throughout the curriculum. The materials often demonstrate a combination of aspects of rigor within single lessons and even single problems.

All three aspects of rigor are present independently throughout the program materials. Examples include:

Conceptual Understanding:

• Unit 1, Lesson 7, THINK ABOUT IT!, students use number lines to demonstrate conceptual understanding of subtracting integers. “Model and evaluate the addition and subtraction expressions on an open number line. a) 25 + (-37); b) 25 - 37. Explain a generalized rule that you could use to subtract integers without the aid of a number line.” (7.NS.1)

Fluency and Procedural Skill: 

• Unit 4, Lesson 18, Exit Ticket, Question 2, students demonstrate fluency regarding the area of a circle by both estimating and finding the exact measure. “What is the exact and approximate area of a circle with a diameter of 6 feet? For the approximate area, round your answer to the nearest tenths place.” (7.G.4) 

Application:

• Unit 4, Lesson 10, Independent Practice, Question 2 (Bachelor Level), students apply their knowledge about multi-step real world problems to find the winner of the reading contest. “Aaliyah and Yohance are having a competition to see who can read more pages over the coming weekend. Aaliyah has bet Ms. Solomon that she’ll read 50 more pages than Yohance. Both scholars read at an average rate of 40 pages per hour. Yohance says that he’s going to read for 7.5 hours this weekend. How many hours will Aaliyah need to read for in order to fulfill her promise of reading 50 more pages than Yohance?” (7.EE.4)

Multiple aspects of rigor are engaged simultaneously to develop students’ mathematical understanding of a single topic/unit of study throughout the materials. Examples include:

• Unit 2, Lesson 5, Partner Practice, Question 2 (Master Level), students demonstrate both conceptual understanding and procedural skill as they use a number line and an expression to represent division of rational numbers. “A submarine starts at the surface and then descends to a depth of 250 feet below sea level. It took the submarine 5 minutes to complete this dive. How many feet can the submarine dive in 1 minute? Draw a model and write an expression to solve.” (7.NS.2) 

• Unit 5, Problem of the Day 5.1, Day 2, students demonstrate fluency and application with operations on rational numbers. “A water well drilling rig has dug to a height of –60 feet after one full day of continuous use. a) Assuming the rig drilled at a constant rate, what was the height of the drill after 15 hours? b) If the rig has been running constantly and is currently at a height of –143.6 feet, for how long has the rig been running? c) A snake is \frac{3}{4} the current distance underground of the rig and a spider is \frac{4}{5} of the same distance. How far away are the snake and the spider?” (7.NS.3)

• Unit 3, Lesson 2, Independent Practice, Question 8 (PhD level), students apply their conceptual understanding of variables to write and solve equations in real-world situations. “You and your friend made up a basketball shooting game. Every shot made from the free throw line is worth 3 points, and every shot made from the half-court mark is worth 6 points. Write an equation that represents the total amount of points, P, if f represents the number of shots made from the free throw line, and h represents the number of shots made from half-court. Explain the equation in words.”  (7.EE.4)

The materials reviewed for Achievement First Mathematics Grade 7 meet expectations for supporting the intentional development of MP1: Make sense of problems and persevere in solving them; and MP2: Reason abstractly and quantitatively, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. The Standards for Mathematical Practice are identified and incorporated within mathematics content throughout the grade level. The Mathematical Practices are listed in the Unit Overviews as well at the beginning of each lesson. There are instances where the Unit Overview gives a detailed explanation of the MPs being addressed within the unit, but the lessons do not cite the same MPs.

There is intentional development of MP1 to meet its full intent in connection to grade-level content. Examples include:

• Unit 1, Lesson 11, THINK ABOUT IT!, students use any estimation or an integer operation strategy to solve a problem and determine if their solution makes sense. “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.”

• The Unit 2 Overview outlines the intentional development of MP1. “ In lessons 4, 10, and 11, students apply their understanding of rational number multiplication and division to multi-step problems and persevere in solving them. Students focus on identifying the appropriate starting point and appropriate rule to solve the problem. Students make sense of problems and persevere in solving them in most lessons, but MP1 is specifically emphasized in Unit 2 by pushing students to apply and evaluate their rules for rational number multiplication and division in multi-step and challenging problems to push understanding.”

• Unit 4, Lesson 19, Independent Practice, Question 7 (PhD Level), students solve an unrehearsed and unfamiliar problem by decoding information to work backwards. The problem, “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.”

There is intentional development of MP2 to meet its full intent in connection to grade-level content. Examples include:

• The Unit 2 Overview outlines the intentional development of MP2. “In lessons 6, 7, 8, and 9 students build on their abstract understanding of the connection between multiplication and division by identifying how to rewrite a division problem as multiplication, rewriting as multiplying by the reciprocal (multiplicative inverse) of the divisor, and rewriting numbers as terminating or repeating decimals. Lesson 12 concludes the unit by having students apply their understanding of multiplication and division rules to mathematical inequality statements with constraints on p and q. While students reason abstractly and quantitatively in most lessons, Unit 2 emphasizes MP2 with the use of rewriting expressions to deepen their reasoning, such as rewriting division as multiplication and to create rules for rational number multiplication and division.”

• Unit 5, Lesson 3, Partner Practice, Question 2 (Bachelor Level), students determine what numbers and quantities mean in a relationship. “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 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.. “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?”

The materials reviewed for Achievement First Mathematics Grade 7 meet expectations for supporting the intentional development of MP3: Construct viable arguments and critique the reasoning of others, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

There is intentional development of MP3 to meet its full intent in connection to grade-level content. Examples include:

• The Guide to Implementing AF Math describes Error Analysis lessons as one way to address MP3. “Purpose: Through the use of error analysis, guided questioning and discussion students will identify and fix a common misconception related to a skill they learned the previous day. These are sequenced so that after a particularly complex conceptual lesson or a lesson involving a skill that surfaces a common misconception, students get another focused at bat to either fix their misunderstanding or deepen their reasoning around key mathematical concepts and viable strategies to guide them away from making the same error. These lessons start with analyzing fictional student work and are structurally based off of the Standards for Mathematical Practice 3.”

• Unit 1, Lesson 9, Independent Practice #7 (PhD Level), students add and subtract rational numbers. “Using the multiple choice question below, determine which two answer choices that you can immediately eliminate without doing any calculations. Explain how you were able to eliminate those answer choices. Evaluate: -4.5 - (-1\frac{4}{5}) - (-2.1) + 4\frac{1}{2}. a) -12.9; b) 12.9; c) -5.7; d) 5.7.” 

• Unit 4, Lesson 13, Error Analysis Lesson, THINK ABOUT IT!, students use variables to create equations. “Compare and contrast Scholar A’s work and Scholar B’s work on yesterday’s exit ticket question. Is either scholar correct? Use numbers and/or words to justify your answer on the lines below.” 

• Unit 5, Lesson 14, Independent Practice #2 (Bachelor Level), students solve problems involving scale. “Mark claims that he can multiply that area of Rectangle A by 4 to get the correct area of Rectangle B. Do you agree with him? Explain and prove your answer.” 

• Unit 6, Lesson 13, Independent Practice #4 (Master Level), students use proportional relationships to solve percent problems. “Justin wants to buy a new IPod that costs $250. When he gets to the Apple store, he sees that they are having a sale for 15% off all IPods. He then has a coupon that takes an additional 15% off the discounted price. Justin thinks that he can figure out the cost of the iPod by finding 30% of $250 and then subtracting that from $250. Do you agree or disagree with his claim? Explain.” 

• Unit 7, Lesson 8, Independent Practice, Question 4 (Master Level), students approximate the probability of an event. “Tishanna is experimenting with the same bag of pens. She randomly pulls a pen out of the bag 30 times, records the color, and replaces the pen. Her results are shown below. Step A: Now make a prediction for how many times Tishanna would pick a red pen, if she conducted 60 trials of the experiment. Step B: Which prediction are you more confident in – the prediction in question # 3, or the prediction you made in question #4? Why? Explain.”

The materials reviewed for Achievement First Mathematics Grade 7 meet expectations for supporting the intentional development of MP6: Attend to precision; and attend to the specialized language of mathematics, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

There is intentional development of MP6 to meet its full intent in connection to grade-level content. Many problems present students with the opportunity to attend to precision within the mathematics and the reasoning of the answer. Examples include: 

• Unit 2, Lesson 6, Partner Practice, Question 2 (Bachelor Level), students attend to precision when rewriting a division problem as a multiplication problem and solving. “Use multiplication to prove that the quotient of -10 ÷ 5 is negative and justify your reasoning.”

• Unit 5, Lesson 10, Exit Ticket Question 2, students attend to precision as they compute unit rates with ratios of fractions. “$$3\frac{1}{3}$$ lb. of turkey costs $10.50. What is the price per pound of turkey?” 

• Unit 8, Lesson 6, Independent practice, Question 1 (Bachelor Level), students attend to precision as they compute unit rates with ratios of fractions. “Rachel and Molly are in the same science class. Rachel’s scores on her first three science quizzes were 79, 86, and 90. Molly’s scores were 70, 78, and 80. Calculate the means and the mean absolute deviations of the quiz scores.”

The instructional materials attend to the specialized language of mathematics. The materials use precise and accurate mathematical terminology. Examples include:

• At the beginning of each lesson plan, there is a section labeled “Key Vocabulary” for the teacher. Unit 4, Lesson 4, Key Vocabulary,

  • “Equation: two expressions set equal to one another. 

  • Variable: a letter used to take the place of an unknown value.

  • Solution: the value that makes an equation true.

  • Arithmetic approach: the approach to solving a problem that involves arithmetic only; numbers and operations.”

• The teacher is routinely prompted to use precise vocabulary such as Unit 5, Lesson 5, Connection to Learning. “Students should understand that a graph is proportional if it is linear (i.e. forms a straight line) and passes through the origin because every value of x is multiplied by the CoP (constant of proportionality) to produce the corresponding output. Students should understand that the point (0,0) must be a part of a proportional graph because no CoP can be multiplied by 0 to produce a non-zero output.” 

• Unit 7, Lesson 1, Opening, Debrief, FENCEPOST #1, students use a spinner to determine probability. “Probability measures how likely an event is from impossible to certain.” The teacher shows student work who has correctly placed an x on impossible for a.) and certain for b.) and asks, “Do you agree with this scholar?” Students might say, “I agree with this scholar because for the first problem, there is no possible way for someone to spin the spinner and it to land on 5 because there isn’t a 5 on the spinner so it is impossible. For the second problem, if you spin the spinner it must land on 1, 2, 3, or 4 so it is certain that it will happen.” The teacher explains, “What you are calculating is called a probability. Probability is the likelihood of an event or outcome happening. An event is an outcome in an experiment (in this case, the ‘experiment’ is spinning the spinner and each number is an event. If you land on 1, that is an event. If you land on 2, that is an event. Etc.).”

The materials reviewed for Achievement First Mathematics Grade 7 meet expectations for supporting the intentional development of MP7: Look for and make use of structure; and MP8: Look for and express regularity in repeated reasoning, for students, in connection to grade-level content standards, as expected by the mathematical practice standards.

There is intentional development of MP7 to meet its full intent in connection to grade-level content. Examples Include:

• Unit 2, Lesson 6, Connection to Learning and Conceptual Understanding describes the structure of fraction division which teachers guide students through before they have the opportunity to practice independently. “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.”

• Unit 3, Lesson 1, Test the Conjecture, Question 1, students use repeated addition and the commutative property within expressions to understand combining like terms. “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?” 

• Unit 5, Lesson 4, Partner Practice, Question 3 (Master Level), students use structure by creating a table to discern the pattern of repeated addition. “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.”

There is intentional development of MP8 to meet its full intent in connection to grade-level content. Examples Include:

• Unit 1, Lesson 3, Exit Ticket, Question 1, students use repeated reasoning to add integers. “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?” 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, “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.