6th Grade - Gateway 2

The instructional materials for Achievement First Mathematics Grade 6 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 2, Lesson 1, Exit Ticket #1, students develop conceptual understanding of fraction division by modeling problems using a tape diagram. “Draw a model and evaluate the expression \frac{9}{10}\div\frac{3}{10}.” (6.NS.1)

• Unit 4, Lesson 9, Partner Practice, Question 2 (Master), students develop conceptual understanding of equivalent ratios by interpreting data points in a table and a graph. “The table below shows the relationship between the number of ounces in various sized boxes of Cheerios and the number of Cheerios in the box. (Table provides 4 data points.) Using the template below (Quadrant I of a coordinate plane provided), make a graph showing the relationship between the number of ounces in a box of Cheerios and the actual number of Cheerios in the box. a) What does the point (14, 4,500) represent? How do you know? b) Are the ratios in the table equivalent? Provide two reasons for how you know.” (6.RP.3)

• Unit 6, Lesson 10, Test the Conjecture #1, students develop conceptual understanding of equivalence by analyzing an equation. Teacher prompts include, “Is the following equation true? 4m + 12 = 2(m + 6). What is the question asking us to do? How do you know? How can we apply our conjecture to this problem?” (6.EE.3)

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

• Unit 4, Lesson 2, Independent Practice, Question 3 (Bachelor level), students demonstrate conceptual understanding of ratio relationships by creating equivalence models. “Write two ratios that are equivalent to 3:5. Use a model to prove that each ratio is equivalent.” (6.RP.3a)

• Unit 6, Lesson 11, Exit Ticket, Problem 1, students demonstrate conceptual understanding of generating equivalent expressions by using properties of operations to rewrite expressions. “Without substituting a value in for x, prove that 3x+9x-2x is equivalent to 10x .” (6.EE.3)

• Unit 8, Lesson 2, Independent Practice, Question 5 (Bachelor Level), students demonstrate conceptual understanding of finding area by decomposing parallelograms into triangles. “Brittany and Sid were both asked to draw the height of a parallelogram. Their answers are below. Who is correct? Explain your answer. Is there another way they could have drawn in the height? If so, draw the different way to identify the height on one of their parallelograms and explain.” (6.G.1)

The materials for Achievement First Mathematics Grade 6 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 11, Interaction with New Material, students develop procedural skill and fluency by finding common factors and multiples. “Ex. 1) The Ski Club members are preparing identical welcome kits for the new skiers. The Ski Club has 72 hand warmer packets and 48 foot warmer packets. What is the greatest number of identical kits they can prepare using all of the hand warmer and foot warmer packets? How many hand warmer packets and foot warmer packets will there be in each kit? ...Based on our understanding of the problem, what is our plan for solving this problem? ...Note to teacher: Ss will likely struggle to make the connection to the GCF. Push hard on Ss understanding that you are dividing each total up and make sure that students truly understand that the number of groups will be the same for both types of warmer and the size of the group will be different.” (6.NS.4)

• Unit 7, Skill Fluency 7.3, Day 1, Question 5, students develop procedural skill and fluency by using substitution to make equations true. “Which equation is true if x = 5? a) 3x = 8; b) 2x = 10; c) x + 5 = 5; d) 25 - 5 = x.” (6.EE.5)

• Unit 8, Skill Fluency 8.2, Day 1, Question 6, students develop procedural skill and fluency by using properties of operations to generate equivalent expressions. “What is the correct first step to take in order to simplify the expression below? [3.5 × (5 - 4.3)] + 2.7: a) Subtract 4.3 from 5; b) Multiply 3.5 by 2.7; c) Multiply 3.5 by 5.” (6.EE.3)

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

• Unit 1, Lesson 1, Independent Practice, Question 1 (Bachelor Level), students demonstrate procedural skill and fluency by using operations on decimals. “Evaluate each expression: a) 23 – 0.324; b) 9.3 + 19.59.” (6.NS.3)

• Unit 2, Skill Fluency 2.2, Day 1, Problems 1-4, students demonstrate procedural skill and fluency by dividing multi-digit numbers. "1) 1,986 ÷ 60 = ?; 2) Solve: 80.25\div20= ?; 3) Find the quotient: 540\div0.60= ?; 4) 35.2\div5.5= ?” (6.NS.2)

• Unit 6, Lesson 2, Independent Practice, Question 3 (Master Level), students demonstrate procedural skill and fluency by evaluating expressions. “Evaluate each expression: a) 24\frac{3}{5}+(4^3\times(8.2-2)); b) 6^2 + (13.5 - 5 + 2) × 2^3 + 3\frac{8}{10}” (6.EE.2c)

The materials reviewed for Achievement First Mathematics Grade 6 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 6, Mixed Practice 6.3, Day 1, Question 4, students apply skills related to solving routine problems using division of fractions. “Carla wants to know how many batches of birdseed she can make with 3\frac{1}{2} cups of  sunflower seeds. She puts \frac{1}{6} cup of sunflower seeds in every batch. Carla divides 3\frac{1}{2}  by \frac{1}{6}  to find the answer. She says this is the same as multiplying \frac{1}{6}  by \frac{7){2}. a) Is Carla correct? Why or why not? b) Using a correct method, find the solution. Show your work.” (6.NS.1)

• Unit 7, Lesson 12, Partner Practice, Question 2 (Master Level), students represent and analyze routine quantitative relationships between dependent and independent variables. "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)

• Unit 10, Problem of the Day 10.1, Question 2 - students apply skills related to solving problems utilizing decimals in a non-routine application. "Four 6th graders are working on a project. They are going to paint a large banner and need to protect the floor. They measured the floor, which is 3.05 meters by 3.68 meters. Plastic is sold in rolls of  0.5 square meters each. How many rolls of plastic will they need to buy in order to cover the floor?” (6.NS.3)

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

• Unit 3, Lesson 12, Independent Practice, Question 3 (Master Level), students solve real life, non-routine problems such as graphing points on the coordinate plane and using them to find area. "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)

• Unit 4, Lesson 5 Exit Ticket, Question 2, students apply skills related to solving problems using ratio reasoning in non-routine ways. “Josh was solving the following problem: The Superintendent of Highways is interested in  the numbers of commercial vehicles that frequently use the county’s highways. He obtains information from the Department of Motor Vehicles for the month of  September and finds that for every 14 non-commercial vehicles, there were 5 commercial vehicles. If there were 108 more noncommercial vehicles than  commercial vehicles, how many of each type of vehicle frequently use the county’s highways during the month of September? He says that he cannot solve it because \frac{108}{14} = a number with a decimal remainder. What is the mistake that Josh is making? Find the mistake, and solve correctly.” (6.RP.3a)

• Unit 5, Day 2, Problem of the Day, students apply skills related to solving problems using ratio reasoning in a non-routine problem. "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)

• Unit 7, Lesson 7, Independent Practice, Question 7 (PhD Level), students apply skills related to writing and solving routine one-step equations. "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)

The materials reviewed for Achievement First Mathematics Grade 6 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 5, Independent Practice, Question 4b (Master Level), students use models and equations to conceptualize division with the standard algorithm. "For each word problem, draw a model, write an equation, and solve the problem. b. Thomas has 575 pennies that he wants to exchange for quarters. How many quarters will he receive in exchange for his 575 pennies?" (6.NS.2)

Fluency and Procedural Skill:

• Unit 6, Lesson 1, Independent Practice, Question 6 (Master Level), students develop fluency with evaluating numerical expressions that include exponents. “Evaluate the expressions: a) 90 - 5^2 × 3.5; b) 6.4 - 2^2\div2 + 0.3^2.” (6.EE.1)

Application:

• Unit 7, Problem of the Day, 7.3, Day 2, students apply their knowledge about using a variable to represent an unknown in an equation to find out about fast food profits. “Wendy’s has a number of franchises, f, in Brooklyn. Each franchise makes $223 every hour. a) Write an expression to represent, m, the total amount of money all Wendy’s franchises make per hour. b) If m = 7,136 how many Wendy’s franchises are there in Brooklyn?” (6.EE.6) 

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 4, Partner Practice, Question 1 (Bachelor level), students demonstrate both conceptual understanding and procedural skill as they create a model and use the standard algorithm to divide fractions. “Evaluate each expression using a model and using the algorithm: a) 2\frac{4}{5}\div\frac{2}{5}; b) 3\frac{5}{6}\div\frac{2}{3}”  (6.NS.1)

• Unit 3, Lesson 16, Independent Practice #2 (Master Level), students demonstrate procedural skill and application as they meet the requirements to construct a rectangle. “Construct a rectangle on the coordinate plane that satisfies each of the criteria listed below. Identify the coordinates of each of its vertices. 1) Each of its vertices lies in a different quadrant; 2) Its sides are either vertical or horizontal; 3) The perimeter of the rectangle is 28 units; 4) Using absolute value, show how the lengths of the sides of your rectangle provide a perimeter of 28 units.” (6.G.3)

• Unit 4, Lesson 5, Exit Ticket, students demonstrate both procedural skill and application as they use ratio reasoning to solve real-world and mathematical problems. “For every six hot dogs that are shipped to a store, two hamburgers are shipped. Yesterday, 12 hamburgers were added making the amount of hot dogs and hamburgers shipped equal. How many hot dogs and hamburgers were shipped originally?” (6.RP.3)

The materials reviewed for Achievement First Mathematics Grade 6 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:

• The Unit 3 Overview outlines the intentional development of MP1. “In lessons 4 and 12, students make sense of integers in real-world contexts by translating them to a number line. In lesson 16, students solve complex, multi-step problems involving integers on the coordinate plane by making sense of real-world situations and persevering to solve them. MP 1 is a major focus of Unit 3 as students make sense of rational numbers in various contexts and persevere in solving problems and situations involving them.”

• Unit 2, Lesson 10, Independent Practice Question 7 (PhD level), students find an entry point and persevere to solve a multi-step problem involving rational numbers. “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?”

• Unit 5, lesson 12, Independent Practice Question 7 (PhD Level), students make sense of a familiar situation in order to find a solution. “Tricia had a birthday party. During the party, she opened 36 gifts, which was 60% of all of her gifts. After the party, she opened the rest of the gifts and found that 25% of them were the same present, so she returned all but one of the duplicate gifts. How many gifts did she return? Show your work.” 

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 3-5, students reason abstractly and quantitatively about situations involving the division of fractions. By utilizing the context of the problem, students reason to determine the proper division expression for the situation. In lessons 10 and 11, students reason to decontextualize word problems and translate them into expressions and equations for solving. MP 2 is a major focus of Unit 2 as students reason quantitatively about complex situations in order to translate them into mathematical situations they can solve.”

• 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. “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?”

• Unit 7, Lesson 14, Exit Ticket, students reason abstractly and quantitatively as they solve and graph the solution for one-step inequalities. “Exit Ticket: Blayton is at most 2 meters above sea level. Part A: Select all of the statements below that are true about the scenario: a) The phrase at most 2 means values less than two are included; b) The phrase at most 2 means values more than two are included; c) The value 2 would be included in the solution; d) An inequality that represents this scenario is x ≥ 2; e) 2 meters below sea level would be included in the solution set for the inequality representing this scenario. Part B: Represent Blayton’s possible elevation on a graph.”

The materials reviewed for Achievement First Mathematics Grade 6 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 2, Lesson 9, Error Analysis Lesson, THINK ABOUT IT!, students compare exit ticket responses about the magnitude of the fraction quotient in relationship to 1. Teacher prompts include, “Which scholar’s work did you agree with? Turn and tell your partner who you chose and why. Why does this relationship between the dividend and divisor make sense? What error did this scholar make? What did this scholar do to get this correct, and why was that helpful?” 

• Unit 6, Lesson 14, THINK ABOUT IT!, students identify equivalent expressions. “Angela drew a regular octagon (meaning all the sides are the same length) with a side length of 3p + 2. Write two equivalent expressions that represent the perimeter of the octagon. Explain how you know that the expressions that you wrote are equivalent.” 

• Unit 7, Lesson 5, Error Analysis Lesson, Think About It, students investigate 1-step 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 8, Lesson 6, Independent Practice, Question 4 (Master Level), students decompose shapes into triangles to find area. “Explain how you can use triangles to derive the area formula for trapezoids. Use an example to help illustrate the explanation.” 

• Unit 10, Lesson 5, Test the Conjecture, Question 1, students work with measures of center to understand what a single value represents. “Over the last ten days, the temperatures in Miami, Florida have been 80\degree, 78\degree, 80\degree, 76\degree, 85\degree, 84\degree, 82\degree, 79\degree, 76\degree, and 40\degree. The weatherman made an error and forgot to include the 40 degrees when finding the mean and median of the data. Should the weatherman report out the incorrect mean or the median in order to try to hide his mistake and report accurately about the weather? Prove your answer mathematically.” (6.SP.3)

The materials reviewed for Achievement First Mathematics Grade 6 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 1, Lesson 5, Question 4 (Master Level), students attend to precision when they apply the division algorithm to divide a three and four-digit dividend by a two- and three-digit divisor. “For each word problem, draw a model, write an equation, and solve the problem: a) Elise has 572 quarters that she has collected over the last several years. She wants to exchange all of her quarters for pennies. How many pennies will she receive in exchange for 572 quarters? b) b. Thomas has 575 pennies that he wants to exchange for quarters. How many quarters will he receive in exchange for his 575 pennies?”

• Unit 4, Lesson 5, Independent Practice, Question 3 (Master Level), students attend to precision as they solve ratio problems involving comparisons using a tape diagram. “Review the student work below and determine whether or not it is correct. If it is correct, explain why. If it is incorrect, explain why and find the correct answer. For every 2 earrings sold at a store, 6 bracelets are sold. Today they had an earring sale and sold 12 extra earrings, making the total number of earrings and bracelets sold the same. How many bracelets did they sell in total?”

• Unit 7, Lesson 5, Independent Practice, Question 1 (Bachelor Level), students attend to precision as they solve and check the solution of one-step equations using substitution. “Directions: Draw a model to solve each equation and check your answer using substitution: 14 = m - 37.” 

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,

  • “Ratio - A comparison of a pair of non-negative numbers, A:B, which are not both 0. Units can be alike or different.

  • Term - 1 part of a ratio (i.e. in ratio A:B, A is a term and B is a term). Terms can represent like or unlike quantities. 

  • Like Quantities - Two quantities with the same unit (e.g. girls and boys- both people).

  • Unlike Quantities - Two quantities with different units (e.g. dollars per gallon, laps per minute).

  • Equivalent ratio – Two ratios that express the same relationship between two terms.

  • Tape diagram - A diagram used to represent equivalent ratios that have the same units.” 

• Unit 6, Lesson 3, Exit Ticket Question 3, students are directed to use appropriate mathematical language. “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).”

• Unit 9, Lesson 1, Opening, Debrief, FENCEPOST #1, students utilize precise vocabulary about 3D shapes. “The shapes that you cut out of the paper are called ‘nets.’ For the first net, what solid were you able to form? How do you know?” Students might say, “I was able to form a rectangular prism. I know this is a rectangular prism because it has opposite parallel bases that are rectangles.” The next teacher prompt, “For the second net, what solid were you able to form? How do you know?” Students might say, “I was able to form a triangular prism. I know this is a triangular prism because it has opposite parallel bases that are triangles.”

The materials reviewed for Achievement First Mathematics Grade 6 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 6, Lesson 2, Think About It!, students analyze the structure of expressions with order of operations. “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.” Students look for structure in expressions to manipulate them for efficient simplification. 

• Unit 6, Lesson 12, Think About It!, students learn about the structure of distributive property as a quicker route of creating equivalent expressions. “You can represent a number next to parentheses as repeated addition.” During instruction, the teacher, “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 might say, “I agree with the expression because we know that multiplication is the same as repeated addition. If we think of 2 as the number of groups and (a + 4) as the group size, then we have two groups of (a + 4), which can be written as (a + 4) + (a + 4).” 

• Unit 8, Lesson 1, Opening, THINK ABOUT IT!, students look for and make use of structure as they analyze the structure of each shape they are working with and reason about this structure to derive the area of it. “Louis’ teacher asked him to find the area of the parallelogram pictured below. He wasn’t sure what the area formula was for parallelograms, so he cut a right triangle off of the left side of the parallelogram and moved the triangle to the other side of the parallelogram, forming a rectangle. You can see his steps below. Assuming that what Louis did is okay, how should he find the area of the parallelogram? Show and explain in the space below.” 

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

• Unit 1, Lesson 7, THINK ABOUT IT!, Debrief, students have the opportunity to generalize understanding about “multiplying a division expression by a fraction in the form of 1 does not change the quotient.” The teacher prompts, “Based on your calculations, we know that the expressions in set A are equivalent and the expressions in set C are equivalent because you can multiply the divisor and dividend each by 10 and by 100 respectively. Do you agree or disagree with this student’s explanation and work?” Students might say, “I agree with this work because the student showed that the expressions are equivalent by multiplying by a form of 1. For example, in set A, you can multiply the first expression by \frac{10}{10}, which is equal to 1, to get the second expression. For set C, you can multiply the first expression by \frac{100}{100}, which is also equal to 1 and therefore the resulting expression is equivalent to the first. For set B, the student did not multiply by a form of 1. Instead, s/he just multiplied the denominator by 10 and the quotients were not equivalent.” Students express regularity in repeated reasoning when manipulating division expressions involving decimals. They shift place value relationships to make solving these expressions easier for them. 

• Unit 4, Lesson 12, Independent Practice, Question 2 (Bachelor 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. “A publishing company is looking for new employees to type novels that will soon be published. The publishing company wants to find someone who can type at least 45 words per minute. Dominique discovered she can type at a constant rate of 704 words in 16 minutes. Does Dominique type at a fast enough rate to qualify for the job? Explain why or why not.”

• Unit 6, Lesson 1, Think About It, students have the opportunity to make sense of repeated reasoning as they evaluate exponents. “Complete the table below. Explain how you came up with the exponential expressions.” The table includes the multiplication expression, product, and exponential expression.