6th Grade - Gateway 2

The materials reviewed for CK-12 Interactive Middle School Math 6 for CCSS meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings. The materials include problems and questions that develop conceptual understanding throughout the grade level.

Chapters 1 and 2 provide students with opportunities to develop conceptual understanding of understanding ratio concepts and use ratio reasoning to solve problems (6.RP.A) with the use of Interactives and Inline Questions. Examples include:

• In Lesson 1.2, Activity 2: Tape Diagrams, students manipulate a tape diagram to build a conceptual understanding of how two quantities form a relationship in the form of ratios. The Teacher Notes describe what the students do independently by stating, “This gives students a chance to practice visualizing and identifying ratios.” (6.RP.1)

• In Lesson 1.5, students complete tables of equivalent ratios and use values to answer questions (6.RP.3a). In Activity 2, students complete a table on beats in a sample song. The problem states,  “How many beats are there in this 12 second song sample? Use the table to find the total number of beats for 60 seconds of the song.” Once the table is completed, students answer the following Inline Questions: “1. What is the relationship between 24 seconds and 12 seconds? How can you use this to find the number of beats in 24 seconds? 2. For 12 seconds, the ratio of number beats to number of seconds is ___:12. 3. Since 48 seconds is four times 12 seconds, the number of beats in 48 seconds is ___times the number of beats in 12 seconds. There are ___ beats in 48 seconds. This works because the ratios 25:12 and _____ are equivalent.” Examples of practice questions for students to complete are problem 2, “If there are six campers per tent, how many tents for 30 campers?” and problem 10, “Complete the table 72:48, 36:24, 24:16, ___:12, 12:8.”

• In Lesson 2.8, Activity 2, students further develop their understanding of ratios by using a double number line to fill in the blanks based on a ratio and answering questions. For example, Item 2 states, “There are 4 thousand (4,000) pet tarantulas in the US. The number of turtles is 150% the number of tarantulas. How many pet turtles are there?” (6.RP.3)

Chapter 3 has multiple opportunities for students to work independently to build conceptual understanding of applying and extending previous understandings of multiplication and division to divide fractions by fractions (6.NS.1) through the use of Interactives. Examples include:

• In Lesson 3.3, Activity 3, students develop understanding of dividing a fraction by a fraction using a visual diagram. The teacher directions state how the students will use the Interactive in the activity to build this conceptual understanding. The materials state,  “Students are given a tape diagram and slider and a fraction (starting at 1). Students can use the slider to divide the diagram and the resulting fraction will appear above the slider.” (6.NS.1)

• In Lesson 3.6, Activity 1, students further develop their understanding of division of fractions through an Interactive where students manipulate a scale of a map to connect division with fractions. The Interactive introduces this to students by stating, “Pirate Captain Jim Hawkins designs a treasure map and draws out a 1 mile by 1 mile map of an island. He divides his map into smaller squares to make it easier to read.” (6.NS.1)

• In Lesson 3.9, Warm-Up, students work with an Interactive to divide fractions in the real-world situation of a water gun fight. The directions for the students read, “Use the Interactive to see how many times you can reload the water gun before you have to run to fill the bucket up with more water. Through this lesson, you will use tape diagrams to model fraction division and find the quotients.” (6.NS.1) 

Chapters 6 and 7 have multiple opportunities for students to work independently to build conceptual understanding of applying and extending previous understandings of arithmetic to algebraic expressions and reasoning about and solving one-variable equations and inequalities (6.EE.A,B) through the use of Interactives. Examples include:

• In Lesson 6.9, Activity 3, students factor expressions using the distributive property. Inline question 4 states, “Write an equivalent expression for 20x + 30 by dividing both terms by 5,” and question 5 states, “Look at the expression 12x + 20. Select the equivalent expressions.” (6.EE.3)

• In Lesson 7.4, Activity 1, students develop understanding of solving equations in the form of 20x + 30 through an Interactive. In the Interactive, students use numbers to try to isolate and solve for x. The student directions state, “Answering the question above will require knowledge of multiplication equations. Multiplication equations have many similarities with addition equations. Use the Interactive below to explore these similarities and to practice solving multiplication equations visually.” (6.EE.7)

The materials reviewed for CK-12 Interactive Middle School Math 6 for CCSS meet expectations for attending to those standards that set an expectation of procedural skill and fluency. The instructional materials develop procedural skill and fluency and provide opportunities for students to independently demonstrate procedural skill and fluency throughout the grade level, especially where called for by the standards (6.NS.2,3; 6.EE.1,2).

In Chapter 4, the materials develop and students independently demonstrate procedural skill and fluency in adding, subtracting, multiplying, and dividing multi-digit numbers and decimals with standard algorithms (6.NS.2,3). Examples include:

• In Lesson 4.2, “There are widget interactives that will guide students through the standard method of adding and subtracting decimals. Students can work an unlimited amount of times, so they should practice the method until they are comfortable before moving onto the real-world examples” (purple text). The first CK-12 Widget Interactive gives students step-by-step procedures on adding decimals, for example “8.153 + 1.535.” Students independently practice with numerous problems before moving onto the second part of the Interactive where they now have to “carry over/borrow” a 1, for example “7.242+1.846,” but the same step-by-step procedures are followed. Again, the student can practice independently as much as needed. The second CK-12 Widget Interactive focuses on subtracting decimals, again giving the same step-by-step procedures. The problems get increasingly more difficult,  for example, “2.972 - 1.141; 6.268 - 1.948,” and students can practice independently with an unlimited amount of problems. (6.NS.3)

• In Lesson 4.4, Activity 2 Interactive, students multiply multi-digit numbers. The teacher directions state, “This Interactive gives students a walk through for multiplying two decimals. Use the text boxes to evaluate the product one step at a time, after a student has typed in their answer they should press the enter key to see if it is correct. If it is wrong, it will turn red, and students can try again. Once a correct answer is entered it will turn black, and a new text box will appear.” In Multiplying Decimals with the Standard Method, students independently demonstrate procedural skill with multiplying decimals in all of the Review Questions. For example, Review Question 7 states, “ 1.7 × 9.691 = ____.” (6.NS.3)

• In Lesson 4.6, the Warm-Up: Practice Long Division “gives the student practice dividing with the standard method.” The Interactive provides the student step-by-step procedures on long division with problems such as 679 divided by 7. In Activity 1, Practice More Difficult Long Division increases the level of difficulty, for example “9460 divided by 43” but still gives the same step-by-step procedures. In Lesson 4.6, Activity 1 Interactive, students demonstrate fluency in dividing multi-digit numbers as students, “Use these Interactives to practice some more challenging and advanced long division problems! Can you answer the most difficult ones? 180482 = ? and 8692505 = ?” (6.NS.2)

In Chapter 6, the materials develop and students independently demonstrate procedural skill in writing and evaluating numerical expressions (6.EE.1) and writing, reading, and evaluating expressions in which letters stand for numbers (6.EE.2). Examples include:

• In Lesson 6.1, Activity 1: Can you make the math?, students write an expression from a word phrase. For example, Inline Question 1 states, “Which of the following correctly displays ‘one-third of the sum of a number and 5’?” answer choices: a. \frac{1}{3}(x+5); b. \frac{1}{3}+x+5; c. \frac{1}{3}x+5; d. \frac{1}{3x}+5.” Practice Questions 2 states, “Choose an expression for the following phrase: Four less than a number.” (6.EE.2) 

• In Lesson 6.2, Activity 2 Interactive, students evaluate expressions involving whole-number exponents using sliders to see how the exponent is used to represent multiplication. The teacher directions state, “For this Interactive, students can experiment with different values raised to an exponent and see the resulting expanded expression. Students can use the red and blue slider to adjust the values of the exponent.” (6.EE.1)

• In Lesson 6.3, Activity 3 Interactive, students demonstrate fluency in writing expressions involving whole-number exponents using the Interactive to determine how many lights are needed and identifying it as an expression with exponents. The student directions state, “Use the Interactive below to figure out how many strings of LED lights you would need to decorate the Christmas tree on an ugly Christmas sweater.” (6.EE.1)

• In Lesson 6.4, Activity 3: What picture does connect the dots make? states, “This Interactive helps students practice evaluating expressions using order of operations with an added bonus of drawing a picture. An expression is given and students can use the buttons at the bottom of the window to choose which operand that should be used next.” For example, Inline Question 1 states, “Which order of operations would you do FIRST in this type of problem? 2(4 + 3)2 ÷ 7.” (6.EE.1)

The materials reviewed for CK-12 Interactive Middle School Math 6 for CCSS 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. Examples include:

• In Lesson 3.1, Activity 1, students develop conceptual understanding of division of fractions with the interactive, Can You Design a Flag?. The materials state, “Students are given an adjustable flag that measures 1\frac{1}{5} m. Students can change the thickness of each portion by clicking and dragging the red points at the bottom of the flag. As the sections are adjusted, the fractions at the bottom will change, showing how much of the flag that portion is taking up. Students can toggle between a flag with 6 stripes and 3 stripes by clicking the button at the bottom right hand corner of the screen.” (6.NS.1)

• In Lesson 4.7, students demonstrate fluency with dividing decimals by decimals. For example, the practice problems include, “Find the quotient 31.93÷3.1. a) 10.3 b) 9.4 c) 12.6 d) 14.7.” (6.NS.3)

• In Lesson 1.5, Activity 3, Inline Questions, students demonstrate application of ratios in the Interactive about bicycles. Some examples include: “1. What is the relationship between 16 teeth in the back gear with 8 teeth in the back gear? How can you use this to find the number of teeth in the front gear for every 8 teeth in back gear?  A. Since 8 back teeth is half the number of 16 back teeth, you can divide 44 by 2 to get the number of front teeth associated with 8 back teeth.  B. Since 16 back teeth is two times 8 back teeth, you can multiply 44 by 2 to get the number of front teeth associated with 8 back teeth.  C. Since 8 back teeth is 16 back teeth minus 8, you can subtract 8 from 44 to get the number of front teeth associated with 8 back teeth.  D. Since 16 back teeth divided by 2 is 8 back teeth, you can divide 44 by 2 to get the number of front teeth associated with 8 back teeth. 2. Since 12 is halfway between 8 and 16, the number of teeth in the front gear will be halfway between 55 and the number of teeth associated with 8 back teeth. True/False” (6.RP.3)

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

• In Lesson 1.3, students develop a conceptual understanding of equivalent ratios through tape diagrams. For example the Warm Up states,” What does “miles per gallon” mean?,” students use the Interactive to adjust miles and gallons to create equivalent ratios. Inline Question 1 states, “Which ratio of miles to gallon would a person driving the truck like to have?” Activity 4 states, “ How can you mix the color brown with paints?,” students apply their understanding of equivalent ratios to mixing paint. Inline Question 1 states, “You mix 12 cups of brown paint by using 3 cups of yellow paint to 4 cups of red paint to 5 cups of blue paint. After painting a portion of a fence you realize that you need more of the same color paint to finish the fence. This time you want to make 26 cups. How many cups of each color do you need?”

• In Lesson 2.6, students develop a conceptual understanding of percentages being a ratio per 100 in the Interactive in Activity 1, How Much of a Century Have You Lived? The materials state, “To start, students are given a number line with dates from 200 to 2100 in ten year increments. Below there is a text box where students can input their birthday (must be between 2000 and the current date) and press the enter key. Students will see a red line on the timeline showing the imputed date to today, and above that, the percent.” Inline Question 5 states, “(Fill in the blank) When you are 20 years old you will have lived __% of a century. When you are 50 years old you will have lived __% of a century. When you are 100 years old you will have lived __% of a century. When you are 101 years old you will have lived __% of a century.” In Practice, students develop procedural skills as they independently determine percentages as numbers out of 100. The materials state, “Write the following percent as a ratio out of 100. 3% a) \frac{4}{100} b) \frac{3}{100} c) \frac{2}{100} d) \frac{1}{100}.”

The materials reviewed for CK-12 Interactive Middle School Math 6 for CCSS 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 materials intentionally identify and develop MP1 in connection with grade-level content by providing opportunities for the students to make sense of problems and persevere in solving them. Examples include: 

• In Lesson 2.3, Price Unit Rates, students are asked to determine how much a veterinarian earns in a day. Since there are multiple variables, students must persevere to find a solution. Activity 2: How much does a veterinarian get paid each day? states, “Veterinarians, or vets, provide medical care to pets and other animals.  In 2015, (6.RP.3b) veterinarians normally got paid about $88,490 per year. Vets often work 6 days a week and get as much as 20 days off of work a year (not including weekends), Use the interactive to determine how much money a veterinarian makes each work day.”

• In Lesson 4.6, Long Division, Activity 3, students are given a long division problem consisting of letters.  Two of the letters are given numeric value as shown below.  The students must figure out the remaining letters’ values.

  ?   ?   ?  O   ?   ?   L   ?    ?   ?  

  0   1   2   3   4    5   6   7   8   9

  “In Activity 3, the students are presented a challenging problem for which they must analyze givens, constraints, relationships and goals.  The students are specifically encouraged to look for entry points and plan a solution pathway rather than guessing and checking values.” The challenge and scaffolding encourage student perseverance. (6,NS.2)

• In Lesson 6.2, Using Exponents, students work with fractal trees to figure out how many branches there are at each step and how it changes from step to step. In Activity 1, “the students use the Interactive to make conjectures about the relationship displayed.”  The challenging nature of this task encourages perseverance. (6.EE.1)

The materials intentionally identify and develop MP2 in connection with grade-level content by providing opportunities for the students to reason abstractly and quantitatively. Examples include:

• In Lesson 3.2, Dividing into Groups, students “use visuals to represent how repeated addition can be used to solve the division of a fraction by a fraction.” In Activity 1: Can you fill the video game health bar?, students experiment with a “health bar” for a video game, dividing it into different fractional parts, reasoning both abstractly and quantitatively.  The problem states, “Every time Maria drinks her health potion, she gets a health boost and a fraction of her health is returned to her health bar.  Find out how many health boosts it takes to fill her health bar.” (6.NS.1)

• In Lesson 5.5, Absolute Value on Number Line, MP2 is intentionally developed throughout the lesson using the Interactives and Inline Questions. During Activity 3: Making a Robot Part 4, students reason quantitatively and abstractly as they “develop a system for using absolute value to write a command which will count the number of steps a robot took.” (6.NS.7) Then students answer such Inline Questions as the following, “2. Look at the robot interactive again, If the robot takes 5 steps forward and 3 steps backward, how many total steps has he taken?” (6.NS.7)

• In Lesson 7.1, Equivalent Expressions, MP2 is intentionally developed throughout the lesson using Inline Questions. Following the Warm Up Activity: How Much Do Beaded Bracelets Cost?, students reason quantitatively and abstractly by answering Inline Questions such as, “2. Create an expression to represent the cost of 5 bracelets with 4 stars and 9 ovals each.” Also following Activity 2: Astronomical Relationship, students answer the Inline Question, “1. How can you describe the relationship between the distance from each planet to the sun and the planet’s orbital period based on the information at the bottom of the Interactive?” (6.EE.3)

The materials reviewed for CK-12 Interactive Middle School Math 6 for CCSS 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.

The materials intentionally identify and develop MP3 in connection with grade-level content by providing opportunities for the students to construct viable arguments and critique the reasoning of others. Examples include:

• In Lesson 5.9, Distance on the Coordinate Plane, Standards for Mathematical Practice: “MP3: In the Introduction and Activity 1, the Discussion Questions provide the students the opportunity to form arguments and to analyze the arguments of their classmates.”  Warm - Up: Maps, Teacher Notes state: “This Interactive introduces distance on the coordinate plane within a real world context. Students can simply click and drag the red point at Dave’s House to count the distance between his house and the school. Either have students look up the distances between two places on Google Maps or use the map above to answer the questions below. Allow the students to construct an argument in small groups. Follow the small group discussions with a full class discussion to allow the students the opportunity to analyze the arguments of their classmates.”  (6.NS.8)  

• In Lesson 7.4, Solving Multiplication Equations, Standards for Mathematical Practice: “MP3: The Discussion Questions in Activity 3 give the students the opportunity to form an argument and to analyze the arguments of their classmates.” Activity 3: Division Equations, in the Discussion Questions, Teacher Notes state: “Allow the students to discuss these questions using Turn and Talk or in small groups to allow them to construct viable arguments. After an argument has been formed, begin a full class discussion to allow students the opportunity to analyze the arguments of their classmates.” Discussion Questions; “1. How did you determine the value of x in the Interactive?  2. If you have \frac{1}{4} of a variable on one side and you add three more fourths to that side of the balance beam, what operation can be used to represent this? 3. How can you check that a value is an answer to an equation? 4. Becky claims that both multiplication and division problems can be solved using multiplication. Is she correct? Support your answer with evidence.”  (6.EE.7)

• In Lesson 8.4, Comparing with an Unknown, Standards for Mathematical Practice: “MP3: In Activity 2, the students develop an algorithm for identifying mystery number as fast as possible. The students share their algorithms and discuss the relative effectiveness, using evidence, of each algorithm.” Activity 2: Guess the Number, Discussion Question asks, “Which strategies were more effective in helping to identify the mystery number in the fewest guesses?  Write out a series of steps that you believe should be followed to find the answer as quickly as possible.” The Teacher Notes state: “Answers may vary. Ask the students to write an algorithm to formalize their steps as much as possible. The students will likely come up with different algorithms. Allow the students to share their algorithms in a class discussion. The students should analyze the different algorithms and use evidence to support which they feel would be most effective.”  (6.EE.6)

• In Lesson 10.5, Measures of Center and Variability, Standards for Mathematical Practice: “MP3: In activity 3, the students are given the opportunity to analyze an argument about the effect of outliers on the interpretation of a dataset.” Activity 3: 100 Meter Hurdles, the Discussion Question states: ”After the completion of Helena’s 18 meet season, her 18 hurdle times are listed: 17.99, 18.25, 17.50, 35.55, 17.42, 17.85, 17.33, 16.98, 32.43, 17.88, 18.10, 17.32, 17.09, 30.45, 17.64, 17.82.  The mean time is 20.475 seconds.  The median time is 17.835 seconds.  Philip claims that the median is the better measure of center because Helena appears to have fallen down making that time an outlier.  Do you agree or disagree? Support your stance with evidence.”  (6,SP.3 & 6.SP.5)

The materials reviews for CK-12 Interactive Middle School Math 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. 

The materials intentionally develop MP6 through providing instruction on communicating mathematical thinking using words, diagrams, and symbols. Examples include:

• In Lesson 1.1, Introducing Ratios, the Teacher’s Edition includes, “In this lesson, students will learn that a ratio is used to describe the relationship between two quantities. They will use ratio language to describe quantities involving recipes, colored blocks, and butterflies. Throughout the lesson, it would be helpful to allow students the time to practice using ratio language to describe things in the classroom” (6.RP.1)

• In Lesson 5.6, The Four Quadrants, at the beginning of the lesson, the Teacher Notes state, “It would be helpful to define the terms horizontal and vertical, so students can use these terms throughout the lesson to describe an object's location on a coordinate plane.” (6.NS.C.6)

• In Lesson 5.7, Points on the Coordinate Plane, the Warm-Up states, “We can describe the position of an object by the location on the x-axis number line and the y-axis number line. The location of the object can be written using the coordinate (x, y) where x is the location of the object along the x-axis and y is the location of the object along the y-axis. Use the Interactive below to practice using coordinate notation.” (6.NS.6)

The materials use precise and accurate terminology and definitions when describing mathematics, and the materials also support students in using the terminology and definitions. There is no separate glossary in these materials, but definitions are found within the units in which the terms are used. The vocabulary words are in bold print. Examples include:

• In Lesson 1.1, Introducing Ratios, Activity 1, Inline Questions 2 state, “Angie wants to bake cookies for a bake sale. The recipe says, ‘for every 1 cup of butter use 3 cups of flour.’” You can use the word ratio to show the relationship between quantities. What is the ratio of butter to flour in one batch of cookies?” (6.RP.1)

• In Lesson 5.5, Absolute Value on Number Line, Activity 1 states, “The absolute value of a number is the distance of that number from zero. The absolute value of 23 is 23, because it is 23 units from zero. The absolute value of -12 is 12, because it is 12 units from zero. The absolute value symbol is written using a straight vertical line on either side of the number or expression. The absolute value of 5 is written \left|5\right|.” (6.NS.7)

• In Lesson 10.4, Mean, Median, Mode, and Range, the Warm Up states: “A measure of center is a single number used to describe a set of numeric data. It describes a typical value from the data set. Measures of the center include the mean and the median. The mean (or what is more commonly referred to as the average) of a data set is the sum of the data values divided by the number of data values in the set. As you saw in the Warm-Up, the mean can be thought of as "evening out" the data values. The range is a measure of spread. You can find the range by taking the greatest data value and subtracting the least data value. In other words, it is the difference between the maximum and minimum data point.” Activity 2 includes, “The median represents the middle value of an ordered data set. It is another measure of center.” (6.SP.2 & 6.SP.5)

The materials reviewed for CK-12 Interactive Middle School Math 6 for CCSS 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.

The materials intentionally identify and develop MP7 in connection with grade-level content by providing opportunities for the students to look for and make use of structure. Examples include:

• In Lesson 2.7, Percentages with Tape Diagrams, MP7 is intentionally developed throughout the lesson as “students break percentages down to increments and use increments of ten percent to find and estimate percentages.” The Warm Up directs students to, “Use the interactive to find the percent of the field a sprinter runs with a parachute. Find the distance she runs at 10%, 20%, 25%, and 50% of the field.” Students use the structure of percentages to determine the answer. (6.RP.3)

• In Lesson 3.5, Finding Rectangle Dimensions, Warm Up,  students use the interactive to create a visual model of fraction division.  The model’s structure is then tied to the equivalent fraction multiplication expression. The Inline Questions help students make the connection between division and multiplication of fractions. For example, Question 3: “What are some ways to rewrite \frac{1}{3}\div5 ? a) \frac{1}{3}\times\frac{1}{5} b) \frac{1}{5}\times\frac{1}{3} c) 3\times5 d) \frac{1}{3}\div\frac{1}{5} (6.NS.1)

• In Lesson 9.5, Area of Polygons, students make use of structure as they “use their existing knowledge of area formulas to derive formulas for the areas of regular and irregular polygons.” For example, in Activity 1, students decompose irregular polygons into known shapes in order to determine their area. (6.G.1 & 6.EE.2) 

The materials intentionally identify and develop MP8 by providing opportunities for the students to look for and express regularity in repeated reasoning. Examples include:

• Lesson 1.3, Equivalent Ratios & Tape Diagrams, In Activity 1, the students are asked to use repeated reasoning to make a conjecture about the relationship between arrangements of objects and the number of objects.” Activity 1: How many boxes of gel pens do we need to buy?  Discussion Questions: “#1 The teacher needs at least one pen each student for a full class of 18 students.  How many different ways can you get 18 pens?  #2 How many different ways can you get 24 pens?  #3 Is there a general method you could use to find the different ways to get a given number of pens if you didn’t know the class size? #4 Are all the arrangements you found ideal in a real-world context?” (6,RP.3)

• In Lesson 4.8, Using Greatest Common Factor, MP8 is intentionally developed as the students play the "mystery number" game more than once and use repeated reasoning to look for general methods and shortcuts when playing the game.”  Activity 3: “Can you guess the number? See if you can guess the mystery number in the game below!  The number is between 1 and 100. After your guess, you will see the GCF between the mystery number and your guess.  Can you find the number in 20 tries?” Teacher Notes “Students are given an empty table with columns labeled Common factors, Guess, GCF. Students can guess the mystery number by typing in a value at the bottom and clicking Guess. The value will be inputted into the table as well as the GCF of the guess and actual value.”  (6.NS.4)

• In Lesson 9.5, Area of Polygons, students intentionally develop MP8. In this activity, students use an interactive to explore breaking a polygon into triangular pieces and, “use repeated reasoning to construct a general expression for the area of a regular polygon using triangles.” The  Discussion Questions ask, “1. How could an expression for the area of the hexagon in the interactive be written as the sum of triangles? 2. How could an expression for the area of a pentagon be written as the sum of triangles? 3. How could an expression for the area of an octagon be written as the sum of triangles?” (6.G.1 & 6.EE.2)