7th Grade - Gateway 2

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

Materials include problems and questions that develop conceptual understanding throughout the grade level. According to the Teacher Resource Program Overview, “Problem-Based Learning The Solve & Discuss It in Step 1 of the lesson helps students connect what they know to new ideas embedded in the problem. When students make these connections, conceptual understanding takes seed. Visual Learning In Step 2 of the instructional model, teachers use the Visual Learning Bridge, either in print or online, to make important lesson concepts explicit by connecting them to students’ thinking and solutions from Step 1.” Examples from the materials include:

• Topic 1, Lesson 1-4, Practice & Problem Solving, Problem 13, students write an equation based on a picture of a number line and use a number line to a different equation with the same difference. “Higher Order Thinking Use the number line at the right. a. What subtraction equation does the number line represent? b. Use the number line to represent a different subtraction equation that has the same difference shown in the number line. Write the subtraction equation.” (7.NS.1c)

• Topic 2, Lesson 2-2, Solve & Discuss It!, students extend their understanding of rates and ratios as they explore real-world problems. “Allison and her classmates planted bean seeds at the same time as Yuki and her classmates in Tokyo did. Allison is video-chatting with Yuki about their class seedlings. Assume both plants will continue to grow at the same rate. Who should expect to have the taller plant at the end of the school year?” (7.RP.1 and 7.RP.3)

• Topic 4, Lesson 4-3, Solve & Discuss It!, students develop conceptual understanding by connecting sorting terms into categories to combining like terms. “How can the tiles be sorted?” Ten tiles are shown with various terms on them, some with and without variables. (7.EE.1)

Materials provide opportunities for students to independently demonstrate conceptual understanding throughout the grade. Practice & Problem Solving exercises found in the student materials provide opportunities for students to demonstrate conceptual understanding. Try It! provides problems that can be used as a formative assessment of conceptual understanding following Example problems. Do You Understand?/Do You Know How? Problems have students answer the Essential Question and determine students’ understanding of the concept. Examples from the materials include:

• Topic 3, Lesson 3-1, Do You Understand?, Problem 3, students develop conceptual understanding as they determine if two different procedures will yield the same results. “Construct Arguments Gene stated that finding 25% of a number is the same as dividing the number by \frac{1}{4}. Is Gene correct? Explain.” (7.RP.3)

• Topic 4, Lesson 4-6, Solve & Discuss It!, students develop conceptual understanding as they determine if a scenario has one solution or many solutions. “The Smith family took a 2-day road trip. On the second day, they drove \frac{3}{4} the distance they traveled on the first day. What is a possible distance they could have traveled over the 2 days? Is there more than one possible distance? Justify your response.” (7.EE.1 and 7.EE.2)

• Topic 6, Lesson 6-1, Solve & Discuss It!, students analyze data from a sample and use it to gather information about the population. “The table shows the lunch items sold on one day at the middle school cafeteria. Use the given information to help the cafeteria manager complete his food supply order for next week.” A table is provided with two columns, one labeled “Lunch Item” and the other labeled “Number Sold”, the following information is in the table, Turkey Sandwich/43, Hot Dog/51, Veggie Burger/14, and Fish Taco/ 27. (7.SP.1)

The materials reviewed for enVision Mathematics Grade 7 meet expectations for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency. 

The materials develop procedural skill and fluency throughout the grade level. According to the Teacher Resource Program Overview, “Students develop skill fluency when the procedures make sense to them. Students develop these skills in conjunction with understanding through careful learning progressions.” Try It! And Do You Know How? Provide opportunities for students to build procedural fluency from conceptual understanding. Examples include:

• Topic 1, Lesson 1-2, Try It!, students use long division to convert a fraction to a decimal. “In the next several games, the pitcher threw a total of 384 pitches and used a fastball 240 times. What decimal should Janita use to update her report?”(7.NS.2d)

• Topic 5, Lesson 5-2, Practice & Problem Solving, Problem 10, students solve an equation that has decimals. “Solve the equation 0.5p - 3.45 = -1.2.” (7.EE.4a)

• Topic 8, Lesson 8-4, Practice & Problem Solving, Problem 10, students solve an equation to find the measurement of an unknown angle. “Find the value of x.” Two vertical angles are shown one measuring 125 and the other measuring (5x + 30). (7.G.5) 

The materials provide opportunities to independently demonstrate procedural skill and fluency throughout the grade level. Practice & Problem Solving exercises found in the student materials provide opportunities for students to independently demonstrate procedural skill and fluency. Additionally, at the end of each Topic is a Concepts and Skills Review which engages students in fluency activities. Examples include:

• Topic 2, Lesson 2-4, Practice & Problem Solving, Problem 7, students find the constant of proportionality of a given equation. “What is the constant of proportionality in the equation y = 5x?” (7.RP.2b)

• Topic 4, Lesson 4-5, Practice & Problem Solving, Problem 8, students factor the GCF from a given expression. “Factor the expression. 14x + 49” (7.EE.1 and 7.EE.2)

• Topic 8, Lesson 8-5, Practice & Problem Solving, Problem 7, students find the circumference of a circle in terms \pi of given the diameter. “Find the circumference of the circle. Use \pi as part of the answer.” An image of a circle is shown with a diameter of 7cm. (7.G.4)

The materials reviewed for enVision Mathematics Grade 7 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of mathematics. Engaging applications include single and multi-step problems, routine and non-routine, presented in a context in which mathematics is applied. 

The materials include multiple opportunities for students to independently engage in routine and non-routine application of mathematical skills and knowledge of the grade level. According to the Teacher Resource Program Overview, “3-Act Mathematical Modeling Lessons In each topic, students encounter a 3-Act Mathematical Modeling lesson, a rich, real-world situation for which students look to apply not just math content, but math practices to solve the problem presented.” Additionally, each Topic provides a STEM project that presents a situation that addresses real social, economic, and environmental issues, along with applied practice problems for each lesson. For example:

• Topic 1, 3-Act Mathematical Modeling: Win Some, Lose Some, Question 14, students predict the winner of a trivia game and the final score, “Construct Arguments If there were one final round where each contestant chooses how much to wager, how much should each person wager? Explain your reasoning." (7.NS.1 and 7.NS.3)

• Topic 5, STEM Project, Water is Life, students research filtration systems, decide which one they would purchase, and plan a fundraiser. Part of planning is writing an equation to represent the amount of money they will earn from a fundraiser to purchase the filtration system, "You have water to drink, to use to brush your teeth, and to bathe. You and your classmates will research the need for safe, clean water in developing countries. Based on your research, you will determine the type, size, and cost of a water filtration system needed to provide clean, safe water to a community. You will also develop a plan to raise money to purchase the needed filtration system.” (7.EE.3 and 7.EE.4)

• Topic 8, STEM Project, Upscale Design, students make scale drawings of existing paths or create plans for new walking paths or bikeways. "Review your survey results on the needs of walkers and bicyclists in your area. Choose an existing path or bikeway and make a scale drawing of the route. Add improvements or extensions to your drawing that enhance the trails and better meet the needs of users. If your area lacks a trail, choose a possible route and make a scale drawing that proposes a new path. How will your proposal enhance the quality of life and provide solutions for potential users?” (7.G.1 and 7.G.2)

The materials provide opportunities for students to independently demonstrate the use of mathematics flexibly in a variety of contexts. Pick a Project is found in each Topic and students select from a group of projects that provide open-ended rich tasks that enhance mathematical thinking and provide choice. Additionally, Practice & Problem Solving exercises found in the student materials provide opportunities for students to independently demonstrate mathematical flexibility in a variety of contexts. For example:

• Topic 2, Lesson 2-1, Practice & Practice Solving, Problem 9, students apply knowledge of solving multi-step problems with rational numbers to solving problems with ratios, rates, and unit rates. “Which package has the lowest cost per ounce of rice?” An image is provided of three bags of rice with various packaging sizes and prices, for example: One type of white rice is 12 punches and costs $4.56. (7.RP.1 and 7.RP.3)

• Topic 7, Pick a Project 7A, students design a game of chance and calculate theoretical probabilities of certain events happening. “Design and develop a game of chance. Find the theoretical probabilities of certain events (winning, losing, winning under certain conditions, losing under certain conditions). Test those probabilities. Have several people play your game. Write a report to accompany your game that compares the actual results to your theoretical results. Explain any inconsistencies.” (7.SP.5 and 7.SP.6)

• Topic 8, Lesson 8-9, Practice & Problem Solving, Problem 15, students solve real-world problems involving the volume of three-dimensional objects. “A cake has two layers. Each layer is a regular hexagonal prism. A slice removes one face of each prism, as shown. a. What is the volume of the slice? b. What is the volume of the remaining cake?” (7.NS.3 and 7.G.6)

The materials reviewed for enVision 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.

All three aspects of rigor are present independently throughout the program materials. Examples, where materials attend to conceptual understanding, procedural skill and fluency, and application, include:

• Topic 1, Lesson 1-3, Explore It!, students extend their conceptual understanding of positive and negative numbers as they use number lines and absolute value to solve problems, “Rain increases the height of water in a kiddie pool, while evaporation decreases the height. The pool water level is currently 2 inches above the fill line. A. Look for patterns in the equations in the table so you can fill in the missing numbers. Describe any relationships you notice. B. Will the sum of 2 and (-6) be a positive or negative number? Explain.” (7.NS.1b and 7.NS.1d)

• Topic 4, Lesson 4-5, Practice & Problem Solving, Problem 14, students develop procedural skill and fluency in finding the GCF and factoring expressions. “You are given the expression 12x + 18y + 26. a. Make Sense and Persevere What is the first step in factoring the expression? b. Factor the expression.” (7.EE.1 and 7.EE.2)

• Topic 8, Lesson 8-8, Practice & Problem Solving, Problem 12, students use application of surface area knowledge to solve real-world problems, “A box has the shape of a rectangular prism. How much wrapping paper do you need to cover the box?” Illustration dimensions provided are h = 3 inches, w = 15 inches, and l = 16 inches. (7.G.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:

• Topic 1, Lesson 1-2, Practice & Problem Solving, Problem 18, students solve real-world problems while developing procedural skill and fluency with rational numbers. “Reasoning Aiden has one box that is 3$$\frac{3}{11}$$feet tall and a second box that is 3.27 feet tall. If he stacks the boxes, about how tall will the stack be?” (7.NS.2d)

• Topic 5, Mid-Topic Performance Task, students develop procedural skill and fluency as they apply their knowledge to writing and solving equations in a real-world scenario. “Marven and three friends are renting a car for a trip. Rental prices are shown in the table. Part A Marven has a coupon that discounts the rental of a full-size car by $25. They decide to buy insurance for each day. If the cost is $465, how many days, d, will they rent the car? Write and solve an equation.” A table is given with the “Item” in one column and the  “Price” in the next column. (7.EE.3 and 7.EE.4a)

• Topic 7, Lesson 7-3, Do You Know How?, Problem 4, students develop conceptual understanding and procedural skill and fluency as they find the theoretical probability for an event. “Kelly flips a coin 20 times. The results are shown in the table where ‘H’ represents the coin landing heads up and ‘T’ represents the coin landing tails up. 4. The theoretical probability that the coin will land heads up is ____.”. (7.SP.6 and 7.SP.7)

The materials reviewed for enVision 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. 

Some examples where the materials support the intentional development of MP1 are:

• Topic 3, Lesson 3-2, Practice & Problem Solving, Problem 11, students examine the relationships between the quantities and solve for the whole, “A restaurant customer left $3.50 as a tip. The tax on the meal was 7% and the tip was 20% of the cost including tax. a. What piece of information is not needed to compute the bill after tax and tip? b. Make Sense and Persevere What was the total bill?”

• Topic 4, Lesson 4-2, Practice & Problem Solving, Problem 17, students are asked to apply understanding of equivalent expressions and look at a group chat message in order to find the amount of money individuals put in. “Higher Order Thinking To rent a car for a trip, four friends are combining their money. The group chat shows the amount of money that each puts in. One expression for their total amount of money is 189 plus p plus 224 plus q. a. Use the Commutative Property to write two equivalent expressions. b. If they need $500 to rent a car, find at least two different pairs of numbers that p and q could be.”  

• Topic 5, Lesson 5-6, Practice & Problem Solving, Problem 11, students must make sense of the advertisement for car rental to ensure that they stay within budget. “Make Sense and Persevere Talia has a daily budget of $94 for a car rental. Write and solve an inequality to find the greatest distance Talia can drive each day while staying within her budget.” ( A chart that includes the rate per day and cost per mile for a car rental is included.) 

Some examples where the materials support the intentional development of MP2 are:

• Topic 6, Lesson 6-3, Practice & Problem Solving, Problem 9, students interpret and compare statistical measures and reason about data sets in both qualitative and quantitative forms. “Reasoning A family is comparing home prices in towns where they would like to live. The family learns that the median home price in Hometown is equal to the median home price in Plainfield and concludes that the homes in Hometown and Plainfield are similarly priced. What is another statistical measure that the family might consider when deciding where to purchase a home?”

• Topic 7, Lesson 7-3, Practice & Problem Solving, Problem 11, students reason about the difference between theoretical and experimental probability. “The theoretical probability of selecting a consonant at random from a list of letters in the alphabet is \frac{21}{26}. Wayne opens a book, randomly selects a letter on the page, and records the letter.  He repeats the experiment 200 times. He finds P(consonant)= 60%. How does the theoretical probability differ from the experimental probability? What are some possible sources for this discrepancy?”

• Topic 8, Lesson 8-1, Do You Understand? Problem 3, students reason about the difference between scaling on a map and in real life.“Reasoning Mikayla is determining the actual distance between Harrisville and Lake Town using a map. The scale on her map reads, 1 inch = 50 miles. She measures the distance to be 4.5 inches and writes the following proportion. \frac{1 in}{4.5 in} = \frac{50 mi}{x mi} Explain why her proportion is equivalent to \frac{50 mi}{1 in} = \frac{x mi}{4.5 in}.

The materials reviewed for enVision 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.

Student materials consistently prompt students to construct viable arguments. These opportunities are found in the following activities: Solve & Discuss It!, Explain It!, Explore It!, Practice & Problem Solving, Do You Understand?, and Performance Tasks. Examples include:

• Topic 5, Lesson 5-3, Explain It!, students use their understanding of the Distributive Property to construct arguments. “Six friends go jet skiing. The total cost for the adventure is $683.88, including a $12 fee per person to rent flotation vests. Marcella says they can use the equation 6r + 12 = 683.88 to find the jet ski rental cost, r, per person. Julia says they need to use equation 6(r + 12) = 683.33. A. Construct Arguments Whose equation accurately represents the situation? Construct an argument to support your response. B. What error in thinking might explain the inaccurate equation?”

• Topic 6, Lesson 6-4, Do You Understand?, Problem 3, students use their understanding of inferences to construct arguments. “Construct Arguments Two data sets have the same mean but one set has a much larger MAD than the other. Explain why you may want to use the median to compare the data sets rather than the mean.”

• Topic 8, Lesson 8-4, Explore It!, students analyze problems and use angle relationships to construct and justify arguments. “The intersecting skis form four angles. A. List all the pairs of angles that share a ray. B. Suppose the measure of \angle1 increases. What happens to the size of \angle2? \angle3?  C. How does the sum of the measures of \angle1 and \angle2 change when one ski moves? Explain. Focus on math practices Construct Arguments Why does the sum of all four angle measures stay the same when one of the skis moves? Explain.”

Student materials consistently prompt students to analyze the arguments of others. These opportunities are found in the following activities: Solve & Discuss It!, Explain It!, Explore It!, Practice & Problem Solving, Do You Understand?, and Performance Tasks. Examples include:

• Topic 2, Lesson 2-5, Do You Understand?, Problem 3, students analyze the arguments of others by interpreting if points contain a proportional relationship. “Construct Arguments Makayla plotted two points (0, 0) and (3, 33), on a coordinate grid. Noah says that she is graphing a proportional relationship. Is Noah correct? Explain.”

• Topic 6, Lesson 6-2, Do You Understand?, Problem 3, students analyze the arguments of others as they make inferences about a population from sample data. “Critique Reasoning Darrin surveyed a random sample of 10 students from his science class about their favorite types of TV shows. Five students like detective shows, 4 like comedy shows, and 1 likes game shows. Darrin concluded that the most popular type of TV shows among students in his school is likely detective shows. Explain why Darrin’s inference is not valid.” 

• Topic 7, Lesson 7-4,  Explain It!, students critique the reasoning of two members of the chess club about their chances of being captain, by using mathematical arguments to justify their answers. “The Chess Club has 8 members. A new captain will be chosen by randomly selecting the name of one of the members. Leah and Luke both want to be captain. Leah says the chance that she will be chosen as captain is \frac{1}{2} because she is either chosen for captain or she is not. Luke says the chance that he is chosen is \frac{1}{8}. A. Construct Arguments Do you agree with Leah’s statement? Use a mathematical argument to justify your answer. B. Construct Arguments Do you agree with Luke’s statement? Use a mathematical argument to justify your answer.

The materials reviewed for enVision Mathematics Grade 7 meet expectations for supporting the intentional development of MP4: Model with mathematics; and MP5: Use appropriate tools strategically, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

The materials allow for the intentional development of MP4 to meet its full intent in connection to grade-level content. Examples of this include:

• Topic 3, Lesson 3-3, Practice & Problem Solving, Problem 15, students identify important quantities, use equations to represent their relationships, and interpret the results using mathematical models in a real-world situation. “Model with Math There are 4,000 books in the town’s library. Of these, 2,600 are fiction. Write a percent equation that you can use to find the percent of books that are fiction. Then solve your equation.” 

• Topic 5, Lesson 5-1, Practice & Problem Solving, Problem 14, students create an equation to represent the scenario of purchasing a pet iguana. “You want to buy a pet iguana. You already have $12 and plan to save $9 per week. a. Model with Math If w represents the number of weeks until you have enough money to buy the iguana, what equation represents your plan to afford the iguana? b. Explain how you could set up an equation to find the amount of money you should save each week to buy the iguana in 6 weeks.”

• Topic 7, Lesson 7-1, Practice & Problem Solving, Problem 15, students determine if a model created by another person is fair and explain how to make it fair. “Model with Math Henry is going to color a spinner with 10 equal-sized sections. Three of the sections will be orange and 7 of the sections will be purple. Is this spinner fair? If so, explain why. If not, explain how to make it a fair spinner.”

The materials allow for the intentional development of MP5 to meet its full intent in connection to grade-level content. Examples of this include:

• Topic 5, 3-ACT Mathematical Modeling: Digital Downloads, in Act 1, students watch a video on digital downloads and must determine how many songs a person can purchase using the balance on a gift card. In Act 2, Problem 7 asks students, “Use Appropriate Tools What tools can you use to solve the problem? Explain how you would use them strategically.“

• Topic 7, Lesson 7-7, Solve & Discuss It!, students explain how tools can be used to simulate events. “Jillian lands the beanbag on the board in about half of her attempts in a beanbag toss game. How can she predict the number of times she will get the beanbag in the hole in her next 5 attempts using a coin toss? Focus on math practices Use Appropriate Tools When might it be useful to model a scenario with a coin or other tool?”

• Topic 8, Lesson 8-2, Do You Understand?, Problem 2, students explain when it is appropriate to use technology strategically. “Use Appropriate Tools How can you decide whether to draw a shape freehand, with a ruler and protractor, or using technology?”

The materials reviewed for enVision 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.

Students are encouraged to attend to the specialized language of mathematics throughout the materials. A chart in the Topic Planner lists the vocabulary being introduced for each lesson in the Topic. As new words are introduced in a Lesson they are highlighted in yellow and students are encouraged to utilize the Vocabulary Glossary in the back of the text (with an animated version online in both English and Spanish) to find both definitions and examples where relevant. Lesson Practice includes questions that reinforce vocabulary comprehension and the teacher's side notes provide specific information about what math language and vocabulary are pertinent for each section.

Examples where students are attending to the full intent of MP6 and/or attend to the specialized language of mathematics include:

• Topic 1, Topic, Review, Use Vocabulary in Writing, students attend to the specialized language of mathematics and precision as they explain how they determined two decimals are equivalent. “Explain how you could determine whether{{\cfrac{21}{3}}\above{2pt}{\cfrac{120}{12}}}and \frac{7}{9} have the same decimal equivalent. Use vocabulary words in your explanation.”

• Topic 4, Mid-Topic Checkpoint, Problem 1, students attend to precision as they write an expression to represent a situation. “Vocabulary If you write an expression to represent the following situation, how can you determine which is the constant and which is the coefficient of the variable? The zoo charges the Garcia family an admission fee of $5.25 per person and a one-time fee of $3.50 to rent a wagon for their young children.” In the Mid-Topic Assessment, Problem 1, students attend to the specialized language of mathematics as they explain the difference between constants and variables. “Vocabulary How is a constant term different from a variable term for an expression that represents a real-world situation?” 

• Topic 6, Lesson 6-1, Try It!, students attend to the specialized language of mathematics as they fill in sentence frames with mathematical terminology. “Miguel thinks the science teachers in his school give more homework than the math teachers. He is researching the number of hours middle school students in his school spend doing math and science homework each night. The ______ includes all the students in Miguel’s middle school. A possible _________ is some of the students from each grade in the middle school.” 

• Topic 7, Get Ready!, Vocabulary, students attend to the specialized language of mathematics as they choose the best term from a box that fit the definition. “Choose the best term from the box to complete each definition. 1. A(n) ___ is a drawing that can used to visually represent information. 2. The number of times a specific value occurs is referred to as ______…” A box is given which has the following terms: equivalent, frequency, diagram, and ratio.

The materials reviewed for enVision 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.

Students are encouraged to look for and make use of structure as they work throughout the materials, both with the instructor's guidance and independently. Examples of where there is intentional development of MP7 include:

• Topic 1, Lesson 1-2, Practice & Problem Solving, Problem 14, students use the structure of a fraction to accurately find the decimal equivalent. “Use Structure Consider the rational number \frac{3}{11}. a. What are the values of a and b in a/b when you use division to find the decimal form? b. What is the decimal form for \frac{3}{11}?”

• Topic 3, Lesson 3-2, Practice & Problem Solving, Problem 15, students use structure to identify and align the part, whole, and percent to set up a proportion to solve real-world problems, “A school year has 4 quarters. What percent of a school year is 7 quarters?”

• Topic 4, Lesson 4-8, Practice & Problem Solving, Problem 13, students analyze relationships between quantities in real-world situations for equivalency. “Use Structure The area of a rectangular playground has been extended on one side. The total area of the playground, in square meters, can be written as 352 + 22x. Rewrite the expression to give a possible set of dimensions for the playground.” 

Students look for and express regularity in repeated reasoning as they are engaged in the course materials. Examples of intentional development of MP8 include: 

• Topic 3, Lesson 3-5, Do You Understand?, Problem 3, students use repeated reasoning to make a general statement about the price being similar or different after a price markdown and then markup. “Generalize When an item is marked up by a certain percent and then marked down by the same percent, is the sale price equal to the price before the markup and markdown?” 

• Topic 6, Lesson 6-3, Focus on math practices, students look for the regularity of real-world data in various display forms to draw a conclusion. “Reasoning Use your data display, what can you infer about the number of siblings that most seventh graders have? Explain?”

• Topic 8, Lesson 8-3, Practice & Problem Solving, Problem 12, students analyze triangles and generalize that its side and angle conditions determine if it results in one triangle, more than one triangle, or no triangle. “Given two side lengths of 15 units and 9.5 units, with a nonincluded angle of 75°, can you draw no triangles, only one triangle, or more than one triangle?”