7.G-Seattle Triangles

Standards:

CCSS.Math.Content.7.G.A.1

Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.

CCSS.Math.Content.7.G.A.2

Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.

Practices:

CCSS.Math.Practice.MP1 Make sense of problems and persevere in solving them.

Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.

In this lesson we will be using drawings of different types of triangles to determine how many different tourist locations we can attend on our field trip. In order to prepare the students for the upcoming field trip we decided it would relate well to our mathematics curriculum to investigate the triangles, and drawing of triangles under the conditions that we have made for the field trip. The conditions that we apply will be “can we travel the distance in the amount of time we have and still visit all of the locations on our itinerary”. Other conditions include:

  1. Type of triangle (scalene, obtuse, equilateral, isosceles, acute, right angle)
  2. Size of the legs, hypotenuse.
  3. Location the students need to travel.
  4. Time students have to travel a certain distance.
  5.  Amount of locations that can be visited to maximize the value of the field trip.

This lesson will model mathematics as more than just arithmetic – it is about problem solving.  Teaching mathematical modelling involves high-order thinking skills in representation of the real world, as well as skills of problem solving.  These are desirable outcomes that as important as getting the “right answers” to “problem sums”. By modeling this lesson in this format the students are engaged, because it is a trip they are going on. They get to make choices, and the lesson is hands on. Because the students are going to be “selling” their itinerary to the class, not only are they doing the work, but they will be explaining their thinking.

Questions to Present to Students:

Using the attached map, have students answer the following questions.

seattlemap (2)

Start at the ferry and from the ferry terminal walk west. The distance of the leg of our triangle is 10 cm (using the ruler given). If we draw an equilateral triangle for the itinerary how many places can we visit along our trip?

Start again at the ferry and walk west. Draw a new equilateral triangle that is 160% larger than your original triangle. With this larger triangle, how many places can we visit?

The class starts walking west from the ferry terminal to reach the Seattle Center. In order to head back to the ferry, what type of triangle would be best suited? (scalene) If you were to draw a scalene what would it look like on your map? What points of interest could you visit on the way back?

Draw an equilateral triangle from the ferry terminal to the Seattle Library.

What points of interest could the class visit if they we took this path?

Now try drawing an isosceles triangle from the ferry terminal to the Seattle Library.

Does this change the points of interest you can travel? How? Can you go to more places or fewer?

Maximizing the Lesson

What type of triangle would be used for the class to get the most educational, tourist, and entertainment options? Try scalene, acute, obtuse, right, and equilateral and consider which triangle if similar in size is best suited to visiting the most locations. Which triangle do you think will be the best? Why do you think this? Can you prove it?

How far can the class travel within 8 hours assuming most people walk at 4 mph(this requires students determine a scale, and the students will need to determine the distance units they will use? Hint: miles makes converting the distance a student can walk easily; otherwise more conversions need to be done)?

What type of triangle did the student draw? (Prompt students to explain their thinking at these points)

Allow students to make presentations to “sell” their tour itinerary.

Our class field trip has 8 hours to spend in Seattle. In order to get the most out of our field trip in the shortest amount of time, and the least amount of walking, draw a triangle that includes:

1 educational site

1 tourist site

1 entertainment site

Using the table given and the time constraint, determine the time it will take to travel to , and the amount of time each location will require to visit to get the most out of the itinerary.

Further Investigations:

Make predictions and determine the area of the triangles. Compare these to google maps to incorporate technology. Discuss whether the area is important for planning the itinerary. How is area going to be different than the perimeter? Which will be the most important for making decisions for the field trip? What about topography, how could this affect the itinerary? What about our model is good? What about our model doesn’t make sense? (Students should be able to determine that the city blocks are not triangles, so buildings may prevent walking in straight lines. What does that mean about our itinerary?

 

Point of Interest Time to Visit
Seattle University  35 minutes
Space Needle 30 minutes
Experience Music Project (EMP) 1 hour
Nordstrom 45 minutes
Seattle Art Museum 1 hour
Underground Tour 1 hour
Ye Olde Curiousity Shoppe 25 minutes
Seattle Aquarium 1 hour
Seattle Pacific University 30 minutes
Pike Place 25 minutes
Seattle Center 1 hour
Pacific Science Center 2 hours
International Fountain 15 minutes
Carousel 15 minutes
Safeco Pro Shop 20 minutes
Benaroya Hall 30 minutes
Seattle Library 45 minutes
Pirates Plunder Souvenirs 15 minutes
Tillicum Village 4 hours

– Eric Kress, Nicole Kraght

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