Gallery Problems Exercise

Gallery Problems Exercise

Map Reading

Work Time

Map Reading

This is a street map of the city of Provo, Utah.

The street map can be thought of as a coordinate system. At the letter "A," you see the center of town, the origin of the street map coordinate system. The street map coordinates that are used here consist of names starting with W, E, N, or S, and number likes 100, 200, 300, etc., that are followed by W, E, N, and S.

  1. What does W 200 N Street mean? Find the street on the map.
  2. Imagine placing over the street map an x–y coordinate grid, whose origin is labeled A. The coordinates of the streets are given by the numbers 1, 2, 3, etc. Center Street can be seen as the x-axis, Route 189 as the y-axis.
    • Where is the point with coordinates (3, –1)? And (–4, 1)?
    • What is the smallest number of blocks you would need to walk to get from one of these points to the other?
  3. Create a problem for someone to solve by giving a route based on coordinate clues to find a specific location.
  4. How can you tell which streets are west of Route 189? What type of integer would you use on the x–y coordinate grid to represent these streets? What coordinate, x or y, represents east-west streets?
  5. How can you tell which streets are south of Route 89? What type of integer would you use to represent these streets? What coordinate, x or y, represents north-south streets?

Burning Candles

Work Time

Burning Candles

Part 1

  1. Watch the video. Compare and contrast how the candle in the video burns compared to a candle in real life. How do you think this video was made?
  2. Create a graph that shows what happened to the height of the blue candle over time. (Height of the candle means height of the top of the candle.)
  • Represent time on the horizontal axis and height on the vertical axis.
  • To make your graph, first use the given information to calculate the candle height at different times. Then make a two-column table with time in one column and candle height in another column. Use your table to help create your graph.

 

Part 2

Before the candle on the right starts burning, it has a height of 120 mm. Its burning time is 75 hours and it burns at a constant rate

  1. Draw a graph that shows what happens to the height of the candle on the right over time.
  2. Use the graph to find the height of the candle after 30 hours of burning.
  3. If you graphed the height of each of the candles in the picture as the candle burned, do you think all of the graphs would be straight lines? Why or why not? VIDEO: Time-Lapse Birthday Candles

More Directions

Work Time

More Directions

Your friend Doug gave you directions for how to bike to his house from his favorite ice cream shop in town. You are not sure which ice cream shop he meant.

Doug’s house is at the corner of H Street and 9th Avenue.

Here are the directions Doug gave you:

  • Bike straight for 4 blocks and turn left.
  • Bike straight for 5 blocks and turn left again.
  • Bike straight for 3 blocks and you are at my house.
  1. How can you use a street map to help you figure out where his favorite ice cream shop is?
  2. Draw a street map showing the streets from 1st through 17th Avenue, and A through M Street.
  3. Add the following features to your map.
    • A river runs between 4th and 5th Avenue.
    • There are two bridges for getting across the river, one at C Street and one at K Street.
    • City Park is "L"-shaped and has corners at 7th and G, 7th and D, 16th and D, 16th and I, 11th and I, and 11th and G. You are not allowed to bike across City Park.
  4. Where is Doug’s favorite ice cream shop located? Show all possible locations and the possible paths you will take to Doug’s house on the map.

Leagues

Work Time

Leagues

A top division has 22 teams. Each team plays all the other teams twice, once at home and once away. Games are usually played on Saturdays, but sometimes on Wednesdays, too. The season lasts about 35 weeks. There is a proposal to expand this top division to 30 teams.

  1. How many matches in all would be played and how many matches would each of the 22 teams play?
  2. What would the effect of expanding to 30 teams be on the length of the playing season?
  3. What operation can help you “share” the matches evenly between the weeks?

Traffic

Work Time

Traffic

Situation A:

Monique stands by the highway. She notices that one car goes by every second. She knows that these cars are going 60 miles per hour.

  1. How many feet apart are the cars?
  2. How many cars are there on 1 mile of highway?

Justify your answers mathematically.

  • Think about 60 miles per hour as a ratio of 60 miles to 1 hour. Use this ratio to write a proportion that you could use to calculate the distance the car travels each minute and second.

  • Remember, there are 5,280 feet per mile.

Situation B:

One car goes by every second, but the cars are going 30 miles per hour.

  1. How many feet apart are the cars?
  2. How many cars are there on 1 mile of highway?

Justify your answers mathematically.

  • When the cars are traveling more slowly, what would happen to the distance between each car? How can you use the relationship between the speeds and the answer to Question 1 to help you calculate the distance between each car?

  • If you are still seeing a car every second and the cars are traveling more slowly than before, would there be more or less cars along a given length of the highway? How can you use the relationship between the speeds and the answer to Question 1 to help you calculate the number of cars on the 1-mile stretch of highway?

Situation C:

One car goes by every 2 seconds and the cars are going 75 miles per hour.

  1. How many feet apart are the cars?
  2. How many cars are there on 1 mile of highway?

Justify your answers mathematically.

  • How does the increase in time from 1 second to 2 seconds affect the problem?

  • How does this increase in time affect the distance between each car?

  • How does the increase in time affect the number of cars on the road?

  • How can you use the relationship between the original speed and the increased speed to help you calculate your answer?

Leaky Faucet

Leaky Faucet

How long will it take for the sink to fill up?

  1. Watch the video Drips. Write down a estimate of how long it will take for the sink to fill up based on what you see.
  2. Watch the video 5 Drops. Write down an estimate for filling the sink that you know is too high and an estimate that you know is too low.
  3. What information do you need to know in order to find the amount of time it will take the sink to fill up at the given drip rate?
  4. The sink holds 3 gallons. Use the information in the video 85 Milliliters to find how long the sink will take to fill.
  5. Watch the video Leaky Faucet Timelapse.
    • How does your answer compare to the time recorded in the video?
    • What might cause the similarities or differences between your calculations and the actual time?

VIDEO: Drips
VIDEO: 5 Drops
VIDEO: 85 Milliliters
VIDEO: Leaky Faucet Timelaspe

Frog Jump

Work Time

Frog Jump

These two frogs can change places in three moves. The yellow frog starts in the leftmost box.

Rules:

  • A frog can either hop onto an adjacent square, or jump over one other frog to the vacant square immediately beyond it.
  • The yellow frogs can only move from left to right.
  • The green frogs can only move from right to left.
  1. The frogs shown can be interchanged in 15 moves. Explain how.
  2. Now suppose that there are an unequal number of green and yellow frogs. These frogs can be interchanged in 11 moves. Explain how.