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Math in Motion – Activity Directions

 
Activity Info Activity Directions Complete Module Lesson Plans Activity Sheets Activity Sheet Keys Origami Jumping Frog Instructions T-Table and Grid Paper 1/4 inch Graph Paper
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Through a series of hands-on activities, students develop proportional reasoning and are able to create tables, graphs, and equations.

Activities on this page:

Jump, Jump, Jump!

Students complete three jumping activities and measure, record, and graph their results. Students also create ratios with their results and learn about equivalent ratios.

Materials:

Directions:

Students should complete each of the tasks below (or other measurable tasks of your design).

  1. Use masking tape to make a line on the floor. Put your toes on the line and jump foward as far as you can (you must be able to stay standing and in one place for the jump to count). Use the masking tape to make another line on the floor at the tip of your toes. Now measure the distance between the two lines in metric and English standard units. Record your data.
  2. Use masking tape to make a line on the floor (or use the same line from part 1). Put the side of your feet on the line and jump sideways as far as you can (again, you must be able to stay standing and in one place for the jump to count). Use the masking tape to make another line on the floor at the side of your feet. Now measure the distance between the two lines in metric and English standard units. Record your data.
  3. Tape a long piece (or multiple small pieces) of paper on a wall. Use the ink pad on the tips of each of your fingers and/or hand. Stand flat-footed next to the wall, reach your arm up the wall as far as you can, and leave your finger/hand print on the paper. Now jump as high as you can and smack the paper on the wall to leave another finger/hand print. Measure the distance between the top of your lower prints and the top of your higher prints in metric and English standard units (you may need to use a footstool to help reach). Note: If you don't have an ink pad or don't want to risk leaving marks on a wall, you may use tape or sticky notes to leave spots on the wall.

On a large piece of chart paper, graph your data for each activity as a group (a bar graph is recommended). Now, measure each participant's height in metric and English standard units and graph the data as a group.

Study the graphs and answer the following questions
  1. Did anyone jump as far forward as they are tall?
  2. Who jumped the longest distance forward?
  3. Who came the closest to jumping as far forward as they are tall? Is it the same person who jumped the longest distance?
  4. Did anyone jump as far sideways as they are tall?
  5. Who jumped the longest distance sideways?
  6. Who came the closest to jumping as far sideways as they are tall? Is it the same person who jumped the longest distance?
  7. Did anyone jump as far up as they are tall?
  8. Who jumped the longest distance up?
  9. Who came the closest to jumping as far up as they are tall? Is it the same person who jumped the longest distance up?

Write your result for each task compared to your height (in metric and English standard) as a ratio in three ways: using a colon, as a fraction, and using the word "per" or "for every." For example, if I jumped 45 inches forward and I am 75 inches tall, I would write: 45 in:75 in ; 45 in⁄75 in ; and,"45 inches forward for every 75 inches tall." If the first number and the second number of your ratio can be divided by the same number, the ratio can be simplified (just like a fraction). Re-write your ratios in simplest form. For example, 45 and 75 can both be divided evenly by 15, so I can re-write the ratios as: 3 in:5 in ; 3 in⁄5 in ; and, "3 inches forward for every 5 inches tall." These new ratios are equivalent to the previous ratios because they are equal to each other even though they have different numbers.

Now that your ratios are in simplest form, focus on the ratios written as fractions. Are any the fractions greater than 1? What does this mean about whether you are able to jump as far as you are tall? Place the fractions in order from least to greatest (be sure to include both metric and Englist standard measurements). Make at least three conclusions based on your ordering of the fractions.

Share your results with the rest of your peer to find out if you answered the previous questions correctly.

Note: If you are doing this activity with less than 6 students, you may want each student to complete each task more than once so there is more data to graph. You may use the average or the maximum of the jumps to find the ratios.

If You Could...

Students use measurements of their own bodies and their knowledge of ratios to find out what super human feats they could accomplish if they could... like various animals.

Materials:

Directions:

Make the following measurements and record your data.

  1. Your height in inches and centimeters.
  2. Your weight in ounces and pounds. (Note: You will most likely have to weigh yourself in pounds and convert it into ounces and kilograms.)

Answer the following questions based on your measurements and your knowledge of ratios.

  1. ...Hop Like a Frog

    The frog is an excellent jumper. On average, a 3-inch frog can hop 60 inches.

    1. Write that as a ratio.
    2. A frog can jump _?_ times farther than its length. How many inches could you jump if you could jump like a frog?
    3. How many centimeters could you jump if you could jump like a frog?

  2. ...Be as Strong as an Ant

    3. Ants may be tiny, but they are great weight lifters. For example, an ant that weighs 1⁄250 of an ounce, can lift a bread crumb that weighs 1⁄5 of an ounce.

    1. How much larger is 1⁄5 than 1⁄250?
    2. If you were as strong as an ant, how much weight could you lift?

  3. ...Scurry Like a Spider

    4. Considering its length, a female spider is incredibly fast. It can move 33 times its body length in only 1 second!

    1. How many inches could you travel in one second if you were as fast as a spider?
    2. How many feet could you travel in one second?
    3. How many miles could you travel in one hour?

  4. ...Eat Like a Hummingbird

    5. Due to its incredibly high metabolism and energy expendaiture, a hummingbird eats a lot of food (for its size). A 3 gram hummingbird has been recorded to eat 14 meals an hour for 12 hours a day and 3.6 grams of food per meal.

    1. If you ate as often as a hummingbird, how many meals would you eat in a day?
    2. Write the amount a hummingbird eats in a meal and its body weight as a ratio.
    3. If you ate like a hummingbird, how many pounds of food would you eat in a meal?
    4. If you ate like a hummingbird, how many pounds of food would you eat in a day?

What's My Rate?

Students find their heart rate before and after exercising and compare the two using ratios, proportions, and graphs.

Materials:

Directions:

  1. Count your heart beats for 10 seconds. It is easiest to find your heart beat by placing two fingers on your wrist or neck. Get a friend to help you if you're having trouble counting the heart beats and watching the clock.

    Record you heart beats in 10 seconds of the first row of the table below. Imagine that this heart rate will remain unchanged. Complete the table below using this rate until you are able to identify your beats per minute.

    Heart Rate Trial 1
    Time
    (seconds)    		 Beats
    10
    20
    30






    What is your first heart rate (in beats per minute)?


  2. Do 20 jumping jacks. Then count your heart beats for 10 seconds and record your findings on the table below. Assume that this rate is constant and fill out the rest of the table until you find your heart rate in beats per minute.

    Heart Rate Trial 2
    Time
    (seconds)    		 Beats
    10
    20
    30






    What is your heart rate after the jumping jacks?

    How does this compare to your previous heart rate? Write this comparison as a ratio.


  3. Graph the data for each of your "Heart Rate Trials" in the same coordinate plane. Use the Time (in seconds) as your x-coordinate and the Beats as your y-coordinate. Use different colors to connect the points from each set of data.

    What do you notice about the graph? Make at least one observation about the points on the graph. Also make at least one observation comparing the two sets of data.

    Are the points on the graph proportional? Support your answer.


  4. Optional: Complete another physical exercise. Count your heart beats in 10 seconds. Create and a graph similar to the ones in Part 1 and 2. Graph your data on the same coordinate plane (remember to use another color).

    Make comparisons of your new data to the existing data. What conclusions can you make based on all your data?

    Some suggestions for more exercises are: jog in place for 15 seconds, do 25 sit-ups, jump rope 50 times, etc.

Frog Race

Students complete five activities, starting with racing origami jumping frogs, and concluding with participating in an actual race. As students progress through the activities they gain understanding of rations, proportions, rates, tables, graphs, and linear equations.

Materials (for all activities):

Part 1 - On Your Marks...

Students create origami jumping frogs and find their jumping ability using ratios, proportions, and graphs.

Materials:

Directions:

Each student should complete each of the following steps. Students should feel free to share information, ideas and results, and also help each other with questions.

Note: If three or less students are completing this activity, you may want each student to create more than one frog and a set of data for each frog.
  1. Create an origami jumping frog according to the instructions. Feel free to decorate and name your frog.

  2. Use masking tape to make a starting line on the floor or on a long table top. Place a meter stick perpendicular to the starting line and tape it to the surface.

  3. "Jump" your origami frog (by pushing down on its tail and releasing) from the starting line along the meter stick for 15 seconds. Record how far your frog jumped to the nearest centimeter (or 1⁄4 inch if you are using a yard stick).

  4. Make a T-table to record your time and distance. Assume the frog would continue at the same rate. Use 10 second intervals on your table to find your frog's rate in centimeters per minute.

  5. Graph your data in a coordinate plane. Use Time (in seconds) for the x-coordinate and Distance for the y-coordinate.

  6. Use your and other students' graphs to answer the following questions:

    • What is the relationship between the rate, time and distance?
    • What variable is the independent variable?
    • What varibale is the dependent variable?
    • Why do the coordinates on the graph form a straight line?
    • What would the coordinate (10,40) mean on this graph?
    • Why are some lines steeper than others?
    • What is the relationship between the steepness of the line and the frog jumping rate?

  7. Use the formula d = rt (or distance = rate × time) to write an equation for your graph.

  8. Based on your equation, how far would your frog jump in 20 seconds? 50 seconds? 1 minute 11 seconds?
Note: Save your frogs, tables, and graphs. You will need them in Part 3.

Part 2 - Tale of Two Frogs

Students further their understanding of the distance, time, and rate relationship by helping complete two stories of racing frogs.

Materials:

Directions:

Read each frog tale; complete each table; and answer the questions

The Story of Hip Hop

Marty is preparing his frog Hip Hop for a big race. But before Hip Hop makes his mad dash for the finish line, Marty wants to find out about how long it will take Hip Hop to finish the race. Marty races Hip Hop for several seconds. After measuring the time and distance, Marty wrote this equation: d = 5.4t.

  1. Using the equation, help Marty complete the table below.

    Hip Hop's Practice Run
    Time
    (in seconds)    		 Distance
    (in centimeters)
    10
    15
    18
    30

    		270

  2. Explain how the rate of 5.4cm⁄s (centimeters per second) can be found using the data in the table.

  3. Explain how the table demonstrates that the graph of this relationship would be a straight line.
The Story of Frogger

Gisella is preparing her frog Frogger for a big race. But before Frogger lets it loose on the track, Gisella wants to find out about how long it will take Frogger to complete the race. Gisella lets Frogger jump for 20 seconds, and finds that she reaches a distance of 140 cm.

  1. Gisella wants to create a table based on Frogger's average jumping rate. Help her complete the table below.

    Frogger's Practice Run
    Time
    (in seconds)    		 Distance
    (in centimeters)
    20 140
    40 280
    50
    55

    		420

  2. What is Frogger's jumping rate in centimeters per second?

  3. What equation would Gisella write for Frogger?
The Race is On!

Hip Hop and Frogger are set to face each other in a 500 cm race.

  1. Which frog is likely to win the race? Why?

Part 3 - Get Set...

Materials:

Directions:

  1. Create teams of 3 or 4 frogs. Using the frogs' tables and graphs from Part 1, create a graph showing all the team's frogs on one coordinate plane. Be sure to use a different color for each set of data and label to which frog each set of data belongs.

    Use your team's graph to answer the following questions.

    • Which frog is the fastest? How do you know?
    • Which frog is the slowest? How do you know?
    • How can you tell how fast or slow a frog is based on a time/distance graph?
    • What is the jumping rate for each frog on your team? Place them from least to greatest.

  2. The slowest frog on your team will now be given a 20 cm head start.

    How will this change this frog's t-table? Create a new t-table for this frog after being given a head start. (Hint: Where is this frog compared to the others on your team at 0 seconds?)

    How will this change this frog's graph? Create a new graph for this frog after being given a head start and add the data to your team's graph.

    Answer the following questions based on your team's t-tables and graph.

    • How does the graph of the slowest frog with a head start compare to the graph of the slowest frog without a head start? How are the two graphs similar? How are they different?

    • What is the rate of the slowest frog when it is given a head start?
    • What is the rate of the slowest frog when it is not given a head start?

    • Write an equation for the slowest frog when it is not given a head start (remember: d = rt).
    • Write an equation for the slowest frog when it is given a head start. (Hint: Try making a small change to the equation for the frog when not given a head start. Think about its distance before it starts racing (t = 0).)

    • When the slowest frog is not given a head start, does its distance/time data form a proportional relationship?
    • When the slowest frog is given a head start, does its distance/time data form a proportional relationship?

  3. Imagine that your team's frogs all raced against each other. Examine your team's graph and answer the following questions.

    • If the slowest frog is not given a head start, which frog would most likely win a race 30 cm long? How did you use your graph to find the answer?
    • If the slowest frog is given a head start, which frog would most likely win a race 30 cm long? How did you use your graph to find the answer?
    • If the slowest frog is not given a head start, which frog would most likely win a race 50 cm long? How did you use your graph to find the answer?
    • If the slowest frog is given a head 20 cm start, which frog would most likely win a race 50 cm long? How did you use your graph to find the answer?
    • If the slowest frog is not given a 20 cm head start, which frog would most likely win a race 200 cm long? How did you find the answer?
    • If the slowest frog is given a 20 cm head start, which frog would most likely win a race 200 cm long? How did you find the answer?

    • If the slowest frog is not given a 20 cm head start, which frog would most likely win a race 5 seconds long? How did you use your graph to find the answer?
    • If the slowest frog is given a 20 cm head start, which frog would most likely win a race 5 seconds long? How did you use your graph to find the answer?
    • If the slowest frog is not given a 20 cm head start, which frog would most likely win a race 15 seconds long? How did you use your graph to find the answer?
    • If the slowest frog is given a 20 cm head start, which frog would most likely win a race 15 seconds long? How did you use your graph to find the answer?
    • If the slowest frog is not given a 20 cm head start, which frog would most likely win a race 1 minute long? How did you find the answer?
    • If the slowest frog is given a 20 cm head start, which frog would most likely win a race 1 minute long? How did you find the answer?

    • What is the smallest head start the slowest frog would have to be given in order to win a 1 minute race? How did you find your answer?

  4. Take a look at your team's graph. Are there any points on the graph where two lines intersect? Use your team's graph to answer the following questions.

    • What do the points of intersection mean? Think about what would be happening at that time if the frogs were racing.
    • Write an equation for each line on your graph and label to which frog the equation belongs. Be sure to indicate which equation is the slowest frog with and without a head start.

    • Make a list of any two lines that intersect. Write their equations one after the other.

      For example:

        Green Guy's line and Marvin's line (after being given a head start) intersect. Green Guy's equation is: d = 11.1t. Marvin's equation (after being given a head start) is: d = 5.7t + 20. Therefore, one line of my list looks like:

          d = 11.1t , d = 5.7t + 20

    • Find the each point of intersection. You may want to start by using your graph to estimate the values of the point of intersection. Try these values in each of the equation. Does the point work for both equations? If not, refine your guess using your graph and your knowledge of algebra. Continue this process until you get within five tenths of the point of intersection (in other words, when you put your value for time into each eqaution, the two resulting distances are within five tenths of each other). Note: You will likely need to use decimals.

      For example:

        I look at my graph and notice that the point of intersection for Green Guy's and Marvin's lines is close to (5, 40). So I try the point in each equation and get:

          40 = 55.5 , 40 = 48.5, which is not correct.

        I look at my graph again and notice that the point of intersection is slightly to the left of 5 seconds, so I try (4, 40). I plug the values into each equation and get:

          40 = 44.4 , 40 = 42.8, which is closer, but not close enough.

        I notice that the right side is still greater than the left side, but not by as much, so I need a value for t in between 3 and 4. So I try (3.5, 40). I enter the new values into the equations and get:

          40 = 38.85 , 40 = 39.95, which is very close for the second equation, but not quite close enough for the first.

        I now realize I must have a value for time slightly greater than 3.5, but less than 4, so I choose (3.7, 40). Now I get:

          40 = 41.07 , 40 = 41.09.

        Now I notice that the right sides are not nearly as close to left sides as before, but both right sides are very close to each other. So, I keep the value for t the same, but change my value for d to match the right sides, and try (3.7,4 1). Now I get:

          41 = 41.07 , 41 = 41.09, which is very close to correct.

        So, my final answer is: (3.7, 40).

    • Now that you've found the point of interception, what new conclusions can you make about a race between your team's frogs?
    • How long would you make a race if you were racing all your team's frogs and wanted the finish to be very close (after giving the slowest frog a head start of 20 cm)?

    Note: When we talk about a specific group of more than one equation, we call the group a system of equations. A point that satisfies every equation (in other words, when the values given by the point are put into the equation, the equation is true) in the system of equations is called a solution to the system of equations.

Part 4 - More Frog Tales

Students hone their skills learned in Part 3 by using tables, graphs, and equations to describe more frog stories.

Materials:

Directions:

Read each frog tale; complete each table; and answer the following questions.

The Story of Jumper

Patrick is preparing his frog Jumper for a big race. Before Jumper takes to the track, Patrick wants to find out about how fast Jumper really is. He finds that Jumper hops at an average rate of 4 cm⁄s.

  1. Create a table using 10 second intervals to describe Jumper's trial run.
  2. Write an equation describing Jumper's trial run.
  3. Create a graph describing Jumper's trial run.
  4. If Jumper could continue at this rate, how far could Jumper travel in an hour? Give your answer in meters.
  5. How long would it take Jumper to travel 300 centimeters?
  6. Patrick decided to change Jumper's equation so that t represents time in minutes and d represents distance in meters. What equation did Patrick write?
  7. Using Patrick's new equation, how long would it take Jumper to travel 12 meters?
  8. Verify whether or not the relationship shown in the table is proportional.
The Story of Cutie

Dana is preparing her frog Cutie for a big race against Jumper. Before Cutie hops her way into history, Dana wants to find out just how fast Cutie is. She finds that Cutie averages 2 cm⁄s.

  1. Create a table using 10 second intervals that describes Cutie's trial run.
  2. Create an equation describe Cutie's trial run.
  3. Graph Cutie's data from her trial run on the same coordinate plane as Jumper's trial run.
The Story of a Good Sport

Patrick and Dana take a look at their graphs based on the trial runs and quickly realize that Cutie is not match for Jumper in a normal race. Patrick, being a good sport, decides he wants to give Cutie a head start to make the race more exciting. Patrick and Dana agree on a 50 cm head start.

  1. Create a new table for Cutie based on her trial run and a 50 cm head start.
  2. Create a new equation to fit the table.
  3. Graph the new data on the same coordinate plane as the previous data.
And They're Off!
  1. If the race lasts one minute, which frog will most likely win? Prove your conclusion in at least two ways.
  2. In a one minute race, will there be any point in time that two frogs will be tied? If so, at what time? How did you find your answer?
  3. Which of the equations for Cutie was NOT proportional? Explain why it does not form a proportional relationship.

Part 5 - Go!

Materials:

Directions:

Note: You will need at least two race participants for this activity. If you are completing this activity with more than 5 participants, you may need to separate them into groups of 3 to 5.
  1. Time each race participant in a 100 meter dash (if you are not using a standard track, the distance should be one fourth or less of your final race distance).

  2. Find the speed (or running rate) of each race participant in meters per second.

  3. Create a table and graph for each participant. The tables and graphs should be projected to times of at least 10 minutes. Participants who will be racing each other should be graphed on the same coordinate plane.
  4. If you have more than one group of racers, have groups exchange graphs and tables. Each group's goal is to create as close a race as possible. The race will be one lap, 400 meters, around the track (or whatever distance the teacher has arranged). Find the head starts that will be provided to each racer to make the race as close as possible. Also write the new equations and create a new graph (you may want to create new tables to help create the new graph). The group should also state who they think will win the race and why.

  5. Measure the head starts on the track, and place each racer at their mark.

  6. Go! Have the racers run their race. Be sure to time each racer and record the results.

  7. After the race, answer the following questions:
    • Did the results of the race match your predictions? Why or why not?
    • Did the new graph accurately portray the race? Why or why not?
    • If the race were shorter, would you be able to more or less accurately predict the race based on the data collected? Why?
    • If the race were longer, would you be able to more or less accurately predict the race based on the data collected? Why?
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