A Trained Army

Overall Rating

Subject:
Math,Social Studies
Lesson Duration:
60 Minutes
Common Core Standards:
8.F.1, 8.F.2, 8.F.4, 8.F.5
State Standards:
SC Math 8.F.1- .2, 8.F.4- .5/ Social Studies 8-2.5
NC Math 8.F.1- .4/ 8.H.1.1

Objective

How are functions calculated, used, and expressed in real world mathematical situations?

Background

The British army was highly disciplined and known for marching in an orderly way. They had different rates of movement for different conditions of fighting or parading. By following the beat of a drummer, they could easily change the speed of movement. We actually know the pace at which their troops marched and from that, can calculate the distance they covered over given amounts of time. This is very helpful to historians as we can learn just how close the British were to the American patriots at various times in the battle.

Preparation

At the Common Step (normal marching) the British could cover 1.10 yards in 1 second. They could cover exactly twice this distance in exactly twice the time, and 4 times the distance in 4 times that time. This pattern mathematically is a function where one input (an x value) gives an output (a Y value).

Lesson Hook/Preview

How are functions calculated, used, and expressed in real world mathematical situations?

Procedure

The British army was highly disciplined and known for marching in an orderly way. They had different rates of movement for different conditions of fighting or parading. By following the beat of a drummer, they could easily change the speed of movement. We actually know the pace at which their troops marched and from that, can calculate the distance they covered over given amounts of time. This is very helpful to historians as we can learn just how close the British were to the American patriots at various times in the battle. Let’s work with some functions to fill in the chart below:
Common Step                  Quick Step                    Double Quick

 Feet Yards Feet Yards Feet Yards 1 sec. 3.30 1.10 5 1.66 5.45 ---- 2 sec. 6.60 2.20 10 ---- ---- 3.62 3 sec. ---- ---- ---- 5.00 16.35 5.43 4 sec. 13.20 4.40 20 6.66 21.80 7.24 5 sec. ---- ---- ---- ---- ---- ----
1. At the Common Step (normal marching) the British could cover 1.10 yards in 1 second. They could cover exactly twice this distance in exactly twice the time, and 4 times the distance in 4 times that time. This pattern mathematically is a function where one input (an x value) gives an output (a Y value). What are the X and Y values in this problem?

1. There are several missing distances in this chart. If the function for the distance travelled at Common Step is Y=1.1X, solve for the missing yard values and fill them in on the chart?
2. You probably know that 1 yard equals 3 feet. You can fill in the values, in feet, by simply multiplying the yard value by 3. For a very lengthy chart though, it will be quicker to develop another function for the distance marched in feet. What is the function for feet marched per second at the common step?

1. Using these functions, graph 6 points for each onto a coordinate grid. You will need graph paper.
1. Repeat these same steps for the Quick Step and Double Quick paces. Write the functions below. Graph them as well. These functions should be more challenging.

1. Compare the graphs of the Common Step (in Yards) and the Double Quick (in Yards). How do they compare?

1. A historian has calculated that it took the British troops with their bayonets around 55 seconds of marching, after forming up, to reach the front line American sharpshooters. Using the function for the Quick Step, how far could the British soldiers in the first line have travelled in 55 seconds? Express your answer in both yards and feet.

1. Now that you’ve worked with your marching rates, if Tarleton’s men marched 12 miles to the battle that morning walking at the Common Step with no breaks, how long did it take them to reach the field? Also, if they left camp to march at 2 am, what time did they arrive at the battlefield?

1. Lastly, at what speed were the British marching? Express this in miles per hour. Show your work. (Hint: 1 mile has 5,280 feet or 1760 yards)

Vocabulary

Common Step (normal marching)

Assessment Materials

A Trained Army

1.
The X variable or input is the time in seconds. The Y variable or output is the distance covered over that time expressed in feet or yards.

2.
Common Step                  Quick Step                    Double Quick

 Feet Yards Feet Yards Feet Yards 1 sec. 3.30 1.10 5 1.66 5.45 1.81 2 sec. 6.60 2.20 10 3.32 10.90 3.62 3 sec. 9.90 3.30 15 5.00 16.35 5.43 4 sec. 13.20 4.40 20 6.66 21.80 7.24 5 sec. 16.50 5.50 25 8.30 27.25 9.05

3.
Xyd * 1.1 / 3 * Ys = f/s
Or
# of yards * common step / feet in yard * # of seconds = feet per second

4.
Students use either provided or own graph paper to complete task.

5.
Quick Step
Xs * 1.66 / 3 = f/s
Or
Xseconds * distance/sec / feet/yard = feet per second

- Students should show a steeper incline due to rate of speed increase.
Double Quick
Xs * 1.81 / 3 = f/s
Or
Xseconds * distance/sec / feet/yard = feet per second

- Students should show a steeper incline than quick step due to rate of speed increase.

6.
The graphing in the Double Quick shows an increasing difference in distance / time than the
graphing in the Common Step.

7.
55s * 1.66y/s = 91.3 yards
55s * 1.66y/s * 3f/y = 273.9 feet

8.
miles * yards/mile / yards/second  / seconds/minute / minutes/hour = hours
(remainder * 60 = minutes) = hrs & min + starting time = arrived at battlefield
12 * 1760 / 1.1 / 60s / 60m = 5.333 hrs
(r333 * 60 = 19.98) = 5hrs20min = 7:20am arrived at battlefield

9.
distance(miles) / hrs = miles/hour
12 / 5.33 = 2.25mph