Math What’s the problem?

Objectives:

  • Use the Pythagorean theorem in everyday life.
  • Master the basics of the metric system.
  • Find out what order really has to do with it, and how speak with letters instead of numbers.

 

Lessons


Right Triangle Field Dilation 
Throughout much of the 19th century, life in the South was characterized by the prevalence of slavery. As was the custom of that day, slaves were expected by their masters to work long hours under the scorching sun, tending the fields of plantation owners. Follow the directions below to create a series of similar triangles representative of the fields worked by the slaves in the 1800s.

Activity:
A dilation (of a polygon) is a transformation in which a polygon is enlarged or reduced by a given factor around a given center point.

On a piece of graph paper:

  1. Draw the x- and y-axes of a Cartesian coordinate plane.
  2. Graph the points A(-2,-1), B(4,-1), and C(4,4).
  3. Find the length of sides AB and BC.
  4. Apply the Pythagorean theorem to find the length of AC, the hypotenuse. Pythagorean theorem: a2 + b2 = c2
  5. Apply a scale factor of 2 to the right triangle with center as A and graph the resulting dilation. Name the vertices of the resulting triangle A’, B’, and C’. List the corresponding ordered pairs.


To consider:

A scale factor of 2 doubles the length of the sides of the original triangle.

  • If the scale factor is ½ , describe the resulting dilation?
  • What would happen if the scale factor is 1?

Traveling Treasure: Calculating Unit Conversions
Have you ever seen a house move? Moving a house is no easy task. Edgewood was relocated twice. The first time the house was moved by mule-drawn sleds for a distance of 30 miles. The second time the house was divided into sections and trucks pulled each of the sections a distance of 3 miles.

Activity:
Use appropriate unit conversion factors to find the equivalent measures for each time the house was relocated. Be sure to show your work in the space below each section. (Appendix M-1)


The Order Matters
Some things in life must be done in a specific order. For example, if you bake a cake, you must preheat the oven before you can actually bake the cake. Likewise, you cannot pour the batter into the pan until you have combined the ingredients in a bowl.

During the mid-1800s, those working on Southern plantations had to complete tasks in certain order, too. Those working in the fields had to follow a specific routine, or set of steps, between planting the seeds for their cotton and making clothes from the cotton. For example, the seeds could not be planted until the fields had been prepared. Once the seeds were in the ground, they had to be watered and fertilized before the crops could be harvested. Finally, the cotton could be spun into thread and made into clothing.

This same idea is important in math. Two properties that let us change the order of the numbers in a mathematical sentence are the Commutative Property and the Associative Property.

Commutative Property: For any numbers a and b, a + b = b + a and ab = ba.

Associative Property: For any numbers a, b,and c, (a + b) + c = a + (b + c) and (ab)c = a(bc).

Optional: The Distributive Property is another mathematical property that helps us simplify equations.

Distributive Property: For any numbers a, b, and c, a(b + c) = ab + ac and ab + ac = a(b + c).

Summary:
In the commutative property, the order of the numbers changes. In the associative property, the order of the numbers does not change: it’s the parentheses that move.

Critical Thinking:
What can we learn from this? When we use add or multiply, the order of the numbers does not matter. However, when we subtract or divide, the order of the numbers is important.

Activity:
Complete the worksheet. Name the property illustrated, simplify the expression, or solve the multi-step equation as instructed. (Appendix M-2)

Enrichment:
Complete the worksheet. Answer each of the questions in the blanks provided. (Appendix M-3)


At This Rate
During the 19th century, plantation owners frequently not only owned the land but they also owned slaves to work in the fields, harvesting their crops. Not only were the plantation owners dependent upon the efficiency with which their hired-hands worked, but they were also at the mercy of the natural elements and weather patterns such as amount of rainfall, daily temperatures, and wind speed and direction.

Application:
Explore rates and proportions through concept practice problems and word problems correlated with the information from the Edgewood docudrama.

Activity:
Complete the worksheet. Translate the word problems into ratios and proportions and solve for the missing information. (Appendix M-4)

Educational Standards

Grade 8: South Carolina Mathematics (Number and Operations)

 

Standard 8-2: The student will demonstrate through the mathematical processes an understanding of operations with integers, the effects of multiplying and dividing with rational numbers, the comparative magnitude of rational and irrational numbers, the approximation of cube and square roots and the application of proportional reasoning. 

Indicators:

  • (8-2.7) Apply ratios, rates, and proportions.

 

 Standard 8-3:  The student will demonstrate through the mathematical processes and understanding of equations, inequalities, and linear functions.

Indicators: 

  • (8-3.3 Use commutative, associative, and distributive properties to examine the equivalence of a variety of algebraic expressions.
  • (8-3.4) Apply procedures to solve multistep equations.

 

Standard 8-4:  The student will demonstrate through the mathematical processes an understanding of the Pythagorean theorem; the use of ordered pairs, equations, intercepts, and intersections to locate points and lines in a coordinate plane; and the effect of dilation in a coordinate plane.

Indicators: 

  • (8-4.1) Apply the Pythagorean theorem.
  • (8-4.2) Use ordered pairs, equations, intercepts, and intersections to locate points and lines in a coordinate plane.
  • (8-4.3) Apply a dilation to a square, rectangle, or right triangle in a coordinate plane.
  • (8-4.4) Analyze the effect of a dilation on a square, rectangle, or right triangle in a coordinate plane.

 

Standard 8-5: The student will demonstrate through the mathematical processes an understanding of the proportionality of similar figures, the necessary levels of accuracy and precision in measurement, the use of formulas to determine circumference, perimeter, area, and volume, and the use of conversions within and between the U.S. Customary System and the metric system.

Indicators:

  • (8-5.7) Use multistep unit analysis to convert between and within U.S. Customary System and the metric system.