LESSON PLAN #2: Proportional changes, Oil Spills on Land
Name: Noor J. Hoque
Length of lesson: 90 minutes
Name of course: Algebra 1 Grade level: 7 / 8 regular
Source of the lesson:
http://www.learner.org/channel/workshops/algebra/workshop7/lessonplan1.html
Standards: NCTM Algebra Standard for Grades 7-8, 2000:
Students would be able to:
Understand patterns, relations, and functions
- represent, analyze, and generalize a variety of patterns with tables, graphs, words, and, when possible, symbolic rules;
- relate and compare different forms of representation for a relationship;
- Identify functions as linear or nonlinear
and contrast their properties from tables, graphs, or equations.
Represent and analyze mathematical situations and structures using algebraic symbols
- develop an initial conceptual understanding of different uses of variables;
- explore relationships between symbolic expressions and graphs of lines, paying particular attention to the meaning of intercept and slope;
- use symbolic algebra to represent situations and to solve problems, especially those that involve linear relationships
- recognize and generate equivalent forms for simple algebraic expressions and solve linear equations
- Judge the meaning, utility, and reasonableness of the results of symbol manipulations, including those carried out by technology.
Use mathematical models to represent and understand quantitative relationships
- model and solve contextualized problems using various representations, such as graphs, tables, and equations
Analyze change in various contexts
- use graphs to analyze the nature of changes in quantities in linear relationships
- Approximate and interpret rates of change from graphical and numerical data.
TEKS addressed:111.32. Algebra I (One Credit).
(a) Basic understandings.
(1) Foundation concepts for
high school mathematics.
The basic understandings of quantitative reasoning; patterns, relationships,
and algebraic thinking; geometry; measurement; and probability and statistics
are essential foundations for all work in high school mathematics.
(2) Algebraic thinking and symbolic reasoning. Symbolic reasoning plays a critical role in algebra; symbols provide powerful ways to represent mathematical situations and to express generalizations. Students use symbols in a variety of ways to study relationships among quantities.
(3) Function concepts. Functions represent the systematic dependence of one quantity on another. Students use functions to represent and model problem situations and to analyze and interpret relationships.
(b) Foundations for functions: knowledge and skills and performance descriptions.
(1) (D) The student represents relationships among quantities using concrete models, tables, graphs, diagrams, verbal descriptions, equations, and inequalities.
(3) (B) Given situations, the student looks for patterns and represents generalizations algebraically.
I. Overview
This lesson teaches students about direct variation by allowing them to explore a simulated oil spill using toilet paper tissues (to represent land) and drops of vegetable oil (to simulate a volume of oil).
II. Performance or learner outcomes
Students will be able to:
- Describe the primary characteristics of a direct proportion.
- Explain why the relationship between the volume of oil and the area of land it covers represents a direct proportion.
- Recognize situations in other contexts that are direct proportion (such as the relationship between circumference and diameter).
- Give other examples of real-world direct variations/ proportion
- Define direct variation and constant of proportionality.
III. Resources, materials and supplies needed
Supplies needed for each group of four students:
· eye dropper
· large sheet of paper
· 8 small pieces of toilet paper or a paper towel
· ruler
· vegetable oil
· overhead transparency sheet (preferably with gridlines)
· overhead pen
IV. Supplementary materials, handouts.
Oil spill Handout:
http://www.learner.org/channel/workshops/algebra/workshop7/oil_black_gold.pdf
Web Sites Relating to Oil Spills
Exxon Valdez Oil Spill Trustee Council Home Page
http://www.oilspill.state.ak.us/
NMFS Office of Exxon Valdez Oil Spill Damage Assessment and Restoration
http://www.fakr.noaa.gov/oil/default.htm
Environmental Protection Agency - Oil Spill
http://www.epa.gov/oilspill/exxon.htm
V. Safety Issues
None
VI. Accommodations for learners with special needs (ELLs, Special Ed, 504, G&T)
None
Five-E Organization
Teacher Does Probing Questions Student Does
| Engage: Learning Experience(s) Teacher conducts a brief discussion about oil spills, their effect on the environment, and ways that scientists work to clean them up. Encourage students that the discussion should involve specific oil spills.
The discussion should help students to conclude that oil spills are actually cylindrical in shape, not circular. The thickness represents the height of the "cylindrical" shape. Approx. Time: 15 mins |
Critical questions that will establish prior knowledge and create a need to know What do you know about oil spills? When has the last oil spills occurred? What was done to clean things up at the Exxon Valdez spill in How does the shape of the land change the shape of the spill? Would it be possible to estimate the volume of a spill? If you spill a drop of water on the countertop and what is its general shape? Does it have depth/ thickness? How would you best describe this? |
Expected Student Responses/Misconceptions Students’ responses vary. Students’ responses vary. Students’ responses vary. Students’ responses vary. Students’ responses vary. They should note that it is generally circular in shape. They should say drum/ cylindrical shape. |
Decision Point Assessment: Depending on students' prior knowledge, have them solve the following problems involving circles with "messy numbers" (the area of circles occurs repeatedly during this lesson)
| Explore: Teacher explains that: A simulation of a real-world event involves creating
a similar, but more simplified, model. In the introduction, for example,
you simulated an oil spill on the ocean using a few drops of oil in
a pan of water. Explain to students that they
will be conducting an exploration using vegetable oil and toilet paper.
Describe how the oil will be dropped onto toilet paper tissues to simulate
an oil spill on land. Using 8 different samples, students will record
data for oil spills involving from one to eight drops. Once all groups have placed
the oil onto the sheets of toilet paper, reconvene the class Allow students to return to
their experiments, complete their charts using data from the experiments,
and create their scatter plots. Approx. Time: 25 mins |
Critical questions that will allow you to decide whether students understand or are able to carry out the assigned task (formative) How would you stimulate real world events such as the tsunami or the damage by Hurricane Katrina? In this activity, you simulate oil spills on land by placing drops of oil on sheets of paper. Can you describe how data should be best collected and organized? What kind of measurements are you taking of the spill? How to make things more accurate? As you are busy with your plots, be thinking about how you would describe the scatter plot that you are creating? |
Expected Student Responses/Misconceptions Students’ responses vary. Students form groups of 4 with access to an eye dropper, vegetable
oil, eight sheets of toilet paper, a large sheet of paper, and a ruler. Students write their names on the large sheet of paper. They place 8 sheets of toilet paper on the large sheet. Each sheet of toilet paper has a dot at the center of each sheet and marked with a numeral. Students should read and follow the instructions for conducting the experiment: Carefully place 8 drops of oil on the pencil dot on sheet 8. Continue creating oil spills of different volumes by placing 7 drops on sheet 7, 6 drops on sheet 6, and so on. Students should then drop the appropriate number of drops onto each sheet Students quickly create the chart Students will measure the radius and diameter of the spills to the nearest tenth of a centimeter. Students collect transparency and an overhead pen. Students create their scatter plot and reflect on the description. Some students need to repeat the experiment if they experience doubtful results. |
Decision Point Assessment: Circulate through the classroom as students work. Offer some assistance, as necessary, but be careful not to give students too much information. While walking around, take note of the work of various groups that would be useful to share during the class discussion later in the lesson.
| Explain: Learning Experience(s) When groups have completed their scatter plots, call on 2/3 groups to share their work with the class using the overhead projector. Teacher asks: Call on several groups to
explain how they determined the slope for their estimated line of best
fit. Ask students to consider what the y-intercept means in the context of the problem. An important point about direct proportion is that the graph will always contain the point (0, 0). In the context of this problem, 0 drops should yield an area of 0 cm². Each group should complete an additional column on their chart that shows the ratio of area to volume (cm²/drops).
Approx. Time:15 mins |
Critical questions that will allow you to help students clarify their understanding and introduce information related to concepts to be learned What
do the points on the scatter plot represent? Pick a point (x, y) from the graph, and describe its meaning in the context of this problem. As the volume increased, did the size of the spill increase? If the points were connected, what type of graph would result? How do you estimate a line of best fit for you data and determine an equation for that line. What does the y-intercept mean for your line? What does the point represented by the y-intercept mean? What would the area of the spill be if 0 drops of oil were spilled? How much area would a spill of 0 drops cover? What does ratio of area to volume (cm²/drops) show? What does the average of the values in the area/volume column represents? |
Expected Student Responses/Misconceptions Students’ responses vary. Students describe its meaning in the context of this problem. As the volume increased, so did the size of the spill. If the points were connected, we would get a straight line graph. Students may suggest several different methods: calculating the slope using two points on the line, determining the "rise over run" graphically by counting squares on the grid, choosing one point on graph and dividing the y-coordinate by the x-coordinate. If students have a y-intercept of 1, that represents the point (0, 1), which erroneously suggests that 0 drops of oil resulted in a spill area of 1 cm². Students say that it makes sense that the graph reflect zero. This will give the slope of a line that passes through the origin (0, 0), and the point represented by each particular row. Students may say that if row 2 has a volume of 2 drops and an area of 10 cm², the slope will be (10
- 0) = 5 The average of the values in the area/volume column represents the slope of an approximate line. |
Decision Point Assessment:
Using
the average from the last column in your table, what line of best fit did
you find? How well does this new line represent your data?
| Extend / Elaborate: Learning Experience(s) Review the definition of "slope" and reinforce that it should be considered as a "constant rate of change." This is an important concept for students to understand about direct proportion. Teacher asks… Using the data gathered during the lesson, explain that the relationship between two quantities that increase (or decrease) proportionally is known as "direct variation" or a "direct proportion."
Approx. Time12 mins |
Critical questions that will allow you to decide whether students can extend conceptual connections in new situations What
does the line represent in relation to the oil spill? What
is the slope of your line of best fit? What can you tell me about the equation for a line? Why is the second equation better (more advantageous) than the first equation when modeling the situation? How does the constant of proportionality relate to the oil spill? |
Expected Student Responses/Misconceptions Students say that the line of best fit has a constant slope and passes through the origin Students say that this means as the volume increases, the area will increase proportionally. If the volume of oil is doubled, the area of the spill is doubled too. If the volume of oil is tripled, the area of the spill is tripled too. Students
should be able to say, "The area varies directly as
the volume (number of drops.)" Students
say the equation for a line that passes through the origin is Students present their table The
second equation shows a proportion between numbers; that is, as one
quantity doubles or triples, so does the other. In addition, the second
equation contains the origin (0, 0), a necessary condition for a direct
proportion. |
| Evaluate: Lesson Objective(s) Learned Have each group make predictions based on a larger spill and answer questions. Teacher asks… Have
students convert the answer to square meters instead of cm². Approx. Time: 10 mins |
Critical questions that will allow you to decide whether students understood main lesson objectives There are approximately 25,000 drops in a liter of oil. What would the area be if a liter of oil were spilled? How
reasonable does your answer seem? Why? |
Expected Student Responses/Misconceptions Students say that it is Students say that it is Students say that it is |
Supplementary Handout:
To calculate the area covered by the oil spill.
|
Extended lesson:
Choose a group and use the two equations for lines of best fit from that group
to draw a comparison. Use T-tables to show how the values relate, based on
their equations. A sample comparison is shown below:
y = 5.5x + 1
|
y = 5.75x
|
Taking it Further
1. The
diagram below shows the shape of an oil slick that spread to a
thickness of 2.5 •10-3 cm.
├────┤
20
km
Estimate the volume of oil in the spill in gallons.
2. a. Assuming that the spilled oil spreads to a thickness of 2.5 •10-3 cm, complete the following table.
| Area of Slick (km2 ) |
Volume of Spill (gal) |
| 5000 |
|
| 7500 |
|
| 10,000 |
|
| 25,000 |
b. Determine the type of relationship formed by the data collected in the table. Create a graph that displays this relationship.
c. Find an equation that models the graph from Part b.
d. Use this equation to predict the number of gallons of oil that would create a slick of 71,000 km2.