Math is Fun: Patterns, Patterns EverywhereJessica Harris, Cheryl Kilpatrick, Cyril Quatrone

Louis E. Dieruff High School

Diffy Warm-Up Oil Spills

Steps:1) Place any 4 numbers in the corners on the outside of the diagram.

2) Subtract joined pairs (largest minus smallest) and place the difference in the circle at the midpoint. (i.e., 18 –13 = 5)

3) Repeat the subtraction process for all joined pairs.

4) No matter what numbers you start with, you will eventually get all zeroes in the end.

EXAMPLE

* (w – y) – (2w – z – x) = – w + x – y + z# (x + z – 2y) – (w – y) = – w + x – y + z

( – w + x – y + z ) – ( – w + x – y + z ) = 0

You are going to use vegetable oil to simulate oil spills on land.

Purpose: to discover the relationship between the amount of oil that has been spilled and the resulting area covered by the spill

Measurements to be taken: •The amount of oil spilled will be measured in drops.•The area of each oil spill will be calculated by first finding the radius of the resulting circle and then using the radius to calculate the area of the circle.• You will also examine the ratio of the area : volume.

Directions:1.Place each of eight sheets of toilet paper on the card stock. 2.Number the sheets of toilet paper 1 through 8. 3.Carefully place one drop of oil on the sheet labeled 1, then 2 drops on the sheet labeled 2, and so on. 4.Measure the diameter of the resulting circle to the nearest tenth of a centimeter. 5.Use this information to complete the chart below.

Next Steps:6.From the data in your chart create a scatterplot on the grid below that gives the volume along the x-axis and the area of the circle along the y-axis.

Activities & Questions for Students:•Explain why a scatterplot is the most effective way to graph the data.•Choose a point on the scatterplot and describe its meaning in context to this problem •Draw a line of best fit and calculate the equation for your line

• Identify the y-intercept and explain its meaning• Identify the slope and explain its meaning

•What type of relationship does your data show?•Write a direct variation for your graph. How is this related to your line of best fit?

Common Core1) Make sense of problems and persevere in solving them.

• Diffy Warm-Up In this activity, students are asked to try completing the steps using several different values until they “believe” they have an explanation as to why the final result of zeroes occurs.

• Oil Spills In this activity, students repeat a process several times to develop a pattern for the graph and then must explain their findings.

2) Reason abstractly and quantitatively.• Diffy Warm-Up The entire activity was about numbers and basic subtraction.• Oil Spills Students used a hands-on activity along with various representations to show

what happens when oil is spilled. They performed several calculations using the data.

3) Construct viable arguments & critique the reasoning of others.1) In both activities, students must discuss and compare results with their groups and justify

their answers using the data.

• Model with Mathematics• Diffy Warm-Up Students used the given chart to perform the math and model the math

process and skills.• Oil Spills Students used actual oil drops on paper, a data table, a graph, and a line of

best fit to model the real life event.

• Use Appropriate Tools Strategically• Diffy Warm-Up the chart• Oil Spills oil, paper, card-stock, data table, graph, and equations• All these tools were used to perform a task and then display the results.

6) Attend to Precision• Diffy Warm-Up subtraction had to be done in the correct order (largest – smallest)• Oil Spills Calculations had to be estimated and rounded to a given place value

7) Look for and make use of structure• In both activities students were asked to look at the patterns that were found in the data they

were producing.

8) Look for and express regularity in repeated reasoning• In both activities students were asked to look for the patterns and then use them to complete

the rest of the activity and answer questions based on the pattern• Oil Spills Students used the “pattern” they found to write a line of best fit equation and then

use that equation to reason out the rest of the activity.