Embedded Files
Learning Targets
I understand that it can be difficult to measure the quantities in a proportional relationship accurately.
I can examine quotients and use a graph to decide whether two associated quantities are in a proportional relationship.
Notes
Notes
When we measure the values for two related quantities, plotting the measurements in the coordinate plane can help us decide if it makes sense to model them with a proportional relationship. If the points are close to a line through (0,0), then a proportional relationship is a good model. For example, here is a graph of the values for the height, measured in millimeters, of different numbers of pennies placed in a stack.
Because the points are close to a line through (0,0), the height of the stack of pennies appears to be proportional to the number of pennies in a stack. This makes sense because we can see that the heights of the pennies only vary a little bit.
An additional way to investigate whether or not a relationship is proportional is by making a table. Here is some data for the weight of different numbers of pennies in grams, along with the corresponding number of grams per penny.
Though we might expect this relationship to be proportional, the quotients are not very close to one another. In fact, the metal in pennies changed in 1982, and older pennies are heavier. This explains why the weight per penny for different numbers of pennies are so different!
Activities
Activities
1.2 Perimeter of a Square
1.2 Perimeter of a Square
Print the PDF .
For each of the squares, measure the length of the diagonal and the perimeter of the square in centimeters. Record the measurements in the table.
(In Class Only) Check your measurements with your group.
Plot the diagonal and perimeter values from the table on the coordinate plane, using the applet.
What do you notice about the points on the graph?
Math 7 Lesson 3.1 Activity Perimeter of a Square.pdf
Questions to Ponder...
Do you think it would make sense to have (0,0) as a possible point? (Yes, since a square with diagonal 0 cm would have a perimeter of 0 cm.)
The data is not perfectly lined up. Do you think it should be? What is causing the inconsistencies? (Measurement error.)
If you had a square with diagonal 1 cm,what would the perimeter be? (About 2.8 cm.)
Add a third column to your table, titled perimeter ÷ diagonal.
Complete the column by computing the information for each row of data.
What do you notice?
Add to Your Notes
Add to Your Notes
When we collect data through measurement, we usually will introduce small errors into the data. Even though the data will look a little bit “bumpy,” it will often show relationships between two quantities. Analyzing the graph is a powerful method to check visually to see if a relationship looks like it may be proportional. The table helps us confirm a proportional relationship and find an approximate constant of proportionality.
1.3 Area of a Square
1.3 Area of a Square
In the table, record the length of the diagonal for each of your assigned squares from the previous activity. Next, calculate the area of each of your squares.
Graph these values. What do you notice?
How is the relationship between the diagonal and area of a square the same as the relationship between the diagonal and perimeter of a square from the previous activity? How is it different?
Add to Your Notes
Add to Your Notes
When we measure the quantities in a proportional relationship, measurement error may make it look like there is not an exact constant of proportionality.
A graph can be helpful to decide whether the points are close to lying on a straight line.
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