Which three go together? Why do they go together?

Coordinate plane with the origin labeled "O." Nine points are graphed and the trend of the data points move linearly upward and to the right, where the first point begins at the origin.  

Coordinate plane with the origin labeled "O." 11 points are graphed and the trend of the data points move linearly downward and to the right, where the first point begins on the vertical axis, high above the origin.  

Coordinate plane with the origin labeled "O." 11 points are graphed and the trend of the data points move in a curve that moves upward and to the right, where the first point begins on the vertical axis, slightly above the origin.

Coordinate plane with the origin labeled "O." 11 points are graphed and the trend of the data points move horizontally and to the right, where the first point begins on the vertical axis and slightly above the origin.  

Mai is creating a rectangular banner to advertise the school play. The material for the banner is sold by the square foot. Mai has enough money to buy 36 square feet of material. She is trying to decide on the length and width of the banner.

1. What is the width of the banner if the length is:

   1. 6 feet?

   2. 4 feet?

   3. 9 feet?

2. To find different combinations of length and width that give an area of 36 square feet, Mai uses the equation , where is the width and is the length. Compare your strategy and Mai's method for finding the width. How were they the same or different?

3. We can use a graph to show the relationship between the side lengths of various rectangles that have an area of 36 square feet.

   • Use several combinations of length and width to create a graph.

   • Explain how the graph describes the relationship between length and width for different rectangles with an area of 36 square feet.

   

   A blank coordinate plane with the origin labeled "O". The horizontal axis is labeled "length", and the vertical axis is labeled "width". For both axes, the numbers 0 through 38, in increments of 2, are indicated.

4. Suppose Mai used the equation to find the length for different values of the width. Would the graph be different if she graphed length on the vertical axis and width on the horizontal axis? Explain how you know.

A cereal manufacturer needs to design a cereal box that has a volume of 225 cubic inches and a height that is no more than 15 inches.

1. The designers know that the volume of a rectangular prism can be calculated by multiplying the area of its base and its height.

   
   Complete the table with pairs of values that will make the volume 225 in³.

   height (in)		5	9	12
   area of base (in2)	75				15

2. Describe how you found the missing values for the table.
3. Write an equation that shows how the area of the base in square inches, , is affected by changes in the height in inches, , for different rectangular prisms with a volume 225 in³.

4. Plot the ordered pairs from the table on the coordinate grid to show the relationship between the area of the base and the height for different boxes with volume 225 in³.

   

   A blank coordinate plane with the origin labeled "O". The horizontal axis is labeled "height in inches", and the numbers 0 through 15 are indicated. The vertical axis is labeled "area of base in square inches", and the numbers 0 through 80, in increments of 5, are indicated.

1. The researcher said that, for these five days, the number of mosquitoes, , can be found with the equation where is the day in the study. Explain why this equation matches the data.

2. Use the ordered pairs in the table to graph the relationship between number of mosquitoes and day in the study for these five days.

   

   A blank coordinate plane with the origin labeled "O". The horizontal axis is labeled "day of the study" and the numbers 0 through 6 are indicated. The vertical axis is labeled "number of mosquitos" and the numbers 0 through 40, in increments of 5, are indicated.

3. Describe the graph. Compare how the data, equation, and graph illustrate the relationship between the day in the study and the number of mosquitoes.

4. If the pattern continues, how many mosquitoes will there be on day 6?