Slope as a Rate of Change Name: _____________________________________
Tick Tock Toothpick
It takes four identical toothpicks to build a square

How many identical toothpicks does it take to build a line of two adjacent squares, as in this drawing?

Use toothpicks to build lines of three, four, and five adjacent squares. Write the number of toothpicks need and make drawings of your work.

3 squares

4 squares

5 squares

The number of toothpicks needed, y, depends on the number of squares, x. Use your results from Items 12 to complete
Squares (x)

Toothpicks (y)

1


2


3


4


5



Describe any patterns you see in the table in item 3.

Use numbers to complete this statement: Each time the number of squares in the line increases by _______, the number of toothpicks increases by _______.

Use the grid to make a scatter plot of the data from Question 3. Label the axes and title the graph.

Label the leftmost point on the graph point A. Label the other points, from left to right, points B, C, D, and E. Explain what you notice about the points in your scatter plot.

Describe how to move along the grid between each pair of points.

From A to B: Go Up _______ and Go Right _______.

From B to C: Go Up _______ and Go Right _______.

Each movement you described in Question 8 can be written as a ratio in the form
Describe each movement by writing a ratio in the form

A to B

B to C
You can think of units up as a change in the y direction and of units right as a change in the x direction. A movement up or to the right is positive.

Describe moves along the grid between each pair of points.

From A to C: units up = _______ and units right = _______
Ratio form: _______

From B to E: units up = _______ and units right = _______
Ratio form: _______

From A to E: units up = _______ and units right = _______
Ratio form: _______

What do you notice about these ratios?

How do the ratios relate to the number of squares and toothpicks?

When points on a scatter plot lie on a line, a ratio such as or is the slope of that line.

What is the slope of the line in the scatter plot you made for Question 6?

What do you think is true about the slope ratios between any two points on a line?

Use the grid to move from B to A and from E to B.

Describe the movement from B to A and express the movement from B to A as a ratio.

Describe the movement from E to B and express the movement from E to B as a ratio

What kind of numbers did you use when you wrote the ratios in Parts a and b?

How do the slope ratios compare to the slope ratios you wrote in question 10?

Use the slope ratio to add another point to the graph and explain what the point represents.

Suppose that you wanted to find the number of toothpicks needed to build 50 squares.

Would your graph be helpful for finding that number? Explain your answer
Squares (x)

Toothpicks (y)

1


2


3


4


5


6


7


8


10


20


Recall that the variable x represents
the number of squares in a line and
the variable y represents the number
of toothpicks used to build the squares
Complete this table that was made by
adding more rows for more squares to
to the table from Question 3.

Explain how you found the number of
toothpicks needed for 10 squares and
for 20 squares

Use the information from the graph and the table to write an equation that represents this situation.

Use your equation from Item 16 to help answer each part. Show your work or explain how you arrived at your answers.

How many toothpicks are needed to build 50 squares?

How many squares can be built from a total of 94 toothpicks?

How many squares can be built from a total of 62 toothpicks?
