LINEAR EQUATIONS Introduction to



Concepts: Slope/Linear Functions

Tie the Knot Activity

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Description of Activity

In this activity students will:

1. Investigate slope.

2. Find the equation of a line using two points.

3. Find the equation of the line using the calculator regression tool.

Contents Pages

Student worksheets 2-5

Teacher Notes and Solutions 6-8

Expected Outcomes

Students will

▪ Review slope

▪ Determine a line given two points

▪ Create an equation of a line

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NCTM Standards 2000

Algebra Representation

Geometry Communication

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Student Worksheet

Materials

1 Measuring tape

1 Piece of rope

1 Marker

1 TI-84 Graphing Calculator

Procedure:

1. Measure the rope (in inches). Enter the data in the chart below; record it next to 0 Knots (round your answer to the nearest inch).

2. Tie a knot in the rope and measure the length of the rope again (round your answer to the nearest inch). Record your results in the chart.

3. With the marker, mark each side of the knot on the rope.

4. Repeat steps 2 and 3 until the chart has been completed.

Chart

|Knots |Length of Rope |

| |(inches) |

|0 | |

|1 | |

|2 | |

|3 | |

|4 | |

|5 | |

|6 | |

Investigation

1. Examining the data in the chart, what do you notice about length of the rope as you tie each knot?

2. If you were to plot the points in your chart, which variable would be the x? Which would be the y? Explain your reasoning.

3. Slope is defined as the rate of change between two points on a line (∆y/∆x). It is found by calculating (y2-y1)/(x2-x1). Calculate the slope between 0 and 1; 1 and 2; and so on. Indicate the results in the table below.

| |Length of Rope | |

|Knots |(inches) |Slope |

|0 | |--- |

|1 | | |

|2 | | |

|3 | | |

|4 | | |

|5 | | |

|6 | | |

| | | |

4. Using the data from your chart, choose two points and determine an equation of the line. Hint: use the point-slope formula to determine the line, (y-y1)=m(x-x1).

5. Using the equation you found in example 4, predict how long the rope would be after 10 knots.

6. Untie each knot and measure the distance between the marks. Find the average of distances. How does this number compare to the slope of your line? How does this number compare to the slope of the line determined by the regression tool?

Calculator Activity

1. Enter the data from your chart into the list editor of the calculator. Enter the x values in L1 and the y values in L2. (See directions below)

Enter data into lists L1 and L2 by;

1. STAT

2. 1:Edit

3. Move the cursor to the first cell in L1 and enter the numbers, pressing enter after each number (x values)

4. Move cursor to L2 and do the same (y values)

2. Create a Scatter Plot for your data. (See directions below)

1. 2nd STATPLOT

2. 1:PLOT1

3. ENTER

4. ON

5. ENTER

Note: select the type, first is scatter

6. ENTER

Note: Xlist should be L1

Ylist should be L2

7.ENTER

Note: mark should be a square

8. ENTER

9. ZOOM

10. 9:ZOOMSTAT

11. ENTER

3. In the y= editor of your graphing calculator, input the equation you determined in example 4 of the investigation (Input it in Y1). Graph the equation.

4. How well does the graph of the equation fit the data?

5. If you had used two different points to find the equation of the line, would your results be more accurate?

6. Use the regression tool on the graphing calculator to find the line of best fit. Follow the steps below:

1. Press 2nd QUIT

2. Press STAT

3. Highlight CALC

4. Highlight 4:LinReg(ax+b)

5. Press ENTER

6. Press 2nd L1

7. Press ,

8. Press 2nd L2

9. Press ,

10. Press VARS

11. Highlight Y-VARS

12. Highlight 1:Function

13. Press enter

14. Highlight y2

15. Press ENTER

16. Press ENTER (the values for a and for b appear)

17. Press GRAPH

7. Compare the graph of the line you produced to the graph of the line created by the regression tool. Which is a better fit of the data?

8. Using the equation you found with your regression tool, predict how long the rope would be after 10 knots. Compare this answer to your answer for example 5 of the investigation.

Teacher Notes

See the Chart below for a sample set of data.

Chart

|Knots |Length of Rope |

| |(inches) |

|0 |70 |

|1 |67 |

|2 |65 |

|3 |61 |

|4 |58 |

|5 |55 |

|6 |52 |

Investigation

1. Examining the data in the chart, students should notice that the length of the rope decreases as the number of knots increase.

2. The number of knots would be represented by the variable x. The length of the rope would be represented by the variable y. The length of the rope depends on the number of knots that are tied, therefore it should be the dependent variable.

3. Slope is defined as the rate of change between two points on a line (∆y/∆x). It is found by calculating (y2-y1)/(x2-x1). Calculate the slope between 0 and 1; 1 and 2; and so on. Indicate the results in the table below.

| |Length of Rope | |

|Knots |(inches) |Slope |

|0 |70 |--- |

|1 |67 |-3/1 |

|2 |65 |-2/1 |

|3 |61 |-4/1 |

|4 |58 |-3/1 |

|5 |55 |-3/1 |

|6 |52 |-3/1 |

4. The equation of the line using (0,70) and (1,67) is y = - 3x + 70.

5. After 10 knots the rope would be 40 inches long.

6. The number is a close approximation to the slope of my line. Similarly.

Calculator Activity

From the sample data:

1.

2.

3.

4. It fits the data well. See below:

[pic]

5. Not necessarily.

6.

7. My line: Regression tool line:

[pic] [pic]

They are nearly identical.

8.

Using the regression line, the rope would be about 39 inches after the 10th knot. The prediction for my line was 40 inches.

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