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Lesson 2b: Graphing Mass and Volume: How math / algebra can help makes the Connections between Slope and Density Objectives:
Key Questions: Overview:
Middle school students struggle with this idea. According to the California Math Standards, 7th graders should work with linear graphs and calculate slopes. Few of the students will have become comfortable with these concepts, and fewer still, will realize that there are physical applications of the concept. This lesson represents an attempt to help students see those connections and to reinforce (or introduce) the mathematical determination of slope. Time Required: One to two class periods. Materials: Procedure: Have the students find a staircase somewhere in the school, or have them measure a staircase at home the night before the activity. They should use the metric ruler to measure the height of each stair, and its width. They should do this for each step in the staircase. Measurements will vary, depending on the builder, but most stairs have a standard rise (height) of about 18 cm (~7 inches), and a run (width) of about 36 cm (~14 inches.) Ask the students whether there is a practical reason for each step in a staircase to have the same height and width. Ask them if stairs are generally designed for ease of climbing, and what happens if the height of each step is equal to, or greater than, the width of each step. If there is such a staircase, have them walk up and down it, and compare the effort they expend with other staircases. On a sheet of graph paper, have the students use the ruler to draw a straight line from the left hand bottom corner vertically that is at least 20 squares high. The have them draw a horizontal line from the bottom of the vertical line at least 20 squares to the right. Ask them which should represent the height of each stair and which line should represent the width of each stair. Remind them that in a linear graph, the horizontal line is the x-axis, and the vertical line is the y-axis. Students usually select the y-axis to represent the height of each stair. Since each step is 18 cm in height, designate each square as 18 cm on a side. Number each square on the horizontal line 1 to 20, starting at the vertical line. Then number each vertical square 1 to 20, starting at the bottom. Then have them use the graph to place a dot at the correct coordinates of their stairs. For each step, they should go out on the horizontal line two segments, and up one segment on the vertical line, since that is the ratio of 18 cm to 36 cm. Continue to do this until they are 18 to 20 squares away from the intersection of the horizontal and vertical lines. They may even draw in the stairs on the graph paper if it will help them visualize their stairs. When they have gone out 18 or 20 coordinates from the line intersection, have them use the ruler to draw a line connecting the tops of the stairs. This is the visual slope. If they have drawn in their stairs, have them draw a second line through the "bottom - back" points on their stairs. Ask them to comment on any similarities between the two lines. Any differences. Ask them if they can state what the "slope" is as a ratio. Teacher notes: Exploring their stair-step and density graphs should help the students to understand these concepts and the connections between math and science. The following notes may help you in helping the students to think through these challenging ideas. Sample
Stair Step Graph (PDF) In math the students will learn that the equation for a straight line is: y = mx + b and that "m" is the slope of the line and "b" is the y-intercept. The intercept, b, for the mass vs. volume graphs and for the stair-step graphs is zero. Hence, for the stair steps, the slope represents the "steepness" of the steps. In equation form, the students can write: Steepness = height of stair / width of stair For the mass/volume graphs, the slope is the same at every point, since the graph is a straight line. Hence, the slope is the density. Again, in equation form: Density = mass of material / volume of material Slopes can also be used to determine the ratio of other variables: time/distance (to give "speed"), and of course, mass/volume (to give "density."). In time and distance problems, time is shown on the horizontal, or x axis, and distance is shown on the vertical, or y-axis, with coordinates (x, y) equal to where an object is at a certain time and distance. If the speed is constant, the slope will be a straight line. You might wish to use time and distance charts to create slopes as an extension of this activity. Later in the year, when the students study "Forces and motion" they will explore the distance-time graphs in some detail. Ask the students to return to their "Mass vs. Volume graphs for different liquids. Have them determine the slope of their graphs using several different combinations of points (some experimental and some read from the graph.) Some students may come to realize that the ratio of any set of mass and volume from the experimental data for this graph is the same as the slope. This is because b, the y-intercept, is zero. If no students notice this, do not belabor it. Later in the year, when the students explore forces and motion, they may be ready to explore this observation. Assessment: Homework: Oakland/ State of California Science and California Math Standards: Science: Oakland Grade 8, Investigations
and Experimentation: 8. 1. e, f, g. Math: California Grade 7. 1.5
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