Illuminations: Constant Dimensions

Constant Dimensions

Students will measure the length and width of a rectangle using both standard and non-standard units of measure. In addition to providing measurement practice, this lesson allows students to discover that the ratio of length to width of a rectangle is constant, in spite of the units. For many middle school students, this discovery is surprising.

Learning Objectives

Students will:

Critique various units of measure based on their appropriateness for this particular activity.
Use a linear graph to model, analyze and make predictions.
Draw conclusions about the relationship of two dimensions based on collected data.

Materials

Rectangle Measure activity sheet
Rulers (both inches and centimeters)
Alternate units of measure (pennies, paper clips, M&M’s, beads, width of index finger, width of pencil)
Graphing calculator (optional)

Instructional Plan

The activities in this lesson are designed to be completed by students working in pairs. When students arrive to class, the Rectangle Measure activity sheet and a ruler should already be at their desks.

Rectangle Measure Activity Sheet

Ask students to measure the length and width of the rectangle in both inches and centimeters. They should record their measurements in the chart below. At this point, it might be helpful to ask students to share their measurements to be sure students are on the right track.

After students have measured the rectangle in inches and centimeters, distribute alternative units of measure, such as paper clips, M&M’s, pennies, beads, etc. It is best if these are already divided so that they can be handed out quickly. Also, be sure that there are enough for each pair of students to measure both the length and width of the rectangle.

You may wish to discuss how each unit can be used to measure the length and width, or you may prefer that students consider this on their own. It might be helpful to use pennies on the overhead projector to give one example—lay pennies along the length, as shown below. You might also explain that it may be necessary to estimate portions of a unit; for instance, the length below appears to be slightly more than nine pennies, so it might be estimated that the length is approximately 9.2 pennies. Further, you might want to plot the length and width in pennies on the graph, to give students an example of how the graph is used.

Allow students to find the measurements of the rectangle using four non-standard units. When they are done, they should have measured the dimensions in six different ways.

When students have completed the chart, discuss how they will use the ordered pairs of (length, width) to create a scatterplot. Have students graph their six ordered pairs on a coordinate grid. (Note that you may wish to complete a scatterplot on the overhead projector based on student measurements. If pairs of students have used different units of measure, you may be able to display a scatterplot with more than six points by aggregating the measurements from the entire class.)

Have students consider the scatterplot, and ask, "Do the points appear to be random, or do they seem to follow a pattern?" Students should recognize that the points follow a linear path. Then ask, "What might explain the pattern formed by the points?"

[The slope represents the ratio of the change in width to the change in length, which is constant. Although the measurements may have changed because of the units, the ratio of length to width does not change. A function with a constant rate of change is linear.]

Ask students to predict the results using other units of measure. For instance, ask, "The length of the rectangle measured approximately 7.9 nickels. What is the width of the rectangle in nickels?"

[Approximately 5.3 nickels.]

Or ask, "One student used gum balls to measure the rectangle and found that it was 22 gum balls by 10 gum balls. Do these dimensions seem reasonable?"

[No. The length should be approximately 1.5 times the width, and 22 ≠ 10 × 1.5.]

To help with these predictions, students should draw a "line of best fit." Students can estimate this line and draw it with a ruler.

Since the points form a pattern, students should realize that a rule relates the length and width for this rectangle. Ask students if they can determine the rule.

[The length is always 1.5 times the width, regardless of the unit of measure. Written algebraically, L = 1.5W.]

Questions for Students

Although more pennies were used than M&M’s when measuring the width, did the size of the width actually change?

[No. Neither the length nor width changed. All that changed was the units of measure.]

Take a look at your six points. Do they appear randomly or does there appear to be a pattern?

[The points seem to occur in a straight line.]

If someone used gumballs to measure the length and width, and their ordered pair were placed at (22, 10), would we suspect that they made a good measurement? What if the ordered pair had the coordinates (16, 10.5)? What is your reasoning?

[The point (22, 10) seems to indicate an incorrect measurement, because 22 ≠ 10 × 1.5. On the other hand, 16 ≈ 10.5 × 1.52, so (16, 10.5) seems like a reasonable measurement.]

What unit of measure could be used that would have produced a point very close to the origin? Why?

[The actual dimensions of the rectangle are 6¾" × 4½". If a unit much larger than these dimensions are used, the result of each measurement would be close to zero. For instance, if the rectangle were measured in yards, the point would occur at approximately (0.19, 0.13), which is near the origin. To get even closer to the origin, the rectangle could be measured in kilometers or miles.]

What remained constant even when the units of measurement changed?

[Regardless of the unit of measure, the rectangle did not change; likewise, the ratio of the length to the width did not change. That is, the proportions of the rectangle stay the same.]

What algebraic rule is associated with the ordered pairs? Write an equation that shows how the two dimensions are related for this rectangle. Can this rule be written in the form length : width = __ : __ ?

[The algebraic rule for this rectangle is L = 1.5W. Stated another way, length : width = 3 : 2.]

If the length of the rectangle is 13 wooches, determine the width of the rectangle in wooches using (1) your line of best fit and (2) your algebraic rule.

[The algebraic rule indicates that the width should be approximately 13 ÷ 1.5 ≈ 8.7 wooches. The line of best fit should give a similar result.]

Assessment Options

In whole-class discussions, ask students to describe how they discovered their pattern(s). Encourage and validate a variety of appropriate responses.
Ask students to write a paragraph explaining a unit of measure they think would work well for this activity. (One that hasn’t been discussed in class.) Why would this unit work well? What would be your prediction for the measure of the rectangle’s width and height using this unit of measure?
The teacher can provide the students with a graph similar to the one that the students created, but this graph would show measurements in a different ratio (possibly 5:2). Students could than be asked to write a summary of some conclusions that can be drawn from the graph.

Extensions

Using the data that students collected for their rectangle, graphing calculators can be used to perform a linear regression and determine the line of best fit.
You can provide other rectangles and ask students if the ratio is the same for every rectangle. Though it’s not the same for every rectangle, students should see that rectangles with the same ratio have the same "shape," leading to the concept of similarity.

Teacher Reflection

What were some of the problems students encountered when using the different units of measure?
Is this lesson better suited as a review or as an introduction to a particular topic?
Would the lesson have gone better if a particular vocabulary word or idea was reviewed at the start of the lesson?
Are there better items that can be used for measurement? Which ones worked particularly well? Which items would you not use if you taught this lesson again?
Were students focused and on-task throughout the entire lesson? If not, what improvements could be made the next time this lesson is used?
In this lesson, every pair of students worked with the same rectangle. What would be some of the benefits and difficulties of giving each pair of students a different rectangle?
How did students demonstrate that they were actively learning?
How did students show that they had achieved the objectives of the lesson?
Were the students led too much in the lesson? Or did they need more guidance?
Did you find it necessary to make any adjustments during the lesson?
Did the materials that the students were using affect classroom behavior or management?

NCTM Standards and Expectations

Algebra 6-8

Represent, analyze, and generalize a variety of patterns with tables, graphs, words, and, when possible, symbolic rules.
Explore relationships between symbolic expressions and graphs of lines, paying particular attention to the meaning of intercept and slope.
Use graphs to analyze the nature of changes in quantities in linear relationships.

Measurement 6-8

Use common benchmarks to select appropriate methods for estimating measurements.
Select and apply techniques and tools to accurately find length, area, volume, and angle measures to appropriate levels of precision.

This lesson prepared by Kristen Chandler.

1 period

NCTM Resources

Learning and Teaching Measurement (2003 Yearbook)


More and Better Mathematics for All Students