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.
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
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
[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
1. 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.]
2. 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.]
3. 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.]
4. 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.]
5. 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.]
6. 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.]
7. 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.]
- 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?
- 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.
Common Core State Standards – Mathematics
Grade 6, Ratio & Proportion
Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, ''The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.'' ''For every vote candidate A received, candidate C received nearly three votes.''
Grade 8, Stats & Probability
Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.
Grade 8, Stats & Probability
Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.
Common Core State Standards – Practice
Use appropriate tools strategically.