Chapter 6: Number and Operations



Chapter 6: Number and Operations

Calculating with a non-electronic computing tool: The slide rule

Students in high school in the 1950’s and ‘60’s most likely learned how to use a slide rule for finding approximate values of multiplicative, exponential and logarithmic calculations. Slide rules were actually used by astronauts on three Apollo space missions during the 1960’s and much of the nation’s existing infrastructure (such as bridges) were built by engineers using slide rules. A basic slide rule looks something like figure 6.1.

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Figure 6.1: Basic slide rule scales (from )

In figure 6.1, scales A, D and K are on the fixed part of the rule, whereas scales B, L, CI and C are on the slide. Scale L is a uniform linear scale from .0 to 1.0, whereas all of the other scales are logarithmic scales. If you compare scales C and L you can see the base ten logarithm of the number on scale C directly above that number on scale L. For instance, the log of 1 is zero, the log of 10 is 1.0, and the log of 2 is approximately .3. Another way to put this is that 100.3 is approximately equal to 2. Using logarithmic scales converts multiplication into addition. For instance 2 times 5 would be converted into 100.3 times 100.7 and following the law of exponents, the answer would be 100.3+0.7 = 101.0. Thus, using the logarithmic scales, adding the logarithmic length for 2 onto the logarithmic length for 5 gives the logarithmic length for 10. Scales A and B are identical (and half the length of scale C). By sliding the 1 on the B scale under the 2 on the A scale and then finding where the 5 on the B scale lines up on the A scale, we have added the logarithmic length for 5 onto the logarithmic length for 2 and produced the logarithmic length for 10. In this way we can multiply (or divide) any two numbers between 1 and 10. The numbers on the second half of the A and B scales would represent the multiples of ten. Thus the complete A and B scales go from 1 to 100 (the logarithmic length for 100 is twice the logarithmic length for 10 as 100=102). Working with smaller (decimal) or larger numbers can be easily achieved by scaling the scales up or down. The C and D scales provide finer scale points for better accuracy.

Squares and square roots can be read off directly by comparing the numbers on the fixed A and D scales. The A scale provides the squares of the numbers on the D scale and the D scale provides the square roots of the numbers on the A scale. Explain why!

The K scale repeats the C and D scale three times, thus providing a means of finding cubes and cube roots of numbers by directly reading corresponding values between scales K and D. Note that scale K represents three powers of ten (from 1 to 1000).

The CI scale is the reverse of the C and D scales and can be used for finding reciprocals and quotients. Reciprocals can be found directly by comparing the numbers on the C and CI scales and adjusting the decimal place. For instance, the reciprocal of 8 (1/8) can be found by reading the value on the CI scale immediately above the 8 on the C scale (1.25) and moving the decimal point one place to the left (.125). The quotient 7/5 can be found by placing the 1 (right end) of CI over the 7 on the D scale and reading off the number on the D scale that corresponds to the 5 on the CI scale (thus subtracting the logarithmic length for 5 from the logarithmic length for 7). The result is 1.4.

A dynamic slide rule in Sketchpad

A dynamic slide rule can be constructed in Sketchpad to multiply and divide numbers. This slide rule, however, does not rely on fixed-length logarithmic scales. Instead, it makes use of the dynamic scaling available in Sketchpad. Open the sketch dynamic_sliderule.gsp in the Arithmetic Operations folder (see Figure 6.2).

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Figure 6.2: Dynamic Slide Rule in Sketchpad

Activity 6.1

Experiment with this dynamic slide rule. Use it to find the results of the following calculations:

1. 1.5x3.0

2. 9.4÷2.3 [Note: If necessary, adjust the scale on the upper number line]

3. -2.0x3.5

4. (-7.0)÷(-3.5)

Check your results with an electronic calculator. How accurate were your slide rule results? If the results differed by more than .01, can you explain why?

What difficulties did you have in performing the calculation in #3 (-2.0x3.5) with the dynamic slide rule?

Why can’t you find the quotient of two numbers with different signs (one positive and one negative) using this slide rule? How did you find the product of two numbers with different signs? Can you find the product of two negative numbers with this slide rule? Why or why not?

Reflection 6.1

The logarithmic slide rule worked by adding lengths representing exponents of the base ten. How does the dynamic slide rule work? (Hint: What are you doing to the numbers on the bottom number line when you change the length of the blue unit point relative to the top number line?)

Activity 6.2: Constructing a product of two numbers on a number line using dilation

One way of describing how the dynamic slide rule works is to regard the bottom number line as being dilated (about the zero point) according to the scale factor indicated by the position of the blue unit point on the upper number line. In other words, every numeric position on the bottom number line is a dilation of the corresponding numeric position on the upper number line by the ratio of the bottom unit-length to the top unit-length. We can also use dynamic dilation to locate the product of any two numbers on a single number line, rather than using the manipulation of the unit point on one number line relative to a second number line. The following steps will create a number line in Sketchpad:

• Open a new sketch.

• Choose Define Coordinate System from the Graph menu.

• Click on the vertical axis to select it and choose Hide Axis from the Display menu.

• Choose Hide Grid from the Graph menu.

• Label the origin and unit points as 0 and 1 on your number line.

• Place two free points on your number line and label them A and B.

Your sketch should now look like figure 6.3:

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Figure 6.3: A number line in Sketchpad

• Select points A and B and choose Abscissa (x) from the Measure menu.

• Using the Label tool re-label the two measures (XA and XB) A and B respectively.

• Change the precision of these two measures to tenths by selecting a measure and choosing Properties from the Edit menu and the Values tab from the properties panel.

Move your two points around on your number line and check that their measures change accordingly. The following steps will create a new point on the number line that will represent the product of your two measures, A and B:

• Select the points 0, 1 and B IN THAT ORDER and choose Mark Ratio from the Transform menu.

• Double click on point 0 to mark it as the center of dilation.

• Select point A and choose Dilate from the Transform menu.

[At this point, a new point should appear on your number line. If you do not see a new point then move either A or B closer to zero.]

• Label this new point C and measure its abscissa (x-coordinate).

• Change the label of this measure from XC to C.

Your sketch should now look something like figure 6.4.

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Figure 6.4: Product of A and B using dilation on a number line

Move points A and B around on your number line and test to see if the measure of point C gives the product of the measures of points A and B.

In figure 6.4, my sketch shows the product of –2.0 and 4.0 as –8.08. Why might this be so? Does it make mathematical sense to have the product’s precision set to hundredths when the precision of the factors is set to tenths? What would the measure of C in figure 6.4 indicate if its precision was set to tenths? What could you do with these measures if you wanted your students to work with integer arithmetic rather than with rational numbers? Modify your sketch to show integer calculations only.

Activity 6.3: Representing arithmetic operations dynamically on a number line

In the above example, the product of two numbers was constructed using dynamic dilation. Sums and differences can be constructed using translations on the number line rather than dilations. Each number is represented as a horizontal vector from the origin (zero), and the sum (or difference) is constructed by translating the end-point representing one number by the vector (or its reverse) indicated by the zero point and the point representing the other number. For example, in figure 6.4 above, the sum of A and B could be constructed by marking the points 0 and B as a vector using the Transform menu and then translating point A by this marked vector.

Construct the sum of A and B on your number line using the “marked vector” method described above. Call the new point D and investigate relations between the product point, C and the sum-point, D.

• When is the sum greater than the product (i.e. when is D to the right of C)?

• When is C to the right of D?

• When is D between A and B?

Activity 6.4: Investigating arithmetic relations dynamically

In the previous two activities you constructed the product and sum of your two independent points on the number line using geometric transformations (dilation and translation). In this section we shall use the GSP calculator to compute arithmetic relations based on the positions of the two independent points. Start with a new number line as in figure 6.3 above and measure the abscissa (x-coordinate) of each point. Re-label these measures A and B as in figure 6.4. The following steps create a new point, C, on the number line based on a defined mathematical relation using the measures of A and B:

• From the Measure menu choose Calculate. The GSP pop-up calculator should appear (you may need to move the calculator in order to see the measures of A and B).

• Create an arithmetic expression in the calculator by typing values, arithmetic operators (+, -, *, ÷) and clicking on the measures for A and B. For example, to create the expression for 2A-B you would click on the 2, the * (for multiplication), the measure of A (the label A should appear in the calculator window), the – key and finally the measure B. You would then see the expression 2*A-B in the calculator window.

• Click on the OK button on the calculator. The measure of 2A-B should appear in your sketch.

• With the label tool, double click on this new measure and change its label to C.

• With the select arrow, select just the origin point of the number line and measure its ordinate (y-coordinate) from the Measure menu. The measure y0=0 should appear in your sketch.

• Deselect everything by clicking in open space and then select IN THIS ORDER the measure C and the y0=0.

• With these two measures selected (and nothing else) choose Plot as (x, y) from the Graph menu. A new point should appear on your number line at the position corresponding to the value of measure C. (Note: you may have to move point A or B to create a value for C that is within the visible portion of your number line.)

• With the label tool, label this new point C.

Investigate the behavior of your plotted point C as you vary your independent points A and B. Does it behave as you would predict? What positional relations can you create with these three points? Will point C ever be between points A and B? To the left of A and B? To the right of A and B?

You can also use the Numberline Tool.gsp sketch for this activity. This tool provides colored tags attached to your independent points and a free tagged point labeled C that you can merge with your plotted point C after following all of the above steps (simply select both the plotted point C on your number line and the free tagged point C and choose Merge Points from the Edit menu). Figure 6.5 illustrates the situation with both points C selected just before merging.

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Figure 6.5: About to merge the free tagged point C with the plotted point C on the number line using the Numberline Tool.gsp sketch.

Figure 6.6 shows the number line after the points have been merged.

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Figure 6.6: Number line with tagged points.

Assignment 6.1: Mystery Relations

Create a GSP sketch for a mystery relation for a classmate to investigate (that is, create a plotted point C based on some relation between measures A and B as in Activity 6.4) and bring this to class. Exchange your sketch with a classmate and try to determine the relation between your classmate’s point C and the independent points A and B without revealing the expression for the measure of C! What strategies can you devise?

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