Math Desk Queries

QUESTION

My son is having difficulty understanding the difference of multiplying whole numbers and proper fractions. When we multiply something we always get more of it and not less. My son can't comprehend why the product of any two whole numbers always is greater than the multiplicand, but when multiplying proper fractions, the products are smaller than their multiplicands. How can I explain this to him?

Abigail

ANSWER

5 X 4 = 20
5 X 3 = 15
5 X 2 = 10
5 X 1 + 5
Fig. a

It is readily seen that multiplying 5 by 4, 3, 2, and 1, and as the the value of their multipliers declines (4, 3, 2, and 1), their products also are getting smaller (20, 15,10, and 5).

Note: when multiplying 5 by declining multipliers 4, 3, 2, and 1 (Fig. a ) their products 20, 15, 10, and 5 respectively also declined . From this we can deduce that if we continue to do so and multiply 5 by the multiplier which has a lesser value than 1, that is, a by a given fraction then their product must be smaller than their multiplicand. Fig. b shows such an proof.

From the example in Fig. c we example we can conclude that if

is multiplied by a number which has a higher numerical value than

, for example, by 4, then the product must also be higher than

, i.e., 2. But when

is multiplied by

(Fig, d ) then the value of the product is less than the multiplicand's because we multiplied it only by

Summary:
When multiplying two proper fractions, your product is smaller than either the multiplicand or the multiplier for proper fractions indicate only a part of the whole.

I need help with stem and leaf plots. My son brought his homework home and it asks him to read a steam and leaf plot. I don't know how to read it myself, so of course I could not explain it to him. Please help! Thanks!

I have never heard of stem-and-leaf terminologies in the subject of math. The words stem and leaf are not mathematical terms. The word stem means stalk of a tree, shrub, or other plant bearing the leaves, flowers, etc. The word leaf means plainly what it says, i.e. a piece of flat, thin, mostly green substance, shaped according to the species of a plant.

I'm baffled as to what "to read a stem-and-leaf plot" really means just as you and your son are. The stem-and-leaf precept is void of the conceptual content that underlie and tie the math structure together. What mathematical concept does the stem-and-leaf "terms" imply? How can one use the stem-and-leaf information in the solution of real life mathematical questions?

It appears that the idea of the number system, its concepts and principles , which are the keys to math comprehension generally are not taught. Rote learning is too prevalent. The method of solution of simple problems often is shown complicated for the student to understand and for the teacher to teach; the solution of problems with a certain degree of difficulty is hardly comprehensible.

What does "teaching" mean? Generally, it means to show or explain how to do something. In teaching math it means not only to show students its "hows" but also its "whys," which is of paramount significance in comprehending the subject matter.

It's difficult to believe that such an absurd notion as stem-and-leaf may in fact exist in contemporary math textbooks. However, perusing math textbooks at the elementary school level I have found the content of Stem-and-Leaf Plots in one textbook published by Addison-Wesley titled Quest 2000 Exploring Mathematics Grade 6, on page 164.

After analyzing its contents I still can't answer your question. But, based on the stem-and-leaf data, and such questions as "What is the median circumference? or "What is the mode" – I see there is an attempt to show how to use the classified set of numerical data to infer a conclusion(s) based on such data. However, without definitions of the essential terms required in the study of this particular segment of statistics – I can't see it as a meaningful teaching method of the subject matter.

So, how to transform an formless Stem-and-Leaf-Plots inception of statistics into a comprehensive lesson?

(a) The title of subject's study should have a purposeful terms, i.e., Elements, Mean, Mode, and not an ambiguous Stem-and-Leaf-Plots.

(b) Students should not be asked to solve the problem before learning the relevant statistical terms that will be required when finding the solution of Josè's question and other exercises of this kind.

"Josè measured the circumference of 25 maple trees.
He made the measurements 1 m above the ground."

Data:
"80.6 cm, 80.4 cm, 79.5 cm, 75.9 cm, 80.3 cm, 79.7 cm, 79.0 cm,
75.6 cm, 77.5 cm, 79.2 cm, 80.8 cm, 79.9 cm, 79.5 cm, 81.4 cm,
77.8 cm, 79.0 cm, 77.5 cm, 79.9 cm, 79.8 cm, 77.2 cm, 83.4 cm,
76.8 cm, 78.2 cm, 82,5 cm, 80.5 cm"

The word statistics means a classified set of numerical data. Actually it's much more than that. It's the study and interpretation of classified set(s) of numerical data from which conclusion can be inferred. To have the ability to correctly draw inferred conclusions based on an examination of such data one must fully comprehend what the terms element, data, mean, mode, median stand for.

element:	a component that belongs to a set
data:	the facts or figures from which conclusions can be deduced obtained
mean:	the sum of a set of elements divided by the number of elements in the set (the arithmetical average)
median:	the middle element when elements are arranged in order of size
mode:	the number that occurs most frequently often in a set of numbers
statistics:	a classified set of numerical data

Example 2
Find the median of the set of elements 40, 18, 45, 30, 24, 36, 12.

Answer
The median is defined as the middle element when the set of elements is arranged in order of size, therefore the median of 40, 18, 45, 30, 24, 36, and 12 is 30 .

Example 3
Find the median of the set of elements 30, 18, 14, 8, 39, 20, 6, 34.

Solution
Step 1. Arrange the elements from the least to the greatest. 6, 8, 14, 18, 20, 30, 34, 39

Step 2. There is no middle element in the set, because the number of elements is even . In that case take two middle elements and find their average. The result will be the set's median. Two middle elements are 18 and 20. Find their average.

Example 4
Find the mode of the set of elements 12, 14 14, 13, 13, 14, 16, 19, 14, 12.

Step 2. Determine the element that occurs most frequently in the set's list. It is 14 .

This will give students the understanding of conceptual notion of basic statistics and the ability to infer conclusions from classified sets of numerical data.

There are two basic concepts of division: (1) the comparison or the ratio concept
and (2) partitive concept.

Ratio:	*A relation between two like* quantities.**
Proportion:	A relation between two ratios.

A building lot measures 75 m by 25 m. The lot's length is larger as compared with its width. We can calculate the difference between the two dimensions

( a ) 75 m – 25 m = 25 m. Subtracting 25 m from 75 m we obtain 50 m. We can state that the length of the lot is 50 m longer than its width.

( b ) 75 m ÷ 25 m = 3. Dividing 75 m by 25 m we obtain 3. The quotient 3 is simply written as 3, and not as 3 m, for it expresses an abstract idea, i.e., it measures the numerical relation between two measures, 75 m and 25 m.

We can state that the lot's width is one-third of the length, or the ratio of the width to its length is one to three. Since a ratio shows the relation of one measure to another by division, and may result as a fraction, simplifying (dividing) both terms of the fraction by the same number does not change the value of the fraction, and thus the ratio (Fig. 1).

When two ratios are written as equal ratios, the equation is called a proportion. The proportion concept is in fact an extension of the ratio concept. It shows a relation (equality) between two ratios.

The equation 2 : 3 = 8 : 12 or $twothird$ = $eight twelve$ states an equality between two ratios and therefore is a proportion. In other words, the ratios 2 : 3 and 8 : 12 or $twothird$ and $eight twelve$ (when $eight twelve$ is simplified to its lowest terms it equals $twothird$ ) are equiva l ent and consequently in the same proportion.

The statement of equality between two ratios is written with the double colon, ( :: ), called a proportion sign, and it is read " as. " The sign of equality ( = ) is read " is equal to. " Thus, the proportion 2 : 3 :: 8 : 12 is read "two is to three as eight is to twelve,"
$twothird$ = $eight twelve$ is read "two-thirds is equal to eight-twelves."

Terms
The word "part" (noun) may be defined as a division or portion of a whole. As a verb it means to divide into parts . The word "partitive" is an adjective derived of the word "part" and it means restricted to a part of a whole.

A chocolate bar is divided into 4 parts (1 ÷ 4 =

) . The quotient

represents1 of 4 equal parts into which the chocolate bar has been divided . The fraction

represents the size of equal parts.

Conclusion
Fractions indicate division. The fraction line (—) means exactly the same as the division sign ( ÷ ) and thus it can be replaced by a fraction line, or vice versa. 1

The numeral

, in the foregoing example, expresses the partitive concept of division: its denominator (4) shows the quantity of parts (4 equal chocolate parts), and the numerator (1), the quantity of these parts into which the chocolate bar has been divided. Thus, the results of a partitive divisions are fractions that imply parts of a whole.

While ratios and proportions may be written in the form of fractions, however, the expressions do not imply the form of part to whole . The terms of ratio $sevenfive$ (Fig . 1) indicate how many times as large the measure of 75 m is as compared with the measure of 25 m. In the comparison or ratio concept of division fractions express the notion of part to part. Hence, the term "partitive proportion" is meaningless. It can't imply a relation of one quantity to the other. There are simple proportions, inverse proportions, compound proportions, and direct proportions – but not partitive proportions .