This section introduces the two main types of useful representations of quantities and measurement: a) analog and digital, and b) positive and negative. The way we see, represent and manipulate quantities will affect the things we can do and problems we can solve in real life.

Although there currently seems to be no strong focus on the academic standards in mathematics at the lower grades regarding the specific implications, advantages and disadvantages of different representations of quantity, the topic becomes central as we begin to deal with the subject of Algebra, when we introduce the concepts of variables, equations and their resolution.

• CCSS.Math.Content.3.NF.A.2 Understand a fraction as a number on the number line; represent fractions on a number line diagram.

• CCSS.Math.Content.4.NF.B.3 Understand a fraction a/b with a > 1 as a sum of fractions 1/b.

• CCSS.Math.Content.4.NF.B.4 Apply and extend previous understandings of multiplication to multiply a fraction by a whole number (including first: "a whole number by a fraction").

• CCSS.Math.Content.4.NF.C.6 Use decimal notation for fractions with denominators 10 or 100. For example, rewrite 0.62 as 62/100; describe a length as 0.62 meters; locate 0.62 on a number line diagram.

• CCSS.Math.Content.5.NF.A.1 Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators. For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.)

• CCSS.Math.Content.5.NF.B.4 Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.

• CCSS.Math.Content.5.NF.B.5 Interpret multiplication as scaling (resizing).

• Science and engineering practices in the NGSS

• 1. Asking questions (for science) and defining problems (for engineering) 2. Developing and using models 3. Planning and carrying out investigations 4. Analyzing and interpreting data 5. Using mathematics and computational thinking 6. Constructing explanations (for science) and designing solutions (for engineering) 7. Engaging in argument from evidence 8. Obtaining, evaluating, and communicating information.

ANALOG REPRESENTATION

Analog representations use imitations, analogies, "analogical" (or “analog”) like forms, shapes, sounds, etc. to give us a highly instinctual, perceptual sense feeling of the amounts and changes of something. Things like big and small, close or far objects or separate (visual) images, pictures or graphs, brilliant, strong or faint colors or tones, loud or quiet, high or low pitch (auditory) sounds, high or low voltage (electrical signals) for example, can make our senses and/or brain instinctively or intuitively perceive a degree of difference in quantity among things being represented and communicated.

These instinctive, imitative forms have been and are especially helpful for reliable, casual, first-glance sensing for immediate human awareness and responses, particularly in situations where humans take part in critical or important processes, such as human operations and control of heavy, fast or dangerous machinery and equipment. Analog scales therefore usually include visual descriptions through the range of values for the quantity being measured. Analog scales can also include a variety of multisensory (attractive color, sound, and physical objects) markers that can indicate the presence of critical conditions requiring direct, immediate attention ("flags", warnings, alarms, etc.). Analog representations are also normally accompanied by a corresponding set of useful numeric indicators (i.e. digital codes - described ahead) describing more accurately the quantities being presented.

Call the Analog Weighting Scale display.

This display presents a weighting scale using the typical arched line segment with ticked numeric marks pointed to by a swinging needle, indicating the range of weights allowed. The display includes a series of objects available for weighting. The gold (brass) colored cylinders on the lower-left corner of the display are “standard”, trusted gram weights that can be used to “calibrate”, or verify the accuracy of the scale.  When properly calibrated, the scale’s needle should point to the correct number in the display scale-line, corresponding to the "official" weight of the objects placed on the scale's platform.

HANDS ON:  Wide-screen/Wireless-mouse or Touch-Screen

1.- Observe the elements of the analog scale display: a) An analog, swinging “needle” and scale, b) a set of standard weights, c) various common objects and toys.

2.- Drag one 50 gram standard weight onto the weighting scale, and observe the needle moving up the scale.  Which number is the needle pointing to? Repeat similar observations for the remaining standard weight cylinders.

NOTE:  Make sure to try using the “zoom” feature of the display, by right-mouse clicking and selecting the proper zoom-in, or zoom-out option, as necessary for accuracy when reading the needle pointing onto the scale line.

3.- Make sure the scale table is clear and then drag two 50 gram standard weight onto it, then read the indicated number.

4.- Repeatedly clear and place each one of the remaining common objects and toys onto (bottom-right) the scale, and read their measured weight, as accurately as possible.

OBJECT	Measured Weight (Grams)
50 Gram standard weight
20 Gram standard weight
10 Gram standard weight
5 Gram standard weight
1 Gram standard weight
Pencil
Blue cube
Red cube
Wooden block
Nut
Apple
Tractor
Puppy

 

DIGITAL REPRESENTATION

 Symbolic and digital, or numeric representations

Digital measurement relies basically on symbols, used to encode information about amounts related to the magnitude of the objects or actions being described. This encoding is meant to provide efficient means of expressing quantity by using simplified forms, objects, scratches, swatches, smears, or scribbles. These simplified forms help focus our attention on the specific magnitude of interest, by stripping other non-essential details. In practice, the difference between the simplified forms or symbols and the quantities they represent is stated by calling the shaped symbols numerals, and the referred quantities numbers. The word digital, originating from the latin word for "digit"s, or fingers, refer to the most common use of numbers as symbols to represent quantity. Although through history we referred to the ten fingers in our hands to construct and describe quantities through number counts (decimal number system), more recent technological progress has shifted to counts referring to two (bi) stable states of electronic devices and circuits symbolized by "one" and "zero" (binary number system).

By their nature, digital representation creations and utilization evolve into and require the use of (often complex) encoding and decoding processes to make sense of the objects and actions in the environment being referred to. Eventually, digital representation protocols (set of rules) have been created to represent almost any object, action or idea imaginable (including text characters and analog representations themselves) through digital representations and a variety of so-called "coding-decoding" (codec) protocols.

Call the Digital Weighting Scale display.

This display presents a weighting scale using the typical digital indicator panel.  A series of objects are shown available for weighting. The gold (brass) colored cylinders at the bottom left, are “standard” gram weights used to “calibrate”, or verify the accuracy of the scale.  When properly calibrated, and items placed on the scale's platform, the scale’s needle will swing to point to the correct weight of the objects (bottom right). The standard "bronze" weights included are for illustration of what a properly calibrated digital scale should display on its screen.

HANDS ON:  Wide-screen/Wireless-mouse or Touch-Screen

1.- Observe the elements of the display: a) The digital, numeric window, b) a set of standard weights, c) a set of common objects and toys.

2.- Drag one 50 gram standard weight onto the weighting scale, and observe the number on the numeric window.  Which number is the display calling?

3.- Drag the second 50 gram standard weight onto the scale, and read the new (total) weight on the scale.

4.- Place one of the available common objects and toys onto the scale, and call their measured weight, as accurately as possible.

5.- Compare the recorded weights recorded when using either the analog or digital scales in the previous activities and write answers to the following questions about the differences between analog and digital representations:

a) Which scale could be more accurate? How would calibration affect the level of accuracy and precision involved?

b) Which could be cheaper?

c) Which could be faster?

d) Which could be safer?

e) Which could be more accurate? precise?"better"?

f) Which is easier for a casual (estimating) glance from far away?

g) Which could be more helpful in uses for example, in coffee pots, airplanes, industrial equipment?

h) In using measurement information for various other related analyses and application purposes?

ANALOG AND DIGITAL REPRESENTATION OF TIME
Call the displays below to observe and practice the representation of time through the day on common analog (needle) and digital (numeric) displays.The representation of time elapsed through the day in these common displays is made either using a full analog circle scale that repeats two consecutive half-day 12 hour periods through the day. The hour number in turn is marked by the position of the shortest and thicker needle rotating from the center of the circle. In turn, each of the 12 hour period segments in the circle is used to sub-divide and represent five-minute periods that elapse to represent the total of 60 minute period times inside each hour. The minutes within the hours are then marked by the longer, thicker needle that also rotates from the center of the circle. Although not included in the model presented in this guide, many clocks also include a third, thinnest needle that will similarly hinge and turn faster to point to the seconds during each minute around the 60 point markings of the circle.

The “SETTING TIME” display in this guide contains two numeric (digital) displays corresponding to the markings of the analog clock, one using a repeating 12-hour sequence, and the other adopting a "24-hour" ("military" or industrial time) sequence format.

HANDS ON:  Wide-screen/Wireless-mouse or Touch-Screen

1.- Practice sliding the time scale in the first clock display, using either the "minutes needle", or the slider below through the 24 hours of the typical day. Observe the daily events from starting midnight to ending midnight.

2.- Practice telling the time ("What time is it?") from at least 25 trials ("Number Correct") in the lower clock display. Make sure to also enable the "Show minutes" and "Quarter to... and quarter past ..." options of this display.

3.- What and when is better for practical use of the different analog or the digital clocks?

QUESTION: What would it take for a newcomer “alien visitor” (or a new student) to be able to interpret the symbols (for example: “238.7” ) as the magnitude of the weight of an object on the scale’s platform?  

Due to its informational and usability advantages, digital measurement has become the most favored form of measurement driving today's industrialized and developed societies. Creating digital measurements typically involves the use of a material called transducer, which is made of a material that opposes electrical current differently when heated, lighted, or in the case of weight, compressed with different intensities. The transducer, placed inside the scale, is set to feel the pressure of the weight and signal quantity by the proportional degree of reduction in the electrical current allowed to pass through it. Electronic circuits are then used to sense the analog intensity of the current, converting it to a digital coded number. This process is called analog-to-digital conversion. The coded information is then used to drive electronic digital display panels, and/or computer calculations and other information processes.

The illustration shown in Figure 6 illustrates the steps of Analog-to-digital conversion of weight information in a digital scale.

Now call the electrical function display to observe corresponding analog and digital meters of electrical voltage and current.

Digitally generated multimedia supporting digital and analog representations

With the growing power of electronic digital technologies, an additional option has become feasible and practical: the simulation of analog or digital representations of quantities, including objects, actions and symbols through " digital multi-media” .  

Today, digital tools and methods can produce practical, visual-friendly, "first-glance" as well as detailed analog-simulations of images, sounds, and movement that can effectively communicate quantities, and improve the efficiencies of their human users. Be it at home, school, or work, commercial and industrial appliances and machinery elements can be simulated electronically, allowing for cheaper (no-wear-and tear), more flexible and safer instruments and operations. Figure 7 shows such examples in a screen of a digital computer simulation of the analog and digital instruments of the dashboard of a plane in a computer-based flight simulator. At the present time, open access-play-flight-simulator software could also be available through a web-connected personal computer (sample)

Figure 7.- A computer screen showing a digital simulation of an airplane analog and digital instruments  

The digitally generated analog and digital measurement instruments used throughout this instructional guide, are also examples of the power of digital electronics.

POSITIVE AND NEGATIVE QUANTITY REPRESENTATIONS

This section contains an introduction to the concept negative quantities, representation, and use. As described, the concept of "negative quantity" arises when we deal with objects and actions that rival, balance, compete with, or cancel one another. Things like opposing forces, heating and cooling, positive and negative electrical charge/voltage, owned goods (assets) versus owed ones (liabilities), constructing and destroying, etc. are examples where quantities can be described using positive and negative representations.

It is important to point out here that due to common confusions in mathematics, it is necessary to emphasize that although equivalent in resulting effects, the essential nature of negative numbers is different from the operation of subtracting a positive quantity: In real life, the process and resources required when adding a negative quantity are often dramatically different than those required when subtracting a positive quantity. Naturally, this could make a great difference in the things we do and how we do them.

Also, and as described in the INTRODUCTON section of this document, the nature of the ratio scales of measurement would usually be most appropriate for applying the concept of negative numbers. The addition of a positive number or real quantity to a negative one of the same magnitude in the scale would yield a cancelled, null, or real zero amount point result in a ratio scale.

CREATING AND REPRESENTING POSITIVE QUANTITY

Call the Seeing the difference between -(-1) and +(+1) display.

This display uses an animation to illustrate the two different ways of creating (and representing) positive quantity.

First, the concept of positive and negative quantity, as existing opposite substance matching and canceling each other through canceling tokens.  Real examples of this are hot and cold temperature, left and right direction, owning and owing resources, good or bad attitudes, yes or no responses.

Next, the display presents an animation showing the creation of a positive unit of quantity, either by removing the negative counterpart (subtracting negative), or by acquiring a new positive unit from the outside (adding positive).

The display animation ends by presenting a list of relevant real-life examples of the use of creation of positive amount, either by subtracting positive, or by adding negative quantity.

HANDS ON:  Wide-screen/Wireless-mouse or Touch-Screen

1.- Step through the display sequence (pressing the CONTINUE button), and ask anyone else in the class to comment on what is happening to the net quantity.

2.- Describe the physical differences involved in each case, in the context of possible real situations by subtracting positive, or adding negative to address a problem.

     CREATING REAL POSITIVE VALUE

CREATING AND REPRESENTING NEGATIVE QUANTITY
Call the Seeing the difference between -(+1) and +(-1) display.

This display uses an animation to illustrate the two different ways of creating (and representing) negative quantity.

The concept of opposite positive and negative quantity is shown through opposite, canceling tokens, as they balance (or not) the effect of each other.

In the animation, the creation of a negative unit is done by removing the positive counterpart (subtracting positive), or by acquiring a new negative unit from the outside (adding negative).

The animation ends by presenting a listing of relevant real-life examples of the creation of negative amount, either by subtracting positive, or by adding negative.

HANDS ON:  Wide-screen/Wireless-mouse or Touch-Screen

1.- Make sure the students identify the two matching but opposite tokens in the initial neutral combination.

2.- Step through the display sequence (pressing the CONTINUE button), and observe what is happening to the net quantity at each step.

3.- Identify the positive and negative quantities and actions involved in each real-life application case, and the difference when subtracting negatives, or adding positive quantities to address a problem.

     CREATING REAL NEGATIVE VALUE

Quantity	Subtracting NEGATIVE	Adding POSITIVE
Heat/Temperature in room or machine	Reduce cooling in a room or in a car engine	Increase heating in room or machine
Fat in human body	Reduce intake of fat and food (lean diet)	Increase fat burning exercise, or remove fat from body (e.g. pills or liposuction)
Monetary value	Reduce spending	Increase earnings
Electric Voltage	Reduce positive voltage	Increase negative voltage
Workforce/people Discipline	Reduce behavioral penalties	Increase behavioral incentives

 

ADDING AND SUBTRACTING POSITIVE AND NEGATIVE VALUE
Call the Play Add and Subtract Negative Tokens display.

This display shows the effects of adding and subtracting positive and negative tokens, as they represent the manipulation of positive and negative quantities in real life.

The tokens can be put in, or taken out of a mat, where their combination takes place.  The mat also contains an initial set of five positive-negative cancelled-out pairs.  Removing the counterpart token of any of these pairs (subtracting from zero) will yield a corresponding, free positive or negative unit left behind.

HANDS ON:  Wide-screen/Wireless-mouse or Touch-Screen

1.- Focus on the first (A) question on the display, and propose some answers.

2.- Focus on question (B), and after manipulating corresponding tokens, answer the question.

3.- Focus on question (C), and after manipulating corresponding tokens, answer the question.

4.- Focus on question (D), and after manipulating corresponding tokens, answer the question.

5.- Focus on question (E), and provide explanations.
( the first MINUS sign means SUBTRACTION, while the second means NEGATIVE QUANTITY  )

ADDING AND SUBTRACTING POSITIVE AND NEGATIVE VOLTAGE
Call the Adding and Subtracting Batteries display.

This display presents a simulation of a battery pack, which provides various voltage levels through a combination of positive and negative oriented batteries. Battery packs are commonly used in portable electric and electronic devices.

In the display, up to 20 batteries can be set all to supply each a given uniform voltage (set using the “Push to reset..” button, often 1.5 volts – AA, AAA), and then combined to produce various final voltages. Naturally, the positive or negative nature, and magnitude of the final voltage will depend on the unit voltage, and the number and orientation of the batteries installed.

HANDS ON:  Wide-screen/Wireless-mouse or Touch-Screen

1.- Demonstrate the functionality of the display, by showing:

1. The installation of positively, and negatively oriented batteries on the pack.  The default voltage for each battery is 1.0 Volts.

2. The availability of a set of neutralized batteries (0 Volts net) inside the covered “BATTERY CASE” section at the top.

3. The ability to change the voltage for each and all batteries by using the entered value in the “Push to Reset” button. Use the 1.5 Volt batteries to seek total the common voltages of 6, 9, and 12 Volts.  Batteries from the BATTERY CASE will be needed to produce the last two values.

4. Return the voltage back to 1.0 Volts.

 2.- Ask six students to focus each on one of the first 6 expression in the “Try the following expressions”, section, to be represented with batteries.

3.- Ask the class to cooperate in figuring out how to represent the last expression. The teacher or a student can execute the visual operations.

ADDING AND SUBTRACTING POSITIVE AND NEGATIVE ALTITUDE CONTROL PULSES
Call the Flying a Remote Controlled Airplane display.

This display illustrates the use of positive and negative “control tokens” (or pulses) to control the flying altitude of a remote-control airplane.

The number and positive/negative tokens in the Commands on Ground Control and the Commands on Board AIRPLANE areas is changed by clicking and dragging them through. Positive tokens added to the Commands on Board AIRPLANE area raise the cruising altitude, and negative tokens lower it. The initial cruising altitude in the scale is set to (relative) ZERO.

Because the display can be noisy, a sound VOLUME control is provided to limit the sound level.

HANDS ON:  Wide-screen/Wireless-mouse or Touch-Screen

1.- Describe the basic functionality of the display, which imitates the control of a flying remote control plane.  Explain the “Commands On Ground Control”, and “Commands On Board Airplane” areas, and the function of the tokens.  Point out to the ground antenna used to send electromagnetic coded pulses to the airplane, requesting to raise or lower its elevation.

2.- Explain why the initial 10 tokens in the “Commands on Board AIRPLANE” area cause the plane to fly at a cruising altitude of ZERO.

3.- Propose two ways to raise the elevation of the airplane, using the remote control tokens (pulses). Ask a student to propose two ways to lower the elevation of the airplane, using the remote control tokens (pulses).

4.- Propose a way to obtain the highest and lowest altitudes of the airplane.  And then have one student each to implement each strategy proposed.

5.- How many different combinations of control tokens in the Commands on Board AIRPLANE area that can be used to make the plane fly at an altitude of ZERO?  (HINT: don’t forget zero!)  
( very many!! )

USING POSITIVE AND NEGATIVE VOLTAGE IN CIRCUITS
Call the Measuring electricity display. This display illustrates the application of positive and negative electrical voltage over a basic electronic circuit.  The slider bar and knobs of the “power supply” on the left control the value of positive or negative voltages that are applied over a (variable) resistor, and a light bulb (an incandescent resistor) in the circuit.

The voltage In the display can be changed from a positive value of 5 volts, to a negative value of -5 volts.  The value of the variable resistor can be changed from 0.23 to 20 Ohms, and the values of resistance of the light bulb and the (direct) current meter are negligible.

The electrical current flowing through the circuit can be computed using the formula I = V/R (Ohms Law).  I is the value of intensity of the current (Amperes, or MilliAmperes one-thousandths of an Ampere), V is the value of the voltage applied (power supply), and R is the value of resistance (Ohms) opposed to the current flow through the resistor.

NOTE:  For very low values of resistance, the current can become large enough to overflow the range of the meters, and resulting on a warning and flashing message: “Watch Your Range”.  When the value of the resistance is negligible, the condition is commonly known as a “short circuit”.  In real circuits, where components are not capable of, or protected from handling the heat caused by sustained large currents, the components, including meters, could easily get damaged, permanently.

HANDS ON:  Wide-screen/Wireless-mouse or Touch-Screen

1.- Describe the functionality of the display, which imitates the setup of a basic electrical circuit: POWER SUPPLY, a RESISTOR, a LIGHT BULB, and ANALOG / DIGITAL meters.  Explain the common representation of the circuit and its elements in the lower middle section, and generally, the graphic representation of Ohm’s Law in the upper middle section.

2.- Demonstrate the effects of supplying a positive or a negative voltage to the circuit, pointing out the reversing of direction of the current flow.

3.- Set up the values of Voltage and Resistance as given below, compute and observe the value of the corresponding currents:

a)    V =    2  Volts      R =  5  Ohms         I =  __________  mA ? (1000 mA)

b)    V =  - 2  Volts      R =  5  Ohms         I =  __________  mA ? (-1000 mA)

c)    V =  - 2  Volts      R =  1  Ohms         I =  __________  mA ? (-2000 mA)

d)    V =   3.5 Volts     R =  5  Ohms          I =  __________  mA ? (700 mA)

e)    V =    5  Volts      R =  1  Ohms          I =  __________  mA ? (200 mA)

f)    V =  0.3  Volts     R =  0.23  Ohms      I =  __________  mA ? (1304 mA)

REPRESENTING POSITIVE AND NEGATIVE NUMBERS IN (DIGITAL) COMPUTERS
Call the Addition and Subtraction in Base-2 display.

This display simulates the representation of positive and negative numbers, and their  addition and subtraction in a binary, base-2 electronic adder, a centerpiece of today’s electronic computing.  The numbers are shown using images of pebbles and lights.  Negative numbers are coded in the base-2 “twos complement” form.

Real-life implementations of binary adders normally use electronic semiconductor switches built in microcircuits, operating at speeds measured in billionths of a second.

 HANDS ON:  Wide-screen/Wireless-mouse or Touch-Screen

1.- Describe the functionality of the display, which imitates the functioning of an electronic, digital, binary adder in a computer: “Enter” windows for the two numbers to be operated upon, the “+ / -“ button to request addition, or subtraction, the “GO” button, and the “Pace” button.

Investigate about the “twos-complement” representation of negative numbers in the base-2 binary counting system - how each of the digits is replaced by its “complement”, simply reversing each digit from ones to zeros, and vice versa.  Also, point out to the fact that the value of 1 unit is added to the resulting form, to compensate for the reversal. 

2.- Use the display to answer the following:

Base-10 Number	Base-2 Positive  Representation	Base-2 Negative Twos-complement Representation
0	00000000	11111111+
1	00000001	11111110+
2	00000010	11111101+
50	00110010	11001101+
100	01100100	10011011+
200	11001000	00110111+

 

3.- Write each of the numbers in the following operations in binary form, add them manually, and then use the display to verify the answers for the following:

a)  2 – 1 =

b) 100 – 50 =    

c) 100 – 200  =