Measuring things accurately when things are small or very small requires dealing with parts of whole (entire) objects. Because of the need for consistency in describing parts of whole things, there is a need for shared understanding of smaller, uniform portions of the wholes. That is the case, for example, when we refer to "one-half", "three-fifths", or "seven-thousandths" of some object, a meter, foot, or any other standard unit of measurement. These expressions refer to convenient and accurate descriptions of incomplete quantities, which can be small or extremely small parts of something. This is the topic of this chapter.

In the school curriculum for the K-12 grades, the topic of fractions appears in various themes of different subjects, but more specifically, in the most current standards for the subjects of mathematics and science (CCSS "Common Core" - Mathematics Standards , and NGSS "New Generation" - Science Standards):

• 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.

Representing fractions
Call the Representing Pieces of One Whole display.
This display features the visual representation of fractions as a collection of a number of equal parts of a whole unit (“EVEN BREAKUPS”). Both the total number of equal parts of the whole, as well as the selected partial number of them, are set by using corresponding slider bars (left and top). The fractions are shown in both circle and rectangle forms, offering visual displays of how portions of "a whole thing" can be represented using even, manageable breakups of a “unit”, as convenient for practical uses.

The concept and methods of FRACTIONS are the way we implement higher accuracy and precision into our practices of measurement.

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

1.- Identify the parts of the display, including the CIRCLE and RECTANGLE image forms of fractions, the DENOMINATOR and NUMERATOR numbers, and their size-control slider bars (left and top).

2.- Use the slider bars of the display to represent the following fractions, and then draw, by hand, on a piece of paper, the resulting images (drawing pages can be obtained and printed from: www://animath.net/drawfracts1.pdf):

The same fraction examples of this activity can be used to help develop mastery in visualizing fractions, by measuring the time taken to obtain each representation on the screen (e.g. multiple students in a computer lab) in the shortest time possible, using the timer located in the blue section ("TIMER ENABLE ") at the bottom of the display : When the yellow button is first pressed, the timer for each student will be set from OFF, to a READY mode, waiting for the screen actions begin and the counting will start as soon as any of the controls of the display are touched in each attempt. The timer will keep counting elapsed seconds during each attempt, and will stop only when the yellow button is pressed again, signaling OFF, or "this attempt is finished". The count will then freeze and can be recorded in a separate log.

The counts of different students can then be compared to determine "the fastest" attempt from a group participant.

The yellow button will then need to be pressed again to make the timer reset ready for a new attempt or case.

QUESTION: What is the smallest part possible in this display (Representing Pieces of One Whole) that can help us be most accurate? one- onehundredth 1/100

PAPER-FOLD FRACTIONS
Have some 4 sheets of blank paper available to fold at your desk in different ways, as requested next.

1.- Use a standard paper sheet and a paper-plate to fold it in equal two/half parts to represent: a) one-half, b) one-quarter, and c) one-eight. d) if folding the one-eights stack in half once again, what would the resulting fractions be?

2.- Can you try folding another page or paper-plate to represent (cleanly and accurately) one-third, or one-fifth? Why is this more difficult?

QUESTION: If we take three equal sized sheets or pieces of paper, then join them all together to become "one whole" thing or amount, could this give us a good representation for one-third fractions of any unit amount? Can we do the same for 5, 7, or any other fractioning that is not multiple of 2? Can we do the same with any other object or thing that has multiples in a group?

COMPARING FRACTIONS
Ca ll the EQUIVALENT REPRESENTATIONS display.
This display allows side-by-side visual comparison of common fractions in rectangular, strip-tile form.

The left part of the display presents a set of common fractions in strip form. The strips can be slid horizontally, and placed near others of interest for comparison. A dotted red-line-and-shade cover is also provided, sliding up and down to highlight comparisons. At the center of the diplay, the images of breakups by 10 and 100 pieces are available together, to highlight their belonging in the uniform "Base-10" or Decimal numeric, or Hindi-Arabic system that has prevailed in counting, measurement, communication and computational activities activities during most of the previous centuries. The fractioning system based on the number 2, Base-2 or Binary system currently prevailing is also illustrated further ahead in this chapter.

The images of a foot, and an inch, and a “batch of apples” strip are also available (right side of the display) for visualization of group partitioning and common length measurement units (feet, inch).

The digital-numeric representation for each of the pieces in each fraction-strip is shown at the top. For example, every one of the 4 equal parts of the green strip is named “1/4”, “one of four”, or “one-fourth.

There are strips for halves (1/2, blue), thirds (1/3, orange), fourths (1/4, green), fifths (1/5, red), sixths (1/6, purple), sevenths (1/7, yellow), eights (1/8, pink), tenths (1/10, white), twelfths (1/12, dark blue), sixteenths (1/16, brown), hundredths (1/100, white).

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

1.- Describe the parts of the display, including the sliding fraction strips, the one-foot ruler, the one-inch, and the batch-of-apples strips. Show how the dotted-line-shade slides up and down to assist comparisons and equivalences across.

2.- Do the following, using the dotted-line-shade:

a) Mark one half (1/2). Which of the other fraction pieces shown can add up to show exactly the same?

1) _____ 2) _____ 3) _____ 4) _____ 5) _____ 6) ______ 7) ______

( 2/4, 3/6, 4/8, 6/12, 8/16, 5/10, 50/100 )

b) Mark one quarter (1/4). Which of the other fraction pieces shown can add up to show exactly the same?

1) ______ 2) ______ 3) ______ 4) ______ ( 2/8, 3/12, 4/16, 25/100 )

c) Mark two thirds (2/3). Which of the other fraction pieces shown can add up to show exactly the same?

1) ______ 2) ______ ( 4/6, 8/12 )

d) Mark three quarters (3/4). Which of the other fraction pieces shown can add up to show exactly the same?

1) ______ 2) ______ 3) ______ 4) ______ ( 6/8, 9/12, 12/16, 75/100 )

e) Mark seven eighths (7/8). Which of the other fraction pieces shown can add up to show exactly the same?

1) ______ ( 14/16 )

3.- Describe the two strips in the “Decimal” box, to the right of the 1/16 fraction strip. These columns are the first two breakdowns used by the decimal number system, which only considers breakdowns and accumulations whose number of unit-parts are multiples of 10. The decimal breakdowns for fractions are therefore in tenths (1/10), one-hundredths (1/100), one-thousandths (1/1000), etc. The representation using exclusively 1/100 unit-fractions is widely used and referred to as “percents”.

4.- Use the red-dotted line across the strips by clicking-and-dragging it, and numerical calculations to show and determine exactly the following:

a) One half (1/2). What are the decimal and percent equivalents of one-half? ( 0.5, 50% )

b) One quarter (1/4). What are the decimal and percent equivalents of one-quarter? ( 0.25, 25% )

c) Two thirds (2/3). What are the decimal and percent equivalents of two-thirds? ( 0.66.., 66.66..% )

d) Three quarters (3/4). What are the decimal and percent equivalents of one-half? ( 0.75, 75% )

e) Seven eights (7/8) . What are the decimal and percent equivalents of seven-eights? ( 0.875, 87.5% )

5.- Describe how the rainbow-colored column on the right side of the display is used to show the number of items that will fall in a fraction of that unit package. For example, if you enter 12 units at the top of the rainbow colored column, 12 units (apples) will show to fill the column, but only 6 of them will fall within a half of the unit package.

6.- Use the display and numerical calculations (multiplication for "fraction of...") to answer the following questions:

a) Approximately how many of 24 units are in one half (1/2, 0.50, or 50%) of the total? ______ ( 12 )

b) Approximately how many of the 24 units are contained in one quarter (1/4, 0.25, or 25%) of the total? ______ ( 6 )

c) Approximately how many of the 24 units are contained in two-thirds (2/3, 0.66…., or 66.6….%) of the total? ______ ( 16 )

d) Approximately how many of the 24 units are contained in three-quarters (3/4, 0.75, or 75%) of the total? ______ ( 18 )

e) Approximately how many of the 24 units are contained in ten-twelfths (10/12, 0.838…., or 83.3….%) of the total? ______ ( 20 )

7.- Focus on the yellow one-foot ruler at the extreme right of the display is available to show the number inches and/or parts of an inch contained in fractions of it. The whole foot has 12 inches.

8.- Answer the following questions:

a) Approximately how many inches are in one half (1/2, 0.50, or 50%) of a foot? ______ ( 6 )

b) Approximately how many inches are in one quarter (1/4, 0.25, or 25%) of a foot? ______ ( 3 )

b) Approximately how many inches are in two-thirds (2/3, 0.66…., or 66.6….%) of a foot? ______ ( 8 )

b) Approximately how many inches are in three-quarters (3/4, 0.75, or 75%) of a foot? ______ ( 9 )

b) Approximately how many inches, or parts of an inch are in one-sixteenth (1/16, 0.0625, or 6.25%) of a foot? ______ ( ¾ or 0.75 ).

THE SIZE OF TOOLS
Call the MONEY FRACTIONS display.
Here we can see how the unit of money (Dollar) can be broken down into parts (change) for buying and selling transactions. The common fractions are: "half-dollars", "quarters", "dimes", "nickels", and "pennies). Observe how coins and dollar values can be made (or replaced) by using (equivalent to) other smaller coins. This can be seen by moving around and placing side-by-side the columns of the dollar and fractional coins. When paying prices that are smaller than a dollar, change can be produced for the excess value of the bill or coins.

a) How many quarters make one dollar? 4

b) How many dimes make a dollar? 10

c) How many nickels make a dollar? 20

d) How many pennies is a dollar worth? 100

e) If you buy something costing 75 cents, and you have one dollar, how many quarters would you spend from it? 3 How many pennies would you receive back for change? 25 How many nickels would you spend from your dollar, and how many nickles would you receive for change? 15, 5

THE SIZE OF TOOLS
Call the MEASURING TOOL SIZES display.
This display presents two sets of common nut-and-bolt wrenches, one using the metric, the other using the English measurement system to specify wrench size. The size of each wrench can be verified by using the moveable ruler and small (yellow) strips provided, and visualized better by using an appropriate zoom-in and zoom-out screen magnification.

The display takes the opportunity to help students become familiar with the size markings of typical wrench sets.

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

1.- Observe the functionality of the display, by dragging each the “English Set (inches)” ruler strip, onto the open end of the first ("5/16") wrench, next to it, and the “Metric Set (inches)” ruler strip, onto the open end of the first ("10") wrench, next to it. Use the zoom-in feature, and further positioning of the enhanced image, to observe the size scale for each wrench.

2.- Verify the size of each of the remaining English set (top), and metric set (bottom) wrenches.

MIXING AND MATCHING FRACTIONS
Call the MIXING AND MATCHING FRACTIONS display.
This display contains a set of fraction strips with detachable fraction sections. The ability to produce visual combinations of fractions, mixing and matching as convenient can be used to present visual representations of fraction operations:

1.- Adding is done by stacking pieces on top of other pieces

2.- Subtracting (or "taking away") is done by overlapping and reducing (negative direction or canceling) quantity pieces down from the top height of the amounts affected.

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

1.- Observe the functionality of the display, by bringing one of the 1/2 (blue) strips to the left staging area, and then A) adding 1/4 to it, setting it over its top to 3/4, and B) subtracting 1/4 to the 1/2, overlapping it down to 1/4 (remaining). Use the sliding dotted-line marker to compare with other fraction strips on the right side.

2.- Demonstrate, visually, that 1/2 + 1/4 is approximately the same as 3/4 .

3.- Demonstrate, visually, that 1/2 – 1/4 is approximately the same as 1/4 .

4.- Answer, visually, the first question shown on the upper part of the display: 1/2 + 1/3 + 1/6 = ? (visually seems like = 1)

5.- Answer the second question shown on the upper part of the display: What is the meaning of a common denominator? (a piece size that can exactly replace all other pieces in a common group)

6.- Answer the Third question shown on the upper part of the display: How is the LEAST common denominator (LCD) useful? (we can then use the the least number of common-size pieces for operations with the group)

7.- Use the LCD (6) to show visually that 1/2 + 1/3 + 1/6 is most exactly = 1.

8.- Represent visually, and compute the result for each of the following questions, using the dotted-line marker of the display:

MULTIPLYING FRACTIONS
Call the Fraction of a Number display.

This display produces a visual representation of the operation of multiplication of a fraction by a whole (positive) number. The operation is shown as extracting "a fraction of a TOTAL number..." by breaking it into an equal number of pieces (denominator), and selecting only "some" of them (numerator), to contain the portion/fraction of interest of the total number.

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

Try the following operations:

Call the Pizza and Olives display.

This involves dispersing evenly (same amount) a number of olives throughout a pizza broken down into an even number of fraction pieces. Each olive is dragged in turn to the pizza to fill each and all parts smootly, that is, by equal numbers of olives on each so everyone sharing ot gets the same amount of olive flavor. This is done by placing each new olive in turn over each piece, around and round until they are all exhausted.

The activity allows also for thinking of what will happen when there are no even amount of olives left to disperse, to fill each part of pizza equally with the same amount of olive content. In this situation, each part already has the same number of whole olives, but there is still one or more left, but not enough of whole olives to apportion into each part of the pizza. What can it be done then to strictly give each part the same amount (i.e. taste) by spreading the leftover olives?

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

1.- Select and drag each of the fraction circle shades from the right and place one at a time over the whole pizza. Observe how the the pizza is fractioned for each selection. How many people can be fed equally with each fractioning choice? Which ones give more quantity to each person? Which ones give less? Which choice serves more people? Which serves less people?

2.- Choose the 1/4 fractioning shade, then select the lower green box by the numeric keyboard below, to enter the number 12 for the number of olives available in the group above. The image of the olives will be updated. Click and drag each olive in the group to place them sequentially onto the pizza parts evenly (that is, the same amount of olives for each piece of pizza).

NOTE: Could use initially the "Click to sprinkle" green button and then adjust individually to speed up the distribution of olives on the pizza.

3.- Now choose the 1/3 fractioning shade, then select the lower green box by the numeric keyboard below, to enter the number 11 for the number of olives available in the group above. The image of the olives will be updated. Click and drag each olive in the group to place them sequentially onto the pizza parts evenly (that is, the same amount for each piece of pizza). What can happen to the 2 leftover olives, so we can get an even distribution of them on each and all the pieces of the pizza?

What would be total final the amount of olive ingredient on each of the 3 parts of the pizza?

COUNTING FRACTIONS IN DIFFERENT NUMBER_BASE SYSTEMS
Call the Count and See Decimal Fractions (Base-10) display.

This display features the animated collection of fraction pieces in the decimal system, which considers representing all quantities exclusively by grouping or units by tens, and similarly, breaking down or partitioning units exclusively in numbers of pieces that are multiples (fractions) or sub multiples of 10.

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

1.- Try the following fractional decimal numbers, making sure to stop-suspend the counting periodically (using the "Pace-step" button) to observe current counts at different times during the accumulation:

a) 0.25 b) 0.75 b) 1.32 c) 12.475

Call the FRACTIONS IN BASE-2 display.
This display features the animated collection of fraction pieces in the binary, or base-2 system, which considers representing all quantities exclusively by grouping or units by twos, and similarly, breaking down or partitioning units exclusively in numbers of pieces that are multiples (fractions) or sub multiples of 2. Because only three "binary digits" are used to the right of the “binary point” when representing fractional quantities, this permits a breakdown of units up to one-eight, and therefore the format is called “binary point eight”. This format has been used extensively for remote measurement and communication of quantity (telemetry), in automated industrial operations10.

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

1.- Use the Fractions in Base 2 display to count to the following fractional decimal numbers. Make sure to stop-suspend the counting periodically (using the "Pace-step" button) to observe current counts at different times during the accumulation:

a) 0.25 b) 0.75 b) 1.32 c) 12.475

1.- Write a response to the following question:

How is the binary point eight format accuracy limited?

Use the BINARY BASE-2 NUMBER KEYPAD or the Try it! displays to practice expressing various numbers with fractions in the Binary sy stem.
This display allows practice visualizing the effect of collecting place value integer and fraction pieces as they are called by enabling or disabling each "bit" light/button being pressed or unpressed.

Call the FRACTIONS 4,000 years ago display.
Th is display features an ancient Babylonian clay tablet presenting the relative size of the sides and diagonal of (2) right triangles. using the Base-60 number system prevalent at that time and place in history. One group of numbers (upper) is for triangles is of size 1 and the other (lower)is 30 units. Small, enclosed and sealed clay tablets were used in ancient times to cast formal deeds among people. The display therefore illustrates how base-number uniform fractioning with increasingly smaller uniform breakdown pieces was already in use since the beginning of our civilizations.

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

1.- Observe the square shape engraved in the inch-wide clay tablet and the sequence of multiples-of-60 place-value positions both for integers and fractions, below it. Observe now the small clusters of symbols around the diagonal in the middle of the square, representing the number of integer and fraction pieces (1/60, 1/3600, ... ) for each of the two corresponding diagonal sizes (use the button for each given size). Question: How many 3600'ths are needed to represent the size of the diagonal of a square of unit (side) 30 units?

a) 1 b) 25 b) 35 c) 60

2.- Looking at the "REFERENCE" section on the bottom-left section of the display, we can see the decimal calculation of the diagonals (hypothenuse) using the Pythagorean Theorem (stated some 2000 years later). Compare the value obtained in this section with the one obtained from using the fraction numbers engraved on the tablet (right section): How different are they?

3.- For the sides of 30 units, notice that there is no number of 1/216000 fractions requested for the diagonal, which appears to be the cause of a greater difference between the two calculation forms.

Using the up/down fraction number controls (red arrow-tips) for the 1/216000'ths fractions determine the possible additional number in the sequence that could have been engraved on the tablet to show the greatest accuracy yet, using 1/216000 slices. (4)

Call the ANALOG to DIGITAL (binary) electrical meter conversion display.
This display illustrates the process of converting an analog voltage measurement value, to a digitally encoded one, using the "successive approximation" method involving binary fraction parts.

The analog value is shown on the left, within a 10 Volt range controlled by a knob-slider and displayed in a swinging-needle scale. The digital binary value is constructed by successively adding and comparing reference half-range fractions (from left to right) to differences, keeping each fraction part if not reaching, or skipping it if they overflow as they approach the target value.

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

1.- Set (slider) the analog voltage on the left exactly to 5 volts (half range). What would be the digital value needed to match the 5 volts?

2.- Set (slider) the analog voltage on the left exactly to 9.5 volts. What would be the digital value needed to match the 9.5 volts?

3.- Set (slider) the analog voltage on the left exactly to 6.24 volts. What would be the digital value needed to match the 6.24 volts?

3.- Set (slider) the analog voltage on the left exactly to 1.5 volts. What would be the digital value needed to match the 1.5 volts?

Call the INTRABASES CONVERTER display.
This display allows comparisons of representation of numbers and fractions involving various different base systems. A decimal target number at the top and two optional number base representations are entered by click-selecting a number entry window and keypadding it correspondingly.

The analog value is shown on the left, within a 10 Volt range value controlled by a knob-slider and displayed in a swinging-needle scale. The digital binary value is constructed by successively adding and comparing reference half-range fractions (from left to right) to differences, keeping each fraction part if not reaching, or skipping it if they overflow as they approach the target value.

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

1.- Enter the number 20.5 base-10 and the bases 2 and 16 to observe (CONVERT button) the binary (base 2) and hexadecimal (base 16) equivalents.

2.- Enter the number 100.75 base-10 and keep the bases 2 and 16 to observe the binary (2) and hexadecimal (16) equivalents.

3.- Enter the number 42.426407 base-10 and bases 2 and 60 to observe the binary (2) and Old-Babylonian (16) equivalents.

Can you recall the place value numbers scribbled (for side 30) on the 4,000 year old Babylonian clay tablet ( FRACTIONS 4,000 years ago ) for the diagonal of squares? (42, 25, 35, +approx 4) - Comment: Isn't that ancient clay tablet an amazing sample of the great mathematical abilities of our "primitive" ancestors, 4,000 years ago!!?

Question: What kinds of nice things we do today (e.g. regarding accuracy and precision that help us design and produce the nice things we have now such as microcircuits, sky-scrappers, medical diagnosis and treatment, mega-cities and supercomputers, conduct atomic and space exploration, etc.) that allow us to advance our science, industry commerce and research using what we learn here in our mathematics classes?