This disclosure is protected under United States and International Copyright Laws.© 2005 Huong Nguyen. All Rights Reserved. A portion of the disclosure of this patent document contains material which is subject to copyright protection. The copyright owner has no objection to the facsimile reproduction by anyone of the patent document or the patent disclosure, as it appears in the Patent and Trademark Office patent file or records, but otherwise reserves all copyright rights whatsoever.
The present invention relates to a method of teaching mathematics, and in particular, to a method of teaching mathematics using visual aids.
Current methods of teaching mathematics using manipulatives may not be effective in providing a concrete, simple, and in-depth learning experience that promotes a successful rate of learning among school children. Using the typical manipulative techniques, students may have problems recognizing numbers, constructing numbers, adding, subtracting, etc. The explaining process frequently is so complicated that children get lost and may not remember the process the next time they are asked to recall the information. Because children have to rely heavily on memorized mathematical facts and road map memorization, their performance on annual academic tests have been relatively low. Currently, the United States is ranked 42nd amongst the world in mathematics.
Generally, traditional teaching methods do not provide stimulating and engaging experiences in learning mathematical concepts.
Disclosed herein are systems and methods for teaching mathematics using the Numero Cubes and/or Whole Number System. In one embodiment, the invention provides an effective and logical solution to teaching mathematics. Students become engaged and active thinkers in the process of seeking out solutions to their given challenging math problems. In one particular embodiment, the Math Logic teaching method may promote self-esteem, resiliency, and teamwork.
In another embodiment, the invention utilizes the base 10 number system. Students may touch, examine, count, compare numbers, develop mathematical patterns, add, multiply, divide, and/or perform simple fractions visually. Students may actively engage in concrete and sequential learning experiences that help them retain information in their short- and long-term memory. Students may think, analyze, evaluate, and construct their solutions to given challenging math problems. The Numero Cubes and/or Whole Number system may offer visual tools to help students accomplish mathematical goals and learning objectives. For example, students may be asked to analyze the number one hundred. In one embodiment, one hundred may be assembled from 10 ten units using two rectangular bars of magnets. These magnets may hold the 10 ten units together. Students may collaborate to create a one hundred unit or may work independently. This may provide an integration of math (i.e. the numbers) and science (i.e. the magnets) and students may learn how science can be used to solve a math problem.
In another embodiment, the invention may permit students to build and/or take apart their creation. For example in subtraction, students may be asked to remove a number of cubes from a peg. The answer to the subtraction problem is what remains on the peg. In another embodiment, students may remove the top peg off of 1 ten unit to have 10 individual cubes when they need to borrow 1 ten. Students may also remove the magnetic bars to have 10 tens when they need to borrow 1 hundred. Therefore, learning may become a visual and/or logical task.
Math Logic comprises an inductive teaching method that may provide students (not shown) with a learning tool to learn mathematics successfully and effectively using cubes 120, pegs 100, placement panel 150, and/or dividers 125. One will appreciate however, that other suitable embodiments of the invention may vary the sizes and/or shapes of the individual components. For example, the pegs 100 may comprise other digit holders, including fasteners and/or security devices such as pins and/or plugs. The pegs 100 may further comprise adhesive or attractive patches or plates, such as magnets and/or Velcro®. In other embodiments, the cubes 120 may comprise any suitable geometric shape, including cube-shaped, rectangular and/or cylindrical.
In another embodiment, students may be able to compare numbers and/or predict a pattern of numbers. This may allow students to perform addition and/or subtraction. Students may be engaged in authentic learning experiences through constructing, building, analyzing, and/or evaluating their processes in finding solutions to challenging and difficult math problems. Generally, young children's′ textbooks and counting books introduce the number 1 as the first number, not zero. In one embodiment of the present invention, zero is the first number of the whole number system. Under the typical method of learning, children may not understand the concept of the number 0 and may not comprehend what zero means as a place holder in numbers such as 10, 100, 1000, etc.
An embodiment of the present invention may show students and young children the importance of the number zero. In one embodiment, zero is the first number of the base 10 whole number system. The peg 100 may be black and each cube 120 white, although any suitably contrasting colors may be applied. Where there is no cube 120 placed on the peg 100, children may clearly visualize the number zero. In one embodiment, zero indicates that there is no cube on the peg. In another embodiment, the base 10 number system comprises 10 basic numbers, 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.
Other numbers may be formed based on basic 10 numbers, 0 through 9. In one embodiment, the one digit number reaches 9 and returns back to 0, thereby forming a pattern. In another embodiment, a second digit, ten, for example, is formed.
Using an embodiment of the present invention, students may compare numbers. Students may compare numbers using the ‘>’ sign. For example, to compare 34 and 43 students may construct and/or visualize 3 tens and 4 ones in 34 and 4 tens and 3 ones in 43.
Still referring to
Students may separate a given number into a sum.
Cubes 120 and dividers 125 may provide students with a systematic method of separating a given number into different sums, although other suitable configurations for separating may be applicable including rope, twine, and/or wire (not shown). By doing this, students may begin to see that a number may be the sum of several combinations of numbers. For example, the number 8 may be a sum of different combinations of 1 through 7. In one embodiment, students may be taught summation before learning addition. In other embodiments, students may be taught summation and addition simultaneously. This process may decrease the time students may take in learning how to add. For example, students may learn that 8=1+7=7+1=2+6=6+2=3+5=5+3. This may comprise a commutative property that students may learn later in algebra.
Students may learn to find consecutive odd and even numbers using cubes 120 and dividers. Students may work in groups, discuss and collaborate with each other over the meaning of even numbers and/or how to find the next several consecutive even numbers. Even numbers may be explored beginning with the number zero. Students may determine the next succeeding even number by using cubes 120. Students may be asked to determine what these numbers have in common. Students may be asked to find the next even number. By being asked directed questions, students may discover a pattern in determining even numbers. Students may direct questions at one another or explore questions cooperatively.
Thus, to find the next even or odd number, students may add two cubes 120 to the current number of cubes 120 already on the shaft 115. Students may start with a first even number, add two cubes 120 to the shaft 115 and determine the next even consecutive number, although one will appreciate starting at any even number and working up or down from there. Students may analyze the differences and similarities between even and odd numbers.
Students may explore number theory before moving onto addition. Students may learn to understand number structure and how to manipulate digits before adding and subtracting. Students may think, analyze, compare and evaluate their work and their learning may become authentic and engaging.
Using an embodiment of the present invention, students may learn addition. By using the cubes 120, students may visualize the process of adding numbers.
Students may learn subtraction by visualizing the subtraction concept using an embodiment of the present invention. To perform the subtraction, students may remove a number of specified cubes 120 from the existing number of cubes 120 on a shaft 115. The remaining number of cubes 120 left on the shaft 115 is the resultant number.
Traditional methods of teaching and learning multiplication require students to memorize math facts. Students who do not learn basic multiplication math facts may not learn advance multiplication, division, fraction, and/or other advanced mathematical concepts. Math Logic teaching method may provide students with a method to explore multiplication without having to recite the multiplication table. Students may determine a product of a multiplication equation in terms of connection between multiplication and addition. Using cubes 120 and shafts 115, students may learn why the product of 5×0=0 and why 4×3=3×4. The student's addition skill may be reinforced during the process of finding each product.
For example, students may be asked to analyze and/or write down a mathematical observation. Their job may be to write some kind of equation to express what they see visually and how they may connect what they see to addition. Students may have to answer questions while going through the analysis phase. For example, they may ask themselves: Is there a pattern here? How many total cubes 120 do I have? How may I write an equation to express the given information? How may I write an equation to show some form of addition here? A sample scenario is illustrated in
Math Logic may provide students and young children an effective method of finding the answer for each product without memorizing the multiplication tables. Students may learn to add in groups. Young children's addition skill may be reinforced as they try to find the answer for each product, as illustrated in
Similarly, current methods of teaching and learning division require rote memorization. Students who do not learn these math facts in elementary school may struggle with more advanced mathematical concepts. The Math Logic teaching method may provide students with concrete examples and/or algorithm to perform division. For example, students may be asked to divide 8 by 2. Students may be asked to determine the following equation: 8÷2. Students may be asked to determine the following equation: 8/2. As illustrated in FIGS. 22 and 23A-23C, division is the reverse process of multiplication. Students may split a given item of the same kind into groups with the same number of items in each group. To determine the answer, students may have to determine how many cubes 120 in each group.
The present embodiment may be taught to young children beginning at approximately 2½ or 3 years of age but may be appropriate to alternative types of students of any age, including elementary students, English as a second-language students, and/or students with mental disabilities. Students may begin learning numbers using the cubes 120. Students may learn a base 10 whole number system logically and sequentially. They may learn that the number zero is one of the most important numbers of the number system. Students may learn that numbers may be built and constructed from the 10 basic numbers, 0 to 9. The present invention may enable students to compare numbers or to find numbers that precede and/or follow a given number using cubes 120 and shafts 115. Numero Cubes and/or Math Logic may provide students with a method of learning mathematics that is relatively easy, simple, logical, systematic, and accurate. The present embodiment may be taught by a teacher, an instructor, a parent, a sibling, a tutor, and/or by peers. Further, embodiments may be incorporated into a computer software program or written publication. For example, a 3D Numero Cube video game illustrating the principles of the Numero Cube system above could be used to accomplish some of the same purposes. This might be especially helpful for students with motor difficulties or handicaps.
The system 10 component kit parts provide for student manipulation and number construction exercises that improve the student's ability to learn several mathematical and algebraic concepts. By constructing number representations, the students learn place value, number decomposition, counting principles related to odd and even numbers, number comparing, addition, subtraction, multiplication and factoring, division, and solving for unknowns utilizing pre-algebraic equations.
The Numero Cube System 10 provides an innovative math teaching tool recommended for preschool to fourth grade. It can also be adapted to the needs of special education students and homeschoolers. The system is simple, friendly, and versatile. The pieces of this kit are easy to handle even for young children. With these bright color cubes, both teachers and parents can successfully provide their students with a solid foundation of basic math concepts such as place value, expanded form, sequences, arithmetic, and factorization. The Numero Cube System 10 uses a hands-on approach based on first constructing and building, and then analyzing and evaluating to find solutions to math problems. Through this process, children develop critical thinking skills, and learn to visually justify their answers and check their own work. Learning with the components of the Numero Cubes System 10 is enjoyable, and engenders an early enthusiasm and appreciation for math. The system 10 provides teaching tools that can be used with any math curriculum. It is aligned with the National Council of Teachers of Mathematics (NCTM) and meets the Washington state math standards. The system 10 is designed to be effective with both individuals and groups, and allow students to work at their own rates in a multi-level classroom.
Discussed below are exercises to help students master the concepts presented. The exercises presented for one child or student can be applied to a group of children or students and vice versa. Many of the exercises provided by the system 10 can be adjusted to fit the needs of both the young child or special education students and the older more advanced learner. At any level, it is recommended that the teacher first guide the student through an example problem and to allow the student ample time to experiment and arrive at the solution on his or her own endeavors. It is suggested to allow the child to wait in order to understand each concept before going on to the next concept. It is good to provide positive feedback while the student endeavors with each exercise of the Numero Cube System 10.
All work may be done on top of the place value mat 150, or nearby the place value mat 150 when supported in a vertical position or other position in view of the students. The shaft 115 is also referred to as a stick 115, or a peg 115, or cube holder 115 in which said shaft, stick, peg, or cube holder 115 includes the base 100 or 300 to which cube holder 115 is inserted to allow upright positioning on horizontal surfaces or when place on the place value mat 150 that is similarly overlaid upon a horizontal surface. The base 100 or 300 may be fitted with ferrous metal plates responsive to magnetic forces. The place value mat 150 shows a ones place column, a tens place column, and a hundreds place column. Representation of single-digit numbers zero to nine (0-9) is possible in which a zero is represented by an empty cube holder 115 which is insertable in the base 100/300 and neither holds Numero Cubes 120 nor 220. Whereas numbers 1-9 have from one to nine Numero Cubes 120 or 220 held by the shaft, stick, or peg 115 that is already inserted into the base 100 or 300. That is, each yellow-colored cube 120/220 denotes a value of 1 or unity. To form the numbers from 1 to 9, the appropriate numbers of cubes 120/220 are slidably engagable with the shaft/stick/peg/cube holder 115. Thereafter, an uprightly positioned cube holder 115 is placed on the value mat 150 in the ones (cubes) column.
Exercises may be varied. For example, a series of Numero Cube representations can be created showing the numbers 0 to 9. A game can be played to see how fast the child can name the number. In another exercise, a child is given cube holder 115 and 9 cubes 120. One child or student may build a number between 0 and 9, and another to say the number. Each child or student can write the number down in both numeric and alphabetic form (e.g. 4, four).
In another exercise a list of numbers between 0 and 9 in numeric and alphabetic form is utilized. The students can be asked to use the Numero Cubes 120, cube holder 115, and the place value mat 150 (not shown) to construct each given number. Other student groups can check for correct answers.
In alternate embodiments of the system 10, the place value mat 150 may include a thousandths place column and/or columns beyond the thousandths place. Correspondingly, differently shaped and/or colored Numero Cubes 120 can designate number groupings in the tens, hundreds, thousands and beyond. Similarly, the hundreds tray 340 with magnet plate can be differently shaped and/or colored and used to represent a thousandths tray, or alternatively, a ten-thousandths tray to correspondingly house cube holder 115 to which Numero Cubes 120 having values other than unity are slidaby placed.
Consecutive numbers after 10 are represented by continuing to add cubes one-by-one into the ones column and transferring the cube holders 120 into the tens column as soon as they are filled with ten cubes 120 and a tens cap 350.
Suggested exercises for students include building the numbers from 0 to 19 on the mat 150, in which the students are encouraged to count aloud at the same time. The student may be encouraged to notice that the pattern of the ones column repeats from 0 to 9.
The same number in the expanded form would be written as the sum of the total number of cubes in each column.
Students may perform exercises to build numbers from 0 to 19 on the place value mat 150, counting aloud at the same time. The students may be encouraged to notice the pattern of the ones column as it repeats from 0 to 9.
The students obtain the kit components of the Numero Cube System 10, such as the mat 150, some cube holders 115, and an arbitrary number of loose cubes 120. Younger students, such as small children, may prefer to perform manipulations with a fewer number of the cubes 120. The students can be reminded that when a cube 120 holder 115 is filled with 10 cubes 120, a tens cap 350 is placed on the top of the shaft portion of the peg 115 and the whole peg assembly is transferred into the tens column on the card 150. Likewise, if there are ten sticks or pegs 115 are located in the tens column, a tray 340 is acquired and placed within the hundreds column. After the number has been successfully constructed on top of the mat, have the students say the number of digits, saying, and writing each digit in the appropriate column on the mat. For example, get the students to construct the number 95 using the Numero Cubes, the cube holders or peg 115, and the mat 150. After students are done with forming number 95 using the Numero Cube System, they may say out loud, “95 is a 2 digit number, 9 tens (90 cubes) and 5 ones (5 cubes).”
In another exercise each student group may be asked to construct the smallest and the largest two-digit numbers from 4, 2, 6, 9, and 3 digits. Each digit is used only once. Determine whether the students can construct a number that is 10 larger than the smallest number and 10 smaller than the largest number using the Numero Cubes 120. Encourage the students to verify their results. In another student group a fairly large number of cubes 120 that are under 100 may be assembled. Students can estimate the total number of cubes 120 that they manipulate and ascertain whether they can find out the exact number of cubes 120 they possess and compare their results. Representations of the above exercises are schematically illustrated in the following figures.
Counting whole numbers with Numero Cubes 120 may progress from right to left. Constructing numbers, on the other hand, can progress from left to right. When constructing a number, let's say 235, first use a dry eraser marker to write a “2” in the hundreds column, a “3” in the tens column, and a “5” in the ones column.
This shows that in the number 235, there are 2 hundreds, 3 tens, and 5 ones. Then fill up 2 trays and place them in the hundreds column. Next build 3 sticks and place them in the tens column. Lastly, slide 5 cubes onto a cube holder and place it in the ones column. Exercises may include writing a 3-digit number down on the mat, with its digits in the proper columns. Have the students say how many trays 340, sticks 115, and Cubes 120 would be needed to construct the number. After building the number on top of the mat, ask the students to say the value of each digit (the number of cubes) in each column. This is good practice for learning expanded form.
Other exercises provide for giving groups of 4 students over 100 cubes 120 and ask them to construct the number using all of the cubes 120. Students may be reminded in the event that when a cube holder is filled with 10 cubes, a tens cap is installed on top, and the whole stick is be transferred into the tens column. Likewise, if there are ten full sticks 115 in the tens column, a tray 340 is acquired and the 10 full stick 115 each having ten cubes 120 are placed within the tray 340 and the filled tray 340 transferred into the hundreds column of the place value mat 150. After the number has been constructed successfully on the top of the mat 150, the students may say or recite the number of digits, vocalizing, and writing each digit in the appropriate column. For example, the number 240 would look like this on the place value chart. The student would say, “3-digit number, 2 hundreds (200), 4 tens (40), and 0 ones (0 cubes)”.
In another example a student draws a number down on the mat 150, with its digits in the proper columns.
The student may also say how many trays 340, how many sticks 115, and how many cubes 120/220 would be needed to construct the number. After building the number on the top of the mat 150, the student is encouraged to say or speak the value of each digit (the number of cubes 120/220) in each column. This is good practice for learning expanded form. In yet other exercise students acquire a mat 150 with 1 tray 340, 3 sticks 115, and 3 cubes 120/220 on the Mat 150. Ask the student how to use the existing number to build a new number 150. This provides the student further challenge in developing the critical thinking skill involved in solving a more challenging math problem.
Exercises: The students may be queried to construct each number below using the Numero Cubes 120. Students can write down the standard form of the number on the place value mat 150, with the digits in the proper columns. Lastly, the students are encouraged to write the expanded form on a separate sheet of paper as the sum of the number of cubes in each column.
The same number in the expanded form would be written as the sum of the total number of cubes in each column. Thus, 124 cubes=100 cubes+20 cubes+4 cubes. per below.
The students are discouraged from writing on the value placemat 150 as it could me a another other than 124, for example it could be construed that 10204 is the number that arises from the combination 100(100)+20(10)+4(1). The students are made aware that each column cannot hold more than nine items each.
The students can practice using the components of the Numero Cube System 10 to visualize the following numbers:
The students write down the standard form of the number positioned on or visually near the place mat 150, with the digits placed within or framed in the proper place columns. Then the students write the expanded form on a separate sheet as the sum of the cubes 120 in each place value column.
8=1+7=7+1; 8=2+6=6+2; 8=3+5=5+3
In so doing, the students learn that the order of the parts does not matter, i.e., 8 can be written as either 1+7 or 7+1, or 2+6 or 6+2, etc. The student can further view or interpret the wedged shaped number separator 125 as a translation to a plus (+) sign, i.e., the number separator functions as a plus sign in addition operations.
In accord with the illustrations in
8=1+1+6=6+1+1; 8=4+2+2=2+2+4
The students are encouraged to place two separators 125 to create more patterns with three number sums for the Number 8. Students similarly write down equations on paper with the number separator 125 visualized to function as a plus (+) sign and instructed to notice any patterns that develop with the addends. The commutative property is shown here. By moving the order of the number, each sum results remain the same, that is, a value of 8.
To count forward is to say a sequence out loudly, starting with a chosen number, and then adding a fixed number to get to the next term in the sequence. Counting backwards using the same principle, except that a fixed number is subtracted each time to get to the next term. In this counting forward exercise the students are reminded to use the tens cap 350 and the move the completed stick 115 in to the tens column on the value mat 150 as soon as there are 10 cubes 120 on the cube holder 115. The students are encouraged to see how high they can count.
From this exercise the student learns that all odd numbers' ones digits end with 1, 3, 5, 7, and 9. When students produce an odd number, the one's digit obtained is one of those numbers. The student realizes that any number can be chosen for each of the other digits (i.e., 211, 305, 477).
Other exercises include having the student count out loud using the following rules, and simultaneously adding or taking away to construct the corresponding Numero Cube structure for each tem of the sequence:
Another exercise provides for querying the students to make up their own sequence, noting the starting pint and the counting rule. The students can challenge other students to construct the Numero Cube representations of the other student's sequences. Conversely, the students are encouraged to build sequences of their own using Numero Cubes and to see if they can write down the starting value and applying the counting rule for each one.
In yet another exercise the students build sequences for the following starting values and counting rules.
Students can be queried whether the sequences generate odd or even numbers and to develop a rule about the occurrence. Students note or observe the ones column each time a new number is generated, and the fact that each of the above counting rules is a multiple of two.
Students note that the largest place value digit is the same in both numbers, such as in the numbers 198 and 107. Then the next highest place value digit is considered; in this case, 9 tens would be compared to 0 tens. Thus 198 is larger than 107, and the ones place digit does not need to be considered. If there are the same numbers of hundreds and tens in each number, then the ones digits is compared.
Using the components of the Numero Cube System 10 allows construction of numbers that visually reinforces the concept of the value of each item in a place value column of the mat 150 and explains why the place values need to be considered in order from the highest to the lowest. Even though 39 has 2 more tens and 6 more ones in the tens and ones columns than 213 (a total difference of 26 cubes 120), 213 has 2 hundreds (2 trays 340) while 39 has 0 hundreds (no trays 340 mean 0 hundreds). Hence, 213 is larger than 39.
Suggested exercises to compare number pairs for students to visually construct and demonstrate are listed below:
As the students visually compare the numbers above, they can be encouraged to consider comparing the numbers of sticks 115 in each construction to determine which number is larger. If there is no difference, the students can compare the number of cubes 120 in the ones column to find the larger number.
The students can be partitioned into groups, say one to four in a group, and instructed to construct larger numbers, say 120, 210, 126, and 216. The students then place their Numero Cube constructions under the place value column headings of the same mat 150. The last two constructions will be off the mat 150. Each student group is instructed to write down the digits of their constructed numbers onto a table of place value columns, as indicated below:
Students can be encouraged to discuss the algorithm behind ordering these numbers by size from largest to smallest.
Other exercises may be cumulative:
1. Students can construct each number using the Numero Cubes 120/220. As they proceed with number construction using it components, learning is enhanced as they write down both the standard form and the expanded form of the number.
a. 102 b. 97 c. 119 d. 135 e. 228 f. 163
2. Students may also construct both given numbers in each problem using the Numero Cube to compare the numbers. Use “>, <, or =” to indicate the correctly relationship between the numbers.
In addition the terms adding, combining the terms, finding the total, and finding the sum have the same meaning. These terms mean combining things into one set and finding out how many there are in this set. Addition without regrouping simply involves combining the digits of the addends in each place vale column. This concept can easily be expanded to three digit numbers, though students conveniently learn with simple addition of smaller sums first before progressing to larger numbers.
As shown in
Use the Numero Cube System 10 to find each sum. Students may be encouraged to ascertain and explain whether patterns occur and repeat.
Exercise 1: Use the Numero Cube System 10 to find each sum. Are there patterns here that you can find? Explain.
Exercise 2: Use the Numero Cube System 10 to find each sum. Find all patterns.
Exercise 2: Students can use the Numero Cube System 10 to find each sum and any and all patterns. Students can also be encourage to explain step-by-step each pattern that appears from the following numbers:
Students can evaluate whether special patterns exist below.
Exercise 3: Similarly, the Numero Cube System 10 can be utilized by the student to find each sum and any special patterns arising within the following numbers:
Example: Take away 15 from 18.
CHECK: Use addition to make sure the subtraction process was done correctly. Simply put back the cubes that were removed to obtain the original number.
15+3=18
Exercise 1: Students can use the Numero Cube System 10 to find each difference in the number sets below:
Exercise 2: Students can use the Numero Cube System 10 to find each difference in the number sets below:
Beginning with step 1, a student removes the 3 cubes in the ones column first. There are only 2 tens sticks left over. He/she needs to remove 4 more cubes and the subtraction process is done. This step is shown in
For step 2, there are only 2 tens sticks left over. The student needs to remove 4 cubes and the subtraction process is done. In order to do this task, the student has to shift 1 tens stick to the ones column and take the tens cap off. Now the student can remove four cubes from the 10 ones.
20−4=20−4=16
In step 3, the student checks to make sure the subtraction process was done correctly by putting back all the cubes 120 that the student took away. If the student got the total number of cubes to be 23, then the subtraction process was done properly.
The check procedure employs three steps illustrated in
Method 2 includes four steps.
Beginning with step 1, the student removes the tens cap 350 from the two sticks 115 in the tens column and shifts this to the ones column. At this point, there are exactly 1 tens and 13 ones.
In step 2, the student removes 7 cubes form the 10 ones to leave 3 left on the cube holder 115.
In step 3, the student transfers the 3 left over cubes onto the cube holder 115 that already has the original 3 cubes. The empty cube holder is then set aside or discarded. At this juncture there are totally 16 cubes left over comprise 1 tens and 6 ones.
Step 4 employs a check procedure in that all cubes taken out are put back. Upon doing this, the student should notice that if the sum is exactly equal to 23 cubes, the subtraction process was done correctly. The student can determine this from the table below that conveys a complete mathematical subtraction process.
Procedurally, step 4 further includes taking the tens cap 350 is taken from one of the two sticks or pegs 115 and shifted to the ones column. At this point there should be 1 tens and 13 ones. This is equivalent to the calculations shown on the left half of
For step 2, 7 cubes 120 are removed from the 10 ones. There are 3 cubes left over on the cube holder 115. This is equivalent to the calculations shown on the right half of
For step 3, 3 cubes 120 are transferred onto the cube holder 115 that already has 3 cubes. The empty cube holder 115 is set aside or discarded.
Exercise 3: Students can use the Numero Cube System 10 to find each difference in the number sets below:
Exercise 4: Students can use the Numero Cube System 10 to find each difference in the number sets below:
Example: Rewriting the following sum in the product form is
2+2+2+2+2=2×5=10
The above sum also can be rewritten as following:
5+5=5×2=10
Multiplication commutative property:
2×5=5×2=10
Exercise 1: Students can use the Numero Cube System 10 to rewrite each problem below as a product form:
Similarly, in the lower half of this illustration, a single number separator 125 is inserted between two five-cube 120 groups. The addition operation 5+5 is shown to be the multiplicative equivalent of the product of factors 5 and 2, or equivalently expressed as 5×2. Removing the single number separator 125 reveals a stack of ten cubes 120 are shown held on cube holder 115.
The logic exercises arising from the manipulation of the kit components as shown in
Exercise 1: Use the Numero Cube System 10 to find each product and further illustrate how multiplication serves as a shortcut for addition. Follow the example as illustrated in
Exercise 2: Use the Numero Cube System 10 to find large products. Follow example of number consolidation as illustrated in
Similarly, as shown in example 2 in the bottom portion of
Exercise 1: Student can find all factors of each number below to ascertain patterns of prime and composite numbers using the Numero Cube System 10. Algorithms depicted similar to the number sequencing or consolidation depicted in
Exercise 2: Find all factors of these large numbers without using the Numero Cube System 10. Algorithms depicted similar to the number sequencing or consolidation depicted in
Exercise 1 provides a representative series of algebraic terms involving division that will have no remainders. The students may be encouraged to come up with other algebraic terms employing division without remainders.
Exercise 2 provides a representative series of algebraic terms involving division that will have remainder groups. The students may be encouraged to come up with other algebraic terms employing division with remainders.
This describes that the number “n” in the box of the algebraic equation 3+n=9 can be determined. One side has 3 plus the blank box and the other side of the equation is equal to nine. To solve for this unknown “n”, the students start with 9 cubes 120 on the shaft or peg 115. The students can use the number separator 125 to acts as a plus sign and insert it right after the three cubes. The unknown is n=6, the number of cubes 120 located above the number separator 125.
Exercise 1 provides for solving “n” in each equation among a representative example of equations involving addition using the components of system 10.
Exercise 2 provides for solving “n” in each equation among a representative example of equations involving addition using the components of system 10.
In the series above the students can be allowed to explore and solve the problem.
Exercise 1 provides for solving “n” in each equation among a representative example of equations involving addition using the components of system 10.
Exercise 2 provides for solving “n” in each equation among a representative example of equations involving subtraction using the components of system 10.
In the series above the students can be allowed to explore and solve the problem.
The students are encouraged to explore the product equations below and solve for “n” using the components of the Numero Cube System 10.
Exercise 1 below provides a representative example of product equation series for the students to solve for “n” in each equation using the components of system 10
Exercise 4 below provides a representative example addition equation series for the students to solve for “n” in each complex equation using the components of system 10.
Exercise 5 below provides a representative example of subtraction equation series for the students to solve for “n” in each complex equation using the components of system 10.
While the preferred embodiment of the invention has been illustrated and described, as noted above, many changes may be made without departing from the spirit and scope of the invention. Accordingly, the scope of the invention is not limited by the disclosure of the preferred embodiment.
This application is a continuation-in-part of and incorporates by reference in its entirety to U.S. patent application Ser. No. 11/955,315 filed Dec. 12, 2007, a continuation-in-part of U.S. patent application Ser. No. 11/381,964 filed May 5, 2006, now U.S. Pat. No. 7,309,233 that in turn claims the benefit of priority to and incorporates by reference in its entirety U.S. Provisional Application No. 60/678,048 filed May 5, 2005. All patent applications incorporated by reference in their entirety.
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Number | Date | Country | |
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Parent | 11955315 | Dec 2007 | US |
Child | 12942829 | US | |
Parent | 11381964 | May 2006 | US |
Child | 11955315 | US |