TRIANGLE COUNT

Here is a very classic math puzzle/problem. Just simply count the number of triangles in each figure. However, many still find it confusing since you have to count all the smallest triangles, all bigger triangles, until the largest triangle. What is confusing in this part is that some triangles may be inside another triangle (e.g. 4 small triangles are inside the bigger triangle). 

This problem requires patience, time and accuracy in counting the triangles. Here are the figures. Count the triangles well. Good luck!

Figure 1

Figure 2

Figure 3

Do you think you correctly counted all the triangles in the figure?

Let me see how good are you at counting! Leave comments here for your answer in each figure.  

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MATH CLOCK

This is my math clock. Use it everyday...

It could remind you of the abundance of mathematics around us - that the world is full of mathematics. A reminder that mathematics can be used by anyone, anywhere and anytime. A reminder that mathematics is important. A reminder that mathematics is necessary and sufficient!

Try to solve each of the mathematical expressions! Good luck!

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MULTIPLYING A NUMBER BY 5

There are a lot of shortcuts and techniques in multiplying numbers, you just have to choose what is easy for you. This time, let us try to multiply numbers by 5. 

If a number is multiplied by 5, you could divide the number by 2 then multiply it by 10. This is easier because when a number is multiplied by 10, just simply append zero(0) to the number. This technique is very effective if the number being multiplied by 5 is divisible by 2.

The reason behind this technique is that 5 is the result when 10 is divided by 2 or 10/2 = 5. It means that if you multiply a number by 5, you can multiply it first by 10 then divide it by 2 or you can divide it by two then multiply by 10. The latter is easier. 

Let us use it in the following examples:

    

      The first thing to do is divide the number by 2. That is

      After that, multiply the result by 10 or simply append zero (0) on the result. That is

      Therefore, 

    

      Divide 186 by 2. 

     Multiply the result by 10.

    Therefore,

Now try these:

1. (56)(5)

2. (102)(5)

3. (258)(5)

4. (364)(5)

5. (273)(5)

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CURVES FORMED FROM STRAIGHT LINES

Series of straight lines could form a curve when the edges are smoothed out. It does not actually form the curve but approximates its form. The curve being formed is usually a parabola - that is why they sometimes call it as a parabolic curve. 

Let us discover how these curves are formed from the straight lines starting with the basics. 

What you need are:

1. paper (you can have a white paper or a colored paper of your choice)

2. pencil, marker or any colored writing material

3. ruler

STEP 1: The basic curve formation starts with two lines intersecting at a point. One is vertical and the other is horizontal. The lines resembles the positive x- and y- axes of the Cartesian plane that bound the first quadrant. 

STEP 2: Put dots or marks on the lines. Make sure that they have equal intervals. You may use the a rule to determine the intervals. Make sure also that there are equal number of dots for the two lines. The dots will be paired so there will be a dot left out if the number of dots are unequal.  

STEP 3: Pair the dots by connecting them using a line. Start with the uppermost dot in the vertical line paired with the leftmost dot in the horizontal line. You may also start with the rightmost dot of the horizontal line paired with the lowermost dot of the horizontal line. Any of these would yield to the same sets of lines. 

STEP 4: Continue with the same process until you paired all of the dots

The final output is this figure. Observe that a curve is approximated by the lines.

You may also use other colors for the set of lines.

You can also try combining the figures formed above like these:   

 You can also form figures like these:

 

The above figures used perpendicular lines. You can make the curve narrower by decreasing the angle being formed by the two lines. 

Aside from the perpendicular lines, you can also use other shapes or polygons like a square or a circle. You can try using a triangle, a pentagon, a hexagon and so on. 

                      

                 

I have tried combining the figures and formed this:

You can be creative in forming figures like this using the curves. Share it here after creating one. Have fun creating your work of art!

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HOW TO DIVIDE A SQUARE INTO FOUR (4) EQUAL PARTS

There are different ways of doing this. I'll show some of the solutions then you'll find the rest.


Solution 1: The easiest way to divide the square into 4 equal parts is to draw three vertical lines or three horizontal lines forming four (4) equal areas. This can be shown in the figure below, where each part is shaded with a different color.  

Solution 2: Another way of dividing it is to draw one vertical and one horizontal line intersecting at the middle part of the square. That is

Solution 3: Another way of dividing it is to draw the diagonals of the square. 

Solution 4: You can also cut the square by a horizontal line or a vertical line, then cut each half into two equal triangles. 

There are still other ways of dividing a square into 4 equal parts. Find them and share them here... comment below.

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CUBE COUNTING

Let us test your spatial ability by cube counting. Just simply count the number of cubes (1x1) in the figure. The figure is a 3-dimensional object and shows only a side view. That means you also need to count the cubes that are hidden, which you can only see in another perspective / view.

Let us start with the basic Rubik's cube. How many cubes are there?

I guess you know the answer already... it is a 3x3x3 cube meaning there are 27 cubes in all.

Now try an irregular figure. Count the number of cubes (1x1) in this figure. 

If you are through with the counting, then write your comment here stating your answer. Good luck!

  

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SQUARING NUMBERS NEAR 100 (Case 2)

Here is the second part of the method of squaring a number near 100.

 Number Near and Greater than 100

Example 1: Find the square of 101.
          step 1. Find the difference of the given number and 100. That is 101 - 100 = 1.
          step 2. Since the number is greater than 100, add the result in step 1 to the given number.

       step 3. Square the result in step 1. The square of 1 is 1.

         step 4. Combine the result in step 2 and 3. The first three digits from the left will be from   

                        the result in step 2 and the other two digits will be from the result in step 3.

                 Since the result in step 3 is a one digit number, add zero (0) to make it a two digit     

                       number. The number 1 becomes 01.

                      Therefore, the square of 101 is 10,201.

           Example 2: What is the square of 109?

                      step 1. Get the difference between the number and 100. That is 109 - 100 = 9.

                      step 2. Add the result in step 1 to the given number. That is 109 + 9 = 118.

                      step 3. Square the result in step 1. The square of 9 is 81.

                      step 4. Combine the results in step 2 and 3. That is 11881. 

                      Therefore, the square of 109 is 11,881.

         Try to find the square of the following numbers:

             1. 103

             2. 105

             3. 106

             4. 107

             5. 108  

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SQUARING NUMBERS NEAR 100 (Case 1)

Last time, I have shared squaring numbers ending in 5. This time let us try to square numbers near 100 - those numbers less than 100 or numbers greater than 100.

Number near and less than 100.




Example 1: Find the square of 98.
      step 1. Find the difference of 100 and the number. That is 100 - 98 = 2.
          step 2. Since the number is less than 100, subtract 2 from the given number.

         step 3. Square the result in step 1. The square of 2 is 4.

         step 4. Combine the result in step 2 and 3. The first two digits from the left will be from   

                      the result in step 2 and the other two digits will be from the result in step 3.

                     Since the result in step 3 is only a one digit number, we need to add 0 to make it a 

                     two digit number. The result which is 4 becomes 04.

       

                     Therefore the square of 98 is 9,604.

    Example 2: Find the square of 91.

         step 1. Find the difference of 100 and the given number. That is 100 - 91 = 9.

         step 2. Subtract 9 from the given number. That is 91 - 9 = 82.

         step 3. Square the result in step 1. That is the square of 9 which is 81.

         step 4. Combine the results in step 2 and 3. The result is 8281.

                      Therefore, the square of 91 is 8,281.

  

   Try to find the square of the following numbers:

     1. 93

     2. 94

     3. 96

     4. 97

     5. 99

                        

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