Advanced Math

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By now you have probably taught your child all of the basic math essentials: Addition Subtraction, Division and Multiplication. If your child is under 6 and understand any of these concepts well then you are way above the power curve. In general children do not begin learning addition in school until 1st grade and they do not teach multiplication until the end of 2nd grade. Division often doesn’t enter the fold until the second semester of 3rd grade.

Being an expert in addition, multiplication and division concepts not needed for the following lessons but comprehension of them is necessary. To clarify, my son did not understand the concepts of two digit multiplication, long division or how to conduct division with remainders before I taught him these lessons and he still learned them just fine. I advise that you individually assess your child according to his or her abilities and teach according to this.

 

Advanced Math Concepts
  • Decimals
  • Fractions
  • Geometry and Measurement
  • Resources

Base Ten is a method of using blocks to put numbers in their respective places. This helps children to understand the true value of the number and its placement. Usually base ten is used for understanding numbers greater than or equal to one.

base10blocks

Once your child understands base ten and place values greater than one, you can introduce them to place values lower than one. Understanding place value is simple but a critical element to understanding decimals.  Below is an example of place value break down you can begin using to help your child understand placement.

place value

Use place value and base ten as a method to provide a detailed introduction to the placement of decimals. Then consider using worksheets to teach simple addition and subtraction.

Our discussion of Money, for the purposes of this lesson, involves focus on teaching children the value (or denominations) of coins and dollar bills and how it relates to decimals.  The dollar bill and all US coins are variations of terminal values of decimals to the hundredth place. Or more plainly stated their values are simple to understand and remember in decimal form. Using visual formats such as the chart below and worksheets that allow children to calculate money values based on a combination of coins is the best way to memorize and understand them.

coin-value-chart

As you already know, there are many levels to understanding and working with fractions. Identifying them, adding/subtracting, multiplying/dividing and simplifying. Here my intent is to help you find an easy way to introduce them and convert them to decimal form.

Like Decimals, Fractions help us to count ‘part’ of something. Specifically, fractions help us understand parts of a whole (like a pizza) or parts of a group (dividing toys between friends). Fractions consist of a numerator, which represents a section or part, and the denominator, which represents the whole.

To help your child understand fractions, present them visually with well known items that can easily be divided (if you have started instructing them with division and maybe them the ‘fairness expert’ you are already doing this).  Commonly recognized items are the best way to do this.

Use shapes to determine the parts that are shaded compared to the parts that are there.

fractions_shapes

Another way to teach fractions is by using the fraction chart.

fractions

Like Decimals, Fractions help us to identify and present parts of a whole. They hold similarities when we are using them in base ten place value format.  Remind your child that when  she verbally identifies a number in decimal form she describes its place value.

For Example:

  • The number 343.25 in written form is three hundred forty-three and twenty-five hundredths

Similarly when she identifies a fraction she identifies it by its numerator and denominator.

For Example

  • The number 25/100 in written form is twenty five hundredths

These two ideas can easily come together when you want to convert a decimal to a fraction OR when you convert a fraction with a denominator of 10, 100 or 1000 to a decimal.

GRID FORMAT

One way to illustrate this is by using the grid format.

deci_fract_grid

WRITTEN FORMAT

The other way is by using the similarity of the written form.

Decimal Written Form Fraction
0.1 one tenth 1/10
0.01 one hundredth 1/100
0.001 one thousandth 1/1000

Geometry is not a favored subject of many high school students because often the idea of proofs and theorems tend to make very little sense, often overtaking the more important ideas of measurement operations. I will not be attempting to provide instruction on these things. Although ultimately your child will take courses on theorems, these concepts have very little value in the long term or are useless on standardized tests. What is important however is understanding shapes, measurements, and angles, and how to derive the area, volume and perimeter of them.

This Top Ten list from Math Work Sheets Center, helps identify some very important and practical reasons why Geometry is important in your everyday life and even in future development.

Angles represent the 2D planar relationship of a turn of two rays from a certain point called a vertex. But what is important is not so much that your child understand the definition at this point but rather than he or she understands the various types of angles. In school I learned that there were four types of angles: acute, right, obtuse and straight. Today however students also learn the concept of the ‘reflex’ angle, which is an angle greater than 180 degrees but less than 360 degrees.

angles

Types-of-Angles

Since, by now, your child should have learned most of his shapes this should merely be refresher material. You can begin to add more depth by identifying the properties of each shape. For instance, a Square has 4 equal sides, while a rectangle has 2 equal long sides and 2 equal shorter sides.

The mathematical necessity of learning shapes is to perform various functions such as area, volume and perimeter.  These measurements help you determine measurements of the surface, depth, and boundary of an object using a common formula.

  • Polygons are two-dimensional shapes with 3 or more sides, specifically triangles, squares, rectangles, etc. and have a length and width. You can measure these objects using area and perimeter.
  • 3-D Solids on the other hand are 3D shapes, which have a length, width and height. You can measure these objects using volume.
POLYGONS

shapes_posters_1

3-D SOLIDS

Solidfigures

The perimeter is the distance around a planar shape or object.  Perimeter can be used to identify the distance around a 2D object, e.g. Polygons. The best way to describe this to your child is to compare it to running around a track during recess or running around the entire house. To visually express this locally use measuring tape around the door.

To find the perimeters one just needs to add the sides of an object.  The below graphic, while silly illustrate the idea of this concept in ways children can understand easily.

Perry the Perimeter Bug

The area of an object is the measurement of the objects surface or the number of square units inside an object. For most polygons area is measured using the formula length x width or A=l *w. 

Because your child understands multiplication and shapes, it will be easy for him to remember that area simply requires that he multiply one side by another. But to understand the concept, I like to illustrate it using visual aids like the below, which help illustrate what the area actually calculates which is the total number of smaller units which can fit into a larger object.

4by8area

The volume of an object is the measurement of the amount of space it occupies inside or the number of cubic units it can hold. Volume is found by multiply the length x width x height or V=l *w*h.

In order to find the volume of an object it must have three dimensions in order to show depth. Visually the difference can easily be seen in comparison to objects with a 2D surface.

volume_rect

Teaching Advanced Math concepts can be a challenge but with the right resources it can be made easy for any parent. At 7 my son understood all of the above and now that he is 8 we are continuing to reinforce them. Most comprehensive workbooks contain mini-tutorials about the concept being taught and numerous practice problems to reiterate the lesson.

My favorite books so far are below, but I am still exploring.

In addition to these books I tried laminated learning guides. I suggest getting ONE but not both.

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