Kindergarten, Number and Operations in Base Ten

 

Opinion – CCM Language Matters

CC Math asks children in Grade K to learn a lot, and language, not normally associated with our youngest students, is used to group specific skills. While Grade-K students need to learn the skills, they may neither be ready for nor need to be aware of the more general categories.  However, since they are being set up for further learning in these areas, it makes sense to me that we, their parents, became familiar with those terms because first, our kids just might ask us about them and second, we will begin to see the big picture ourselves, that math topics such as counting, arithmetic, algebra, geometry, etc., are closely related. If we show no fear, our children are less likely to be afraid.

Repetition has an important place in education, so let’s remind ourselves of a couple of definitions that we have already encountered. Keep in mind that we adults want to know what these terms mean; our K-level students are unlikely to know or care.

The cardinality of a set is the number of elements in that set.  The set {dog, cat, chair} has a cardinality of 3.  The set {3, 6, 9, 12} has a cardinality of 4.

An algebra consists of symbols and the operations indicated by those symbols.  For example, a simple algebra might consist of {+, -, *, /, superscript, =}

Where  A+B = the sum of A and B

A-B = the difference between A and B,

A*B = the product of A and B

A/B = the quotient of A divided by B

A^B = A raised to the power B (e.g., 3² = 3 squared = 9) and

A and B are whole numbers (e.g., 1, 0, -7, 22, etc.).

Here are two new terms that will help us think about numbers but will probably elude all but the most precocious K-Level students, base and place. The base of a number system refers to the number of symbols in that system.  A decimal system uses ten symbols {0, 1, …, 9}; an octal number system uses eight symbols {0, 1, …, 7}; and a binary number system uses two symbols {0, 1}.  If we write those symbols in a string, place refers to the position of each symbol.  A number written in any base assigns a weight of one to rightmost place, the base itself to the place second from the right, the base squared to the place third from the right, etc.  Confused? See the examples in today’s lesson below.

CCM Lesson of the Day: Number and Operations in Base Ten (K.NBT)

Work with numbers 11–19 to gain foundations for place value.

Compose and decompose numbers from 11 to 19 into ten ones and some further ones, e.g., by using objects or drawings, and record each composition or decomposition by a drawing or equation (e.g., 18 = 10 + 8); understand that these numbers are composed of ten ones and one, two, three, four, five, six, seven, eight, or nine ones.

Comment. I am fairly sure that this makes sense to you, but it is a sophisticated concept. Instead of having to mark off eighteen 1s, I can simply write 18 (in base ten) to represent eighteen items. Therefore, I would not expect every K-level student to grasp it immediately. Moreover, while the number “eighteen” does capture, in a single word, the combination of eight and ten, numbers like 11 and 12, which we refer to as “eleven” and “twelve” are more obscure. Once again, K-level students are being asked to learn counting and the concepts of base and place at the same time. Repetition and memorization must be part of this learning process. I believe that it is worth allowing children the time to learn these initial lessons because they will be important throughout their math education. I suggest that you not be impatient to move ahead before they appear comfortable with these ideas.

Definitions

Base: The base of a system refers to the number of symbols in that system.

Place: The place of symbol refers to its position (in the string of symbols which comprise a number)

Decimal: A decimal system has base 10 and uses the symbols 0,1,2,3,4,5,6,7,8,9.

Examples

In the examples which follow, a subscript is used to indicate the base.

Example 1: 234₁₀ = 2 * 10² + 3 * 10 + 4 = 200 + 30 + 4

Example 2: 234₈  = 2 * 8² + 3 * 8 + 4 = 128₁₀ + 24₁₀ + 4₁₀ =  156₁₀

Example 3: 10110₂ = 1 * 2⁴ + 0 * 2³ + 1 * 2² + 1 * 2¹ + 0 * 2⁰

                                = 16₁₀ + 4₁₀ + 2₁₀ = 22₁₀

                                    = 2 * 8 + 4 + 2 = 20₈ + 6₈ = 26₈