What are decimals?

In this lesson, we will learn:

• How to understand decimals using number lines
• How to understand decimals using place values
• How to represent decimals using standard form, expanded form, and written forms
• How to use base ten (block) models to represent decimals

Notes:

• When we look at a number that has multiple digits, each place value is 10 times MORE than the place on its right, and 10 times LESS than the place on its left.
• For example, when we look at the ones place value, we start counting until 10 which starts the tens place value:



• Then, we group ten 10s to get to the next place value of hundreds:



• Continuing, we can then group ten 100s to get to the thousands place value and ten 1000s to get to the ten-thousands place value…etc.


• The same rule applies for decimals! Decimals are place values that are even smaller than the ones place (to the right of the decimal point), and the numbers (place values) are further split into ten parts (divisions).
• This can be shown on a number line, splitting into tenths:

• If we split the number line even further, each tenth can be divided into hundredths and thousandths:

The names of the place values for decimals mirror the names of the whole number place values, but you need to add the “-ths” suffix to the end.

Decimals can be written in either standard form, expanded form, or even written from:

• Standard form shows all the numbers written from left to right, with a decimal point after the ones place
• Ex. 165.407
• Expanded form shows the value of the digit multiplied by how much each place value represents.
• Ex. (1×100) + (6×10) + (5×1) + (4×0.1) + (0×0.01) + (7×0.001)
• Place values that hold a zero can be left out of expanded form
• Written form shows how to write out decimals as words! Name the number of each digit followed by the name of the place value:
• Ex. one hundred sixty-five and four tenths, zero hundredths, and seven thousandths

We can use base ten models (place value blocks) to show decimals too—not only whole numbers

• Our whole numbers will always refer to the ones place value (or greater)


• If our number only represents until the hundredths place, then one whole is one-hundred block:

If our number only represents until the thousandths place, then one whole is one-thousand block: