What are decimals?

Lessons

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:

• Introduction
  Introduction to decimals:
  a)

  What are decimals and how do you read decimals on a number line?
  
  
  b)

  What are decimal place values?
  
  
  c)

  How do we write decimals in standard, expanded, and written forms?
  
  
  d)

  How can we use base ten (block) models to represent decimals?
  
  

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• 1.
  Reading decimals on number lines
  What are the decimal numbers given by each point shown on the number lines below in:
  
  
  a)

  Tenths
  
  
  
  
  
  b)

  Hundredths
  
  
  
  
  
  c)

  Thousandths
  
  
  
  
  

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• 2.
  Naming decimal place values
  Name the place value that is underlined in the numbers below:
  a)

  1.83
  
  
  b)

  7.88
  
  
  c)

  9.801
  
  
  d)

  0.845
  
  
  e)

  83.271
  
  

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• 3.
  Decimal place value word problem
  Take a look at the two times the number "5" appears in this number: 5,124,368.057
  a)

  Name the two place values that the two 5s sit inside?
  
  
  b)

  Compare these two place values; how many times do you have to multiply/divide by 10 to get to the other place value? How many times will you have to group ten into a bigger place value (or the other way, split into ten parts for a smaller place value) to get to the other place value?
  
  

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• 4.
  Converting decimals between standard, expanded, and written forms
  a)

  Rewrite the decimal 0.126 (given in standard form) into

  1. Expanded form and
  2. Written form
  
  
  b)

  Rewrite the expanded decimal of (4×0.1) + (5×0.01) + (3×0.001) into standard form.
  
  
  c)

  Rewrite the written decimal of seven ones, eight tenths, three hundredths, and nine thousandths into standard form.
  
  

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• 5.
  Base ten (block) models for hundredths
  Using base ten models where one whole is represented by a big square (made of 10 tiny squares), answer the following:
  
  
  a)

  What decimal is represented by the picture below?
  
  
  
  
  
  b)

  Draw the blocks you would need to show the decimal 6.29
  
  

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• 6.
  Base ten (block) models for thousandths
  Using base ten models where one whole is represented by a cube (made of 1000 tiny squares), answer the following:
  
  
  a)

  What decimal is represented by the picture below?
  
  
  
  
  
  b)

  Draw the blocks you would need to show the decimal 9.645
  
  

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