Multiplying decimals (no standard algorithm)

CCSS Math: 5.NBT.B.7
We'll start with simple problems like 0.9 x 0.2 and build to more complex problems like 3.4 x 6.1.
In this article, you'll learn how to multiply decimals by jumping in and giving it a try without being shown how to do it first.
The problems go from easier to more difficult. Along the way there are examples and explanations to help you out if you get stuck. If you get a little confused, just think of it as a chance to learn!
Let's start by multiplying two tenths together.
Example: 0.9×0.20.9 \times 0.2
Let's convert each number to a fraction:
=0.9×0.2\phantom{=}\blueD{0.9}\times\greenD{0.2}
=910×210=\blueD{\dfrac{9}{10}} \times \greenD{\dfrac{2}{10}}
=9×210×10= \dfrac{\blueD{9} \times \greenD{2}}{\blueD{10} \times \greenD{10}}
=18100=\dfrac{18}{100}
=0.18=0.18

Problem set 1:

Problem 1a
0.1×0.5=0.1 \times 0.5 =
  • Your answer should be
  • an integer, like 66
  • a simplified proper fraction, like 3/53/5
  • a simplified improper fraction, like 7/47/4
  • a mixed number, like 1 3/41\ 3/4
  • an exact decimal, like 0.750.75
  • a multiple of pi, like 12 pi12\ \text{pi} or 2/3 pi2/3\ \text{pi}

Beautiful, let's move on to multiplying whole numbers by tenths and hundredths.
Example: 3×0.63 \times 0.6
There are many ways to solve this problem. Let's take a look at two student solutions.
Student A's solution
I thought in terms of tenths:
=3×0.6\phantom{=}3\times 0.6
=3=3 × 6 \times ~6 tenths
=18= 18 tenths
=1.8=1.8
Student B's solution
I used fraction multiplication:
=3×0.6\phantom{=}\blueD{3}\times\greenD{0.6}
=31×610=\blueD{\dfrac{3}{1}} \times \greenD{\dfrac{6}{10}}
=3×61×10= \dfrac{\blueD{3} \times \greenD{6}}{\blueD{1} \times \greenD{10}}
=1810=\dfrac{18}{10}
=1.8=1.8
The answer
3×0.6=1.83 \times 0.6 = 1.8

Problem set 2:

Problem 2a
0.8×7=0.8 \times 7 =
  • Your answer should be
  • an integer, like 66
  • a simplified proper fraction, like 3/53/5
  • a simplified improper fraction, like 7/47/4
  • a mixed number, like 1 3/41\ 3/4
  • an exact decimal, like 0.750.75
  • a multiple of pi, like 12 pi12\ \text{pi} or 2/3 pi2/3\ \text{pi}

Great, let's finish with a few more challenging problems.
Example: 3.5×0.213.5 \times 0.21
=3.5×0.21\phantom{=}\blueD{3.5}\times\greenD{0.21}
=3510×21100=\blueD{\dfrac{35}{10}} \times \greenD{\dfrac{21}{100}}
=35×2110×100= \dfrac{\blueD{35} \times \greenD{21}}{\blueD{10} \times \greenD{100}}
=7351000=\dfrac{735}{1000}
=0.735=0.735

Problem set 3:

Problem 3a
0.4×0.62=0.4 \times 0.62 =
  • Your answer should be
  • an integer, like 66
  • a simplified proper fraction, like 3/53/5
  • a simplified improper fraction, like 7/47/4
  • a mixed number, like 1 3/41\ 3/4
  • an exact decimal, like 0.750.75
  • a multiple of pi, like 12 pi12\ \text{pi} or 2/3 pi2/3\ \text{pi}