Cassie has a wood board that is 3/4 foot long. She wants to cut the board into pieces that are 1/8 foot long. Which expression will help Cassie find how many pieces she can cut?

This question requires division. The expression used to find how many pieces Cassie can cut is:

#3/4 -: 1/8#

A fraction consists of two parts the top number and the bottom one.

Think of the top number as the count and the bottom as the size indicator. So you have:

#("count")/("size indicator")" "vec("proper names")" "("numerator")/("denominator")#

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Consider the action example of dividing 3 into 6.

We can do this division directly because their size indicators are the same. That is, we have: #6/1-:3/1#. People do not normally write it this way but it is correct.

#color(brown)("To directly divide the counts the size indicator must be the same")#

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So we need to see how many of #1/8# will fit into #3/4#

We can not do this directly in one operation as the size indicators are not the same.

#color(green)("Lets make the size indicators (denominators) the same")#

#color(brown)("Multiply by 1 and you do not change the value but")#
#color(brown)("1 comes in many forms.")#

#[3/4color(magenta)(xx1)] -: 1/8" "->" "[3/4color(magenta)(xx2/2)] -:1/8#

#[(3xx2)/(4xx2)]-:1/8" "->" "6/8-:1/8#

Now that the size indicators (denominators) are the same we can just divide the counts (numerators)

#("count")/("size indicator")-> [6/8-:1/8] = 6-:1 =6#

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I could explain more showing the connection between this and the shortcut method but what I have shown you will do for now.

If you wish further explanation leave me a message on https://socratic.org/users/abr-1