Professor Triangleman posed an interesting question a few weeks back:

If 15/10 is an improper *fraction*, then shouldn’t 1.5 be an improper *decimal*? Or is 1.5 a *mixed *decimal, having more in common with the mixed fraction 1 5/10? Both? Neither?

One definition of decimal:

A fraction whose denominator is a power of ten and whose numerator is expressed by figures placed **to the right of a decimal point**.

Thus, in 1.5, the implied denominator is 10 and the implied numerator is 5, the figure to the right of the decimal point. We read 1.5 as “one and five tenths,” a mixed decimal. The whole number part is treated separately, making an improper decimal an impossibility.

But what if we didn’t just look at the figures to the right? Nested tenths don’t stop/start at the decimal point. What if we looked at *all* the figures? We’d read 1.5 as “fifteen tenths,” an improper decimal.

Maybe the improper vs. mixed comparison is throwing me off track. Fractions can be classified as either *proper* or *improper*. Why not decimals? Decimals less than one, such as 0.5, would be proper; decimals greater than or equal to one, such as 1.5, would be improper (or, in Britain, *top-heavy*).

Christopher Danielson wasn’t trying to introduce new vocabulary to the world of math(s). Probably. Rather, he was making a point about place-value.

When we teach decimals using ten-frames we do.

If the whole is one full ten-frame, students may build 3.7 like this:

Students will describe 3.7 as “3 ones and 7 tenths,” “37 tenths,” or even “2 wholes and 17 tenths.” This mirrors what students know about place value and whole numbers: 37 can be described as “3 tens and 7 ones,” “37 ones,” or even “2 tens and 17 ones.”

Just like with whole numbers, thinking about place value makes calculations with decimals easier. For example, consider 4.8 + 3.6:

- 4 and 3 make 7
- 0.8 (“8 tenths”) and 0.6 (“6 tenths”) make 1.4 (“14 tenths”)
- 7 and 1.4 (“1 and 4 tenths”) make 8.4 (“8 and 4 tenths”)

Note the shift in thinking, not notation, from 1.4 as “14 tenths” to 1.4 as “1 and 4 tenths.” With fractions, it’s a shift in thinking *and* notation. Probably why we know about improper fractions but not improper decimals.

**Blackline Masters:**

Ten-Frames – Full

Ten-Frames – Less-Than-Ten

Ten-Frames – Place Value Mat

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I’ve been thinking about this issue a little bit myself as well. It would certainly make understanding decimal place value a bit easier, I think, if we spent a bit more time on different representations than many of us do currently. The 0.05 = or > 0.5 misconception I’ve seen many times is probably symptomatic of this problem.

I like the different representations! I don’t really get to teach decimals but those ten-frames are cool. To me they really highlight the connection between the fraction and decimal.

I’m going to recommend that our instructors share this post with all of our pre-service teachers. Thanks for including the blackline masters. Group discussion paired with a hands-on activity and exploration–what could be better than that?