# Math Operations: understanding the words will help with understanding the concept.

I originally put this information together for a fourth grade teacher that I was mentoring at the time.  I thought that if she understood these words beyond a dictionary definition, she could find ways to incorporate that understanding into her teaching.  I went beyond the basics with this information so the teacher could make a professional decision about what to include in her teaching.

We have evidence of this word existing in English as far back as the late 14th century.  At this time it meant “action of adding numbers.”  Prior to that it was in Old French (early 1400) spelled as adition and meant “increase, augmentation.” Even earlier than that (1300) it was in Latin and spelled as additionem and meant “add to, join, attach.”

Before the printing press was invented, spelling was similar place to place, but certainly not consistent.  For example, in Middle English, this word was spelled as addissyoun and also as addycyoun!  Those are only two of many variations.

A deeper dive that you may or may not wish to share with students …

Ultimately it comes from the Latin verb dare meaning “give.”  This is not the modern word ‘dare’ as in I dare you to do something.  This would have been pronounced as ‘dar’ (as in darling) +’ey’ (as in they).  The thing about this is that the <are> on the Latin word dare is a Latin suffix and can be removed!  That means that what came into English is the single letter base <d>!  We never see a single letter base on its own.  It is always attached to a prefix or suffix.  That’s why it’s called a bound base.  So what that means is that the structure of the word ‘addition’ is <ad + d + ite + ion>!  The prefix <ad-> is seen on many words like adverb, adjective, advocate, adjacent.  It carries the sense of “to.”  In this word, the base means “give,” so the word means “give to.”  That makes sense when we think of what addition is.  The suffixes <-ite> and <-ion> are often used together and the final <-ion> indicates this is a noun.

If you’re wondering where else we see this single letter base <d>, one word that your students will know is the word ‘condiment.’  The structure of this word is <con + d + i + ment>.  The prefix <con-> has a sense of “together,” the <d> means “give,” the <i> is a connecting vowel, and the <-ment> is a suffix indicating this is a noun.  Condiments are a variety of things we put together with our food (ketchup, mustard, relish, etc.).

Another word is ‘mandate.’  The structure of this word is <man + d + ate>.  The first base <mane> means “hand” (think manicure “professional care of the hand”, manuscript “written by hand,” and manual “done or used by hand”).  The second base <d> means “give” and the suffix <-ate> indicates this can function as either a noun or a verb.  This word means a judicial or legal order but we can think of it literally meaning “give into one’s hands.”

Yet another word is ‘tradition.’  The structure is <tra + d + ite + ion>.  The prefix <tra-> is a shortened form of <trans-> meaning “over.”  The base <d> means “give.”  The suffixes <-ite> and <-ion> indicate this word is a noun.  The idea of a tradition is that it is something that is “given over” from one generation to the next.

So, what to tell students?

There are several ways to share some of this with your students, and only you can make the best decision about which is most appropriate!

1) You could say that the base is <add> and have them help you think of words built off <add>.  Then show them the following matrix and see if they recognize the words they suggested.

2) You could show them the structure of <add> and connect the <d> base to words they know such as condiment and tradition, but show the matrix that is totally related to math.  The advantage of this is that they will be aware of the <ad> prefix and might begin to recognize it in other words.

3)  You could show them the following matrix, although it might be better to begin with one or both of the others.  YOU, however, will no doubt be intrigued by the words included here.

Subtraction-

We have evidence of this word existing in English as far back as 1400.  At that time it was used to mean “withdrawal, removal.”  It wasn’t until the early 1500’s that it was used as a mathematical term.  Before it appeared in English, it was in Late Latin and spelled subtractionem “a drawing back, a taking away.”  In earlier Latin it was subtrahere “to pull away, draw off.”

The structure of this word is <sub + tract + ion>.  The prefix <sub-> has a sense of “from under,” the base <tract> means “pull, draw,” and the suffix <-ion> indicates this is a noun.  The idea that we pull or draw a small number from “under” a bigger number is clear in the way we write a subtraction problem, lining up by place value.

There are many words that share this base that also share its sense and meaning.

tractor – something that pulls.

distract – when one’s attention is pulled away from something.

attract – One’s attention is drawn or pulled toward someone or something.

extract – something is withdrawn or pulled out. (flavorings in cooking, water from a rug)

retract – something is withdrawn or pulled back.  (a statement, an awning)

contractor – someone who pulls a project together and makes sure all parts are completed.

Here’s a matrix with <tract> as the base that includes the above words.

In case you are wondering about the word ‘subtrahend,’ I will once again point out that the furthest back relative of this word was the Latin word subtrahere “to pull away, draw off.”  If we replace the Latin suffix <-ere> with the modern English suffix <-end>, we’ll have the word ‘subtrahend!’   In modern English, subtraction and subtrahend don’t share a base so they can’t be on the same matrix.  You can tell the students that they share a great grandparent!  Even in the Oxford English Dictionary, I was unable to find any other words with a <trah> base.  Most of the words besides ‘subtrahend’ that derive from subtrahere have a <tract> base.

Multiplication-

We have evidence of this word existing in English as far back as the mid-14th century.  At that time it meant “any increase in size, number, or amount; act or process of increasing in number.”  Earlier, in the 12th century it existed in Old French as multiplicacion and meant “duplication, multiplicity.”  Earlier yet, it came from Latin multiplicare “to multiply, increase.”  As you may be noticing, the spelling of this word is not much different than it was in Latin!  If we remove the Latin suffix <-are> and replace it with the modern English suffixes <-ate> and <-ion>, we will have <multiplication → multi + plic + ate + ion>.  The prefix <multi-> has a sense of “many,”  the base <plic> means “folds,” and the suffixes <-ate> and <-ion>, often used together, indicate this word is a noun.  Well, it is the final suffix that is indicating this is a noun.  When <ate> is final, it is usually indicating the word is a verb (illustrate / illustration), although as we saw in the word ‘mandate’ it can indicate a noun as well.

Think of the words duplicate and triplicate.  Their structures would be <du + plic + ate> and <tri + plic + ate>.  The sense comes from the idea that something in triplicate is threefold.  Not necessarily in the sense that it has been actually folded three times, but rather that it is as if it was replicated three times.  Oooh!  Look at the word ‘replicate!’  <re + plic + ate>!  I’m also seeing that complicate and implicate share this same base as well!

replicate – repeat – as if folded over for an exact copy

complicate – folded together, confused, intricate.

implicate – act of entangling.  Inference drawn from what is observed.

With your students, you may want to stick to words you know they know.  Here is a simpler matrix, comparing the spelling of ‘multiplication’ to ‘complication.’  You’ll notice that the word ‘multiplicand’ is there.  The <-end> suffix we saw in addend and subtrahend is an <-and> in this word.

The word ‘multiply’ has existed in English since the 12th century when it was spelled as multeplien and meant “to cause to become many; cause to increase in number or quantity.”  As with ‘multiplication’ this word comes from Latin multiplicare.  But further back, it comes from Latin multiplex “having many folds.”  There are a few words that share the base <ply> meaning “fold.”

multiply – many folds

imply – entangle and involve something unstated

Looking at the words with the <ply> base gives you a perfect opportunity to talk about when we do and when we don’t toggle that final <y> to an <i> when adding the suffix!

Here’s a matrix focused only on words directly related to multiply.  Again you have an opportunity to look at the <y> to <i> toggling convention.

The word ‘multiple’ first entered English in the 1640’s.  The structure of this word is <multi + ple>.  The base <ple> is from Latin multiplus and means “fold.”  How about that!  So the words multiplication, multiply, and multiple all share the literal meaning of “many folds.”  Because they have differently spelled bases, they can’t appear on the same matrix.  Here’s one for <ple>.

Division –

The word ‘division’ has existed in English since the late 14th century.  At that time it was spelled divisioun and meant “act of separating into parts, portions, or shares.”  Prior to entering English, it was in Old French and Latin before that.  When it was in Latin, it was the verb dividere which meant “to force apart, distribute.”

As is often the case with Latin verbs, two modern English bases have derived from the same Latin verb:  <divide> and <divise>. The second base here (<divise>) is a bound base.  We don’t see it as a word on its own.  It is always bound to either a prefix or a suffix.  Because they derived from the same Latin verb, they have the same meaning, “separate, break up, share, distribute.”  These two bases from the same Latin verb are sometimes referred to as twin bases  (I always told my students to think of them as fraternal twins who may not look alike but came from the same parent).

Here is a matrix for each. In the first one you will note the opportunity to replace the final <e> on the base when you are adding a vowel suffix.  There is another great opportunity to talk about the word ‘indivisible’ which is included in the Pledge of Allegiance.

In the second matrix you’ll see the suffix <-end> to form the word ‘dividend.’  This matrix also gives you the opportunity to talk about the common practice of replacing the final <e> on a base when adding a vowel suffix (suffix that begins with a vowel).

Resources used –

Etymonline

Oxford English Dictionary online

Neil Ramsden’s Mini Matrix Maker

# When People Say, “Just Start,” What Does That Look Like?

New opportunities have opened up for me! I am meeting with almost all of the 5th grade students at my elementary school (half the group on Tuesday, the remaining half on Thursday) for about 25 minutes each week. I am coming into their classroom and one of three grade level teachers is observing the lesson. I am also working with an 8 year old for 6 hours a week in my home!

This week I talked about the spelling of ‘two’ with both groups. Since I was meeting the 8 year old for the first time, I had him use manipulatives. In this way, I could both check his math understanding and also his understanding of the spelling of some numbers. While he was having a bit of play time with the superhero figures we were using, I asked him if he could spell the number ‘two.’ He paused and slowly spelled out, “t.o.w.” I said, “That’s great. You have all the right letters! But the ‘t’ and the ‘w’ need to be together. Let me show you how I know that.”

I wrote the word ‘two’ on my paper and showed it to him. Then I asked him if he noticed whether or not there were any twins in the superhero collection. He did. There were two sets of twins. I wrote the word ‘twin’ on my paper and asked him if ‘twin’ had anything to do with ‘two.’ Then I asked him what the words (spelling) ‘two’ and ‘twin’ had in common. I asked the same questions of ‘twice.’

Next I asked him to count out ten of the superheros. I asked him how many more we would need in order to have twelve. He said, “two.” I replied, “So twelve is two more than ten. It is ten plus two.” He grinned. I added the word ‘twelve’ to our list. Then he laid out two rows of superheros for a total of twenty. He knew that two rows of ten would give us twenty. I added the word ‘twenty’ to our list. Then I wrote the word ‘between’ and asked him to name the superheros that Batman was between. From there we connected the meaning of ‘between’ to the meaning of ‘two.’ Then I went back through the list and underlined the ‘tw’ in each word and asked him why I did that.

From there, I asked him to write down the word ‘ten.’ He didn’t have any trouble. I asked him to write down ‘six.’ Then I said, “If we add ten to six, we’ll have sixteen. What will we add to the spelling of ‘six’ to have the word ‘sixteen’? He wrote ‘ten.’ Perfect opportunity to talk about the <ee> digraph versus the single <e> grapheme in the context of these words! Then we talked about the meaning of ‘ten’ and the meaning of ‘teen’ in the word ‘sixteen.’ He noticed right away that they shared meaning, but not spelling! (He seemed to have an established awareness that that can happen – cool!)

Now that he had an understanding of <teen> in ‘sixteen,’ I asked him to write the number ‘five’ and then ‘fifteen.’ He started writing *’fiveteen,’ but realized that wouldn’t represent how we pronounce ‘fifteen.’ In the context of these two words, we could focus on the voiced <v> in ‘five’ and the unvoiced <f> in ‘fifteen,’ and how the two spellings share meaning but not spelling. As he was thinking about this, he said, “fifteen and fifty both have the /f/!

Before we stopped with the superhero figures, skip-counting and number words, I asked him once more how to spell ‘two.’ Without hesitation he said, “t.w.o.”

The 5th graders were fascinated. They were engaged and quickly recognized the meaning connections between words like ‘two’ and ‘between.’ They helped brainstorm many words with an initial ‘tw’ and we discussed the meaning of each. I made sure the word ‘twilight’ came up because I knew they knew its meaning, but might not have thought of it since the ‘tw’ spelling isn’t initial in the word. At least three students came up afterwards to tell me how cool the discussion was!

Day Two

My copy of Mona Voelkel’s new book, Stanley and the Wild Words arrived in the mail, so I shared it with my 7 year old friend, Michael, and also with the 5th grade students.

I began by reading the book aloud. I paused at times to encourage the students to share their understanding. For instance, I asked what they thought ‘enormous’ meant and then asked for examples of things that could be considered enormous. We talked of whales, dragon teeth, and mountains but also of appetites and loads of wash. Then we talked about the denotation of <norm> being “rule.” I wondered what they thought about when they thought of ‘rule.’ So I asked. The fifth grade students could name several rules they follow at school. One boy defined a rule as a condition that everyone followed. In other words, following the rules is considered normal. When something is outside of what we think of as normal – is bigger than normal – it can be considered enormous.

Below are pictures of what I wrote down as I was reading the book to Michael. As you can see, I began with a word sum for ‘enormous.’ I labeled the morphemes as ‘prefix,’ ‘base,’ and ‘suffix.’ After we had a list of words that shared the base <norm>, I asked Michael to draw a box around the base in each word.

When talking with the 5th grade students I added the term “analytic word sum.” I explained that with an analytic word sum, we begin with the fully spelled word and then loosen it into its morphemes. I pointed out that the bound base <lyt> had a denotation of “loosen.” The prefix <ana> brings a sense of “throughout.” If we begin with the morphemes and join them to form a complete word, that is a synthetic word sum. Having brought up the words ‘synthetic’ and ‘analytic’, I wanted to expand the students’ understanding of them by mentioning other situations in which we use these words. We talked about synthetic materials being put together by man and how analyzing a problem requires us to look closely at each component.

With the large group of fifth graders, I did the boxing of the base and then had them tell me what the word sums would be. As they hypothesized the word sums, I wrote them on the board. When we got to the word ‘ginormous,’ I explained that ‘gi’ isn’t a prefix – it represents the word ‘gigantic’ in this portmanteau word (gigantic+ enormous gives us ginormous). You’ll notice we didn’t include ‘gi’ on our matrix – again, because it isn’t a prefix. This may be the first time I am mentioning portmanteau words to these students, but it won’t be the last. At some point, I’ll ask the students to choose ten of their favorites. Until then, I used the example of ‘brunch’ being a combination of breakfast and lunch. I pointed out that when the two words join to become a portmanteau, letters from each word might be lost. That makes this different from a compound word, where two bases join (intact) to form a new word.

With the fifth graders, I followed this activity by having them write the word ‘help’ at the top of a piece of paper and then writing as many related words as they could think of. Then I walked them through creating a matrix with ‘help’ as the featured base. One of the students thought of ‘prehelpfully.’ I look forward to talking about this invented word and also about matrices next time. Although, before I even began reading the book today, one student said, “I thought of a word for you to explain – one!” Perfect! Doesn’t look like I’ll run out of topics, does it?

Day Three (Days Three through Five focus on my time with Michael)

Today Michael and I read the story of Ibis, a whale who gets caught in netting and almost drowns. It is based on a true account, although the author added details that gave us an opportunity to talk about authors and story writing. Why do authors sometimes embellish the facts? Why did this author give Ibis human characteristics?

While we were discussing the main character’s interest in humans, we noticed that the following illustration gave us the whales’ perspective from deep in the water. Michael had used the word ‘perspective’ earlier, so I pointed it out here. “Isn’t it interesting to see the boat from the whales’ perspective? What do you suppose that word ‘perspective’ means?”

Michael said, “My perspective might be different than yours. Because of where I am standing, it might look different.”
“That’s a great way to define that word!”
I wrote the word down and showed it to Michael. I boxed out the <spect> base and wrote down the denotation “see, look”. Then I pulled up Etymonline and wrote down the Latin root specere above where I had written the word ‘perspective.’ I typed specere in the search bar and looked for other words Michael might be familiar with that had this base (or a variation of this base). When I came across ‘spectator’ I asked him if he’s ever been part of a large crowd – at a sporting event perhaps. He said, “Like at a baseball game?”
“Yes. The people who come to watch, to see the game, those are the spectators.

Next I found the word ‘spectacle.’ Michael said he was not familiar with it, so I told him that I can refer to my glasses as my spectacles. They are what I look through! Then I described another way to use that word. Let’s say I was with a group of people who were all walking slowly, but then one of the people jumped out in front and started dancing and singing. We might say they were creating a spectacle. They were making a spectacle of themselves. In other words, their actions were drawing attention and people couldn’t help but look at them. Then we thought of a few other situations in which people could make spectacles of themselves.

The word spectacular is used to describe things. A sunset might be described as spectacular, but so might a tricky catch in a game of football. I asked Michael to tell me how either of those were related to the denotation of <spect>. He said they were cool to see. He was making the connection between the meaning of the base and the meaning of the words it shows up in! It was around this time that he said, “What about ‘expectation’? Does that word work here?”

I’m sure he noticed my smile at that suggestion. I said, “Let’s see if it comes from the same Latin root. That’s a way to know for sure that the base in ‘expectation’ is related to the base in ‘perspective.’

I read the entry for ‘expect’ to Micheal.

“1550s, “wait, defer action,” from Latin expectare/exspectare “await, look out for; desire, hope, long for, anticipate; look for with anticipation,” from ex- “thoroughly” (see ex-) + spectare “to look,” frequentative of specere “to look at.”

I said, “Look at that. We still use the word in the same way the Romans used it 500 years ago! When we expect something, we are looking forward to it with anticipation. In this word we are combining the base <spect> meaning “look” and the prefix <ex> meaning “thoroughly.” At that point we noticed that even though ‘expect’ has the <spect> base, we no longer include the ‘s’ in that spelling. Etymonline shows us that two spellings were used in Latin. One of those has the spelling we currently use.

It was around this time that he asked about the word ‘despicable.’ I said, “It doesn’t have the <spect> base, but it still might be related. I know there are some alternate spellings for this base. We won’t know for certain until we look!” As luck would have it, it was on the same page as some of the other words related to ‘perspective.’ It also derives from Latin specere. The base in this word is <spice> (which is not at all the word ‘spice’ ). I said, “We might find more words in which the base is spelled this way. Let’s start another list.

When we talked about the word ‘inspect’ meaning to look into something, I also pointed out that if we add an <or> suffix, we would have the word ‘inspector.’ That is the person who is doing the looking. Then I slide my finger back up to the word ‘spectator’ and said, “There’s the <or> suffix! A spectator is the person who is spectating! It was our first look at agent suffixes.

Michael was familiar with the words ‘inspect,’ ‘respect,’ and ‘suspect’ and was able to describe how they connected back to the denotation of <spect>.

Towards the end of our search, we found the word ‘suspicious.’ I listed it beneath ‘dispicable’ and pointed out the spelling of the base. I asked him what being suspicious of someone might have to do with looking or watching. Michael quickly replied that when you are suspicious of someone, you watch everything they do.

At this point, we had spent about 30 minutes talking about these related words and their connection to “see, look, watch.” As a final way to look at what we collected, I made a word web. As I was writing, Michael noticed that some of the suffixes could be used on several of these words (expecting, spectating, suspecting, inspecting, etc.) He also used some of the words in sentences which reinforced his understanding of their meanings.

My goal with this activity wasn’t so much that Michael would be able to walk away knowing how to spell these words, but rather that he notice the meaning connections within word families. I was helping him broaden his vocabulary by taking a familiar word and connecting it to unfamiliar words that share that base. I was showing him that words come in families and that he can expect this kind of familial relationship not only with spelling, but with meaning. I was showing him how the English language is structured.

Day Four

On this day I took pieces of paper and wrote suffixes on some, bases on others, and prefixes on yet others. I took two long pieces of paper and created a matrix. We started with the base ‘like.’

The object was to see whether these particular suffixes and prefixes could work with this base. As Michael moved each affix towards the base and read the word, he described what the word meant and sometimes even used it in a sentence. A cool thing happened when he was fixing the <un> prefix to the base <like>. Instead of announcing the word ‘unlike,’ he announced the word ‘unliked.’ I immediately said, “Oh, neat! As you said the word, you added not only the <un> prefix, but also the <ed> suffix! You created the word ‘unliked’! He smiled and said, “Yup. The villain was unliked by the superhero!” The benefit of having these suffixes on movable pieces of paper is that Michael is already recognizing that the <ed> suffix will sometimes replace the single final non-syllabic <e> on the base. He moved it into place himself. This suffixing convention will be revisited many times. This is the second time we’ve talked about it.

He decided that all of these prefixes and suffixes could be used with this base.
Next I moved the base <like> out of the way and replaced it with <hope>. Now Michael had to decide if these same prefixes and suffixes work with this new base.

He started with the prefixes and grabbed <dis>.

He scrunched his nose and said, “Nope!” And moved <dis> to the side. Then he tried <un>. He was undecided about this one. When he tried to talk about what it would mean, he used the word ‘hopeless.’ I said, “Ooooh! We can add the suffix <less> to our group of suffixes. I’m so glad you thought of that word!” Michael grinned.

As he pulled the suffixes <ing>, <ed>, <er>, and <s> next to the base to see if they formed a word he recognized, we noticed that the suffix didn’t always replace that final non-syllabic <e>. We talked about vowel suffixes and consonant suffixes. The suffixes that replaced the final non-syllabic <e> were <ing>, <ed>, and <er> – all vowel suffixes.

Next he matched up the base <hope> with the suffix <ly>.

Hmmm. He said the word a few times, looked at me and said emphatically, “Nope! I can’t think of how to use that one!” We laughed. Then he slid the <ly> suffix out of the way and we looked at the matrix full of morphemes that worked together. We slid the base <hope> out of the way and pulled in the base <rope>.

By this time, Michael knew what to do. He moved the affixes next to the base and thought about whether it was a word that made sense to him. He slide <dis> and <ly> to the side. He noticed, too, that when he paired up the base <rope> and the suffix <er>, he was talking about a person who did the roping. I quickly slid <hope> back into place to see if a ‘hoper’ was a person who did the hoping and then if a ‘liker’ was a person who did the liking. On another day we’ll test that suffix some more! Does it always refer to a person?

After this activity, I read the book From Wolf to Woof by Hudson Talbot. Michael is very interested in the evolution of living things, so when I found this book, I knew he would be interested. Besides, I saw in the title the opportunity to talk about /f/ and /v/. We talked about these two when we talked about ‘five’ and ‘fifteen’ and I was looking to reinforce that concept.

I wrote down the word ‘wolf’ and asked what word we use if we are talking about more than one wolf. Michael said, “Wolfs.”
I asked, “Do you notice how it feels in your mouth to go from the /f/ to the /s/?” We tried it a few times. “Now let’s see how it feels when we go from /v/ to /s/ as in ‘wolves.’ We agreed that was a smoother transition. I wrote down the spelling of ‘wolves’ and showed Michael the switch from <f> in ‘wolf’ to <v> in ‘wolves.’

I had a list of other words where this happens and added them to the paper. We worked one at a time and Michael wrote the spelling of the word as a plural. We toggled the <f> to a <v> and added an <es> suffix. We noted that the word ‘leaf” had an <ea> digraph and the word ‘loaf’ had an <oa> digraph. We paused to talk about what a digraph was and thought of at least two other words that had each of those same digraphs. Then we came to the last two words. They were slightly different than the others. They didn’t have a final <f>. They had a final <e>. Michael recognized that the final <e> wasn’t being pronounced. That meant that it was a grapheme whose job was something other than representing pronunciation. But what? Well, it was signaling that the previous vowel (<i>) would have a “long” pronunciation. Does that change what we’ve been noticing about the /f/ to /v/ in these words? No. Not at all. We will still toggle the <f> to a <v> and add an <es> suffix. In this case, the <es> suffix will replace the single final non-syllabic <e> as it does in many other words.

Because we are also focused on meaning, we had a great discussion about the pronunciation of ‘live’ when it functions as a verb and when it functions as an adjective (There were live snakes in the exhibit. I live down the street.) With this word (and many others) we need to see the word used in a sentence to know how to pronounce it. There are other words like this and we will consider them at another time.

Day Five

I began with the matrix again – the one in which morphemes were written on slips of paper and we could move them around. Michael was enthusiastic. The base I chose today was ‘do.’ (There is no significance to the two colors. It was just the paper I had on hand.)

Interesting things we noticed.
– The base is pronounced the same in ‘doing’ and ‘doer,’ yet different than in ‘does’ and ‘done.’
– If we were to strictly rely on pronunciation to spell ‘does’ we might think it is spelled as *duz. But then we wouldn’t see that it is built from the base <do>.
– Even though there is a shift in pronunciation, there is not a change to the spelling of the base.
– The base represents the meaning.
– Instead of adding an <ed> suffix to show past tense, there is a different spelling of the base.
– The original spelling of ‘did’ in Old English was dyde. According to Etymonline, the final <de> functioned as a suffix and has become our current <ed> suffix.
-When the <er> suffix is added to the base, we get the word ‘doer.’ In this word, the <er> is an agent suffix. A ‘doer’ is a person who is doing something. We can compare this suffix to the <er> in hoper, roper, and liker.

After exploring and noticing things in this <do> matrix, I pulled <do> and replaced it with the base <go>. Michael rejected both the <re> and the <un> prefix, but grabbed a piece of paper and wrote ‘by.’ Then he put it in front of the base and pulled the <ne> suffix to the end of the base. “Bygone! That’s a word, right? Let bygones be bygones, meaning something happened already.”

“Wow! You’re right. In this case, the <by> is not a prefix, it’s another base. So bygones is a compound word. Nice going!”
Michael was familiar with what a compound word was and was able to give me a few examples, so we were able to move on.

We noticed that ‘go’ was similar to ‘do’ in that neither could show past tense by adding an <ed> suffix. They had different spellings to represent that.

Just as we were finishing up, Michael grabbed two more pieces of paper and wrote a sentence. Perfect!

# Here’s What’s Wrong With Teaching a List of Prefixes

I ran across a fascinating article recently called “Anumeric People:  What Happens When a Language Has No Words for Numbers?”  While I immediately noticed the word ‘anumeric’ in the title, I set it aside while I read the article and imagined a life without words for numbers.  What are the advantages/disadvantages?  It’s quite likely that there are people in remote areas of the world whose lives don’t revolve around clocks and other numbered things.  But is the ability to distinguish by number the difference between 3 and 6 items crucial to one’s existence?  Obviously not, for the people who only have words to name “some,” have lived for generations.  The interesting focus in this article is “how the invention of numbers reshaped the human experience.”  The article is not particularly long, but certainly gave me something to think about!

Now.  Back to the word ‘anumeric.’

Right away I connected it to the following.
numeric
numeral
numerous
innumerable
numerology

If you compare the spelling of these words, you’ll notice (as my students would) that they each have <numer> in common.  If given the opportunity to write a word sum hypothesis for ‘numeric’, I might see students write both <numer + ic> and <num + er + ic>.  They are both logical.  The first includes the letter string that is consistent among the words and might be the base.  The second includes prior knowledge of <er> being the suffix in baker, teacher, and colder.

Once we have discussed the hypotheses and the fact that both are based on what we already know to be true about word construction, it is time to find evidence that will support one more than the other.  If I look in either Etymonline, Chamber’s Dictionary of Etymology, or the Oxford English Dictionary, I find that all the words on our list derive from Latin numerus “a number.”  Once the Latin suffix <us> is removed, we see the Latin stem that came into English as the base <numer>.  This evidence shows that the <er> was part of the word’s spelling in Latin and is part of the base in English.  I like to compare this situation to the <ing> in ‘bring.’  We know there to be an <ing> suffix, but that doesn’t mean that every time we see that letter string we are looking at a suffix.  It’s logical to wonder about it, and scholarly to check with a reference!

Once I had looked closer at the base of ‘anumeric,’ I thought more about the prefix <a>.  Thinking about its use in the article where I found it, it obviously has a negativizing sense.  It has a similar use in the following.

apnea – without breathing
amnesia – not remembering
atheist – without a god
apathy – without feeling or emotion
atypical – not typical
aphotic – without light

The prefix <a> that incorporates a sense of “not, without” is sometimes spelled <an>.  According to Etymonline, it is “a fuller form of the one represented in English by <a>.”  You may recognize the <an> prefix in the following.

anarchy – without a ruler
anonymous – without a name
anomaly – not the same
anesthesia – without feeling
anhydrous – without water

So does this mean that every time we see a word with an <a> or <an> prefix that it contributes a sense of “not, without?”  No.  No it doesn’t.  There are a number of words like asleep, awash, aside, and aflame that originated in Old English and in which the prefix <a-> contributes a sense of “on, in, into.”  That <a> prefix can also be an intensifying prefix as it is in ashamed.  An intensifying prefix is one that doesn’t contribute a separate sense to the base, but instead intensifies the action of the base.  (More about intensifying prefixes to come.)

An unexpected sense

As I began a deeper dive, looking at words with an <a> prefix, I came across afraid, award, and astonish.  The word ‘afraid’ was derived from Anglo-French (afrayer) and further back from Old French which influenced the spelling (affrai, effrei, esfrei) and further back from esfreer “to worry, concern.”  The first part of this word is actually derived from Old French es-; Latin <ex-> prefix “out” and the second part is from Vulgar Latin *exfridare “to take out of peace.”  Please note that the asterisk in this ancestor means that the spelling is unattested.  This spelling is thought to be a likely spelling by those who study languages.  Beyond that, just think about the denotation of this word!  To be afraid is to have been taken out of peace!  Don’t you love it?

Looking at ‘award,’ this is another word that was derived from Old French.  It is from Old French (awarder) and further back from Old North French (eswarder).  Do you notice the initial <es> spelling?  To award something to someone is to give one’s opinion after careful consideration.  As with ‘afraid,’ the first part is actually from the Latin <ex-> prefix “out” and the second part is from Germanic warder “to watch.”  So the person choosing who will receive an award is the one who watches out for which person will be deemed most worthy!

That brings us to the word ‘astonish.’  This word, too, was influenced by its use in Old French.  It is from Old French estoner “to stun, daze, deafen, astound.”  If you noticed the ‘es’ in the Old French word estoner, you may be expecting that the first part of this word is from Latin <ex-> “out,”  and you’d be right!  The base is from Latin tonare “to thunder.”  If something astonishes you, it leaves you a bit stunned or dazed, as if you were shook by thunder!

So the question with afraid, award, and astonish is whether or not they have an <a> prefix.  The etymology clearly reveals that the prefix sense here is from <ex> even though we see an <a> prefix.  The story of how the <ex> prefix came to be spelled as <a> can be found in the influence of Anglo-French and Old French spellings!   So here we have evidence of words with an <a> prefix that represents Latin <ex>.

Assimilated forms of other prefixes

The prefix <an> can also be an assimilated form of the prefix <ad> “to” as it is in announce, annul, and annexation.  You’ll notice that the <ad> assimilates to <an> when the next element in the word begins with an ‘n.’   The <ad> prefix can reduce to <a> in words like ascend, ascribe, avenue, and avenge.

In the word ‘avert,’ the <a> is a reduced form of the <ab> prefix “off, away from.”

If you’re wondering, “How will I know which prefix it is or which sense it brings to the word I’m investigating?”  Fear not!  A quick check with a reliable source like Etymonline will clear up which <an> you are looking at as well as which sense it brings to the base or stem!

What about other prefixes?  Are they all like this?

Once I got thinking about <a> and <an> as a prefix, about all the different ways it can contribute sense to a word, I thought about all the other prefixes that I have been similarly surprised at.  You see, prior to SWI, my understanding was that prefixes contribute a consistent meaning to each word they are attached to.  For instance, in books that I was using to understand prefixes, suffixes, and “root words,” the prefix <re> was listed as meaning “again.”  The examples given were similar to remarry, reuse, and resupply.  Every prefix that was mentioned had a specific definition.   Examples of some of those are below.

de – down
dis – away
ex – out
in – not, without
pre – before
un –  not
con – with

I bet you’ve seen lists like this.  Taking a close look at the English spelling system by incorporating Structured Word Inquiry into my teaching and learning has made me realize so much!  For instance, the way in which a prefix steers the meaning of the base isn’t as “set in stone’ as we have been led to think.  We’ve already had a glimpse of that with our look at the <a> prefix!

Recently the International Dyslexia Association presented a live Facebook chat featuring Sue Scibetta Hegland, who spoke on the topic of incorporating morphology in spelling instruction.  The presentation was recorded and you can watch it below.  In this talk, Sue uses the prefix <dis> to address the very point I am making in this post. I encourage you to watch it.  Besides her point about prefixes, she makes many many others that are so eye-opening!  In the paragraphs following the video, I have elaborated on the point she made with <dis>.

If you think about words in which you’ve seen a <dis-> prefix, you might think of words like disapprove, disappear, and disable.  In all three of these words, the prefix brings a sense of “opposite of.”  If you disapprove of something, that is the opposite of approving.   When something disappears, it does the opposite of appearing.  When a machine is disabled, it is the opposite of when it is able to do its intended job.

In the words distract, disrupt, and dismiss, the <dis-> prefix contributes a sense of “away” to the denotation of the base.  In all three of these examples, the prefix is paired with a bound base.  Looking closer at ‘distract,’ the base <tract> is from Latin trahere “to draw.”  When someone is distracted, their attention has been drawn away from where it was.  Looking closer at ‘disrupt,’ the base <rupt> is from Latin rumpere “to break.”  When a meeting is disrupted, everyone’s attention is broken away from what it had been focused on.  Looking closer at ‘dismiss,’ the base <miss> is from Latin mittere “to send, let go.”  When you dismiss your students, you send them away!

A third sense that the <dis-> prefix might bring to a base or stem is “not.”  This is the case in the words displease, dislike, and dishonest.  When you are displeased, you are not pleased,  When you dislike something, you do not like it.  When you are dishonest, you are not being honest.

There are other senses as well.  In the word ‘distribute,’ the base is from Latin tribuere “to pay, assign, grant.”  The prefix <dis-> contributes a sense of “individually.”  When you distribute materials, you are assigning those materials to each individual in the group.  In the word ‘distort,’ the base is from Latin torquere “to twist.”  The prefix <dis-> contributes a sense of “completely.”  When something is distorted, it is completely twisted (whether physically or metaphorically).  In the word ‘dissension,’ the base is from Latin sentire “to feel, think.”  the prefix <dis-> contributes a sense of “differently.”  When there is dissension within a group of people, they no longer are in agreement.  Some or all think differently than the leader of that group.

Intensifying prefixes

I spoke earlier about prefixes that act as intensifiers.  The example I gave was ashamed.  In ‘ashamed,’ the state of feeling shame is intensified.  There are others, of course.  Once you begin finding them for yourself, you’ll experience a new kind of fun!  Until then, here are a few I’ve discovered.

Let’s compare the words ‘reunion’ and ‘refine.’   A reunion happens when people are coming back together again to become one group with something in common.  The main sense and meaning of that word, “the act of joining one thing to another,” has been consistent since it was first attested in the early 15c.  The prefix ‘re’ adds that the act of joining one thing to another is happening again. These people have come together before and now they are coming together again.  According to Etymonline, the word ‘refine’ was first used with a reference to metals (1580) and later to manners (1590).  It has to do with reducing something to its purest form (or as close to it as one can get).  The main sense and meaning of that word is “make fine.”  In this word, the prefix <re-> does not indicate that a thing is becoming fine again.  Instead, the <re-> prefix is an intensifier.  It is intensifying the action.  Whatever it is that is being refined is being made super fine.

Another example of a prefix that can intensify the action of the base is found in the word ‘corrode.’  The sense and meaning of the word since it was first attested in the late 14c is “wear away by gradually separating small bits of it” according to Etymonline.   You might recognize the base as <rode>.  It is from Latin and has a denotation of “to gnaw.”  We see it in rodent and erode as well.  The meaning connection is pretty obvious, isn’t it?  That leaves <cor-> as the prefix.  It is an assimilated form of <com->.  We often think of <com-> or one of its assimilated forms (<col->, <con->, <cor->, or <co->) as bringing a sense of together to the base’s denotation.  But that’s not what is happening here.  Instead, the <cor-> of ‘corrode’ is intensifying the “wearing away.”

One more example of a prefix being an intensifier is found in the word ‘complete.’   The Latin bound base <pl> has a denotation of “to fill.”  If you think about how you use the word ‘complete,’ you’ll realize that the <com-> doesn’t bring a sense of “together” to this word.  The act of finishing or concluding something can be done together with others, but it can also be done alone.  The prefix <com-> in this word is intensifying the “filling of something.”  Check out the entry at Etymonline to see for yourself.

Concluding thoughts

I hope I’ve made it obvious that when we teach children that <con> means together and <re> means again, we are teaching them only one possible sense when the truth is there are many.  There’s nothing wrong with saying that <re> typically incorporates a sense of “again” to a word it is part of as long as we also say, “but let’s check to be sure.  It could be doing something else as well!”

People who are hesitant to use SWI with their struggling students often say it is because their students don’t find dictionaries friendly.  Mine didn’t either.  That is, until they had a reason to use them.  I remember the days when my dictionaries sat unused on the shelf.  If I sent a student to grab one so we could look up a word, the student often said, “Nevermind.  I’ll use a different word.”  Since the students and I started asking questions that we were genuinely interested in exploring, those same dictionaries have become dog-eared and in come cases the pages have popped out.  I couldn’t be happier!  Once there was an authentic need to use the dictionaries, the students picked up the skills necessary more quickly than when we used to make up a fake scenario so they could practice.  “Let’s check to make sure,” became the quick look it’s supposed to be.  Students like knowing whether they’re on the right track or not, and using a dictionary lets them do that for themselves.  They learn confidence by not needing to run every hunch they have by the teacher.   When you avoid using dictionaries with your students because they are uncomfortable with them, you lose a huge opportunity to show them how to use reference materials and how to find out things on their own.  In effect, you are helping them stay uncomfortable with them.

So do your students a favor.  Make, “Let’s check to be sure,” a common practice in your classroom.  Let them discover the value and worthiness of a great reference material!  Thank goodness we have dictionaries and solid etymological resources like Etymonline, Chambers Dictionary of Etymology, and the Oxford English Dictionary!  That is where you and your students will be able to distinguish which sense a prefix is contributing to a word!  You don’t want your students to sort-of, kind-of understand the words they read and use in their writing.   A quick “check to be sure” will create a solid definition of a word as well as a scholarly habit.

# Grouped by Number

When I read about a woman in Mali having nonuplets a month ago, naturally my first thought was, “Yikes! Nine babies born at one birth?”  But just this morning that news was topped when I read of a South African woman giving birth to decuplets two days ago!  Most of us have heard of triplets and quadruplets, but having nonuplets and decuplets is so rare, they may be world record setting!  The more I thought about the words ‘nonuplets’ and ‘decuplets,’ the more I was reminded of how many times I’ve come across familiar morphemes in words that help indicate a particular number.  I also thought about the different ways we use these elements in different contexts.  In thinking about these different ways to group things and the common morphemes we use to represent those numbers in words, I learned some interesting things!

### Numbered sets – usually at one birth

The headline read, “Nonuplets: Woman From Mali Gives Birth To 9 Babies : NPR.”  Nonuplets.  How many is that?  Well, obviously, the headline tells us it is nine.  Nine babies born in a single birth.  We recognize this word ending as connected to numbers of babies born at one delivery.  And now there is this new report out of South Africa that a woman has delivered decuplets!  Etymonline explains that the spelling of ‘quadruple’ came from ‘quadruplet,’ and the spelling of the ‘plet’ came from the ending on ‘triplet.’   Are these words uniquely fitted to babies?  Not at all.  Quadruplets are sets of four.  They don’t have to be four babies.  Triplets are sets of three.  They don’t have to be three babies.   But my guess is that most people picture sets of babies when they hear these.

twins
triplets
quintuplets
sextuplets
septuplets
octuplets
nonuplets
decuplets

According to Wikipedia (and this will come as no surprise to you either), twins are the most common type of multiple birth.  Without fertility treatments, the chances of having twins is 1 in 60.  The possibility of having fraternal twins runs in families.  The possibility of having identical twins does not.  The chances of having identical twins is more like 1 in 250.

Having triplets is much less common with the possibility being 1 in 1000.  Triplets can be identical (least common), fraternal (most common), or a combination of those.  According to Etymonline, in 1831 another name for triplet was ‘trin’.  As you can probably guess, it was modeled on ‘twin’.  Quadruplets are even less common than triplets.  Although as fertility treatments become more widely used, the possibility is increasing.  Quadruplets are usually a combination of fraternal and identical.

Quintuplets occur once in 55,000,000 births (without fertility treatments).  The most famous set of quints to survive infancy were the Dionne sisters who were born in 1934.  I have watched documentaries about these sisters.  Unfortunately, the government feared the parents would exploit the quints and took custody of them. In the end, these girls were exploited by everyone.  While they were at play each day, some 6,000 visitors stopped to watch them.  Yikes!  My mother-in-law shared a first name with one of the quints and was given this spoon at some point in her life.  It was one of many souvenirs being sold.  If you are interested, here is an article about the life of these five identical sisters.

###### Unknown authorUnknown author, Public domain, via Wikimedia Commons

A well-known set of sextuplets are the Gosselin siblings.  They were born in 2004 and became well-known when their parents agreed to be part of a reality tv show that chronicled what life was like with a set of six babies!  It was a very popular show for a while, but it took a toll on the family.  Must be difficult to have a camera recording so much of your life.

When the McCaughey septuplets were born in 1997, they received a lot of media attention.  One of the more interesting letters they received was from the three surviving Dionne quintuplets.  The Dionne sisters offered their congratulations, but also warned the parents to keep the children out of the public eye as much as possible to avoid what they themselves experienced.

The first confirmed set of octuplets was born in 1969.  Unfortunately, all eight babies died within 13 hours.  It wasn’t until 2009 that a full set of octuplets (Suleman) survived infancy.  It illustrates how risky multiple births are, and yet also how the field of premature infant health care keeps improving.

With the birth of nonuplets to a couple in Mali, it appears another world record has been set.  Interestingly enough, the couple and their doctors thought they were having septuplets.  Apparently two of the babies were hidden during the ultrasounds.

As of June 9th I read of a woman in South Africa who gave birth to decuplets!  Like the couple who thought they were having seven but had nine, this couple thought they were having eight and had ten!  They were delivered at 29 weeks.  Guinness World Records is the group that officially verifies these things and determines world records.

### Numbered sets of legs/feet

If you want to group creatures by the number of legs/feet they have, you’ll work from this short list.

unipeds
bipeds
pentapeds
octopus
centipedes
millipedes

Snails and slugs are obviously unipeds. You can see the one foot they use to move with.  The organisms that belong to this class Gastropoda were previously called univalves.  That descriptive term referred to the fact that they have one valve or shell.  The name Gastropoda is equally as descriptive and revealing.  It comes from the Greek γαστήρ (gastér “stomach”) and Greek πούς (poús “foot”).  Its stomach is positioned above its single foot.

###### Grapevinesnail_01.jpg: Jürgen Schoner derivative work: Tim Ross (talk) – Grapevinesnail_01.jpg

Other unipeds are marine and freshwater mollusks, also known as bivalvia.  I bet that after learning about univalves, you can hypothesize the meaning of ‘bivalvia!’  These are mollusks with shells that have two valves (hinged parts).

###### Fernando Losada Rodríguez – Own work Tridacna gigas (Giant clam) in Aquarium Finisterrae (House of the Fishes), in Corunna, Galicia, Spain.

Much of the information in this section of the post is coming from Wikipedia which also has some interesting information about bipedalism.  They describe it as “a means of moving forward by means of two legs and feet.”  Picture a moving kangaroo or ostrich for a clear idea of an organism that uses bipedalism.  Of course, humans are bipeds too!  Some animals like bears and some lizards who are quadrupeds move bipedally when needing to move quickly or get to a food source.  Can you picture it?  Here’s a pretty cool video from National Geographic that shows a lizard running bipedally across the surface of the water!

Isn’t it interesting to see the use of biped, bipedally, and bipedalism in the same paragraph?  Once you understand the structure of biped (<bi + ped>), you can also understand the suffixes that have been added to change how the word might be used.  As an adverb, we would use <bi + ped + al + ly –> bipedally>  and as a noun describing the condition of moving on two feet we would use <bi + ped + al + ism>.

If bipedalism is a means of moving by the use of two legs, then quadrupedalism is a means of moving by the use of four legs to bear the weight of the body.  The word ‘quadruped’ can also refer to a machine.  It simply means anything “that usually maintains a four-legged posture and moves using all four limbs.”  Most often we use this word to refer to terrestrial mammals and reptiles, but there are also aquatic quadrupeds such as turtles, amphibians, and pinnipeds.  If you’re wondering what a pinniped is, I’m right there with you!  At Etymonline, the entry provides us with this information.

Word investigations lead to such interesting unintended discoveries, don’t they?  So seals, sea-lions, and walruses are quadrupeds in a similar fashion to zebras, dogs, and giraffe’s!  A look at the entry for ‘quadruped’ reveals more information.  This word is from Latin quadrupes “four-footed, on all fours.”  In contrast to the word ‘quadruped,’ there is also the word ‘quadrumane.’  That refers to an animal that is four handed or with four hands and feet with opposable digits.  Merriam-Webster describes quadrumanes as having hand-shaped feet.  I bet you are already picturing monkeys or other animals that are primates.  This word was once used more commonly in the field of zoology, even being the name of the order of mammals Quadrumana, which included non-human primates.  It is now considered obsolete.  As an adjective, someone might refer to another creature as quadrumanous.  In that instance, they are describing that creature as ape-like.

###### By Dave59 at the English-language Wikipedia, CC BY-SA 3.0

While reading about quadrupeds at Wikipedia, I learned that while the word ‘tetrapod’ literally means four-footed in the same way that quadruped does, there is a very specific difference between how the two words are used.  When comparing the structure and etymology of these two words, you may have guessed that ‘quadruped’ has two elements from Latin (quandri- “four” and pes “foot”), and ‘tetrapod’ has two elements from Greek (τετρα “four” and πούς “foot”).  Tetrapods descended from a four-limbed ancestor.  Quadrupeds use all four limbs to walk/run.

“The distinction between quadrupeds and tetrapods is important in evolutionary biology, particularly in the context of tetrapods whose limbs have adapted to other roles (e.g., hands in the case of humans, wings in the case of birds, and fins in the case of whales). All of these animals are tetrapods, but none is a quadruped. Even snakes, whose limbs have become vestigial or lost entirely, are nevertheless tetrapods.”

It wasn’t until I published this post that I heard about pentapeds.  It is one of the reasons I am so grateful for the broad audience my posts reach!  Not having much exposure to kangaroos, I always picture them as moving fast and in that case moving bipedally.  But check out this video of the walking movement of a kangaroo.  Its tail is like another foot!

Another fascinating creature that I initially forgot to mention is the octopus, an eight limbed mollusc.  There are 300 species of octopuses.  According to Wikipedia, the largest octopus ever recorded weighed 600 pounds with an arm span of 30 feet!  It was a giant Pacific octopus.  The octopus wolfi, on the other hand is the smallest known.  It weighs less than 1 gram and is about 1 inch in size.

###### albert kok, CC BY-SA 3.0 <https://creativecommons.org/licenses/by-sa/3.0>, via Wikimedia Commons

There is a character in the movie Finding Dory called Hank the septapus.  Apparently, Hank was born with eight limbs, but one was pulled off by a human child who played too roughly.  So in a sense, he might now be referred to as a septapus.  But the reality of it is that he is an octopus.  He was born with eight limbs, and if one was cut off, another would grow in its place.  (I hope I’m not ruining the movie for anyone.)  I found this information at Scientific American.  It is quite a fascinating article!

“Like a starfish, an octopus can regrow lost arms. … Rare is the octopus with fewer than eight—at least partial—arms. Because as soon as an arm is lost or damaged, a regrowth process kicks off to make the limb whole again—from the inner nerve bundles to the outer, flexible suckers.  Aug 28, 2013″

That being said, while I was looking in Wikipedia for information about the octopus, I came across a species known as the seven-arm octopus, Haliphron atlanticus.  One of its earlier names (1929) was heptopus!  You might recognize that hepta is Greek for “seven” whereas <sept> derives from Latin septem “seven.”   Seeing as the second base in the word heptopus is from Greek pous “foot,” it makes sense that this species was once named using all Greek elements.  As I read further about this specific octopus, it was revealed that it only appears to have seven limbs.  One of males’ limbs has the specific function of helping with egg fertilization.  Because of that specific function, that particular limb is kept coiled in a sac beneath the right eye.  So the reality is that it has eight limbs like all octopuses; we just don’t see them all!

Whoever created the character Hank, knew about the Latin element for “seven” and knew of the spelling of ‘octopus’ and blended the two.  You might call this a hybrid word because it combines elements from two languages.

I bet you think you know about centipedes.  Well, at least you think you know how many legs they have.  According to Wikipedia, centipedes always have an odd number of legs.  That means they would never have exactly 100!  Surprising, isn’t it?  In fact they can have anywhere from 30 to 354 legs!  I bet they were named centipedes because it seemed like they had a hundred legs when they were seen moving.  A few more interesting facts are that they are carnivorous and range in length from a few millimeters to 12 inches.  Wow.

###### Centipede: Kingsley, J. Sterling (1890) Popular natural history: a description of animal life, from the lowest forms up to man – Vol. 1

The name ‘millipede’ comes from Latin and means “1000 feet.”  If you’re going to guess that they don’t actually have one thousand feet based on what you just learned about centipedes, you’d be correct.  There is one species of millipede (Illacme plenipes) that holds the record for having 750 legs.  That is more than any other animal in the world!  Millipedes are detritivores (eat dead plant matter) and are found in central California.

Grouped by months

There are four months in our year that also contain one of these word elements that indicate a number.

September
October
November
December

Many people might guess that October might have something to do with “eight,” but they’re not sure what.  Well, it used to be the eighth month, that’s what!  In Ancient Rome, March was considered the first month of the year.  Interestingly enough, July was originally named Quintilis “fifth” and August was originally named Sextillia “sixth.”  While March, April, May, and June were named with other ideas in mind, July, August, September, October, November, and December were named for their order.  January and February were added to the end .

Julius Caesar brought about changes to the calendar when he aligned it to the earth’s revolutions around the sun.  January and February were moved to the beginning of the calendar.  This caused some of the months to be out of alignment with their numbered names.  That’s why September, October, November, and December are no longer the 7th, 8th, 9th, and 10th months.

It was in remembrance of Julius Caesar (and to honor him for his adjustments to the Julian calendar) that Quintilis was renamed as July.  In a similar way, the month of Sextillia was renamed August to honor Augustus Caesar, the first Roman emperor.

Final adjustments to the calendar were made in 1582 by Pope Gregory XIII who corrected some inaccuracies with the Julian calendar.  We refer to the calendar we use today as the Gregorian calendar!   Look at these two links (Wonderopolis and The Old Farmer’s Almanac)  for more information about the naming of the months that you and your students can read together.

### Grouped by millions

The following is a list that I have purposely shared with my students each year.  There are so many great morphemes to know here!  Let’s start with the word ‘million.’  After all, all the rest of these words share part of its spelling.  According to Etymonline, it was first attested in the late 14c.  At that time it was spelled milioun and used to mean “a thousand thousands.”  Further back it was from Old French and spelled million.  Further back yet it was from Italian millione and literally meant “a great thousand.”  And the furthest back ancestor we find is Latin mille.  You can see that the structure would be <mille/ + ion>.  According to the Century Dictionary, “The French, who like other northern peoples, took most if not all their knowledge of modern or Arabic arithmetic from the Italians.”  (I found that quote at Etymonline in the entry for ‘billion’).

Now when you look at the entry for ‘billion,’ you see that it was created from Latin <bi> “two” and million.  When you look at the entry for ‘trillion,’ you see that it was created from Latin <tri> “three” and million.  It is described as the third power of million.  There is a pattern developing here.  we can see the structure of ‘million,’ but can’t see the same structure in ‘billion’ and ‘trillion.’  Interesting, isn’t it?  Every once in a while I come across words like this that are modeled on another spelling which makes them hard to analyze on their own.  We can know how they came to be that way, but we can’t analyze them as we might like to.  Instead, in a list like the following, we might underline the morphemic element that indicates a number.  An example would be to underline the <quint> in ‘quintillion’ and mark that it means “five.”

Families of million  (1-20)

million
billion
trillion
quintillion
sextillion
septillion
octillion
nonillion
decillion
undecillion
duodecillion
tredecillion
quattuordecillion
quindecillion
sexdecillion
septendecillion
octodecillion
novemdecillion
vigintillion

The following link takes you to Sbiis Saibian’s Large Number Site.  It is a web book on large numbers.  The link takes you to the specific chapter called “The -illions Series.”  I found that this chapter thoroughly tells the story of large numbers beginning with ‘million.’  I found much of the same information in my research, so I trust that this information is accurate.  What’s nice is that this author has the story, along with the different versions of what to call the numbers larger than million all in one place!  It’s quite fascinating, and I encourage you to take a look!  The part of the chapter that deals with how these numbers came to have these names is under the the article 2.4.2 – Origin & Development of the -illions . Here’s a brief excerpt to whet your appetite!

[The term “million” doesn’t seem to exist at any time before the 13th century (1200’s). Apparently it is an augmented form of the latin word “mille” meaning thousand. By dropping the e and adding the -ion as a suffix one could translate “million” as literally “Great thousand”. It is not known who first coined this term. It was used sparingly in the centuries to follow and was sometimes regarded as a kind of slang and not legitimate language (perhaps much the same way neolisms today are regarded as unofficial ), and writers more often than not preferred the non-ambiguous “thousand thousand.”]

A particularly interesting fact is that the list doesn’t continue beyond the 20th family of million.  But then, it is actually quite rare that any of these number names are used with any regularity beyond the use of trillion!

### Grouped by shape

The following list is no doubt very familiar to anyone who has studied geometry.  It is a list that I’m sure many school children have seen before and perhaps struggled with figuring out how to remember what each word means.  To make that task easier,  I usually put it side by side with the list you were just reading about – the list of numbers beyond millions.  It is so interesting when you compare the two lists.  Instead of me pointing out similarities and differences, the students can do it for themselves.  I’ve made a chart so that you can see at a glance how all the numbered groups I’ve mentioned relate to one another in meaning.  You’ll find the chart below this list of shapes.

What I have found interesting in looking at the list of shapes is that the first one (triangle) pretty clearly refers to a shape with three angles.  The second shape listed is a quadrilateral which in math books is also referred to as a quadrangle.  But why the two names?  The word quadrangle originated in Latin as quandrangulum and was used to mean a “four-sided figure.” What’s interesting to me is that the word literally means “four angles” and yet it is defined as a “four sided figure”.  Perhaps it is nothing to get hung up on since a four-sided figure will have four angles and a shape with four angles will have four sides.  The word quadrilateral originated in Latin as quadrilaterus and was used to mean “figure formed of four straight lines.”  That makes sense to me because I’ve seen ‘lateral’ in other words and it has always had something to do with “side.”

Sometimes the word is used generally.  In the following pictures you are looking at a lateral view (side view) of a goat skeleton and also a building.

###### ProtoplasmaKid, CC BY-SA 4.0 <https://creativecommons.org/licenses/by-sa/4.0>, via Wikimedia Commons

Other times the use of lateral is more specific.  Notice the darkly pigmented lateral line on this fish?  Many fish have a lateral line.

###### Brian Gratwicke, CC BY 2.0 <https://creativecommons.org/licenses/by/2.0>, via Wikimedia Commons

Here is an example of a lateral pass in football.  According to Wikipedia, “The ball carrier throws the football to a teammate in a direction parallel to or away from the opponents’ goal line.”  So the ball is moved to one side or the other.  In Canadian football this is more commonly known as an onside pass and in American football it is known as a backward pass.

###### U.S. Navy photo by Chief Warrant Officer 4 Seth Rossman., Public domain, via Wikimedia Commons

Getting back to the mathematical use of these words, I also recall seeing ‘equilateral’ in math texts that I have used with children.  This indicates that a shape has equal sides.  And just as I have seen quadrilateral and quadrangle referring to the same shape, I have also see equilateral and equiangular referring to the same shape.  Again, when we have a shape with equal sides, it will have equal angles.

Now let’s look at ‘pentagon’ which will help us understand the spellings of the rest of the words on this list.  According to Etymonline, its Greek root is πεντά–γωνος (pentagōnos) meaning “five-angled or five-cornered.”  The Greek root is a compound made up of pente “five” and gōnia “angle.”  From this information we can see that hexagon, septagon, octagon, and the rest are words for a shape with a particular number of angles.

An interesting relative of  Greek gōnia is Greek gony “knee.”  Do you see what a knee and an angle have in common?  The Latin equivalent to Greek gony is genu.  You may be familiar with that base in the word ‘genuflect’ which is when someone bends their knee in worship or out of respect.  The Old English word for knee was cneo or cneow.  In Old English, the initial <c> would have been pronounced as /k/.  Now you can see where our <kn> digraph spelling came from!

3    triangle
5    pentagon
6    hexagon
7    septagon
8    octagon
9    nonagon
10   decagon
11   undecagon
12   dodecagon

I brought together all of the words I’ve mentioned that have something to do with a numbered group.  Might be a great discussion starter!  Perhaps someone will think of another kind of group that gets numbered in this way that I haven’t thought to include.  How exciting!

Can you and your students spot instances in which the same element is used in the different lists?  Do you notice that The Hellenic people and the Romans had different names for elements that represented the same amount?  Do any of the word elements we see remind you of words from other languages?  What does the rest of the word (besides the element that indicates a number) in each list mean?

Here’s an idea for those of you in classrooms.  Split your class into groups and pair up each group with one of these categories.  Let the students find what they can about the words, their origins, and the way we use these words in our society.  This would obviously be a project they work at each day for a week or two depending on your students. Your job is to circulate between the groups to offer guidance and celebrate what they are finding.  Then let each group rehearse in a corner of the room before present their findings to the rest of the class.  In my experience, the groups may not finish at the same time.  That’s fine.  Let them present when they are ready.  Have some new investigations ready for the groups who finish first.

The sources I used today were Wikipedia, Merriam-Webster, Etymonline, Liddell and Scott Greek Lexicon, and the OED.

# Whispered Words of Wisdom

The students were not quite seated before the whispering began.

“I’ll ask if you don’t want to!”
“No, Ben should be the one to ask.  He’s the one who brought it up to begin with.”

As you can imagine, my interest was piqued.  I looked at Ben whose cheeks were bright red.  “Ben? Do you have a question for me?”

I hope you can picture just how big my smile was at that moment!  This was the first orthographic question of the year that was inspired by something happening outside of our classroom!  I was delighted, and I hoped my smile conveyed that!  “What a great question!  Tell me more.  What were you talking about when this question came up?”

Ben began by explaining that in math  class they were discussing polygons.  Specifically they were talking about shape families.  When they got to quadrilaterals, the teacher asked if students knew any other words with <quad>.  As students named words, it was the consensus that words with <quad> have something to do with “four.”  When Ben asked whether or not <squad> was related to <quad>, the teacher suggested they bring that question to me.  Perfect!

The first thing we did was to recreate the list of words the students had thought of earlier in math.  They included:

Then I asked, what is the spelling they have in common?  What specific string of letters do you see in each and every word?

The first response was <quad> (no doubt because that was what they had been talking about earlier).  I asked them to look again and more carefully.  That was when several hands shot up at once.  “I see q-u-a-d-r!”

Great!  Now I underlined the <quadr> in each word so we could look at the rest of each word.

Before I could even ask a question about this word, a student raised their hand to say, “The <i> could be a connecting vowel!”  Awesome!  I didn’t expect that, but it is true!  It could be!  Next I asked if anyone recognized any suffixes.  Someone called out <er> and <al>.  Great!  Those might indeed be suffixes.  They often are.  (Notice that instead of saying, “You’re right,” or “Sorry, you’re wrong,” I’m using words like “might” and “could.”  At this point we are doing some out-loud thinking about this word.  We will consult a resource when we have had a chance to think through our observations.)

At this point I asked if anyone knew what <lateral> meant.  No one did.  So I said, “What if I told you that a fish has lateral fins?  Does that help?”
There was a moment of hesitation as students mulled over this idea.  Then someone said, “Side fins?”
“Yes!  Do those of you who love to play football know what a lateral throw is?”
“Yes. It’s when you throw the ball in a backwards or sideways direction.”
“Right.  So we’re seeing a sense of “side” in both when we refer to a lateral fin and a lateral football throw.  So now tell me what a quadrilateral is.”

Several students at once responded with, “Four sides.”

Right away I wanted someone to tell me what quadruplets were.  Everyone seemed to know that it was when four babies were born in a single birth.  None of us knew much about the <uplet> part, but had heard it as part of <triplet>, <quintuplet>, <sextuplet>, <septuplet>, and <octuplet>.

Having identified the base as <quadr> made the rest of this word recognizable.  I could just ask, “What is a quadrangle?”  And several students replied that it was a shape with four angles.  Instead of quickly moving on, I wondered aloud whether a quadrangle and a quadrilateral could refer to the same shape.  Hmmm.  After a bit of thought, the students agreed that a shape with four angles would also have four sides, and a shape with four sides would also have four angles.

The students quickly named <million>, <billion> and <trillion> when thinking of the second part of this word.  I went on to name <quintillion>, <sextillion>, <septillion>, <octillion>, <nonillion>, <decillion>, <undecillion>, and <dodecillion>.  (I love knowing this list because I can see the same <sept> in <septillion> as I do in <September>, the same <oct> as in <October>, and the same <dec> as in <December>.)

The students weren’t as familiar with the use of this word.  I explained that if an area were to be split into four areas, one of the areas would be called a quadrant.

At this point a boy raised his hand and stated, “I don’t think <squad> fits with these.  None of these words begins with an <s>.”
I loved knowing that the original question sat in his head as we were discussing all the words with <quadr>.  I replied by saying, “You might be right, Sam.  But then again, we can often be surprised by what we find.  I don’t know the answer, but it’s almost time to look.”

But there was still something the students were wondering about.  “Isn’t quad a word all by itself?”
“Yes.  I think you’re right.  I wonder if it isn’t a clip of one of the words we’ve looked at.”

Then I went on to explain that there are other words that had been clipped from a longer version – words like auto from automobile and flu from influenza.  This was the perfect time to go over to my desk and pull up Etymonline on the Smartboard.  I looked up <quad>.  The entry was very interesting.  It seems that <quad> has been a shortening (or clip) of several longer words over the years.  In 1820 it was a shortening of <quadrangle>, which at the time referred to a building on a college campus.  In 1880 it was a shortening of <quadrat>.  In 1896 it was a shortening of <quadruplet>.  We were all fascinated to read that a quadruplet originally referred to a bicycle for four riders!  It was only later on that it referred to four young at a single birth.  Lastly, <quad> was a shortening of <quadraphonic> in 1970.  I remember my older brother talking about wanting quad speakers to go with his stereo!  One of the students brought up one other more recent use of <quad> as a clip.  They mentioned quads as in  leg muscles.  We decided that in that sense, <quad> must be a clip of <quadriceps>.  This is an example of a word that needs to be in a context in order for us to know what it is referring to.

Once we had looked at <quad>, it was time to look at <squad>.  This was really fascinating!  In 1640, this word was used to mean a small number of military men.”  That was a familiar use of the word for everyone.  It is kind of what we were expecting.  As we read on, we noticed this word had been in French as esquade, Middle French as escadre, and Spanish as escuadra or Italian squadra where it meant literally “square.”  Notice how the spelling in French, Middle French and Spanish began with <es> and the Italian spelling began with <s>.  The next interesting information was this word was from Vulgar Latin (the Latin spoke by the everyday people) and possibly spelled (not for sure – notice the asterisk next to the spelling) *exquadra meaning “to square” from Latin ex “out” and quadrare “make square.”  Ben, the boy who originally asked the question noticed the connection between a square and four right away.  Another student pointed out the <quadr> spelling in the Latin word quadrare.

All in all, this glorious discussion took about 25 minutes.  I enjoyed identifying what we knew already, and what things we could relate to other things without running immediately to a resource.  There is such value in recognizing the connections one already knows.  This is how the students will strengthen their confidence in their ability to connect one word to another.

What an opportunity to point out that both<quadr> and <squad> began in Latin, but had different journeys into Modern English.  Both were used in French, but <squad> was also used in either Spanish or Italian and that different journey has been reflected in their spellings.  It turns out that they ARE related!  They are related etymologically, but because they do not share spelling, they are not morphologically related.

Now isn’t that something worth whispering about?

# Pi, Pi, Mathematical Pi

Every day this week we took a look at Pi.  On Monday and again later in the week we listened to the Mathematical Pi Song (to the tune of American Pie). I gave each student a paper plate and assigned them a number.  They were to write that number on the plate and decorate.  We hung the plates in a “Pi Number Line” down the hall.  That way we could think about this irrational number all week!  I gave each student a copy of the Greek Alphabet and we practiced reciting it.  I wanted them to see the letter Π and know where it came from.

On Tuesday we reviewed the words <radius>, <circumference>, and <diameter>.  Earlier in the year we had discovered that the word sum for <circumference> is  <circum> + <fer> + <ence>.  The prefix <circum> means around and the base <fer> means to carry.  The circumference is the distance carried around the outside of the circle.  The word sum for <diameter> is <dia> + <meter>.  The prefix <dia> means through and the base <meter> means to measure.  You measure the distance through the circle to find the diameter.    The word sum for <radius> is <radi> + <us>.  The base <radi> means rod, spoke of wheel, and ray of light.  I like to picture the radius of a circle as a spoke of a wheel.  I brought miniature donuts.  We multiplied the diameter of a donut by Pi, and in doing so calculated the circumference of the donut.  We watched another video having to do with Pi.  This one was the  ‘The Dance of the Sugar Pi Fairy’!  How fun to find creative ways to memorize digits of Pi.  (Certainly not a necessary thing to do, but certainly a tempting thing to do)

On Wednesday I brought cookies.  We measured the circumference and divided that by Pi to calculate the diameter of the cookie.  We also listened to some very creative math/musicians who put digits of Pi to music!  The first one is called Song from Π.  It is an enchanting piano piece with interesting facts about the number Pi.  The digits of Pi float by as they are being played on the piano.  Quite honestly, we loved the music, but found it difficult to follow the digits.  The second one was fascinating for other reasons.  The artist repeats the first 31 digits of Pi.  He uses different instruments and different tempos.  It’s called What Pi Sounds Like.

On Thursday I brought peanut butter cups and we measured both the circumference and the diameter.  We then found Pi by dividing the circumference by the diameter.  Several students came up with 3.166.  That’s pretty close!  We watched a video about art created from the digits of Pi.  It is called Pi is Beautiful.   We were inspired to see what kind of art we could create using the digits of Pi.  I found a circular graph.  We numbered around the outside using the digits of Pi.  If the number was a 3, then 3 squares toward the center were colored in.  Some students assigned a specific color to each number 1-9 to see what that would look like.  Some students stuck to using 1, 2, or 3 colors.  We were very pleased with the results and hung them in the hall above our Pi number line.

Then came Friday – March 14!  Students brought in pies – 17 in all!  We ate pie several times throughout the day and shared with adults around the school.  We held a contest to see who had been able to memorize the most digits of Pi throughout the week.  Our winner memorized 100 digits of Pi!  We were amazed!  Our second place winner memorized 61 digits and our third place winner memorized 24 digits!  Wow!  What fun.  Enjoy the following video.  It’s kind of a medley of our day!

# Celebrating Our Thousandth Day of School!

If our school year is approximately 180 days long, and we multiply that by the five years my students have been in school (kindergarten, first grade, second grade, third grade, and fourth grade), at the end of fourth grade they’ve been in school for 900 days.  So that means that on the 100th day of fifth grade they are celebrating their 1000th day of school!

I decided to center our celebration around the number 1000.  Mid-morning I mixed up a snack stew.  In the stew were 100 each of 10 different ingredients.  There were things like animal crackers, marshmallows, cheese crackers, pretzels, and conversation hearts.  Yum.  Throughout the day we jumped for joy in ten sets of 100 jumps.  Whew!  Students answered questions such as, “How old will you be in 1000 days?  1000 months?  What are the factors of 1000?”

Take a peek and just see how much fun we had!

# It’s Time for a Math Break!

Today was such a fun Friday.  My math students have been improving all week when it comes to  staying focused during work time.  Staying focused means finishing work in class and making less errors.   Making less errors means less fix ups and a better chance of building a deep understanding of the skills being practiced.

Today we decided to take a break from learning math and instead focused on teaching math.  We are currently learning how to find the fraction of a number.  For example, do you know what 5/9 of 27 is?  If the answer doesn’t come quickly, watch this video and then see if you can figure it out.

I know.  Entertaining, wasn’t it?  But besides all that, do you think you could figure out what 5/9 of 27 is now?   Leave a comment, and we’ll let you know if you’re right.

# Celebrating Pi Day with Pizazz!

Pi.  Pi.  Mathematical Pi!   We had so much fun today!  We sang songs.  Some were based on familiar Christmas tunes, one was a rap, and one went to the tune of American Pie.  We learned about William Jones who first recorded the symbol for Pi.  We  read about Gaurav Raja who at one time held the record for reciting 10,000 digits of Pi.  We learned about Ludoph van Ceulen who had the first 35 digits of Pi engraved on his tombstone (until his wife swapped it out for something more proper)!

We felt that in order to really understand Pi, we needed to really understand circumference, diameter, and radius.  The class divided into three groups and the word investigations began!  With limited time, no group quite finished, but they made great progress.  I find it interesting that many students still fall back on old habits of dividing words by syllables instead of beginning with their lists of tried and true affixes.  Patience and practice.  They must discover the logic of that for themselves.   In one part of the video, the group investigating the word <diameter> started laughing.  I had just asked if the words they found that begin with <dia> had to do with the definition of <dia> which is through.  You see, diarrhea was on the list and they definitely saw the connection!

Next we held a Pi Digit Contest.  We were looking to see who could memorize the most digits of Pi with only two days of preparation.  Our first place winner recited 65 Pi digits!  In second and third place, students recited 46 and 42 places respectively.  The fourth, fifth, and sixth place winners recited 36, 28, and 27 places.  What an amazing accomplishment for all who gave it a try!

Then there were the pies!  Yum!  We had cherry, lemon meringue, apple, turtle, peanut butter, chocolate, toasted coconut creme, pecan, banana creme, and brownie pies.  Heavenly!

During math, we actually measured circles of all kinds and calculated Pi for ourselves by dividing the circumference by the radius.

Lastly, we sang our favorite Pi song just one more time.  Love Pi Day!

# Divisibility Rules!

Simplifying fractions becomes easier when you can recognize what both the numerator and denominator are divisible by. Hope you enjoy our video on the divisibility rules for 2, 3, 4, 5, 6, 9, and 10.