I’ve been working on my next *Playful Math Singles* book, based on the popular Things to Do with a Hundred Chart post.

My hundred chart list began many years ago as seven ideas for playing with numbers. Over the years, it grew to its current 30+ activities.

Now, in preparing the new book, my list has become a monster. I’ve collected almost 70 ways to play with numbers, shapes, and logic from preschool to middle school. Just yesterday I added activities for fraction and decimal multiplication, and also tips for naming complex fractions. Wow!

Gonna have to edit that cover file…

In the “Advanced Patterns” chapter, I have a section on math debates. The point of a math debate isn’t that one answer is “right” while the other is “wrong.” You can choose either side of the question — the important thing is how well you support your argument.

Here’s activity #69 in the current book draft.

### Have a Math Debate: Adding Fractions

When you add fractions, you face a problem that most people never consider. Namely, you have to decide exactly what you are talking about.

For instance, what is one-tenth plus one-tenth?

Well, you might say that:

of one hundred chart

+ of the same chart

= of that hundred chart

But, you might also say that:

of one chart

+ of another chart

= of the *pair* of charts

That is, you started off counting on two independent charts. But when you put them together, you ended up with a double chart. Two hundred squares in all. Which made each row in the final set worth of the whole pair of charts.

So what happens if you see this question on a math test:

+ = ?

If you write the answer “”, you know the teacher will mark it wrong.

Is that fair? Why, or why not?

CREDITS: Feature photo (above) by Thor/geishaboy500 via Flickr (CC BY 2.0). “One is one … or is it?” video by Christopher Danielson via TED-Ed. This math debate was suggested by Marilyn Burns’s blog post Can 1/3 + 1/3 = 2/6? It seemed so!

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If you are given 1/10 + 1/10 =?

There is no reason to believe that it means anything other than add two fractions with the same denominators. So you must give the answer 2/10. If you were told to give your answers in reduced form, 2/10 =1/5.

It’s fair for the teacher to mark it wrong if your answer is 2/20 because you don’t have any context given; if the 2 rows of ten from 2 hundred charts was given, you would not have 1/10 + 1/10.

It would be 1/20+1/20= 2/20 = 1/10.

1/10 + 1/10 does not equal 2/20, ever. Why confuse a student with this kind of wording,” 2 pairs of charts”?

I do not think this is a good example for use of hundred charts!

I do think math debate might be interesting, though.

The point of this debate is to think more clearly about the units we choose to count in. What we define as “one whole thing” determines the meaning of a fraction, or of any number. If our units are muddled, we can’t communicate our ideas to another person.

Perhaps using hundred charts for this debate is too far a stretch, since charts are normally not doubled up. I wrote it that way to fit the theme of my book.

But there is nothing inherently wrong with a “pair” being treated as a unit. The main problem with my 2/20 example is that the unit we’re counting in

changesmid-equation.But whether you like my example or not, you do need to deal with the idea behind it. This sort of thinking about fractions is VERY common for elementary students, and it can even confuse an experienced teacher. See Marilyn Burn’s article: http://www.marilynburnsmathblog.com/can-1-3-1-3-2-6-it-seemed-so/

As teachers, we have to help our students learn the importance of being clear and consistent with units, while still being flexible enough in our thinking to understand someone else — especially a child — who uses units in a way we didn’t expect.

A pair of charts is 2 hundred charts.

So, you have H.C.1 and H.C. 2 is the same as the pair of hundred charts.

So, what part of the first hundred chart did you add to the pair of hundred charts? You have 1 row out of the total # of rows in h.c.1 and h.c.2 which is 20.

I am

Ok. I thought that this was about units, and I get that it can get confusing. Would you please give me an example of a time it happened to you with a student, with a real situation when the units got confused.

I saw Marilyn’s case before, I think, on your blog. I will check.

I agree we need to be flexible. It’s just that at some point you, especially with scientific work, you need to comply to universally accepted units, like meters, liters, etc. I do get your point,though. 🙂

I don’t have a specific story like Marilyn’s. I wish I had kept a more consistent journal when my kids were growing up.

But I remember so many times in working with my kids when they would give an answer that was wrong according to the standard way of understanding math, but when I asked them to explain their thinking, it was right from their point of view.

The kids do need to learn the standard way of reading fractions. But the most important thing I want my kids to learn is that

math makes sense. That it isn’t just rules handed down from on high, but something they can think and reason about for themselves. And that means I need to listen to and respect their intuition, even when they do something nonstandard, like change units midstream.As a master teacher once wrote:

“Most remarks made by children consist of correct ideas very badly expressed. A good teacher will be very wary of saying ‘No, that’s wrong.’ Rather, he will try to discover the correct idea behind the inadequate expression. This is one of the most important principles in the whole of the art of teaching.”

— W. W. Sawyer

Vision in Elementary Mathematics

One thing that was mentioned, in the comments on Marilyn’s blog, was ratios, written like fractions, but added the way you said for the test question.

So, maybe you can add 1/10 + 1/10 and get 2/20. What do you think? I can’t say I’ve done this activity with the hundred chart. We used it mainly for addition and doubling until we were off the chart.

I know your children have read books, so I’m sure you will agree sometimes it’s good to let them have some math time that is just exploring. I also just have math discussions and we used to play games, from Scholastic Professional Books. I also found NCTM to be helpful .Math forum just joined with them. I’m sure you know about these resources. I just wanted to let you know what I’ve actually used personally.

Marilyn’s blog is very interesting.Thank you for sharing it with me. I see ratios being written like fractions, but being added differently. That is very confusing.

How would you add 2 ratios together?

Adding two ratios is sort of like doubling a recipe. Say you want to make pink lemonade. Your lemonade recipe calls for 1 cup sugar to 6 cups liquid (water and some lemon juice). And your cranberry juice recipe calls for 1 cup sugar to 6 cups water (after boiling cranberries in it).

When you combine the two, you have 2 cups of sugar to 12 cups of liquid in your pink lemonade. Or perhaps you prefer to use only half as much cranberry juice as lemonade, so you’d have 1 1/2 cups sugar to 9 cups liquid.

Ratios can be written as fractions, but working with them involves thinking about the situation. You can’t just apply the standard fraction rules blindly, because it may not make sense that way.

I will try to find an example. Someone on Marilyn’s blog said you can add both parts of 2 ratios together, when written as a fraction without a common denominator. There was no example. Maybe it’s just confusing what the comment was referencing.

I think your example shows this:

1/6 ratio of sugar to liquid for lemonade; 1/6 ratio of sugar to cranberry juice. Put them together and you have 2 cups of sugar to 12 cups of liquid. Does that mean 1/6 + 1/6 = 2/12?

Maybe I’ll get it if I write it out. I see now why it can be confusing. Thank you for the example.

I think the doubling threw me off,so your example helps. ☺ Thanks. You can write ratios with colons to show it, right?

Yes, I prefer to write ratios using colons, but I remember that MathCounts preferred fractions, back when my math club students used their materials.

I can’t imagine writing ratios as an addition problem, though, since they don’t behave like standard fractions. They behave like the changing-units-in-mid-equation examples above.

Thank you, Denise. I looked at the MathCounts website and thought it might be helpful to me with my family.

I watched this ratio video:

https://www.mathcounts.org/resources/video-library/mathcounts-minis/mini-59-ratios

He made a chart and put the parts and the totals in the boxes. I think that helps, as did your lemonade example.

A chart can be very helpful in working with ratio problems. I have an example on my blog, based on an old brain-teaser rate problem. denisegaskins.com/2010/06/28/rate-puzzle-how-fast-does-she-read

Hi,

This is my attempt to better word your original post. I don’t think you can write the ratios for this because if 1/10 is a ratio and we add the labels of what we have, it would be one row out of 10 rows, and 1/10 of a chart which is a fraction, not a ratio.

Does that make any sense? You write it differently depending whether you have fractions or ratios.

Then, if you write the ratio of 2 rows to 20 rows, that is equal to 1/10 of 2 charts. I think the number of rows is necessary to make it ratios. We don’t add ratios. We compare them. One row out of ten is same ratio as two rows out of twenty.

When we say fractions are numbers, we can put them on a number line. A ratio cannot be found on a number line, though. And that is why ratios do not “behave like” fractions unless the fractions are equivalent. and that’s what cause confusion in my opinion, in Marilyn’s example.

When you ask:

“So what happens if you see this question on a math test:

1/10 + 1/10 = ?

If you write the answer “2/20”, you know the teacher will mark it wrong.

We don’t add ratios, so it must be fractions, and that answer 2/20 would be wrong.

Does this make the example clearer? Still useable?

I think I learned from thinking about all this. I got a little help from a book called, Common Core Math for Parents for Dummies, by Christopher Danielson, PhD.

I guess he is a math blogger and a curriculum writer.

I hope this helps.

Note for last post:

Sorry, I made a few mistakes I can’t edit hopefully, you’ll figure out what I’m trying to say.

I understand how ratios and fractions work. This was more of my attempt to clear up the confusion than it was to rewrite what you are putting in your new book about hundred charts. I find writing very challenging and I would never dream of writing math books. Keep up the good work. You make me think & for that I say Thank you🐈

Hi, again. This is my 3rd post in a row. I was just thinking I must not have answered in debate form. Am I overstepping the rules for debate? I did pick a side. I am attempting to explain why these are ratios you are working with, just written in fraction form. Since ratios are comparing 2 amounts of things, the ratios, written to look like equivalent fractions are actually proportions, right? The ratio of rows to charts is 1:10 or 1/10. You don’t add ratios, you change the numbers in the positions for rows and charts and get 2 rows per 2 charts. The way you write it with pairs would have to be something I can’t think of right now- a fraction of a pair? No:(

Since this is a debate and you said,”The point of a math debate isn’t that one answer is “right” while the other is “wrong.” You can choose either side of the question — the important thing is how well you support your argument.”,

I would please like to know if 1. Did I choose a side of a question?

2. How well did I support my argument?

3. Does anyone else want to take up the other side of the argument that it was NOT fair to mark the answer wrong?

I think you made a good argument for the “Yes, it’s fair” side of the debate. If I’m understanding right, you’re saying this: When we see an equation with fractions, without any explanation, we have to assume they are simple common fractions and work with them according to the standard fraction rules and definitions. To do anything else is to introduce nonsense.

I do think you’re wrong to dismiss the idea of using “pair” as a unit, but your argument doesn’t rely on that.

To make an argument from the other side, I would focus on the importance of units. Perhaps that’s my physics degree showing since we spend a LOT of time working with and converting units in physics. If a student added unit labels to make a different answer make sense, I would be inclined to give them the credit. In my mind, it shows that they understand fractions — and that they’re feeling the contrariness of gifted-student boredom with a too-simple problem. But a student who writes 2/20 without any justification deserves to be marked down.

In the original post, Marilyn Burns was dealing with elementary students who hadn’t yet mastered fractions. In that case, I would want to focus the class’s attention on what the fractions are measuring. Talk about how the units (the things they were counting) changed in mid-equation. Play with some of Christopher Danielson’s “How Many?” pictures. Discuss the importance of labeling numbers, especially when you are changing the way you count. And then talk about what it means when we write numbers by themselves, without units, and why being consistent matters.

And if I could remember the name of it, I’d read them the picture book with counting riddles like “1 + 2 = 1.” [Possible answer: One large front wheel plus two small back wheels equals one tricycle. Or one baby plus two parents equals one family.] We’d probably take some class time to make up counting riddles of our own. Anything that makes kids laugh helps with learning.

I want to respond to your side of the debate. You are saying you wrote a fraction answer with no labels after them. If there are units that these fractions represent tell me what they are, we can see if the 2 fractions add up to your answer. Since the hundred charts are there, it could be:

Well, you might say that:

1/10 of one hundred chart

+ 1/10 of the same chart

= 2/10 of that hundred chart

But, you might also say that:

1/10 of one chart

+ 1/10 of another chart

= 2/20 of the

pairof chartsMy problem with this is not with the words but with the use of the math symbols to represent it mixed with the words.

Let’s say we want to write a true mathematical statement for the parts of the pair of charts we are adding. Let’s say we want to use 1 pair of hundred charts as the unit.

Is it true, then, that the parts that we are adding to get the 2/20 of the pair of charts must be converted to how much they are in pairs of charts?

One-tenth of a chart is how much, in fraction form, of the pair of charts?

If you follow what I am saying, do you come up with a different equation with the answer 2/20 of the pair of charts?

If any of this is correct, I think we have rid ourselves of the idea that there are any ratios involved and are only looking at units and adding fractions.

Let me know if this is unclear. I am trying to write it out on paper.

Thank you.

In my puzzle, there are no ratios involved. It’s just a question of units.

But the idea of ratios comes in when we try to make sense of the student thinking in Marilyn Burns’s blog post.

Those students were mixing the concepts of fraction and ratio. They weren’t understanding *any* of the fractions as numbers, only as part/whole comparisons.

Within that “ratio” way of thinking, their equations made sense

to them.They hadn’t mastered fractions enough to see “1/3” or “2/6” as actual numbers. The kids would never have thought to write “2 + 2 = 2.” Numbers don’t act that way.

But they didn’t recognize that “1/3 + 1/3 = 2/6” was a parallel sort of nonsense — adding two numbers to make a sum that’s the same as just one of the original numbers.

Ok. I mixed up the 2 examples.

http://www.mathantics.com/section/lesson-video/ratios-and-rates

I want to share this video because it explains fractions, ratios and rates. Sometimes, I think it is necessary to give information to students that help them see what is going on, so they can recognize ratios when they are written as fractions.

Units are part of rates which are different than ratios. It might be best to watch this relatively short and entertaining video before we continue the debate, though.

Thank you.

I think I may have it: you write it terms of a pair of charts. Each chart is half of a pair of charts. So 1/10 of 1/2 of a pair of charts plus 1/10 of 1/2 of the other 1/10 of 1\2 of the other chart equals 2/20 of the whole pair of charts.

So, it may be out symbols are wrong, maybe.

Yes, that would be a valid way of writing the equation. One-tenth of a half plus one-tenth of a half together are two twentieths. In that case, you are still keeping the same unit throughout your equation. The unit is the pair of charts.

But when the unit changed within the equation (switching from single charts to the pair), that was a different thing. It would be more like saying that two dimes plus a nickel equal a quarter of a dollar.

You would never write an equation “2 + 1 = 1/4” without the units. But when the units are put with each number, the equation makes complete sense.

2 dimes + 1 nickel = 1/4 dollar

In any real-world situation, the units are a vital part of the meaning of any number.

It took me awhile to get that equation with 1 pair of charts being the unit.

Equation,MATHEMATICS

a statement that the values of two mathematical expressions are equal (indicated by the sign =).

I was satisfied that I didn’t have words mixed with symbols and that the numbers in the equation by themselves we’re a true statement.

I’m not satisfied to write

2 dimes + 1 nickel = 1/4 dollar

To show it’s true, you write

2( 10 cents) + 1(5 cents) = 1/4(100 cents)

Then, 20 cents + 5 cents=(100/4) cents

And 25 cents = 25 cents

I’m sure we could explain it so children could understand it.

In physics, the units have to be converted to all be the same. The word equations are necessary; we express our ideas in words and write and rewrite the numbers until they show the relationship is true.

Just because you write numbers and words with a plus sign and an equal sign doesn’t make it true.

So, if you write a mathematical equation, it needs to be true whether their are units or not.

I found a true equation for the 2nd word equation you wrote about the pair of charts.

So, maybe I would be able to help a child learn about adding fractions and units, just ia a different way.

I guess I just disagree that this is an equation: 2 dimes + 1 nickel = 1/4 dollar.

So, I guess I will leave it at that since I showed you how I would write the units and the equation.

I’m not sure if you like how I wrote a new equation; it used the pair of charts for the unit AND it makes a true statement that can be written as a true equation.

Hi Denise,

I forgot that this is an activity to do with your children or another group of children.

I’m sorry I got so carried away.

Thank you for all the interesting resources you mentioned which make it clear that you are focusing on what units you are using when describing how much of the hundred chart and charts you have.

I think maybe the book is called “1 + 1 = 5 and other unlikely additions”

By David LaRochelle.

Ah, yes, that’s the book!

I’ve enjoyed the discussion. It’s always interesting to hear how someone else thinks about a puzzle. 🙂

Yes, I’m glad you are interested in what I am thinking. I started writing out things on paper so you would get less stream of consciousness thoughts from me. I need to keep pen and paper with me as much as possible.

Your adding of nickels and dimes gave me an idea for the hundred chart. I’m not sure if it’s a separate activity or just an alteration of this activity above. I’m not done with it yet. ☺

My idea is to put a penny(1 cent) on each square of both of the hundred charts.

Then, proceed with your questions. Since this is about adding fractions, using labels that are a fraction of a dollar and the fact that 100 cents = 1 dollar would be as far as I would go with it to keep it about adding fractions. I’m trying to work out what the pair of charts is.

To be continued…..

Yes, I’m glad you are interested in what I am thinking. I started writing out things on paper so you would get less stream of consciousness thoughts from me. I need to keep pen and paper with me as much as possible.

Your adding of nickels and dimes gave me an idea for the hundred chart. I’m not sure if it’s a separate activity or just an alteration of this activity above. I’m not done with it yet. ☺

My idea is to put a penny(1 cent) on each square of both of the hundred charts.

Then, proceed with your questions. Since this is about adding fractions, using labels that are a fraction of a dollar and the fact that 100 cents = 1 dollar would be as far as I would go with it to keep it about adding fractions. I’m trying to work out what the pair of charts is.

To be continued…..

Ok. That’s all I’ve got. I thought that making it about money could add some interest to it(very funny).😺

I wonder what someone who is learning fractions would think of these activities.

It would be very Interactive.

I do have a couple of money activities in my hundred chart book, though they mostly focus on simple counting (and one logic puzzle). I hadn’t thought about using money with fractions.

Here are some money activities for young children, from Baltimore County Public Schools: https://www.bcps.org/parents/pdf/Money-Activities-on-a-Hundred-Chart.pdf

Hi Denise,

Thank you.

We were meeting with a homeschool group locally which does math-related activities. I will let them know about these resources you mentioned.

😺