Tag Archives: fractions

Represent! Part 2

In my previous post, I shared the first few questions I asked at a recent #ElemMathChat I hosted. Today I’d like to continue talking about using and connecting mathematical representations with a focus on fractions.

Let’s start with this question from the chat:

07

Before reading on, pick one of the models yourself and analyze it.

  • How does it represent 2/3?
  • Where is the numerator represented in the model?
  • Where is the denominator represented in the model?
  • Can you answer these questions with all three models?

First it might help to differentiate the three models. The top left corner is an area model, the top right corner is a set model, and the bottom middle is a number line.

If you look at the area model, you’ll see that the whole rectangle – all of its area – has been partitioned into three equal parts, each with the same area. When we divide a shape or region into three parts with equal area, we actually have a name for each of those parts: thirds. Those thirds are countable. If I count all of the thirds in my area model, I count, “1 third, 2 thirds, 3 thirds.”

rep2-02

Two of them have been shaded orange. So if I count only the parts that are orange, “1 third, 2 thirds,” I can say that 2 thirds, or 2/3, of the whole rectangle is shaded orange.

If you look at the set model, you might think at first that this model is the same as the area model, but this representation actually has some different features from the area model. In the set model, the focus is on the number of objects in the set rather than a specific area. I used circles in the above image, which are 2D and might make you think of area, but I could have just as easily used two yellow pencils and one orange sharpener to represent the fraction 2/3.

two-third-pencil-sharpener

I can divide the whole set into three equal groups. Each group contains the same number of objects. When we divide a set of objects into three groups with the same number of objects in each group, we actually have a name for each of those groups: thirds. Those thirds are countable. If I count all of the thirds in my set model, I count, “1 third, 2 thirds, 3 thirds.”

rep2-03

Two of the groups contain only pencils. So if I count only those groups, “1 third, 2 thirds,” I can say that pencils make up 2 thirds, or 2/3, of the objects in this set.

Finally, we have the number line model which several people in the chat said is the most difficult for them to make sense of. While we have a wide amount of flexibility in how we show 2/3 using an area model or set model, the number line is limited by the fact that 2/3 can only be located at one precise location on the number line. It is always located at the same point between 0 and 1.

In this case, our whole is not an area or a set of objects. Rather, the whole is the interval from 0 to 1. That interval can be partitioned into three intervals of equal length. When we divide a unit interval into three intervals of equal length, we actually have a name for each of those intervals: thirds. Those thirds are countable. If you start at 0, you can count the intervals, “1 third, 2 thirds, 3 thirds.”

rep2-04

However, what’s unique about the number line is that we label each of these intervals at the end right before the next interval begins. This is where you’ll see tick marks.

  • So 1/3 is located at the tick mark at the end of the first interval after 0.
  • 2/3 is located at the tick mark at the end of the second interval after 0, and
  • 3/3 is located at the tick mark at the end of the third interval that completes the unit interval. We know we have completed the unit interval because this is the location of the number 1.

rep2-05

This quote sums up what I was aiming for with this discussion of representations of 2/3:

“Helping students understand the meaning of fractions in different contexts builds their understanding of the relevant features of different fraction representations and the relationships between them.” – Julie McNamara and Meghan Shaughnessy, Beyond Pizzas and Pies, p. 117

The bold words are very important to consider when working with students. What is obvious to adults, who presumably learned all of these math concepts years and years ago, is not necessarily obvious to children encountering them for the first time. What children attend to might be correct or it might be way off base. One common problem is that children tend to overgeneralize. A classic example is shared in Beyond Pizzas and Pies. Students were shown a model like this:

Rep2-06.JPG

They overwhelmingly said 1/3 is shaded. The relevant features to the students were shaded parts (1) and total parts (3). They weren’t attending to the critical feature of equal parts (equal areas).

I’ll close this post with a Which One Doesn’t Belong? challenge that I shared during the #ElemMathChat. (Note: I revised the image of the set model from what was presented during the chat.) As you analyze the four images, think about the relevant features of the area model, set model, and number line; look for relationships between them; and then look for critical differences that prove why one of the models doesn’t belong with the other three.

rep2-07

 

Weighty Matters

This year I won a grant from our district’s Partners In Education Foundation. (Yay!) With the money, I was able to purchase quite a few platform scales for every third grade team in our district. Today I got to visit a class using the scales, and I got to see the amazing Julie Hooper teach a lesson I developed with my partner Regina. It was so much fun!

The class started with a computation warm-up which made my math heart happy. It was so amazing to listen to Julie’s students solve the problem in so many different ways. They were so comfortable doing it, too. You can tell they have internalized the idea that they are able to solve problems in ways that make sense to them.

After the warm-up, the class dove into the day’s lesson. Julie started by asking the students to name things that are heavy and things that are light.

She asked some thought provoking questions after they had compiled their list.

  • Is 100 pounds heavy to you?
  • Do you think it’s heavy to a weight lifter?
  • Are big things always heavy?

I love how the conversation got the students thinking about their current conceptions of weight.

Next, the students had the opportunity to explore two different scales. Julie asked them to notice and wonder as they tried out the scales. I noticed that 3rd grade students *love* to put as many items as they can on the scale all at once. They couldn’t believe how much it took on the larger scale to make the dial move.

After having some time to explore, Julie asked the class to think about which scale they would use to measure different objects in the room. The reason for this is because one scale can measure weight up to 11 pounds while the other can only measure up to 2 pounds. She was curious to see if students had already started noticing that the bigger scale would measure heavier things while the smaller scale would max out unless the objects were lighter.

After all of this exploring, Julie brought the class together to focus on the scale and to make connections between the scale and the number line. The class talked about whole number connections first, but then she drilled down to fractions and mixed numbers.

Finally, Julie asked the students what unit of weight they thought the fractional parts might represent. Someone volunteered ounces. Then she asked a wonderful question: “How many ounces do you think are in a pound?” Many students thought there must be 8 ounces in a pound, which makes sense given the number of parts between 4 and 5, but then she transitioned to the other scale to see what students would notice.

She wants the students to figure out that there are 16 ounces in a pound, but unfortunately she ran out of time for the day. I did like that the final comment from a student was, “That scale goes up to 4 pounds.” Just wait until they continue their work tomorrow!

Thank you to Julie for letting me spend an hour learning with her students!

 

Inspiration

Tonight I hosted #ElemMathChat and our topic was inspiration. Specifically, what inspires you as you’re planning for and teaching math?

One place I’ve found a great deal of inspiration is the seasonal aisle at Target. Honestly, inspiration can be found at just about any store, but the seasonal aisle is a particularly rich source of inspiration because it taps into the novelty and appeal of holidays.

My closing thought for the chat was this:

Q0-FinalThought

“What can I do with this?” That’s the question I carried with me as I wandered the Easter aisle this week, wondering what mathematics I could draw out of the colorful assortment of products around me. I shared a few examples during #ElemMathChat tonight. I’ll share those here along with several more examples I couldn’t squeeze into the hour-long chat.

If you’d like even more examples, check out these posts I wrote around Halloween and Valentine’s Day:

As you’re reading this post, I challenge you to continually ask yourself “What can I do with this?” because you might notice something I didn’t and be inspired to ask a different question or draw out different mathematical ideas. If that’s the case, I’d love to hear about it in the comments!

Let’s get started!

Jelly Beans

How many jelly beans are in this bag? What is an estimate that is too HIGH? Too LOW? Just right?

Q1

When estimating, our goal is to come up with a reasonable guess. The reasonableness comes from our guess lying within a particular range of numbers that makes sense. You could easily say that your “too low” guess is 1 because you know there is more than 1 jelly bean in the bag. You could also say your “too high” guess is 10,000 because it is unlikely there are 10,000 jelly beans in this one bag. But those are just cop out answers, not reasonable estimates. They don’t demonstrate any understanding of what makes sense given the picture of the bag and the window showing some of the jelly beans.

If you share this picture with your students, see if you can get them to take risks as they estimate. For example, I can count about 12 jelly beans in the bag’s window. I’m going to guess there are at least 10 groups of 12 jelly beans in the entire bag for a low-ball estimate of 120 jelly beans. However, I don’t think there’s enough room for 25 groups of 12 jelly beans in the bag, so my high-ball estimate is 300 jelly beans. I think the actual number is somewhere in the middle around 200 jelly beans.

See how much more narrow my range is? I think the number of jelly beans is somewhere between 120 and 300 jelly beans. In some ways that’s still a fairly broad range, but it’s so much more reasonable (and riskier!) than saying there are between 1 and 10,000 jelly beans in the bag.

And now for the reveal:

Q1-Answer

Notice I didn’t give the actual answer. I’d want my students to use the information provided to find out about how many jelly beans are actually in the bag. Depending on the grade, this could be a great impromptu number talk to find the product of 23 × 9.

We’ve talked about one bag of jelly beans, but let’s compare that to some others. Which of these bags do you think has the least jelly beans? The most? How do you know? (Click the pictures to enlarge them.)

After some discussion and estimating, reveal this image for the SweetTarts bag. How does this bag compare to the Nerds jelly beans? Can you compare without calculating?

Some students will likely calculate the products regardless, but I would want to make sure it also came out that both packages have 9 servings. The serving size in the SweetTarts bag is larger so the total amount of jelly beans in that bag is greater than in the Nerds bag. In other words, 31 × 9 > 23 × 9 because you are multiplying 9 by a greater number in the first expression, so the resulting product will be greater.

After that discussion, it’s time to reveal the answer for the third bag. A challenge to students: Can you compare the quantity in this bag to the other two without calculating the actual product?

Q2-Answer2

Which One Doesn’t Belong?

If you’ve never checked out the site Which One Doesn’t Belong?, I highly recommend it. The basic gist is that students are presented four images and they have to choose one and justify why it doesn’t belong with the other three. The twist is that there isn’t one right answer. You can make a case for why any of the four pictures doesn’t belong with the other three.

Look at the four pictures below. Find a reason why each one doesn’t belong.

Q3

And here’s another example, this time involving candy:

Q6

You’ll notice I’m not providing answers, because there isn’t one right answer! To quote Christopher Danielson, “It’s not about being right. It’s about being true.”

Chocolate Bunnies

Take a look at this picture of two boxes of chocolate bunnies and ask yourself, “What questions could I ask about this?”

Q4

Here are some questions that came to my mind:

  • How many chocolate bunnies are left? Can you find the number in another way?
  • How many chocolate bunnies have been sold? Can you find the number in another way?
  • If each bunny costs 75¢, how much will it cost to buy the remaining bunnies?
  • What fraction of each package has (not) been sold?

Peeps

How many Peeps are in this package? What is an estimate that is too HIGH? Too LOW? Just right?

Q5a

The quantity is smaller and you can see so many of them that I would want students to be very narrow in their range of estimates and very clear in their justifications.

We know it’s a number divisible by 3 because there are three rows. We also know there are at least 3 Peeps in each row – we can see those! I would estimate 12 (four per row) is too low and 18 (six per row) is too high. My just right estimate therefore is 15 because I think there’s room for more than 4 in each row but not enough room for 6.

This might be a tad controversial because some folks associate estimating with numbers that end in 0 or 5, such as 25, 75, 100, 900. However, given the facts – three rows – I know the total number has to be divisible by 3. That means estimates like 12, 15, and 18 make much more sense to me than 10 or 20. That’s not to say that 10 and 20 are unreasonable estimates – they’re decent in this example – but I’m not going to limit myself to just those numbers given what I know about the configuration of Peeps.

And here’s the reveal:

Q5a-Answer

But it doesn’t end there! Now that you know the quantity in one package, what can you tell me about the number of Peeps in this case?

Q5b

And to take it another step further, here’s the price of one package. How much would it cost to buy half the case? How many Peeps would I be getting?

Q5c

I love the layering in this example because it starts out so simple – estimating how many Peeps in one pack – but it really takes off from there with a few added details.

Easter Eggs

How many eggs in my hand? What is an estimate that is too HIGH? Too LOW? Just right?

Q7

This one is trickier because the eggs are not arranged neatly like the Peeps. In this case I’m probably going to use numbers like 5, 10, or 20 to make my estimates.

However, this question is also a bit tricky because of how I worded the question. Did you notice?

Let’s take a look at the front of the package.

Q7-Answer

Students might be drawn quickly to 18 as the answer, but that’s not quite it. If you read carefully, it says “18 colored eggs and one golden egg” which brings the total to 19. But that’s not quite right either. I asked how many eggs in my hand, and if you’re noticing the shape of the container, there are actually 20 eggs in my hand. Sneaky!

So, if there are 20 eggs in my hand, how many colored eggs inside these 5 containers? (I would say “on this shelf” but students might get caught up in the fact that you can see there are more containers in the back. I want to focus just on the five up front.)

Q7b

This is another chance for an impromptu number talk. I especially like how it can build off the discussion about the number of eggs from the previous image. You can start with 20 × 5 and back up to remove the 5 large egg containers (I asked about the colored eggs inside) and the 5 golden eggs (I asked about the colored eggs, and the packaging does not include gold as a colored egg. This is semantics though, so I might accept these in the total since gold is a color.)

Now that we’ve talked a bit about this package, let’s do some comparing. Which would you rather buy – one package of the eggs we just talked about or two packs that each have 12 eggs in them.

Q7c

In case you missed it, the price for the package on the left is $5.00. It’s printed on the label. The price for the packages on the right is 89¢ each. (I would probably ignore the Buy One, Get One 50% Off unless you wanted to take into account that wrinkle.)

Notice I didn’t ask, “Which is cheaper?” I asked, “Which would you rather buy?” On cost alone the two dozen eggs is significantly cheaper, but there are some definite perks to the $5.00 package. Again, it’s about being true, not correct. So as long as students are able to defend their choice, that’s what matters.

For this next one I would probably change up the question and ask, “Which is the better deal – 1 pack of 48 eggs or 4 packs of 12 eggs?”

Q9

The price you see in the left picture – $2.50 – is the cost of 1 pack of 48 eggs. Ignoring the buy one, get one 50% off, the left picture is a clear bargain. However, this might be a good time to tell students that for every one pack of 12 eggs, you get a second for half off. Then I would challenge them to determine the price of 4 packs given that discount. It’s definitely a closer answer when you take that into account!

Coconut Macaroons

I don’t know that I associate coconut with Easter, but I had to share these packages of coconut that caught my eye in the Easter aisle.

How many cups of shredded coconut in this package? What is an estimate that is too HIGH? Too LOW? Just right?

Q8a

Here’s the reveal, which is why these packages caught my eye:

Q8a-Answer

Such an oddly specific amount! So if I bought all of these bags of coconut, how many cups of coconut would I be getting? How much would the three bags cost?

Q8b-2

There’s a recipe for coconut macaroons on the back of the package. If I bought three bags of coconut, how many cookies could I make?

Q8b-1

I like this because students have to wade through a lot of information to find what they need. Oftentimes in math problems we make needed information stand out or we don’t provide any distractions at all. It’s good to make students work for it a bit like they would have to do in the real world if they wanted to bake these cookies.

Another question I thought of is, “How long does it take to make 3 dozen macaroons?” This provides another opportunity for reading the recipe to search for relevant information. Students might just add 15 minutes and 20 minutes, but that’s only if you can fit all 36 cookies in the oven at the same time. If you only have one baking sheet that can hold 12 cookies at a time – which is about all I can do at home – then how long will it take? What if you could squeeze 18 cookies on a cookie sheet? How much time would you save?

Miscellaneous

I’m going to close out this post with a final set of pictures that might inspire you to share them with your students and prompt some mathematical discussions. (Click the pictures to enlarge them.)

 

Bunnies

I get a kick out of this last one because it’s pretty easy to tell how many candies are in the package.

Q9d

I can foresee some really interesting discussion when you reveal what the packaging says about the number of candies contained within.

Ferrero

Final Thoughts

Please feel free to use these pictures with your students. I’d love to hear about the conversations they spark. If you get inspired to use them in ways I didn’t think about, please share in the comments. That way we can all learn and get ideas from one another!

Is 1/2 always greater than 1/3?

Lately I’ve been reading the book Beyond Pizzas & Pies: 10 Essential Strategies for Supporting Fraction Sense by Julie McNamara and Meghan Shaughnessy.


I posted the following picture to Twitter while I read during my daughter’s swim class.


My colleague, Hedge, replied about being challenged by a middle school teacher on this very issue.


I let her know I was also challenged about this idea several years ago when I was a digital curriculum developer. The argument I heard back then was that using contexts to validate the correctness of fraction comparisons ran counter to the fact that fractions are numbers. As such, 1/2 is always greater than 1/3 regardless of the context. At the time, I wondered about it, but I still felt that bringing context to bear was important.

Flash forward to now and I have been mulling this idea over all day. I think I may finally understand why we have to be careful what we say about the role of context when comparing fractions. I may be completely off the mark, but I’m going to share my thoughts anyway and let you decide in the comments if you’d like to challenge my thinking or share an alternative point of view.

Let’s start with whole numbers. If I told you to compare 3 and 6, you would probably tell me, “3 is less than 6,” or, “6 is greater than 3.” That is how the numbers 3 and 6 are related.

Now, what if I were to show you these two pictures of 3 and 6: (As illustrated by my daughter’s toys.)

Three large dolls

Six small figurines

Technically, the 3 dolls are larger and therefore they amount to more stuff, but does that really mean 3 is now greater than 6? In the end, the number of dolls my daughter has (3) is less than the number of figurines she has (6). The context doesn’t fundamentally change the relationship between the numbers 3 and 6.

In this case, I don’t even know how I’d justify that she has more when referring to the dolls. Sure, they’re bigger, but she may prefer to have more things to play with and choose the 6 figurines even though they are less in total size.
Let’s continue by looking at this from a fraction perspective. Now I’m going to take 1/3 of the dolls and 1/2 of the figurines.

1/3 of the dolls is 1 doll

1/2 of the figurines is 3 figurines

In keeping with the idea that context should dictate when one number is greater than another, I should be convinced that 1/3 of the dolls is greater than 1/2 of the figurines because 1 doll is so much larger than the 3 figurines. Oh wait, or is it that I should be thinking that 1/2 of the figurines is greater than 1/3 of the dolls because I end up with 3 figurines which is a greater number of things than 1 doll? It’s not so clear cut, even though I’m trying to let the context dictate how to interpret the fractions.

What it boils down to is that fractions represent a relationship. If I think about the relationships each fraction represents, then 1/2 is always greater than 1/3 no matter how I try to spin it. Looking back at my examples, taking 1/2 of the group of figurines means I am taking a greater share of that group (that whole) than when I take 1/3 of the group of dolls (a different whole, but a whole nonetheless). The size of the things in my group (whole) doesn’t matter because the relationship represented by 1/2 is greater than the relationship represented by 1/3.

Now, does that mean we should ignore contexts altogether? No. There are still rich conversations to be had about who ate more pizza when one person eats half of a small pizza and another person eats a third of a large pizza. Context is still interesting to discuss and helps students use math to interpret the world around them. However, if our goal is to compare fractions, then 1/2 is greater than 1/3 every time.

That’s the argument I came up with today as I tried to understand the criticisms I’ve heard. Now that you’ve read it, what do you think?

Interpretation Frustration [UPDATED]

[UPDATE] I wrote an email to TEA and heard back from them within three days. I’m very impressed! The person who wrote me went over each of my concerns one by one:

1. The example for 3(3)(E) in the side-by-side document discusses the partitioning of objects and the fraction as a concept of division, not a numerical representation. The goal would be that a student realizes that each student would receive five half-pieces of cookie. With this basis, students can the develop improper fractions and mixed numbers in grade 4.

Saying that the answer should not be a numerical representation sounds like splitting hairs. The text of the standard says: solve problems involving partitioning an object or a set of objects among two or more recipients using pictorial representations of fractions with denominators of 2, 3, 4, 6, and 8;

If I look at the phrase “solve problems” I’m led to believe I’ll get a numerical answer like I get when I solve other math problems. Yet I think he’s focusing on the phrase “pictorial representations” to say that their answer is meant to be less formal. All in all, I feel students can solve problems like sharing five cookies among 2 people, but the avoidance of 5/2 as an answer leaves me scratching my head. What purpose does it serve?

2. Much like with 3(3)(E), the focus of [3(7)(A)] is the division of the line segment. In the given example, the mark is ¼ of the distance between the numbers 16 and 17 on the number line.

This is some shady logic. The standards don’t ever mention mixed numbers in any elementary grades, but apparently they are implying that because a mixed number is composed of a whole number (3rd graders should be comfortable with those by now) and a fraction less than one (introduced in grade 3), they are fair game. I’m not against this interpretation. What I don’t like is the vague language that leaves it open to interpretation in the first place. I feel like I need to hire a lawyer to help me make sure I’m interpreting the language of these standards accurately!

3. You are correct; the last stand-alone measurement standard is 2(9)(D). However, students can be asked to measure the side lengths of a polygon in 3(7)(B).With the process code of 3(1)(A), a ruler could stand in for a number line in 3(7)(A).

And yet another example of relying on implication rather than writing standards that were clear and easy to follow in the first place. And he ends with my favorite line that I’ve heard over the years:

4. Please remember that the Texas Essential Knowledge and Skills are minimum standards and are not intended to limit what is taught.

It’s the “Get Out of Jail Free” card. I don’t think the issue is that teachers are scared of teaching beyond the standards. The problem is teachers trying to get a good grasp of what the bare minimum is in the first place. After reading through the TEKS, which are technically the standards, teachers can walk away thinking they know where the bar is. However, based on supplemental documents and email clarifications, the bar seems to be in a state of flux, leaving teachers unsure of how high their students need to jump. This doesn’t seem like a fair position to put teachers (or their students!) in.

Original post follows.

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I blame the TEKS formy headache today. Specifically the grade 3 TEKS. They are not on my good side right now.

To give you some background, for the past few years I designed curriculum based on the Common Core Standards. I’ve also designed materials for Texas, but lately it was kind of secondary to the Common Core stuff. I’ve grown to love the Common Core standards. There is a lot of thought and care into the progression of topics from grade to grade. They aren’t perfect, but I value how much they do make sense, especially if you read the accompanying progressions documents.

Several years ago, Texas decided to write some new math standards. They didn’t want to adopt Common Core…because Texas…but it was clear the writing team appreciated those standards, too. The first draft of the new Texas standards had so much Common Core language in them, they may as well have been the Common Core. But then the Texas standards went through a round of revisions and what came back looked like someone had hacked off pieces of the Common Core standards, shuffled them around a bit, and called the final product new Texas standards. Needless to say, I’ve been unimpressed.

However, in my new job, I am working squarely in a Texas district in the state of Texas so the Texas standards (TEKS for short) are my focus from here on out. Lord help me.

Today, while putting together assessment materials for a grade 3 unit on fractions, I started to come across some inconsistencies in the language of the TEKS. It started with 3.3A and 3.3B:

3.3A represent fractions greater than zero and less than or equal to one with denominators of 2, 3, 4, 6, and 8 using concrete objects and pictorial models, including strip diagrams and number lines;

3.3B determine the corresponding fraction greater than zero and less than or equal to one with denominators of 2, 3, 4, 6, and 8 given a specified point on a number line;

Remember, I come from a Common Core background. Their standards say this:

Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.

And this:

Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line.

Can you spot the main difference? In Common Core there is no specification that the numerator a has to create a fraction that is less than or equal to 1. You could just as easily make 5/4 as you could 3/4. In the new TEKS, however, there is a clear specification that third graders are working with fractions greater than 0 but less than or equal to 1. (By the way, what’s with the fractions having to be greater than 0? Anything wrong with discussing 0/4?)

Ok. I can handle that. But what’s this grade 2 standard over here say?

2.3C use concrete models to count fractional parts beyond one whole using words and recognize how many parts it takes to equal one whole;

Oh, so in second grade it’s okay to count fractional parts above one whole, but we need to stop in grade 3? Apparently that’s the case because improper fractions aren’t brought up again until this grade 4 standard:

4.3A represent a fraction a/b as a sum of fractions 1/b, where a and b are whole numbers and b > 0, including when a > b;

Weird. Let’s introduce an idea in grade 2, completely skip it for a year in grade 3, and come back to it in grade 4. Well, at least that’s settled…I think.

Let’s look at another grade 3 standard:

3.3E solve problems involving partitioning an object or a set of objects among two or more recipients using pictorial representations of fractions with denominators of 2, 3, 4, 6, and 8;

In the Side-by-Side comparison documents provided by the Texas Education Agency, we see the following example provided to help clarify 3.3E:

Examples of problems include situations such as 2 children sharing 5 cookies.

I can buy students solving this problem. That’s fine, but how do you rationalize the answer? You are either going to end up with 5/2 which contradicts the rigidity of 3.3A and 3.3B, or you’re going to end up with 2 ½ which is a mixed number. By the way, did I mention the term mixed number doesn’t appear in the TEKS at all across grades K-5? At all. Can you see why this might make my head hurt a bit?

My guess is that they are cheating a bit in their interpretation of 3.3A and 3.3B. By having students use mixed numbers, they are really only writing a whole number combined with a fraction less than one. Do you get it? The number 2 ½ doesn’t break their rule because the fractional part is less than 1.

So students are likely going to be held accountable for understanding mixed numbers in grade 3 even though they aren’t mentioned in the standards and several of the grade 3 standards explicitly state students work with fraction less than or equal to 1. (Good luck third grade teachers!)

I’m pretty sure this is how they are interpreting it because of how they interpret another standard. In the old TEKS we had this standard:

Old 3.10 The student is expected to locate and name points on a number line using whole numbers and fractions, including halves and fourths.

On this year’s high stake test (STAAR), the students had to locate the mixed number 16 1/4 on a number line. Do you think they would ask the same thing based on the wording of the new TEK? I sure can!

3.7A represent fractions of halves, fourths, and eighths as distances from zero on a number line;

And that’s not all! Looking at the TEKS related to fractions on a number line got me thinking about measuring to fractions of a unit. Guess what! That’s a whole new can of worms. Here is the linear measurement standard from grade 2:

2.9D determine the length of an object to the nearest marked unit using rulers, yardsticks, meter sticks, or measuring tapes;

In which grade level do you think they specify measuring to the nearest half, fourth, or eighth of an inch? If you guessed “they never specify it”, you’re right! The standard 2.9D is the FINAL linear measurement standard in the TEKS. The only mention I could find about measuring to fractions of a unit comes from the grade 5 Side-by-Side document put out by TEA. Here’s the standard:

5.4H represent and solve problems related to perimeter and/or area and related to volume.

And here’s how the Side-By-Side “clarifies” it:

Because fluency with the addition and subtraction of positive rational numbers is expected within the Revised TEKS (2012), lengths may reflect fractional measures with perimeter.

So the wording of the standards themselves never brings up fractional measures in grades K-5. The only way you would even know this grade 5 standard uses fractional measures is if you happen to cross reference it in the Side-By-Side document which is available on a completely different website from the standards themselves. I’m not even sure they’re available on the Texas Education Agency website.

Can you see why I had a headache today?

I did email someone at TEA today to request clarification. I can’t imagine I’m the only person who finds these particular standards unclear and confusing. If I hear back, I’ll be sure to share! I don’t want anyone, teachers or students, to suffer as I did today.