Tag Archives: measurement

My First Three-Act Lesson

My co-worker Regina and I took a stab at our first three-act math lesson. Well, we took a stab at writing a lesson to provide some motivation for learning about measuring liquid volume, and it sort of morphed into a three-act lesson along the way. However we got there, it was fun to write, and the teachers we shared it with at a PD session in March really enjoyed it. Here’s hoping I get the chance to develop another three-act lesson sooner rather than later!

Writing this lesson came hot on the heels of spending a day with Dan Meyer at the recent Texas Association of Supervisors of Mathematics meeting. He offered some advice for designing engaging learning experiences that I couldn’t wait to try out:

  • Start a fight
  • Turn the math dial down
  • Create a headache

If you’re intrigued by his advice – and I hope you are – I recommend checking out his recent talk at NCTM. You’re only going to get about 45 minutes with his ideas about engagement instead of the 6 or so hours I got, but I guarantee it is still time well spent.

A Gallon of Ice

Standards

  • Texas: 3.7D and 3.7E
  • CCSS: 3.MD.2

Act 1

Watch the video.

  1. What do you notice? What do you wonder?
  2. How long do you think it will take for all of the ice to melt? Estimate – Write an estimate that is too low, an estimate that is too high, and your just right estimate.
  3. How much water will be in the jug after all the ice melts?

I recommend bringing in an empty milk jug so students can draw small mark and their initials on the side of the jug to show their estimate. Start a fight! The students will want to know if their answer is correct. I did this with teachers during a PD session, and they had quite a range of answers. At this point, the math dial is turned down low, so we did not talk about units of measurement, just an estimate of how high the water will fill the jug once the ice is melted.

JugLines

Act 2

Watch the video.

  1. How long did it take the ice to melt? (Sadly, it finished melting while I was sleeping, so the most precise answer we can give is longer than 11 hours but less than 20 hours, since I checked the jug again at 7:00am.)
  2. Whose estimate was closest to the actual height of the water in the jug? (Resolve the controversy!)
  3. How much water is in the jug? Estimate – Write an estimate that is too low, an estimate that is too high, and your just right estimate.

This is where you start to slowly turn up the math dial. Question 3 is a great question to find out what your students already know about units of volume. They might very well be stumped depending on their prior experiences. You might have them imagine other packages and containers that have liquids in them and think if there are any words they know that describe how much liquid is inside. It’s totally fine for the estimates to be sort of weak here.

The whole purpose of this question is to create a bit of a headache – get the class to a point where you (or your students!) can say, “I think we need to know a bit more about measuring liquids so we can come up with estimates we’ll feel confident about,” and then take a break from this three-act lesson to do some explorations of measuring liquid volume. After doing that, which might take a day or two, show the Act 2 video again and then give the students a chance to add on or revise their estimates.

Here are some estimates made by 3rd grade teachers at our PD session:

JugEstimates.PNG

I can tell the teachers were hooked when they reacted in shock when they found out I wasn’t going to reveal the answer right away. Just like with students, we took a detour away from this lesson. We wanted to spend a bit of time sharing ideas for how students can explore measurements of liquid volume. But they wanted to know the answer! One of them was really worried and wanted to make sure we would tell them before they left the PD session.

I couldn’t have been happier.

Act 3

All is revealed! Now that your students have some personal experiences with measuring liquids using various units and you’ve given them a chance to add on or revise their estimates, it’s time to find out the actual volume!

And of course I spilled some water! When I was first filling the jug, I had to cut a flap in the top to make the opening wider for ice cubes to fit. Unfortunately, I forgot about it when I was doing my first pour and water did not come out like I was expecting. Thankfully it was only a small amount.

There’s so much going on in this video! You’ve got quarts, and half gallons, and cups, and fractions of cups. All great stuff to talk about! But I purposefully tallied the number of cups throughout the video so that students could at least come up with 8 2/3 cups. However, this is a great opportunity to talk about how we can read measurements differently depending on our units. For example:

  • 8 2/3 cups
  • 1/2 gallon and 2/3 cup
  • 69 1/2 ounces
  • 2 quarts and 2/3 cup

This is different from making conversions; it’s more about the choices available when reading a measurement off a tool. You don’t have to go here, but I think it is important for students to know that they do have choices in how they read a measurement given the options provided by the tool. Learning that flexibility here is only going to help them when they start encountering questions related to measurement conversions down the road.

And that’s a wrap! If you try out this lesson in your own classroom, I’d love to hear about it in the comments.

 

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!

 

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.