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What’s the Point of Expanded Form?

OK, I’m going back in time to tell you something totally ridiculous that I actually said to another teacher. It was because of my inexperience and my inability to understand math at a deep level during my second year of teaching.  I was frustrated that my students weren’t understanding what expanded form (that 145 is 100+40+5) was, and had no idea how to teach it for them to understand it conceptually. To the other teacher I said, “What is the point of teaching expanded form anyway? There is no point in learning this!”

Insert forehead smack emoji.

Now, 11 years later I am seeing the error of my ways. Expanded form is a way to deeply understand place value, but because back then I was a young and desperate, I thought anything that was hard to teach was worthless. Expanded form isn’t hard to teach with the right tools, and it will greatly enhance students understanding of place value. Place value simply is a way of understanding quantity in parts or chunks.

So what I do now is use my favorite approach, the concrete-representational-abstract string of learning. The concrete part is the tools, the actual quantity of what we are using.  The representational part is a picture form of the tools/number, and the abstract part is the actual number itself.

I STILL learned my lesson the hard way with assuming things of children. I knew they could make these numbers with tools, show them with ten frames and write them with much practice…but get this. They STILL didn’t understand what the numbers stood for when the number was in standard form. In the photo below they made the number 158 with bundles of sticks (in 10s and singles), cubes (in stacks of 10 and singles), place value blocks, and ten frames.  I asked, “What do you guys think that the number 5 stands for in this number?”

“Um…….5?”
“No! It’s for 50 tens!”
“What do you mean what does it stand for?”
“Tens…no…hundreds?”

teaching-place-value-tips-expanded-form

So we did some practicing, I figured if they could make the number, they could identify parts of the number in what they made. So I asked them to put their hand on the hundred, what the 1 stands for:

tips-for-teaching-place-value

Then we put our hands on the 50 (or what the 5 stands for):

tips-for-teaching-place-value-2

We were getting somewhere! After practicing this a bit more, I was able to bring in the idea of expanded form with number cards, and then even write the equation:

tips-for-teaching-place-value1

Notice we kept as many tools in place as we possibly could. Making the links between the tools (place value blocks and bundles), the representations (ten frames), and the number cards finally seemed to solidify their understanding.  Eventually I began to remove the concrete tools, using only the ten frames and number cards…and then I could eventually remove the ten frames.

It was magical! Now I truly understand how wrong I was to teach it without the thinking behind it, just to get through it. I’m still learning, every day.

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Subtracting Decimals Is Not Scary!

Something about that decimal point really freaks kids out. So here we go back again to misconceptions and misunderstandings…

I’m working with a student right now who was taught procedurally how to subtract decimals, but gets all mixed up when regrouping.  The VERY BEST way to intervene with any student is to determine where the hangup is in the Concrete-Representational-Abstract approach. If a regrouping error is being made, you can guarantee that the conceptual understanding is missing, which comes in both the concrete and representational part of this approach.

If you don’t already know about this approach, here it is in a nutshell:

  1. Concrete: When a student is introduced to a new concept or something unfamiliar, you allow the use of tools. Sometimes students become stuck here, and can be moved to the next stage by linking the two together.
  2. Representational: When the student can perform the task using tools, they move on to representing the concept with drawings or pictures of their tools. Again, when students become stuck here, we link the next step in with this one.
  3. Abstract: When the student can master the task with a drawing or a picture they move to using only numbers and symbols to represent their thinking.

* If you have a little more time you can read about this sequence of teaching and it’s accompanying research here, and also here.

So with errors in regrouping, we backed up to the basics of the American currency. We looked at what each piece of currency represents and why:

subtracting-decimals-with-regrouping-strategy

So in America, we have many coins, but to keep it conceptual (whole, tenths and hundredths) I used dollars, dimes and pennies.  A dollar is the same as 10 dimes, and also the same as 100 pennies.  A dime is the same as 10 pennies.  To have a reference guide we made the key above to refer back to. Later this translated really well to using place value blocks when we needed to start using larger numbers, and when I was out of money. (Imagine that…a teacher without a lot of money! Ha!)

So for a while we added two amounts of money. Addition seems to come really easily to students. From K-4, I’ve noticed a pattern that adding two whole numbers seems to be a piece of cake, but subtraction is like HITTING A WALL.  So, I decided to start with subtraction slowly. This was the sequence:

  1. We subtracted over and over when we didn’t have to regroup, and we didn’t write anything down.
  2. Then we subtracted where there weren’t enough pennies, and he had to trade (regroup) a dime in for ten pennies.  We still didn’t write anything down.
  3. Next we subtracted where we had to trade in (regroup) a dollar for ten dimes. We STILL didn’t write anything down.

Why aren’t we writing anything down?

Well, that was the concrete stage. All we do in the concrete stage is play around with the manipulatives to be sure the student is understanding the concept behind the regrouping. It makes the next two steps SO much better.

Now that’s where we talk about noting what we did. We’ll use pictures along WITH the materials, which is how we link to the representational stage.

  1. We just kept subtracting when we didn’t have to regroup, and this time we drew what we did.  Since the student already knew the procedural shortcut to regrouping, we linked that in with the drawing.
  2. Then we subtracted where there weren’t enough pennies, and he had to trade (regroup) a dime in for ten pennies.  We still didn’t write anything down.
  3. Next we subtracted where we had to trade in (regroup) a dollar for ten dimes. We STILL didn’t write anything down.
  4. Once that felt good, we started writing, and then suddenly he wanted to try out the algorithm to see how it might match. Instant connections to the abstract stage…the procedure he had already “learned”. Now, it made sense to him, and he no longer had to memorize which number to cross off because he knew why.

subtracting-decimals-with-regrouping-strategy1

Too often we are pulling manipulatives from our older students.  Whenever I’m working with a student who struggles, it seems to be the first place that I notice problems.  When we pull away those supports (or never start with them in the first place), math really is just a bunch of meaningless numbers and symbols without any connection to the real world.

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Real World Examples For Teaching Fractions on a Number Line

Number lines go wrong in SO many ways. Let us count the ways:

  1. Students struggle to read number lines when labels are missing on the tick marks.
  2. When drawing number lines students do not evenly space the tick marks.
  3. Students count the tick marks instead of the spaces and are often off by one increment.
  4. When a number line is open (without labels or tick marks) it’s difficult for students to imagine a benchmark number.
  5. Not all number lines go up in increments of 1 like most students think.
  6. Number lines are so abstract that students are unsure of what they might represent.

So then for fun…let’s throw FRACTIONS on the number line, that won’t be difficult at all!

All kidding aside, number lines are incredibly important. We use number lines in our lives ALL the time. When we drive, a road is just a gigantic number line. We listen to our GPS and we know about how far 1/4 mile is to our next turn. We use number lines when we measure length, or even as we measure capacity in a liquid measuring cup. I use a mental number line when I’m trying to figure out how much sleep I’ve gotten the night before.  We use number lines in real life without thinking about their mathematical significance, and yet often in many books/math curriculum you will see a number line without any connection to real life with it. No connection whatsoever!

So maybe 4th graders aren’t old enough to drive, but they sure can pretend!  Since they could already identify benchmark fractions on a number line, could they use this idea to learn about equivalence? Could we find two points on the road that could be called two different things?

teaching-fractions-on-a-number-line

So we started our engines, the 4th graders quickly realized that our number lines represented a mile, and that a mile could be marked in different ways. They even talked about how they’ve seen signs for 1/2 mile or 1/4 mile on the highway.

fractions-on-a-number-line1

After working for a bit, they began to see that this mirrored the previous day’s lesson of fraction equivalence with fraction strips.  The connections were forming, and even better they were splitting number lines in two ways without us having to present it that way.  Best of all, they made connections to why 2/8 was the same as 1/4, they were just cut into twice the number of pieces.  This was an awesome link to why equivalent fractions are related to multiplication and division.

Don’t have toy cars to use? Here are some other ways to make number lines real life for kids (all of which can be so precise that fractions are useful when reading the line):

  1. A tiny number line: You are watching an ant crawling on the sidewalk. How far has it gotten?
  2. An eating number line: A snickers bar is being eaten, how much has been eaten if you start at one end and start chomping?
  3. A measurement number line: How tall is the doorway to the classroom?
  4. A time number line: How long can you hang from the playground bars?
  5. A reading number line: How many pages have been read of your book if you stop in the middle of page 39?

When number lines are playful, and connected to real life they are not nearly as scary. Give them a try!

Start driving!

fractions-on-a-number-line

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The Significance of Understanding Equivalent Fractions

I am going to tell you a secret about myself that I’m ashamed of still to this day, and then I will lose all credibility and you will likely unsubscribe from this blog forever and ever.

In the 4th grade, I received a “D” in math during the quarter that we studied fractions.

I know. It’s shocking.

A math coach and math INTERVENTIONIST helping children understand math who received a near failing grade in the 4th grade?!

Before you unsubscribe, let me explain…

It’s no secret that math up until the last few years in the United States has always been about memorization of procedures and algorithms.  If you were really great at memorizing every possible type of math problem, this likely worked for you.  It worked for me actually, I received A’s in math all the way up until the 4th grade. When things went wrong it was devastating for me as a student. I had gone to one school in 3rd grade and then switched to another school in 4th grade.  When I moved, I missed something along the way about fractions…but at the time I didn’t know what it was. Suddenly I was having to add and subtract fractions (numbers I had never seen written that way before) with unlike denominators and it made absolutely no sense to me. I still remember thinking as assignments were passed back to me full of red marks, that I couldn’t believe that I was bad at math, and that I was bad at fractions. The assignments (and the tears) piled up, and I received a “D”.  Shame.

When I look back and think about what I was missing, it’s clear to me. I had never explored or even known that there was such a thing as an equivalent fraction. That I didn’t recognize that it wasn’t a whole number for one thing, and that I didn’t know that two fractions could be the SAME number.  How could two numbers be the same? 7 was 7…21 was 21. How could 1/2 be the same number as 4/8?

So when I teamed up with a 4th grade teacher at my school, we decided to REALLY spend a bunch of time on fraction equivalence.  And we decided to make it as real world as we could. So we put up an inquiry statement, knowing that they have had some experience with equivalent fractions already in 3rd grade…and told them they had to use tools to PROVE what they were thinking.

equivalent-fractions-teaching-tips

Then we let them loose with measuring cups and sand, fraction tiles, fraction towers, cuisinaire rods, diagrams of pie charts and asked to see what they might notice.

It was really kind of awesome. As we walked around, we asked them what they noticed about the numbers.  They began to figure out the relationship between the two numbers without us even saying a word about it!

We came together and shared:

equivalent-fractions-anchor-chart-teaching-tips

We talked through which ones were equal and which ones weren’t. We added some more onto the chart and found out how we can actually use multiplication or division to decide equality if we didn’t have the tools with us.

There were definitely some misconceptions in the room as we worked. Those were noted and then cleared up as we continued on with the rest of the unit throughout the week. I’ll share more lessons soon.

 

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Getting UN-STUCK: A Test Prep Strategy

Testing season is upon us. Yuck.  We all have experience with standardized tests, for some of us starting all the way back in elementary school.  I will admit it…at times when I was tired of taking the test, knew it wasn’t for a grade, or was simply stuck, I just filled in the bubbles. Haven’t we all done that? I heard stories of classmates filling in bubbles in the shape of a fish, or a smiley face, or…

Here’s the thing. There’s no escaping measurement. We are measured in our adult lives at our physician’s office, at our dentist, and by our creditors when we want a loan (just to name a few). In many ways we face measurement, allowing others to see our choices and skills in number form. Surely there are pros and cons to this, but it’s a reality for all of us starting at a young age in our schools.

So what do we do about standardized testing? Well, it’s not about TEST PREP only.  It’s about laying a solid foundation of learning for your students, year after year.  It’s about providing them with rich tasks, and most importantly teaching them how to change their inner thoughts when encountering difficulty.  (See Carol Dweck and Growth Mindset if you’ve somehow missed it.)

I’ve put together this thinking map to show what happens in my brain when I am stuck.  It’s the actions I take to get myself out of those initial negative thoughts when something doesn’t make sense. Your personal thinking map may be totally different.  I want students to know that perseverance is in actions. So I tried it out in a few 4th grade classrooms last week to see if students might be interested in seeing my thought patterns when I’m stuck. I had the teacher give me a “tough” problem and I modeled how I’d work through it, even though I had those negative thoughts.

test-prep-sample-questions-elementary-standardized-tests2

 

Then they tried it with a Fraction Reasoning Puzzle.  At first, the initial reaction when I gave out the puzzle was kind of like a stunned silence. It was so many words to read, with a strange diagram at the top to figure out…kids were STUCK. Then they started to read it, and read it again, and maybe even once more.

test-prep-sample-questions-elementary-standardized-tests

Murmurs of “Oh! I get it!” started to ripple through the room. One student even wrote down her justifications for her thinking.

test-prep-sample-questions-elementary-standardized-tests1

When students got closer to being done, I asked them to walk around and look at others.  The room buzzed with them thinking it through and talking out loud. There were some pretty heated discussions. Some students thought that many of the statements were false because there was no point 0 or point 1 labeled on the line.  At one point one student was SO certain that one of his statements was false (when others thought true), he asked to defend his thinking to the entire class.  He stood up in front of everyone and said, “It’s completely false because there is no such thing as the end of a line, a line goes on FOREVER.”

It was so powerful to see them work through difficulty on their own!  You can find free Reasoning Puzzles by clicking on the image below if you’d like to give it a try in your own classroom:

math-test-prep-standardized-testing

 

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2 Winners!

For the sake of privacy, I’ll announce both the Amazon gift card winner and TpT gift card winner by their first name only. If your first name is Andrea, check your email! You won the $25 Amazon gift card. If your name is Faith, check yours! You won the $10 TpT gift card.

OK, whew! Gimmicks over.

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Gimmicks Are NOT My Thing…

I made a promise to myself when I started, to always blog about teaching tips, and things that would help the greater good of the teaching profession.

I’m not into gimmicks and buy one get one sales or anything of that sort. But when I got an email that Teachers Pay Teachers was giving away one thousand $10 gift cards, only for the purpose of giving away to a follower, I decided to enter my name to see if I could win one for you. Why wouldn’t I pass along the love?

Now would you believe it…I ACTUALLY snagged one of those little gift cards to give to you. So I’ll make it easy for you to try to win. Fill out this form:

Click me! Click me! Click me!

I’ll put your name into a random generator and choose a winner. The thing is, their sitewide sale begins tomorrow, Feb.7th – 11:59pm Feb. 8th, 2017…so I need to unload this thing fast.  I’ll leave it open for 24 hours! You don’t have to do anything except type in your name, email and why you love me.  Just kidding! Just your name and email will do.

If you haven’t seen this ridiculous random generator called fruit picker, check it out by clicking below. It’s really strange…but kids love it.

fruit-picker-random-name-generator-classroom-tools

There, a teaching tip for you after all!