Saturday, October 25, 2014

Fly On the Math Teacher's Wall: Teaching Place Value Through Operations

I am so excited to be participating in the FIRST edition of "Fly on the Math Teacher's Wall". This month we are discussing place value! I will be talking about teaching place value within addition.

If you are on Facebook, surely, you have seen rants about the way that math is currently taught. I want to defend one aspect of this "new math" for a moment.

Take this example:

I get it, believe me, I do! The way that student 'B' above solved the problem seems quite laborious and there is a lot of room for error when you need to break apart one addend, and then follow with two addition problems in order to solve the one original problem. The issue here is that there has been a miscommunication about what is being taught: procedure or concept. 

The activity above, believe it or not, is not designed to teach a math procedure as far as the addition of 19+6 is concerned. The purpose of the activity is to use the operation of addition to teach place value, number sense and properties of operation.

Rest assured, the procedure will be taught. Here is the standard at grade 4:
Fluently add and subtract multi-digit whole numbers using the standard algorithm.

However, prior to grade 4, let's use the operation of addition to teach and learn as much about place value as we can!

In grade 2, students are expected to:
Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.

So what does a "strategy based on place value" look like and how will this benefit our children and students?

Take an expression like 27 + 32 above. In the standard algorithm you would stack the numbers, add the 7 and 2 and then add the 2 and 3. You would get a result of 59 and you would be absolutely right. Is there anything wrong with this algorithm? No! 

... Unless that is you were hoping to teach place value concepts. I am going to walk you through a MUCH LONGER method. The purpose of this method is NOT to teach an addition procedure that a student would be expected to use for the rest of their life. It is to take advantage of addition to teach the concept of place value in a meaningful way. 

 First, a student would decompose 27 into 20 and 7 and 32 into 30 and 2. If you were to do 27 plus 32 in your head, it is likely that this is the first step that you might take. As adults, we flexibly break numbers apart so that we are able to do mental math. A child often thinks the number 27 just simply means "two seven". 

Next, a student would be asked to put the tens together. As adults, we know that 27 + 32 is the same as 20 + 30 + 7 + 2 but a student learning the traditional algorithm learns nothing about the flexibility of numbers! 

The step of putting 20 and 30 together in and of itself is a powerful step in learning about place value. Students can see that adding 20 + 30 is really as simple as adding 2 + 3. If 20 means two tens and 30 means three tens, 20 + 30 is really just putting 2 tens and 3 tens together. 5 tens = 50. 

I'll remind you that I completely understand that the parent in the Facebook post above would right now be beside themselves with the above paragraph. Yes. This is A LOT more steps. Yes. There is significantly more opportunity for error in these steps. Yes. Yes. Yes. But I'm not teaching the procedure of addition. I am teaching place value concepts by USING addition. 

As a side note, I work with students who are struggling with the math in their classrooms. A group of 2nd grade students who had been having a lot of difficulty with place value concepts went through this activity with me. Just working up to this point (in combination with a lesson about bundling groups of tens to create new units) and my struggling students were able to calculate that 60 + 40 = 100 in their heads because 6 tens plus 4 tens = 10 tens and when you have 10 tens together you can call it 100! 

After adding the tens together, students combine the ones together. 

Lastly, students combine their group of tens and their group of ones to find their total. 

How many times have you seen a student add numbers vertically and fail to line the digits up correctly? This type of work lays the foundation for why it is so important to line these numbers up correctly. A student who understands place value would not make a "lining up" mistake because they would know that 5 tens plus 9 ones could never equal 140. It just wouldn't make sense! 

There is so much power in teaching place value through the use of operations. In the above problem you will notice that I did not deal with any regrouping. Working through a similar problem that does involve making a new ten has an additional layer of power in terms of teaching place value concepts. 

I hope this post has been helpful in understanding the "other side" of what we are being asked to teach and why the standards are requiring that we hold off on the standard algorithm! 

I would love if you would follow me on Facebook for additional tips, tricks, lesson ideas, flash freebies, and sales. Click HERE to find The Math Spot on Facebook. To continue the place value blog hop, click below on the icon for "Lessons with Coffee". 

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Monday, October 13, 2014

Numbers are CREEPILY Flexible

I am linking up with 4th Grade Frolics for the fun monthly linky "Monday Made-It"

12 tens
120 ones
1 hundred 2 tens
One Hundred Twenty
50 + 50 + 12
(1x100) + (2x10)

So many different ways to represent the same exact value! Challenge students to see the flexibility in ways we can name a value with this *creepy* craftivity. You will need very few materials which you probably already have around your classroom!

Colored Cardstock (I used purple because who says a spider can't be purple?)
Masking Tape (Or another traceable circle of similar size)
Black Marker
Place Value Chart/Base Ten Blocks or any other manipulative tools

 Start by tracing a large circle to one side of the paper. Had I been thinking, I would have traced at the top or bottom so that I could have fit two circles but I am at home and am only making one spider... not a big deal :)

Next, use the ruler to mark out "spider legs" measuring 3/4" across.

At this point, you could photocopy and print this page for your students. Personally, I have my students complete this step. A bit of fine motor practice with the tracing and ruler work marking out the legs never hurt anyone!

Cut out each piece and fold the "legs" in the center to create a bend.

 Next, assign each student a number. Depending on your grade level or student's abilities, choose a number that works for you. I can see this activity working with numbers as small as teen numbers to large whole numbers, decimals or even fractions.

Students need to come up with a different way to show the value of their number on each spider leg. They could, for example use word form, unit form, expanded form, "powers of ten" expanded form, or any other creative equation.

Students can use their place value mats (or base ten blocks etc.) to come up with creative ways to display their value.

In this example, the student (or me...) drew a representation of the original number on a place value mat. Then, one of the ten thousands was exchanged for ten thousands yielding a new way to represent the number.

This really helps to drive home the standards about the place value to the right or the left being worth ten times less or ten times more than the place value next door. (I'm talking to YOU 4th and 5th grade teachers!)

After a whole lot of experimenting and discussion, your students should have all 8 legs filled out.

Simply tape them to the back of the body and display!

If you need a place value chart for your students, the one I am using is included in my free place value hangman style game. Here is a link if you need it! 

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Saturday, October 11, 2014

5 And Above, Give It A Shove... No Really....

Insert blog post that will NOT make me popular!

I am linking this post up to the "Loved That Lesson" Linky hosted by The Teacher Studio. Head on over to read all about other math lessons that have really worked for teachers! 

"5 and above, give it a shove. 4 and below, let it go"
"5 or more, raise the score, 4 or less, let it rest"

I propose that we let these poems go. While they might seem like a great trick to get students to round small numbers there will, inevitably, be students who misunderstand, don't remember the poem quite right, and, worst of all, DON'T ACTUALLY UNDERSTAND ROUNDING!!

Does the following conversation sound familiar?

Teacher: Which two tens does the number 42 fall between?
Student: 30 and 40
Teacher: Tell me more about your thinking
Student: Well, my teacher last year said "5 or above, give it a shove, 4 or below, let it go" So that means you have to go down.
So you see the problem...

Teaching the skill of rounding is an excellent opportunity to help students to develop number sense, to earn command over their internal number line, and to understand scale and magnitude of numbers.

The following is a protocol I have developed and have used with my students who are struggling with the skill of rounding. I would caution you that, if you would like to use this strategy, you start at the beginning and don't skip a single step. This year I have used this protocol with both 4th and 5th grade students and, although the first few steps seem easy, they aren't necessarily and you will learn A LOT about your students!

Step #1: Numbers 0 - 10 

On tiles, write the numbers from zero to ten. You will also need a "game board" with 11 spaces.

Start by giving your students the tiles for 0 and 10 guiding them to put these numbers in the first and last spaces.

As students work their way though this activity, you will give them 1 tile at a time modeling the thinking about where a tile may be placed but encouraging students NOT to count the spaces.

The purpose of the spaces is to help students to see the even spacing between numbers. Have you ever seen a student working with an unhashed number line pointing and counting the line in a way which does NOT reflect understanding of scale? This step helps students to gain an understanding of this spacing.

I have done the next step in a number of different ways. Sometimes I give students the 5 to see where they may place the number. Sometimes, I flat out tell the students the 5 will be in the middle and allow them to see and use that pattern moving forward. Sometimes, I give them tiles that work from the outside in and then we discuss the number that falls in the middle later. You can use whichever method works for you. Each has their merits at various times.

Last, I give students the other tiles one at a time saying things like "If you know where the 5 goes, you can figure out where the 4 goes". This gets students thinking about the relationship between numbers on a number line rather than counting the spaces.
 Step #2: Numbers 0-10 On A Number Line

This step is almost identical to the first step. You are using the same tiles, 0-10, you are going about the process in a similar order, the only change is that students are placing the tiles on a number line rather than on the game board.

When I go through this step, we usually "play" 2-3 times. The first time though, I don't usually say too much. I can tell you that it is likely that students will put the tiles much too far apart, close together, and in the wrong places on the number line.

As I see students make changes to their arrangement as they get more and more tiles, I ask questions so that they can verbalize their thinking. For example, a student who left room between the 4 & 5 will eventually have to push them together to make room for the other numbers. When they do this, or make other changes, I ask questions like:

"What made you push those numbers closer together?"
"Why don't you need any space between those numbers?"
"What made you put space between those numbers?"
"How did you know how much space to leave?"

As I mentioned above, we usually repeat this sequence a few times. You will find your students become more accurate as you go!

Intermediate teachers, please, please, pretty please, do not skip these first two steps. Although you are working with numbers to 10 this will give you a lot of information about your student's abilities with number lines and scale. 
 Step #3: Numbers 0-100 Counting by 10

At this point you will need to move back to the original "game board" configuration along with tiles for the numbers 0-100 counting by 10.

You will follow the same steps you followed in step one, however, this time, students are skip counting by 10s from 0 to 100.

I generally find that students do quite well on this activity after noticing the patterns from 1-10.

You will notice that I have used different colored tiles for numbers to 10 and numbers to 100. This is purely a materials management preference. I usually do this activity in a small group of 2-3 students at a time. I place all of the tiles for each student in a cup. It is then very easy to get out the tiles I need without having to go through and make sure that I have the right set.

After students have laid the tiles out on a game board, as before, move to the number line. You may have to "play the number line game" a few times through but you will find that students are much more accurate when grounded in the numbers 0-10 activity.
Step #4: Looking for Hidden Numbers

When students have laid out all of the tiles from 0-100, have students go looking for "hidden numbers".

I will tell students "Between which two tens would you find the number...." we start out slowly at first but as students realize that it is between the 10 in the number you say and the next ten up, they will move more quickly. I give them a LOT of examples, probably upwards of 15-20 examples coming quite rapid fire towards the end of the exercise.

It was at this point in the activity that the student in the scenario at the top of my post let me know what he had learned about when he should "round up" and "round down". With his firm understanding based on the first few parts of the activity, he quickly realized his error and was able to move forward- abandoning his rhyme about rounding.
Step 5: Zooming in

At this point, the activity still doesn't really feel like you are teaching rounding. That's ok, it's all about to come together and pay off in a big way!

Teacher: Between which two tens would you find 34
Student: Between 30 and 40
Teacher: Here are tiles for the numbers 30-40. I will give them to you one at a time as I did before and I want you to do your best to put each number in the correct space the FIRST time, without sliding your tiles around. Think about what you know, the distance between numbers, and you will be fine!
Student: {Easily places tiles for #s 30-40}
Teacher: Which number is directly in the middle of 30 and 40?
Student: 35
Teacher: Good, let's push the 35 up a bit so we can see that mid point. Now, the number 34. Which ten is it closer to?
Student: 30!

Your student will get even that first question right a vast majority of the time. I have seen it again and again and again. They understand the number line, they understand the scale of numbers, they understand the mid point and all of a sudden there is no need for a riddle, poem or rhyme to help them to round. THEY UNDERSTAND.

Step #6: Next Steps

In terms of next steps, you want to get your students away from the tiles and able to round on paper or even in their head.

I have developed a number of rounding games for my students that start to get to this piece. They are, basically, a way for students to practice rounding to the nearest 10 on a number line. Over and over and over.

In this game, students pick out a pirate ship, determine the two closest tens (at first, I leave out the 10-100 number line to help, later they won't need it) and then count on by one from the first ten to see where the ship lands. Students record their findings on a sheet I have slipped into a paper protector.

In a 30 minute lesson with 2 students, this is as far as I was able to get. They were able to complete about 5-6 examples before their time was up. The very cool part was that, just before they walked out the door, I showed them a few pirate ships and asked if they could solve the problem in their head and, with minimal subvocalizing... "54 is between 50 and 60, but it's closer to... 50!" ...they were able to do it!

This same protocol (with additional tiles) can be created for numbers rounded to the nearest 100 as well!

Another next step is to move away from ANY manipulatives or game boards and to have students draw their own number line to round. The picture to the left is the work of the student who, just one day before, told me that 42 was between 30 and 40.

It's amazing what can happen when students learn rounding through understanding rather than rhymes!

If you would like the materials needed to teach this lesson, I have bundled the lesson instructions, materials and game (an apple theme rather than the ship theme) into a Rounding Without Rhymes pack on TPT. Click on the picture below to check it out!

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