Where's the Value in Place Value?

I have a love-hate relationship with the New Math TEKS, as I'm sure many of you Math Educators can probably understand. Although, if I truly took your pulse, I'm willing to bet 'hate' would be the more dominant beat than love.

Well I love that the TEKS address the foundational and conceptual basis of Math (which in fact are NOT new at all, just new to us). I hate that there are little to no resources that really instruct teachers on how to disseminate the information to their students. But with the latter, I find it fun (call it the geek in me) to study different strategies and create in my mind the perfect classroom situation and centers for students to engage in to assist with their discovery of these understandings.

Big shout out to my 3rd grade teachers who are now helping students understand value of numbers to the hundred thousands place. Here's a few strategy and center ideas to help you out next school year! And if you happen to teach 4th grade, you may benefit from this as well.

The NEW TEK states:
3.2A (similar 4th grade TEK is 4.2A) -  Compose and decompose numbers up to 100,000 as a sum of so many ten thousands, so many thousands, so many hundreds, so many tens and so many ones using objects, pictorial models, and numbers, including expanded notation as appropriate.

3.2B (similar 4th grade TEK is 4.2B)- Describe the mathematical relationships found in the base-10 place value system through the hundred thousands place.

*Note the only difference in the 3rd and 4th grade TEKS are verbs and the place value focus.

STRATEGY IDEA: So, during your 15-20 minute mini-lesson, have your students sit in cooperative groups and work with a set of Base Ten Blocks (distribute up to 1 thousand block, 3 hundred flats, 5 ten longs and 10 units per group/pair-this should help you keep the numbers you compose/decompose within a certain range).

1. Give your students the task of creating a ten long using only unit cubes. Have them verbally express how we could write this in a number sentence. [They may say 1 + 1 + 1... = 10] Assist them in creating a multiplication number sentence (because up to this point they've only covered multiplication in the contextual sense, the multiplication symbol hasn't been introduced formally).

 

 

Verbal sentence: "1 group of 10 units = a 10 long" 

Number sentence: 1 x 10 = 10

 

 

2. Repeat facilitation of this task by having them line up the ten longs to see how many it takes to create a hundred flat; how many hundred flats stacked create a thousand block. Again, have them create a verbal sentence and corresponding number sentence.

 

3. Students can have a place value chart drawn in their journals. They can draw an arrow from the ones place to the tens place and label the arrow "10 times". Then draw an arrow from the tens place to the hundreds place labeled "10 times" and so on; to show that the progression from right to left across the place value chart is based on multiples of 10. Allow them to extend the pattern to the hundred thousands place.

 

 

4. Finally, demonstrate how to build a number (such as 324) and trade one of the hundred flats for 10 ten longs. Have them count to see if the value remains the same:

 

Say: "One hundred, two hundred, two hundred ten, twenty, thirty..., three hundred, three hundred ten, three hundred twenty, three hundred twenty-one, ....three hundred twenty-four)."

 

*This will help them grasp the concept of 'regrouping' when subtraction!!!

 

**Then see if they can discover other relationships; can they determine a relationship between the units and the hundred flat?

 

Verbal sentence: "10 groups of 10 units = a 100 flat"

Number sentence: 10 x 10 = 100

 

 

CENTER IDEA:

Materials: Number generators (dice), index cards with different numbers written on them, Place value disks.

 

Place value disks

 

Create numbers on index cards (or let students generate numbers using dice) in standard form up to the hundred thousand place. Have students build that number using place value disks or place value blocks.

 

 

 

 

Students can then write the number in expanded notation.

(1 x 1,000) + (2 x 100) + (4 x 10) + 3 x 1)

 

What many don't know is that the standard implies that students must be able to represent the number in other ways. So have students trade out a place value (for its x 10 value equivalent) and rewrite the same numeral a different way.

 

 

Notice here how the thousand block has been traded for 10 hundred flats.

This makes a combination (total) of 12 (10 and 2) hundred flats.

So the new expanded form & notation, respectively, would be as follows:

 12 hundreds + 4 tens + 3 ones

(12 x 100) + (4 x 10) + (3 x 1)

 

OR

 

Notice here how one of the ten longs has been traded for 10 units.

This makes for a combination of 13 total units, which doesn't change the value of the number but does affect how its written:

1 thousand + 2 hundreds + 3 tens + 13 ones

(1 x 1,000) + (2 x 100) + (3 x 10) + (13 x 1)

 

 

Now ordinarily, I would advise one to use a center sheet so students can record their findings. Something like what you see below (but of course for larger numbers):

 





 





No disrespect for those types of sheets, I often use them! But I'm a bit of a "tree hugger" (I like to protect the number of copies I make) so I prefer dry erase boards (unless of course I'm collecting it for a grade). Then in that case, I would put to use the construction paper in my room and have students create a foldable and record their work on it, in chart form.

 

Can't wait for the Fall to roll around again so you guys can try this out and let me know how it works for your students! Happy Educating!



Trick or Strategy?

I remember being in the classroom and prior to beginning a new unit, I would typically have a "pow wow" with my students about what we were about to learn and how it would be useful in their practical lives. As we entered the fraction unit and discussed how to compare fractions, I would always have a students shoot their hand up and yell out "I know! We can "Cross Multiply!"

 

 

 

And each time, my skin would crawl as my Cruella DeVille would rise up and say "No, we will NOT!" I would be infuriated that the kid wanted to use some lame trick to compare fractions." But I soon realized I couldn't be upset at the student. It was the 4th grade teacher I needed to spit venom at. They were making us 5th grade teachers work harder (we had to break the old, bad habit then teach a new habit) when really we should have been teaching the concept from a clean slate (or at least building on a foundational fraction concept).


Does "Cross Multiplication" work? Absolutely! But what does it really represent?
If students could not find a common denominator to turn both fractions into equivalencies, they could easily multiply the two denominators together (In the example above, they would get 75).
Then the numerator of both fractions, respectively, would be the product results of cross-multiplication. But this is a bit advanced for 3rd and 4th graders.

So what happens when teachers teach "Tricks" like "Cross Multiplication" is students lose conceptual understanding essential to their development in mathematical learning. They learn a trick but have no understanding of the "why" behind its' process. In actuality, they learn to be lazy and settle for a drive-thru junk food meals for a quick fix rather than wait for and feast on a home-cooked meal that won't only satisfy their craving, but provide nutritional value.

So what's the remedy? Learn the foundation for concepts yourself (even if it seems new to you).

First let's look at foundational progression of fractions.

Watch the progression:
3rd Grade- 3.3H "Compare two fractions having the same numerator or denominator in problems by reasoning about their sizes and justifying the conclusion using symbols, words, objects, and pictorial models.

4th Grade- 4.3D "Compare two fractions with different numerators and different denominators and represent the comparison using the symbols >, =, <.

Prior to the NEW TEKS this TEK progression applied to 4th and 5th grade, respectively.

 Students can understand fractions on a number line by relating Fraction tiles to number lines. Why? Because in previous grades (even in 3rd grade), the standard encourages open play with fraction tiles! LET.YOUR.STUDENTS.EXPLORE! Take the manipulatives off the shelves and let them build and create and make connections. They're capable!

Once students have grasped that (for example) three-thirds is equivalent to one whole they should be able to see and understand that one-third is close to zero, while two-thirds is closer to one whole. This leads to the understanding of "benchmark fractions"

Benchmark fractions can be seen and taught fairly quickly. Once students know 0, 1/2 (and its equivalencies) and 1 on a number line, they can justify which fractions are closer to zero, which are closer to half and those which are closer to 1 whole. This is more than half way to the learning expectation!!!

*Coach's note: What I would do is have my students build a blank number line. Then take a fraction tile (such as a fourth) off the tray and place it on the number line close to zero. Have them make a tick mark where the 1/4th tile ends. Continue this pattern until you reach 4/4ths = 1 whole.

Using the Number line as a basis, comparing then becomes feasible because students can plot two fractions on a number line or compare them both to benchmarks (five-sixths) would be closer to one whole (six-sixths) which is larger than, say, five-eighths because 5/8 is closer to four-eighths or one half.

Coincidentally, once students understand that the larger the denominator, the small the fractional piece (through exploration)...and vice versa, they will begin to compare fractions easily. They can justify that three-eighths is 3 small pieces, while one-half is 1 large very piece so one half MUST BE larger than three-eighths.

Now THAT'S strategy! And though it takes some doing (or developing) to get students to this point, their understanding goes a long way.

Ever wonder how students will use "cross multiplication" when they must compare (or "order" is the mathematical SE verb) three or more fractions? That's how you know the "Trick" you taught them, was unreliable.