Greg Tang: Place Values

Part two of the workshop is place values and Greg making us compute in different bases. Place values boiled down are groupings of different size units. Although many students learn about place values in elementary school, we don't spend enough time teaching students to really understand place values enough so when they get to adding and most importantly subtracting, that they understand regrouping or borrowing and why they need to borrow from the next place value.

In the number 43, the 3 represents 3 ones, and the 4 represents 4 tens. Our number system is in base 10, which is also called the decimal system.

Base 10 means grouping place values by 10s. A group of 10 ones can be grouped into one group of tens. A group of 10 tens can be grouped into one group of hundreds, etc. Greg simplified it by saying, 10 is too many, so we group things into groups of ten.

We had to switch to a different base, where place values are different depending on different size groups. We did some exercises in base 5, and changing it back to base 10 where we understand the numbers better. Changing bases helped me understand place value a lot better and even helped me understand enough to be able to explain to my students.

----------------

Even in "real life" math, we do deal with different bases, and a firm understanding of place values. For example, in dealing with time, we have to switch to base 60 to figure out elapsed time. Greg's example was finding the total elapsed time for:

1 hour 45 minutes and 30 seconds
1 hour 19 minutes and 50 seconds

When we're adding up the seconds, you get 80 seconds, but of course, no one would leave 80 seconds in the final answer, you have to regroup for a group of 60 seconds, or one minute, and have the remaining 20 seconds.

With our regrouped minute, we add to 45 minutes and 19 minutes, which equals 65 minutes. Again regrouping into 60 minutes (1 hour) and remaining 5 minutes, we add to the hours place value.

Ending up with:

3 hours 5 minutes and 20 seconds

instead of:

2 hours 64 minutes and 80 seconds

Which mathematically makes sense, but no one in the real world would be comfortable leaving an answer like such. It goes to show that anyone who understand time and time conversions have an understanding of base 60, but very few people talk about changing bases.

----------------

Having a firm understanding of place values is incredibly important for doing fraction and decimal equations.

4 and 5/7ths + 3 and 5/7ths = 7 and 10/7ths, but improper fractions are usually frowned upon, so students have to understand that in this case, 7 is too much and you have to regroup to get a proper fraction. Regrouping the 10/7ths into 1 whole and 3/7ths gets us to the "proper" answer:

4 and 5/7ths + 3 and 5/7ths = 8 and 3/7ths

-----------------

Where this comes most handy in most Math Rules! math is subtraction and fractions & decimals. Helping students with regrouping in subtraction can be easier for this problem:

800
-347

Instead of regrouping twice for both zeros, thinking of 80 tens, and 0 ones and regrouping once to get 79 tens and 10 ones. Which is much easier and doesn't involve regrouping twice.

79 tens | 10 ones
- 34 tens | 7 ones

It also works for bigger numbers, to see that
1000
-458

Can also be seen as:

100 tens | 0 ones
-45 tens | 8 ones

And regrouped once to get:

90 tens | 10 ones
-45 tens | 8 ones

Making this problem a lot easier to solve with less work.

------------------------------

I'm looking forward to the third workshop with Greg because like I said yesterday, it's a revolutionary way (for me) of looking at math problems that will help me help my students. Not only do I feel better at math, I have a firm grasp of number sense and Greg's two main concepts of partial sums and place values which will be useful for working with 3rd - 5th graders.

.

Too much candy?

The kids were a little hopped up on an entire day of sugar last Friday. It took a lot more effort for me to keep them focused on their work. They had a substitute so we had a review worksheet the kids had to get through. Adding and subtracting decimals and then comparing decimals ( < , > , = ) that sort of busy work. I managed to convince them that the comparing decimals side was much easier, and they whipped through it. The other side was a lot more tedious and boring for them. I got a lot of "can I go get a drink?" and "can I go sharpen my pencil/can I go get another pencil?" today. I put up with it and let them go as needed, but compromised with their work.

One of my girls is extremely studious and finished the worksheet by herself before the other kids had a chance to finish just one side. One of my other students is a very sharp student but had a lot of trouble focusing today, so I bargained with her. She could finish the rest of the addition problems if she could make up a fraction of what she had finished. At first she balked and said that was too hard. Fortunately for me, I pushed her a little more and worked through it with her. We talked about numerators and denominators and came up with a reduced fraction representing the addition problems she wanted to do. I also used this trick with the boys on Friday because neither one was very intent on finishing a review worksheet.

In summation, it was tough to keep all four students focused on their work, but I managed to trick them into doing "less work" with a challenge question that actually required more work than simple addition and subtraction.

--------------------------

Steven's third installment of his math column discusses negative numbers.


.

Money subtraction

Working with money and math has always been a fairly easy thing for me, just because it's so concrete, it makes sense in the real world, and everyone has had to deal with it in one way or another (unfortunately). I wanted to help my students be able to easily compute money math because it matters in the real world (as opposed to abstract uses of Calculus in the real world).


The class was working on word problems involving subtraction and money and I got to help with a worksheet the kids were working on. Somehow, a few extra students wanted to join the table, so I ended up with five youngsters to help.

Though the extra peer support was somewhat necessary, it got a little crowded at the table, and one of my students asked if he could move to a different place with his math buddy. It really showed initiative and independence which was great. They worked together and when I checked up on them, they said they had finished the front of the worksheet and needed some help on the back.

The example questions the class worked on together invovled multi-part questions such as: "David got $37 from his mother for his birthday and had saved up $48 from his allowance. He wants to buy a $100 video game, how much more money does he need to save?"

While the class could articulate that you add what he already has ($37 + $48 = $85) then subtract it from his goal ($100 - $85 = $15), the worksheet problem was more difficult for my students yesterday. I'm not sure if it was the math problem itself that was distracting them or if the extra students were, but two of my students weren't listening as intentently as I had hoped. I even asked my independent student to explain his reasoning to the others, but they weren't paying attention. I felt sorry for him because he explained it really well.

Dividing your attention between several students is sometimes harder than usual and yesterday was one of those times. I think I'll try to only work with my students next week to give them more personal attention and to keep distracting students out of the group.