Representing Numbers

Today’s Question (Sept 30):

In order to activate their thinking we put out a question of, “What is Place Value”?  We did not come to a concencus, but we had a good discussion.  Then the students were asked…

The idea was to get the students  to think about how and why numbers are ordered. 

Some students mis-interpreted the question and ordered the digits within the number (not the numbers themselves):

Some students had the right idea, but did not inlcude an explanation:

Some students were able to explain, but did not focus on math language:

Some students were able to explain using good math vocabulary (ten-thousandth column):

Some students efficiently used a chart with the place value columns:

Consolidation: Together we developed the explanation that, when ordering numbers you have to look at the place values of the number.  You begin on the left, with the ten-thousands column (and look for the biggest digit).  If the digit is the same as another number, you look to the next place value column (the thousands) to see which number has the largest digit.  If they are the same you look at the hundreds column, then the tens column and then the ones column.

Today’s problem (Oct 3):

Find 3 numbers that together make up…
Grade 5’s: fifty-three thousand three hundred
Grade 4’s: three thousand one hundred fifty

Some students used pictures of base 10 blocks to answer the question:

Some students used a variety of numbers that add up to the answer (in this case 53000+100+200):

Some students used a PLACE VALUE chart to divide up the number:

Some students know that the number could be broken up into its place values (without using a chart):

Consolidation:  We discussed how the digits that make up a number represent a number and we connected that to yesterday’s work (i.e.: a 3 in the hundreds place value column represents 300).  We talked about how in math vocabulary this is called a number’s EXPANDED FORM.

The students were given 5 numbers to convert into ‘expanded form’ for homework (copied off of board).

Today’s problem (Oct 4):

We were focusing on rounding off numbers…

Some students focused on rounding up (if the number was a 5 or greater) or rounding down (if the number was smaller than 5), but had trouble arriving at an answer:

Some students used a number line to help visualize when rounding numbers:

Some students used their math language well when explaining how they were able to round numbers off:

Consolidation:  Together we came up with a rule…

So if you have a number (5 354)…
and you want to round it off to the hundreds place value column…
you look at the digit to the right of the hundreds place value column (5 354)…
If the number is a 5 or greater you round-up (to 5 400)…
Note: the digit in the hundreds column went up from a 3 to a 4.

If you were rounding 5 324 to the nearest hundred…
If the number to the right of the hundreds column was a 4 or smaller (5 324)…
then you round down (to 5 300). 
Note: the digit in the place value column you are rounding would stay the same.  It’s the digits in the tens and ones column that move down to ’00’.

Practice a few orally at home 🙂

Today’s problem (Oct 6):

One of the students asked about the word “about”.  The students were able to conclude that “about” meant that an exact answer was not being looked for, that “rounding” should take place.

Some students decided that Mr. Wendler was correct…

Some students thought Mr. Wendler was crazy… (I mean wrong)…

Some students were able to demonstrate that Mr. Wendler could be both right and wrong, depending upon which place value column you were rounding to…

Consolidation: We had a good discussion around how well people are using math vocabulary in their explanations of their thinking.  We also noted that the ability to answer this question both ‘yes’ and ‘no’ demonstrated a really good grasp of the concept of rounding numbers.  Good work!

Today’s problem (Oct 7):

Some students rounded off the prices using a number line to get the answer…

Some students rounded off the prices and added them (and gave a good explanation)…

Some students demonstrated a thorough understanding of rounding numbers by finding three different answers (rounding to the nearest thousand, hundred, and tens)…

Some students answered the question in a very creative way by add ing the exact prices together and then rounding off the answer to the nearest hundreds and thousands.  Good job!

Consolidation:  We talked about how questions can be approached in different ways.

Today’s problem (Oct 14):

We were focusing on wether the concepts of place value would be connected to numbers that had decimals (in this case the concrete idea of money)…

Some students answered the question without explanation:

Some students included a brief explanation of what they were doing, but not how or why they were doing it…

Some students inlcuded an explanation using the vocabulary of ‘first number’ and ‘second number’ to describe the place value columns (we looked at the first number, if it was the same we looked at the second number…):

Some students used vocabulary such as ‘dollars’, ’10 cents column’, and ‘cents’ to describe the place value column… 

Some students used the math vocabulary of ‘ones’, ‘tenths’, and ‘hundredths’ to describe the place value columns…

Consolidation: We focused on the math vocabulary of ‘tenths’ and ‘hundredths’ when focusing on decimals.  The students made some great connections between decimals and money, as well as between the ending of tenths (ths) and hundredths (ths) and how it is commonly found when working with fractions (e.g.: two fifths).

Today’s problem (October 17):

We continued to focus on representing numbers.  Here we were looking again at using a good math vocabulary.  But also, as the results came out, it was interesting to look at strategies students used to come up with all the possible outcomes…

Some students randomly put the digits in different orders, making sure not to repeat any numbers…

Some students talked about using strategies like writing the same number, but switching the places of only two digits (for example)…

Some students were very efficient, using a strategy of keeping the largest digits on the left hand side, and slowly shifting one digit at a time to the right (55533, slide the last 5 over one space, 55353, move it again one space 55335)…

Consolidation: We reviewed a number of student responses based on the math vocabulary being used.  But then we looked at and discussed the strategies the students used to organize their thinking when they were making up five-digit numbers (How did/do you know if you have all the possible numbers?  Is there a pattern that you can follow?)  The students were able to share some excellent ideas.