Place Value

These last few months we have been focusing on place value. Place value is such an important beginning for any primary student in mathematics. We have started the unit with basic counting. Now this may seem too basic and you may think, “what kid doesn’t know how to count by grade two?” This may seem an obvious skill to many but it is something that many (not some) still struggle with.

 Counting goes beyond being ale to tag each object and say its corresponding number. By grade two students should be seeing groups of objects, especially twos, fives and tens, and be able to count by them efficiently and effectively. Students are still grasping with recognizing fives and tens as they count often still counting by ones till they get to five and then putting that aside. Students should start to see 5′s as 2+3 or 4+1 or even better 10′s as 9+1. 5+5, 2+8, 4+6, 7+3, without having to count.

 To help with this we have been collecting and organizing objects in our classroom. Students have been counting bins, pencils, books, etc. in order to tell me how many is in each basic. We then moved to figure out how many bundles of tens there was in each basket and if there was any patterns we noticed in the numbers. Students soon realized that the number (or numbers) to the left became the amount of groups of tens. I told them that this was because that is called the tens column in the place value system and really it is saying 1 group of 10 or 1 x 10.

 We are now trying to see how many groups of fives and tens there are in the bins. Now again, I thought to myself this should be an easier concept. Obviously if they see the fives then they will see how many tens. I also thought that since we worked on doubling so much in patterning that they would see that there was two fives in one ten. However, I was wrong again. Like many students, we are struggling to see how one group of objects can be called a 1 group but still be 5 or 10 things. Another mistake that my students are making is assuming that the ones place value tells us how many tens we have. They assume that if the left column told us the tens then the right must tell us the ones. We are currently working on this concept by looking at numbers and asking how many tens and how many fives? The follow up questions are simple: What patterns do you notice? Why does this occur? My hope is that students will see that there are two fives for every ten and if the leftovers (after making a group of ten) is greater then five it is just one more group. Example: 76: The number 76 has 7 groups of tens because there is 7 tens in 70 (10+10+10+10+10+10+10=70). We also have 15 fives because there are two fives in one ten and we have 7 tens so you double it; however, we also have 6 leftover which can make another group of five; making the total 15 fives, with one leftover. 

 To help out at home, keep practising the subutizing plates (dot plates) or counting objects in the house and looking for patterns.
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Making conjectures and proving them

In grade two we have been exploring the concept of doubles and what is a double. It’s an interesting concept because we probably assume that by grade two students should know what a double is and why it is called a double, but that was not so. Oh of course, all of the kids could count by twos, but when asked what makes a double all I got was blank stares.  With this in mind we went through some problems exploring whAt a double is.  We started with the story of Madeline, see previous post, and then moved to a discussion about where we have seen a double before.

Today, the class looked at pairs of shoes.  The problem was if each person in your house had one pair of shoes in the front hallway, how many pairs would you have and how many individual shoes would you have?

Most of the students drew out the people and then the shoes, they then counted by ones or twos to get the individual shoes. This alone is a good math problem but I decided to take it a step forward. I asked them to look at their results and make theory, so that I could figure out how many shoes I would have for any number of people?  Once they made a theory they had to test t out to make sure it was true.

I was amazed at how many of the students looked puzzled. It was almost like I asked them to fly to the moon. I am amazed every year at how students struggle with thinking. We often say that’s we are teaching 21st century skills but are we really?

As I look at what my students eventually did I think how they are starting to become real mathematicians. Sure I could have told them the rule was the amount of people doubled would give you the individual shoes because each person has two shoes or mode the rule with pictures, t-charts and then follow up with a question like, “what pattern do you see?” or i could count the shoes with the kids, but then would my students have learned?

By doing this, this way, I have allowed my students to make their own theories and thentestthem out and prove them to the mathematical community. They have thought about the process, they have looked at the numbers in relation to the context and the math became real.

So I ask you, what are you doing to make your students think?

Madeline

Madeline Math Problem

This was a really cool problem.  The kids read the story of Madeline and we discussed all of the math problems in it.  After closely looking at the story the kids saw that the girls always walked in two rows.  We had a discussion about how many kids that was and how they knew.  I then had them go back to their seats and work on this problem:

1) Suppose that there was 7 kids in a row, how many would their be altogether?

2) What if there was 8 kids in a row, how many would their be altogether?

3) What is there was 23 kids in a row, how many would their be althogether?

Here is what the kids said:

Dot Plates

It has been an amazing experience working with Dot Plates.  Such a simple exercise but what rich discussion we had.  In this simple exercise my students learned about subitizing, counting on and one to one tagging.  They also learned that numbers are made up of other numbers and that there are parts to numbers.  This is the foundation for addition and subtraction.  If you want to create your own dot plates all you need are simple 35 paper plates and bingo dabbers.  The patterns are simple here is a youtube video to follow:  http://www.youtube.com/watch?v=POSgVl07Go0.   Our next move is to play a game called part-whole Bingo. I’ll fill you in on how that goes in my next blog.