The Common Core Standards expect students to model with mathematics. They also expect students to solve problems within 100 using place value strategies. If students can only use one representation of the digits to model their thinking, they will be greatly limited in meeting this standard. Today I am introducing how money is another way to represent digits in a hundred's number. It is another tool towards developing the ability to model mathematical thinking with a variety of different models and not always relying on a single representation. By relating money to the blocks students are already familiar with I am providing a scaffold from the known to the unknown.
Today I begin by putting blocks representing 100, 10 and 1 on the Interactive White Board and ask students to write the numbers they see in their math journals. I put up 3 hundred 8 tens and 2 ones. We try several different combinations of 3 digit numbers such as 5 hundreds, 3 tens and 2 ones. We review the term digit and how we know which digit should be in which place. I am careful here not to put up the same amount for 2 different digits (ie 2 tens and 2 ones) because I want to make sure students are identifying the ones tens and hundreds correctly.
Next I hold up a large representation of a dollar, 2 dimes and a penny and ask them to write the amount that they see. I repeat the same amount with the blocks on the white board. I ask students if they notice anything about the two amounts. How did that happen that the numbers are the same?
I repeat with 3 dollars, 4 dimes and 1 penny. The students write the amount. Next I put up 3 hundreds blocks, 4 tens and 1 one and the students write the number. What just happened, I ask students?
We talk about how many pennies in a dollar (100), how many ones cubes in a hundreds block (100), how many pennies in a dime (10), how many ones cubes in a tens rod (10). Can anyone tell me how the money and the blocks might be related? We discuss how we can represent numbers with a variety of objects such as blocks, money, popsicle stick bundles, etc. The objects help us to see what the digits mean.
Next I show the number 246 on the Interactive White Board and ask if anyone could build the same number using money. I let a child try and then I ask for a volunteer to build the same with the blocks. We again compare the dollar to the 100 block, the dimes to the 10 block and the penny to the ones block.
We talk about how the money and blocks are related.
Together we build the numbers 826, and 219 using and comparing money and blocks.
I tell students that today we will be working to see how money and base ten blocks are related.
I want to give students a chance today to work with the concrete before moving to more abstract representations. I have devised a hands on activity that will allow them to use numbers, blocks and money to represent the same amount. I feel that as students have practice doing this, they will clarify their own understanding of how the objects are representing the numbers and each digit within the number.
I tell students that today they will work in pairs to compare base ten blocks, numbers and coins. I show them an example where I turn over 3 number cards (no numbers above 9, so if you are working with traditional cards remove the 10, Jack, Queen, and King). I choose a student to be my partner. They build the number with base 10 blocks, and I take the right number of dollars, dimes and pennies to equal the amount. We put the pennies next to the ones blocks, the dimes next to the tens and the dollars next to the hundreds. We check to see if we have the same number of each and that our amounts match the numbers.
We do at least one more whole group problem and then I ask for questions before pairing the class up and having them practice together.
During the practice, I move among the groups checking for understanding.
I have children practice for about 15 minutes and then ring the bell and ask students to clean up their materials, return them to their proper places and return to their seats.
During independent practice I give students a paper to complete. The paper asks for students to find the amounts using money or blocks, or to match block and money representations. Here I am looking for students to begin to move from the concrete objects to representational drawings. I am looking to see if students have the understandings that will allow them to move in this direction or if I still need to work with the concrete. I am aware of developmentally appropriate practice here and do not want to pull away from the concrete too fast so I check for individual understanding.
For students who are competent with this work, I provide a challenge page asking for students to solve problems involving 3 digit numbers.
Students did not have trouble today drawing the blocks and money to represent the same number. Most students showed a competency with this. They had more trouble writing the numbers when they looked at the money and block amounts. When I presented the drawings, some of the children wrote things like 30 cents for 3 dimes so their number came out 2305 for 2 dollars, 3 dimes and 5 pennies. This did not happen with the dollars suggesting that students may be less aware that a dollar is 100 pennies so 3 dollars would be 300. They are used to counting dimes by tens so they often counted the dollars 1,2,3 but the dimes 10,20, 30 and wrote them that way.
I saw a similar happening with the blocks where they would count 2 hundreds blocks and write a 2 but them count the tens rods by tens instead of seeing them as how many bundles of ten.
This is a misconception that I will need to work with to correct in future lessons. I need to reinforce the idea that 2 bundles of 10 is equal to 20, but when we record it is the number, we record how many bundles, and not what they are worth.
This is a difficult concept for children in second grade and one that should not be overlooked because the students can read the numbers correctly. Place value is present in many of the second grade Common Core standards and so the time spent in clarifying any misconceptions is important.