The Depth of Decimals: Comparing Using A Fractional Model

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SWBAT compare decimals using a fractional model.

Big Idea

Proving that a decimal is greater than another decimal is easy after an in depth discussion about common mistakes in thinking, strategies to keep us thinking correctly and having fun drawing on graph paper.

Warm Up: I want you to write about this...

10 minutes

Yesterday, I taught students to compare fractions on a number line and assigned a set of comparisons for them to draw. We got out our home work to review because I wanted to be absolutely sure they understood the concept and mastered this portion of the standard. I also wanted to add another faction to it; getting them to write an explanation of their proof.

So, after we corrected each problem on the white board, I turned to them and asked;

How would you write an explanation that proves that this fraction is larger than the other? You have drawn your picture, now how would you explain it?

I turned and a student started dictating her ideas and I started writing. We stopped, thought it through, erased and worked at it until it looked like this:Writing to prove thinking

The sentence made sense and was in their words. This is a great mini lesson about writing in math. I loved it because it was short, sweet and to the point. It helped them participate in thinking and held them accountable for their thinking. Later, I will expect them to practice this writing independently.

The Scoop on Comparing Decimals

15 minutes

Thinking about Comparing Decimals SB Lesson

Today's lesson was a direct instruction whole class lesson because my students have not been exposed to decimals enough that I am convinced they truly understand the meaning behind the place value nor the relationship to fractions. It is my goal in this lesson to get them thinking about their thinking and then demonstrate their understanding through simple drawing on a grid.

First we reviewed and talked about the decimal and what it meant. We discussed how that's where we say "and." I wrote several decimals on the white board and we recited them aloud...reading them properly together. That seemed to be smooth!

I turned to the second page and guided them to think about decimals like money. I think any connection to things that matter, like money, really helps master this standard. As we discussed this page, I heard a lot of "Oh's." One student said that a nickel was 1/5 of a dollar and another student jumped in to correct her by saying, "No it isn't it's 1/20th of a dollar because 5 x20 is 100. There are twenty nickels in a dollar!" This discussion was proving to be as rich as I had hoped and we were only on the second page!

I was anxious to talk about common mistakes because I think that if we can point those out right away, my students are then more aware of what mistakes they could make. We talked through the third page and I heard more "Oh's." One student corrected me saying that I needed decimal points in the first part of the sentence. Another student said " No, she is talking about the number 4 ones and 17 ones being like .4 and .17." I noticed all of them were interested and engaged.

I moved on telling them that our final goal today would be to try and compare fractions and do it with understanding of place value. It was at this point a student asked if we could do it on number lines because he thought that was so much fun yesterday. So I added a page to our SB lesson and we tried it. Only to discover, it was too hard for us to draw those hundredths. We came to the conclusion that we could compare if it were all set up and drawn for us. I told them that we would do that another day.

I turned to the next page to talk about another common mistake. I wanted them to understand that in haste, or lack of thinking about place value, students might mistake a decimal for the wrong fraction. They thought the butterfly was cool because we had just been talking about mimicry in science. My one little "Oh er" spoke up again with a big "Oh I get it now!" I think pointing out the common mistakes was good for this group of students.

Moving on to the Nitty Gritty of Comparing

15 minutes

As I turned to the next page, I heard three students squeal " We get to draw again!" I smiled and proceeded to talk about how on grid paper we can easily see the difference in sizes of decimals.

I used the grid paper on the final page to draw out a 100 block and compare .20 and .2. I asked them to predict which was larger. Several piped up and said, "They are the same." From this I could conclude that my prior guidance had registered! So I drew the first .20 and then let a student draw the second, .2.

I erased and started again, this time comparing .34 and .4. This time, they made the common mistake! The majority thought that .34 would be larger! The visual is so valuable to mastering the concept! I really think they will understand using this modeling!

I had sent a copy of the SB lesson to each of them on their emails asking them to put it in their math folder and keep it to use as they drew their comparisons. I sent them back to their desks to work a few minutes on three comparisons. Some had kept some tenth and hundredth comparison gridwork they had done a few days ago and asked if they could start with it. Of course! The next three videos are of the same student's learning process.You gotta love the way she says "same value." in this video. This was an epic moment! I saw a student connect the concept of value to decimals for the first time in my career! Connections to fractions make a huge difference in her understanding of how to compare decimals because she can visualize it much like a benchmark. Finally, she is drawing the comparison and can master explaining it.

Students practiced like this for awhile until it was time to wrap up.

Wrap it up and assign some homework

10 minutes

We quickly closed with students sharing their drawings and I did a quick scoop around the room looking to see if they were accurate. All were! They had done three drawings and they were accurate!

I assigned for more practice in a different medium IXL math Level F ( 4th grade) T.1. They could practice this for 30 minutes. We tried a couple together and I had a hard time getting them to stop!