Fractions: Using Graph Paper to Prove Equivalency of Hundreths and Tenths

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SWBAT prove that hundredths and tenths are equivalent using a fractional model.

Big Idea

Graph paper is the tool of choice as we prove that fractions of 10th and 100ths are equivalent.

Warm Up: Counting by 11's backwards

5 minutes

I was in a hurry to get going with my core lesson today, but always try to take a moment every day to work on numbers sense, even if it's just for a minute or two. I know my students are weak with mental math because I see the hesitancy and difficulty with addition and subtraction. Even their basic facts are not coming a long. Common Core demands that they are fluently adding and subtracting numbers. Since Common Core sets them up to be college and career ready, I don't like to underestimate the value of mental math in that process, regardless of technology! So little goofy games like this really help. I have a 100s chart posted in my classroom and I often pick numbers to add and subtract. Today, we subtracted by 11's starting from 72 and working as close to zero as we could. As we recited and I pointed to start out, one student shouted, " I got the pattern! You just subtract ten and then one!" I  could have told them that, but had hoped someone would catch it. And they did! We are further along with this now than we were in the fall! My hope is that they apply it without a numbers chart and so that is the next step.

Comparing with Graph Paper

15 minutes


Resources Needed:Tenths and Hundredths SB File Sample 

1/2' Graph Paper.

I opened this lesson today with reviewing values of place value blocks. We discussed the value of each block, rod and then square. I used manipulatives and held them up as we talked. I placed the rod over each section as we counted to 100 by tens. I told them that today the place value block of 1 whole is going to represent the number 1 and that all values, the block and rod are less than one.

It was important to establish this so that they didn't confuse it with the past use where the block represented one whole. I told them specifically using the word "value" that today, the block represents the number one. That 100/100 = 1. I told them that it was the same as having a number line  from 0 to 1, divided by 10's and 100's. That was a lot to wrap their minds around, and I could tell many were on board. I gave it a minute to sink in before I moved on.

I opened up the SB lesson and began using the models on the SB.Tenths and Hundredths SB Student Notes. I also referred to the magnetic fractional models using 10ths and the whole, so they could see me manipulate that 10 tenths was one whole. I reiterated that in this case, one whole was 100 pieces and that there were ten in each rod. Therefore if we had one rod, it would be 1/10th of the whole.

They understood the first page without a problem. When I turned to the second page, I was certain they would need a lot of guidance in understanding equivalence of 10th and 100ths. I was wrong. Most of them got it right away.

We drew 3/10s on the SB using the grid paper page. I asked how many hundredths 3/10 would be?  Students' hands shot up. I told them to shout it out on the count of three. They shouted out 30/100ths...with a few 3/100ths in the minority.

I picked someone who could come up to the SB and point and explain why 30 / 100ths was equivalent to 3/10ths.

We did 60/100ths and talked again about the 6/10ths. I ran my finger along the rods on the drawing counting by tens to reinforce those who might be having trouble visualizing it still.

I stopped and told them it was time to draw their own on grid paper. I asked that they draw three equivalent 10th and 100th problems. We picked out together 4/10, 5/10th and then their choice.


Classroom practice

20 minutes

Students got out colored pencils, rulers and each took a piece of 1/4" grid paper. They began drawing out 10x10 blocks next to each other on their graph paper, as we had done. I saw them all work to be on task doing what they needed to do. When they were finished, I checked work. If the work was correct, they could go to Educreations on their iPad and transfer it to technology by creating a grid to practice some more or re- draw what they had done. I told them to record on it by explaining the equivalent fractions they had drawn and then save it to their files for study later.

As I roved, I stopped by one student who needed some help. As I helped him I realized that his 10x 10 was  not quite square. So I worked with him to point it out. The sample above was just fine, so I felt he understood the 10x10 concept. During this conversation I reinforced his understanding of counting the rods and using his finger as a guide to count. He was trying to do it with just his eyes and he was miscounting. However, he could explain the application of arrays strategy to count the 100th's. I think it is good to draw back on old strategies and I really reinforce this in my teaching as we master standards. I understand that CCSS is interdependent and scaffolding, so spiraling strategies and skills is essential. But this kid did it on his own! Wow.

We were done fairly quickly, so I stopped everyone to wrap up. I heard a lot of people say how much fun this was. Through this strategy of drawing the equivalency, they come to own the standard because the visual understanding stays in their minds. Numbers should be easier to manipulate now and equivalency should be understood deeply, making it easy to transfer to decimals next.


5 minutes

We took a minute and had three quick shares. Everyone held up their pictures and I "scooped the loop" to make sure that everyone had mastered this part of the standard: create equivalent fractions of 10th and 100ths. They indeed had! 

Homework: IXL math Level F Q.2  I assigned homework practice that involves just the fractional equivalency written as an equality. 3/10 = 30/100. I told them that they should use their I pad on Educreations and draw it out using the grid paper. They could erase it and use it again and again.