# Fraction Online Excitement!

I’m excited about this fraction lesson on two levels. (1) I taught it to a group of K-8 teachers in our graduate program in an online session. (2) We had a situation occur that confirmed the validity of the activity.

Prior to coming to class, students were given 2 equal sized fractions – 1 divided into 3 equal parts with 2 shaded and 1 divided into 4 equal parts with 1 shaded. (Actually, they were so lightly shaded the students were instructed to re-shade the rectangles. This comes in play later.) They were instructed to have at least 6 of each rectangle cut out and ready to use for class.

We met in an online room and the instructions were to take the rectangle with fourths and fold it so that each strip (or fourth) was divided into equal sized pieces. I didn’t tell them how many pieces or how to fold it. While the students were doing this, I drew rectangles on the white board and told the students to represent their new rectangle on the white board and write the fraction for the new rectangle. Below are the results.

I then asked them to look at the 4 representations and make an observation or statement about what they see. Student A said, “All of our new fractions are equal to the ¼ that we originally had.” I asked her how she knew. She went on to describe how the rectangles were folded differently but each of the fractions were equal to 2/8. I asked for other comments and Student K stated that “if you were to cut off the yellow portions of each of the rectangles and placed them on top of each other, they would all be the same size. That’s how we can say they’re all equal to ¼.” Student S noticed that “every time the numerator goes down one like from 4 to 3, the denominator goes down by 4 from 16 to 12. And then from 3 to 2, the denominator goes 12 to 8.” (I wish I had explored this statement further with a question of “Why do you think this happens?” But I was focused on my goal of this lesson and completely let that nugget pass me by. Maybe I’ll bring this comment back up in another class and we’ll investigate why this happens then.)

At this point I had a goal in mind and they were nowhere near that goal, so I decided to have them try this again. I asked them to get another ¼ rectangle and follow the same instructions as before but to make sure they folded it differently from the first time. I asked similar questions and received similar comments. This was not happening as I anticipated so I decided it was time to move on.

Next I asked them to get the rectangle divided into thirds with 2/3 shaded. I again asked them to fold the rectangle so that each third is divided into equal sized pieces. While they were doing this I set up the whiteboard with 4 rectangles.

This time I asked them to look at their new fractions, the representation of the new fractions and the original fraction of 2/3 and to see if they could relate these 3 to each other. Student N said, “Starting with 2/3, I divided the 3rds into 4 more equal pieces within each third. By adding those pieces it’s almost like I multiplied it by 4 more and 2 x 4 is 8 and 3 x 4 is 12.” Student A stated, “Like Student N, I took my thirds and divided them into 3 equal pieces. So like hers was 3 x 4 is 12. Mine was 3 x 3 was 9. It’s like everyone’s.” Student S said, “They were talking about division I saw mine as 2 times as many. Each third has twice as many so I divided each third into two. So I guess I divided too but I thought of it as multiplication.” YES!! This discussion was just what I had been looking for!

I then asked them to look through the K-5 fraction standard(s) and determine which ones we had been working with up to this point. I wanted to see if they ‘saw’ what I was hoping they would see.

This is what they came up with

3NF.3 – Explain equivalence of fractions in special cases…

4.NF.1 – Explain why a fraction a/b is equivalent to a fraction (n x a)/(n x b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.

Student S explained 4.NF.1 by saying, “We kind of did this with our last activity of 2/3 times a certain amount. We divided our thirds into more, we multiplied the number of parts in each third. So we did something with that kind of.”

I said that I thought I understood what Student S was saying. I then asked if someone could explain in their own words the same standard.

Student A said, “I can see how students can totally GET multiplying the numerator and denominator by the same thing; they could actually see it holding true. It’s not just something that we say, but they can see that relationship working…I think that this activity would explain why we say ‘whatever you do to the numerator you do to the denominator’ because it actually HAS to be that way when you’re folding the rectangles. You changed each third into 4 smaller pieces. You were doing it to each third. I see students get confused and just multiply either the numerator or denominator by 4 but this would help them see that the entire rectangle’s pieces are being changed.”

Bingo! This was what I had hoped would come from this activity! We then moved on to the second goal of the lesson.

They were instructed to take both rectangles – where 1 fourth was shaded and where 2 thirds were shaded – and fold both sheets in such a way that both rectangles were divided into the same number of equal pieces. This took a little longer for some. I went ahead and drew rectangles on the board while they worked.

I then asked them how they decided what they were going to do. Student A said, “I wanted them to be equal. I chose 12 because 12 is a multiple of 3 and 12 is a multiple of 4 as well.” Student T stated, “I thought about our previous problems and I had folded both of them into 12ths before so I knew that I could do it again.” Student K pointed out that one might also do this activity by trial and error. To be honest this is exactly how I solved it because I wanted to see what a child might do who didn’t know about common denominators. I first folded the thirds in half horizontally and the fourths in half horizontally. That gave me 6ths and 8ths. I folded the thirds again (just repeating the first fold) and then one more time horizontally. That gave me 12ths. I looked at the fourths that had been folded into 8ths and realized that I couldn’t turn it into 12ths; but if I started over with a new fourths rectangle, I could fold it into thirds horizontally it would also have 12ths.

After this discussion I then asked If we wanted to compare 8/12 and 3/12, which fractional amount is greater and why or are they equal?

Student A stated that they are not equal. “8/12 is greater than 3/12.” I asked her how she knew. “I’m looking at my models and 2/3 is 8/12 and there is more shaded than the ¼ or 3/12 on my other rectangle. But if you look at my drawing you can see that 8/12 is 5/12 more than 3/12.”

Again, we moved on to another problem. I asked, “Can you fold both rectangles in such a way that the shaded amount on each rectangle is divided into the same number of pieces?”

Then something beautiful happened. Notice that some students already have their drawings up while the 4th student is still folding her rectangles. I notice that Student K doesn’t have 2/3 shaded on her first rectangle so I asked a tricky question, “Hey Student K, did you do the actual folds on your rectangles or did you just start drawing it on the board first?” She used a technique she’s seen me use often – answered my question with a question, “did you ask us to fold it so that you have the same number of shaded pieces? They each have 2 pieces shaded; they’re just not of equal size.” I then repeated my question, to which she confessed that she didn’t fold the papers. After looking at the other drawings she realized that she didn’t shade in 2/3 of the first rectangle. So she went to correct it by shading in the 2nd third. I told her to go back and do the problem again with her paper rectangles.

Notice what happened. Again Student K did not use the paper rectangles. Instead she hurriedly divided the 2 eighths into  shaded equal pieces because she already had 4 shaded equal pieces on the first rectangle. As a result she didn’t think to continue the new horizontal lines. This confirmed for me the importance of students using the actual paper rectangles and folding them. If you fold a horizontal line in the shaded section, it’s going to go all the way across the sections that are not shaded as well.

Once all 4 students completed their drawings and fractions I asked them to use words to discuss their 2 fractions? Student N said, “My fraction 4/6 was orinigally 2/3 and I went ahaead and divided it more so 4 out of 6 parts are shaded. And in the picture on right, it started as ¼ and I divided that ¼ further into smaller parts of fours. So it pretty much mulitplied itself by 4 going from ¼ to 4/16. So you can see how ¼ times 4 on both the numerator and denominator is 4/16.”

Student A stated, “Both of my numerators are 6 and I can see from the shaded parts that I have a common numerator. We can compare the 6/9 to 6/24. We know 6/9 Is greater but typically we compare with the denominators the same but now we are comparing the numerators.”

I then said let’s assume a student has had a week or two of doing similar activities to what we’ve been doing with folding rectangles, representing them on paper, writing the fractions, discussing them, comparing them. How might one know which fraction, 6/9 or 6/24, would be greater and why, without the rectangles?

Student S said, “In one of them you imagine folding the fractions into much more pieces. So they are all smaller pieces than in the other rectangle where you folded it only into 9 pieces. So the pieces are much bigger than the other pieces. So 6 bigger pieces (6/9) is much bigger than 6 much smaller pieces in 6/24.”

What standards are we addressing now?

3.F.3.d. Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, <, and justify the conclusions, e.g., by using a visual fraction model.

I was very pleased with how this lesson progressed with developing fractional understanding of equivalence and fraction comparisons. But I was also delighted that we were able to develop such deep understanding in an online setting. We were once timid about doing any of our program online because we felt that it would not be as ‘good’ as face to face classes. This shows that it can be done. My brain does not automatically think in an online format yet, but it’s getting better and I’m no longer afraid to try new things.

# Why Students in the US need Common Core Math

It seems that everywhere I go, I either read or hear someone’s opinion regarding the Common Core standards. (As I’ve stated before I am ignorant of the English Common Core Standards, so I refrain from making any comment.) Please become accurately educated regarding the Common Core Math standards. Attached is an excellent video produced by Jo Boaler, mathematics educator from Stanford University. She clearly explains the intent of the Common Core Math standards as well as provides evidence to support its existence.

# Frustrated Reflections

The following is a reflection written by a night time student and a day time teacher. She provides thoughtful insights on frustrations.

“…I was incredibly frustrated when I could see the problem, but I was making a mistake that I didn’t notice, causing me to get stuck. At this point, if this problem were being completed by my students, many of them would quit trying. For me, though, there is a satisfaction of knowing I struggled through a problem and triumphed, a sense of accomplishment. This class and this type of experience make me want to be more vulnerable around our students and other teachers. If we share our struggles with our students and show them how to push through that struggle, maybe some of them will have the courage to push through the difficulty they are having when working out something in our classes. One of the biggest complaints I hear from other teachers is the lack of motivation in our students… [T]eachers [are] still learning how to teach conceptually, we [should] show students that there are things we struggle through just like they do.

“The problem that we completed last week with the various routes was interesting in that all of the people at my table attempted the problem in a different manner at first. Once we all got to a place of frustration with the amount of combinations, the pieces that we each had helped us to be able to get organized to make sense of the problem. This type of problem could be taught in school just as it is, but I wouldn’t extend it to the point we did in class unless I was working with high school students. Most middle school students would get extremely frustrated. I even got a bit frustrated, as my peak performance is not in the evening; I am a much better thinker first thing in the morning. This brings me to another observation of my classes lately: we need to remember that students’ peak performance time does not always match with the school day calendar. While I’m not saying we should let students sit back and take a nap during class, I am saying we need to be mindful of students’ energy level and if they are having difficulty seeing a problem through. Maybe they need a hint or to work with the teacher in order to find some inspiration or motivation; maybe a student needs to get up and stretch; or maybe a student needs to think the problem through in a quiet place.”

Heather Isenhardt is a middle school mathematics coach in Henry County Georgia. She is also a graduate student in K-8 mathematics education at the University of Georgia.

# Saturday/Online Hybrid Cohort to Begin August 2014

I hope that you all will help me spread the word!The K-8 mathematics education graduate degree programs are now being offered on the Griffin Campus of the University of Georgia in a hybrid format. We hope to start a new cohort in August.

If you are interested in more information email me, tweet me, facebook me, or write a comment to me here.

Pass it On!
Robyn

# Common Core Clarity

I have a headache! My hands were actually trembling a few minutes ago. Over and over I keep hearing people complain about the Common Core Standards. I know nothing about the Language Arts Standards; I haven’t even read them, so guess what? I’m not going to say anything about them! Undoubtedly there are numerous people ‘out there’ who do not follow the same ‘rule of thumb’.

I came across a post on facebook today. Here is the link: http://www.bizpacreview.com/2013/12/18/arkansas-mom-destroys-common-core-in-four-powerfulw-minutes-89152 In this video a woman is talking about the Common Core State Standards for Mathematics to some people in her state. (I think they were the state school board, but again I will admit that I really do not know.) She claims that the standards are “not rigorous…not college ready…[and] not preparing our students to compete in a global economy.” She also states that the statements to the contrary are “empty sales pitches from corporations and government agencies to profit from our kids and sell them downriver at the name of saving education.” Below is the following 4th grade example she provided.

Mr. [X’s] class has 18 students. If the class counts around by a number and ends with 90 what number did they count by?

She claimed that the students were “expected to draw 18 circles with 90 hash marks solving this problem in exactly 108 steps.”

In her defense she only had 3 minutes to make her point; however, she did not read the standard from which this problem probably was derived. I will include it here for my readers (the notation is included for educators who should be familiar with the format).

CCSS.Math.4NBT6 – Find whole-number quotients and remainders with up to 4-digit dividends and one-digit divisors, using strategies based on place value, properties of operations, and/or the relationships between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays and/or area models.

No where in the Common Core standards would 4th graders be expected to solve this problem in the manner she described. The standard simply states that students should use strategies based on… A student could solve this problem

* using their understanding of doubles: 18 + 18 is 36 (that’s 2), 36 and 36 is 72 (that’s 4) and 72 + 18 is 90 (that’s 5). They counted by 5’s.

* using their understanding of divisibility rules: I know that the only way an 8 in the one’s place will give me a 0 in the one’s place is if I multiply it by 10 or 5. 10 groups of 18 is 180 (that’s too much). Since 90 is half of 180, I know that I need half as many groups. So they counted by 5’s.

The Common Core Standards do not say that students should never learn how to use a traditional algorithm for computation. The standards simply encourage that students develop a conceptual understanding of the operation before being taught the traditional algorithm. Parents (and teachers) are doing their students an injustice when teaching the traditional algorithms too soon. For many children being shown the algorithm too soon hinders their development of conceptually understanding the operation.

Please READ the Common Core Standards before sharing an “uninformed” opinion. I have provided links below to help make this reading more accessible.

http://www.corestandards.org/

http://ime.math.arizona.edu/commoncore/

(Note: I am creating an additional blog site where I will discuss my thoughts on just about anything. It will be titled “Robynisms On Things” http://robynismsonthings.wordpress.com/)

# A New CGI Public Forum

If you use Cognitively Guided Instruction (CGI) questions in your room or if you would like to get started using CGI in your classroom please feel free to go to the following public wiki site that I just created. Please feel free to share and borrow.

# I Might be Famous! Reponses to the Previous Post

I NEVER believed that I would receive so many comments regarding one post. I decided to share the comments I have received. But instead of ‘accepting’ each one I decided to compile them into 1 post. There are several reasons for doing this. The first is that if you follow this post, I am trying to protect you from receiving 50+ emails for each comment I accepted. The second is that I wanted to remove names to protect everyone just in case one didn’t want his/her name shared. The third is so I can respond to them all as a whole.

I want to thank all of you for taking the time to read my post. As you can tell I was discouraged for the teachers I know and I hoped that I could encourage them and (obviously) other teachers that I don’t know by saying that I understand and I support and praise you for all that you do.

I believe that all teachers go in to the profession because they love children and feel called to help educate our future. In addition I believe that that is also the reason why teachers REMAIN in the profession.

The title of my last blog was meant to catch people’s attention. I was NOT saying that everyone in education takes an antidepressant, nor was I saying that if the educational system was perfect NO one in education would be taking antidepressants. That is a personal decision that is between an individual and his/her doctor.

My final comment is that I AM in SUPPORT of the new Common Core Standards as far as the mathematics is concerned. (I say this only because I have not read any of the other content standards, nor would I be comfortable making a judgment about a content in which I am not as familiar with teaching.) There has been a lot of misrepresentation from those opposed to the CCSS by saying that Common Core is the cause for so much testing taking place in schools. This is simply NOT true. Common Core in Mathematics calls for students to develop a deeper understanding of mathematics; to make connections between different ideas in mathematics as well as between the classroom and the real world;  to develop a confidence in thinking mathematically; to make conjectures; to reason abstractly; and to defend and make mathematical arguments. If you are interested in knowing more about the Common Core Standards please click on the following link http://www.corestandards.org/.

And my final response is that I purchased my t-shirt from http://www.signals.com/cgi-bin/hazel.cgi?action=DETAIL&ITEM=CL6271 but it is my guess that it can be found elsewhere. I typed the slogan into a search engine and there are buttons, bumper stickers, etc with the same slogan.

If you have the time, skim over the following comments provided by readers (current teachers, retired teachers, para professionals, pre-service teachers, parents, grandparents, and non-teachers). I have copied word for word (correctly or incorrectly spelled) every post except for one that contained a curse word.

1. Love this Robyn! I am glad to hear you say that you are actually okay with most of the common core standards because that is one of the reasons I have been concerned about having Noah in public school. I agree the testing stuff seems ridiculous. I know I have been out of the classroom for 15 years, but now I wonder, would I even WANT to go back? Or could I even handle it if I did???? I don’t know what the answer is. I wish I did. I am just so abundantly thankful that I have been able to keep my kiddos at home and school them here, climbing trees and all!!! And I am thankful for those teachers who are in the trenches and are now teaching my son. I pray for them each day because my son and others NEED great teachers desperately. Thank you for shedding some light on this for me. I will be thanking his teachers TODAY for a job well done…
2. Thank you!
3. You are so right! How can we change it! I’ve taken my child out of school 3 times because of this crazy and how it is affecting her. I honor every teacher that puts up with this. This crazy needs to change. I’ve been saying this for 5 years every since my daughter started school. How do we take back OUR educational system?
4. Step one…get all of the politicians and school district superintendents OUT of the restaurants where they hold their “so-called” educational conferences/meetings/professional development seminars, and make them SIT IN on classes, ALL DAY, EVERYDAY. Let them FEEL the stress along with the teachers and students, instead of CREATING IT over lunch and coffee…in between all the conversation about their golfing events. Trust me on this. While I’m not a teacher, I have been a waitress who has waited on these Educational and political big wigs. I’ve overheard their conversations. They are, in my own opinion, nothing more than extremely over paid, professional luncheon organizers.
5. Another thing you can do – contact your legislators and school board members!!!!! Speak up!!!
6. Thank you! I needed that today!
7. SUCH A SAD TIME WE LIVE IN….. THIS BLOG IS SO TRUE. WHEN THIS COUNTRY STARTS LETTING THE CHILDREN BE CHILDREN ,THEN THE TEACHERS CAN ONCE AGAIN BECOME TEACHERS. I WILL BE PRAYING FOR EACH ONE OF YOU. AND THANK YOU !
8. Where can I get that cool & totally true tee?
9. Love it! Children are being “shoved into their heads” far too soon!! Great book: “Magical Child,” by Joseph Chilton Pearce. I read it decades ago and concur with 95% of it based on my own life experience, other reading I’ve done, and observation over the years. My grandmother taught 4th grade for 25 years. She was the best teacher in the world! It was a lot of fun and everyone learned what they needed to learn in the midst of enjoying it all. What can be better than that?
10. As a 3rd grade teacher, I am so thankful to you. We are truly over stressed and underpaid, but we love our students dearly. Those who teach, love to teach, no matter how stressful things get. Unfortunately, it is part of the way our educational system works. I just wish we could let our kids be kids, and not be stressed out at such a young age. It makes me so sad to see my 3rd graders worried about these standardized test. I agree, something needs to be done.
11. Thank you Robyn. This was perfect.
12. Perfect!!!
13. As a former paraprofessional, substitute teacher, and now a teacher, I say bravo to your blog!!! Teachers are severely under appreciated, and under paid. Why does a pro athlete make millions of dollars more than the person who teaches you how to read, write, add, and subtract? What message does this send to our nation’s youth? Not a positive one if you ask me.
14. Will pray more for our teachers. Sounds more stressful than ever. Makes me glad to be retired before this new curriculum, even though I loved teaching very much.
15. we can do more than prayer. We can hold protests and become political. If you’re a parent, please ask your teacher what they need help with… Parents are more powerful than teachers. A parent that goes to the district office can get a few things done for that teacher.The parent is more likely to get the air filter changed in the room then the teacher! The parents can protest the test and not send their child to school that day.
16. wonderfully written and VERY truthful!!
17. You hit the nail on the head!! Awesome job!!
18. And we wonder why fewer college students are heading for careers in education. Or why so many first-year teachers don’t continue into second year. My terrific husband retired recently from 30+ years as an elementary music teacher. He had some fantastic years, and he still keeps in touch with some phenomenal colleagues. But everything you said above, about what teaching is like – and the toll it is taking on teachers and on children – is true true true.
19. Thank you… and…political correctness.
20. I’m glad that someone understands. While the rewards are priceless, the journey is exhausting.
21. So very well written and so very right. Thanks!
22. When I retired I had an ulcer, was on Xanax, had high blood pressure and panic attacks. After retirement I was diagnosed with PTSD. I had planned to teach 2 more years but my health could not take it. I felt worthless, incompetent and depressed. Now, a year and a half later I feel much better, I sleep at night, I am no longer on Xanax and haven’t had a panic attack in well over a year. Not only that but I remember when I loved teaching, when I felt like I was making a difference and changing lives. I miss that. However, teaching as it is today was joyless for me and I would not return for anything.
23. Well said, you really captured the environment in which many teachers work in across our nation. Thank you for posting.
24. Many of us speak up and no one listens, which leads us to believe “they” don’t care. Sad isn’t it. So much for the legislation of No Child Left Behind. That’s exactly what “they” are doing!
25. I am a 13 year vet teacher. Every word you have written is true. I stay in teaching because I love it, but over the past few years I have gotten the distinct impression that it doesn’t love me back.. If I hadn’t gotten a job in a charter school this year, where I am given more autonomy as a teacher, I might have thrown in the towel. We still have to adhere to standardized testing, but those tests are not considered the BE ALL of teaching and learning. If this nation is going to retain the teachers it has and attract a new generation of teachers, things are going to have to change..
26. Awesome !!
27. so well stated and I agree 100%~I stopped teaching 8 years ago basically as a result of pretty much everything you have written. I miss it, but not enough to feel that emotional pain.
28. Thank you. I am 27 years in and all I can think of is that I want out. I have been on anti-depressants for over ten years. I can’t do this anymore. I am glad someone understands.
29. Most people have to deal with these type of expectations at their job. The difference is most people didn’t grow up next to someone performing their occupation, allowing them to develop a child’s view of what that job entails.
30. Don’t forget to add in severe behavior problems, behavior plans for individual students, referrals, counseling, before and after school duties, lunch with students, and PLCs.
31. Amen!!! About the only things left out were working all those sporting events (gate, concession stand, crowd control), and our ESL students. I have ESL students who work from 10 pm to 6 am, come to school, go home and nap, and start again. Their learning is limited to classroom.
32. This is simply the truth!
33. Wow! is all I can say…………..
34. Thanks for writing this. It’s nice to know that someone is actually paying attention. When you’re in the situation and can’t say anything for fear of losing your job, it’s nice to have someone else raise their voice and question what in the world is happening…
35. I think you’re exactly head-on! I’ve been teaching for 12 years and I love it but I spent most of yesterday and will spend a good bit of today working on reports that I couldn’t finish during my “work hours” because I was trying to TEACH!!
36. I feel your pain. I taught first grade for thirty four years, retiring 4 years ago. Over the years I taught I went from not having a complete set of books to teach from my first few years, to watching the administration jump on every new program possible if it was recomended by some higher paper pushing “expert”. There is WASTE in education.. We need fewer experts, lots of people with good common sense, more input from the classroom teachers who are there fighting the battles, and if you find something that works don’t throw it out to jump on some glossy program that an “expert” will say solves all your problems. That program does not usually exist. If you have given your best don’t worry about the rest. Education is just a reflection of our society today. Everyone is stressed. When they have a problem the first thing to do is blame someone else. Thus all the pressure on teachers to produce a perfect product at the end of 12 years. Forget all the others who touch the student’s life daily and screw up. Through out my career I felt so sorry for the little kids who were there trying to do the best with so little support from the outside world. Sometimes you as a teacher are the only security that a child may feel Please hang in there and remember when you have done your best then that is all someone can expect of you.