My small rural school needed someone to do summer school for elementary math.
So I volunteered. I teach high school. How hard could it be? We split the large group of 16 into a junior high group and elementary group. Summer school would consist of 8 days total, split into two weeks, Monday through Thursday from 8:30 to 11:30. The idea is that the students would complete their previous grade level using Study Island. Study Island is a great website that would provide an individualized curriculum and pacing for each student.
After day one, I was going out of my mind with boredom. I went around and helped a few that had questions and observed many that chose not ask questions and just guess. This was dissapointing. Not only were they bored and discouraged, but I was not seeing the type of learning I love-engagement, confidence, enthusiasm.
I decided that the next day, I would break up the monotony with some activities. I stopped by HEB and purchased two plastic balls, six foam noodles, and giant markers. When I have fun, I like to go big!
Taking one of the plastic balls I wrote the numbers one through twelve all over it. As a warm-up game, the students and I would get in a circle (harder than I thought it would be) and bounce the ball to each other. When the student catches the ball they would read off the two numbers that their thumbs were on as a multiplication fact and then the answer aloud.
After the multiplication fact warm-up, I had the students go online to Study Island. While they worked, I wrote the numbers 0 to 50 on sheets of colored paper. I laid them on the floor in order and in rows of ten (exception of the first row).
After about twenty minutes on the computers, I called for a break and transision.
We began 'Prime and Composite Exploration'
Initial layout of values 0 to 50.
I want to share the conversation that I had with the students. This is my favorite part of teaching-listening to what the students think. (I think that's what drove me crazy the first day-no interaction.)
T: What do you notice about the values you see?
S: The numbers are between zero and fifty.
T: What else do you notice?
S: They are in rows of ten.
S: Five rows of ten except for the first which includes zero.
T: What else do you notice about the values that you wouldn't notice if I had tossed them on the floor?
T: The following value exist between which two values?
S: Zero and fifty
T: Select a value for me.
T: What are the fact pairs of fifty? (we elaborated a little on the difference between factor and fact pairs)
S: Five times ten, fifty times one, twenty-five times two.
T: Are you sure?
S: (Thinking. When you ask if they are sure, they start thinking and exploring) ten times five, one times fifty, and two times twenty-five
T: Are you sure? Do you think it makes a difference?
S: No, not really.
T: Does anyone know the name of that property ten times five and five times ten?
T: Commutative Property
T: Now, select a value and list the fact pairs for that value.
I laid out their selected values
(There was one incorrect factor pair-twelve times three for sixteen)
T: What do you notice about the values?
S: They are between four and thrity-six.
T: What else do you observe about the following values? What do they have in common?
S: They all have a factor of two.
T: What do you know about numbers that contain a factor of two?
T: Are you sure?
S: (Pause)... Yes!
(Process is repeated. Students are this time asked to select an odd value as their second number and list the factor pairs. One student selected zero. ;) I love when this happens. Oh, the conversations to come.)
(The student that selected zero realized that his list of math fact pairs of multiplication wasn't going to stop. He asked me if he could just list what he had. I told him a mathematician never leaves something unfinished. They, at the very least, summarize with a hypothesis and then give it up to peer review. He liked the idea. He did pretty well: "everything times zero is zero!")
T: The following numbers exist between what values?
S: Zero and forty-nine. (I need to expect complete sentences now that I reflect upon this lesson.)
T: Take thirty seconds to make an observation of the following values. ... Share your conclusion with your neighbor. ... Who would like to share?
S: They're all odd.
T: What indicates an even number?
S: A factor of two. ... Wait, so, zero isn't odd? (Interesting result.)
T: What do you think?
(I love these moments because students begin to think and share. After a few minutes of good spirited debate between the students, I shared my brief research on the number. Some people consider it even for certain reasons, others consider it neither even nor odd, and some consider it both. I told the students to be objective.)
(For my younger group of 2nd through 4th grade, they were not as quick to see even and odd numbers. I was surprised and figured it was due to my type of questioning and vocabulary. So, I took their stack of selected values and split them into to two seperate circles for a Venn diagram. (I created two circles out of pool foam noodles with bright blue duct tape.) I used a think pair share strategy and finally aquired the conclusion of even and odd from them. I asked them why and they described the value of the ones place as the determining factor for even or odd. I then had them label to two circles of the Venn diagram as Even and Odd.)
We continued a combination of computer work, listing factor pairs, and multiplication fact practice.
My classroom is ready for my return!!! I couldn't wait to get into my teaching grove. I'll admit that over the past year, I have gotten further away from integrating technology into my classroom. My first two years, my students each had a laptop and the integration of technology was a breeze. From Khan Academy to My Big Campus to Survey Monkey, class work was full of technology usage.
The kids were intrigued when they found out we were going to my classroom. If I could make so much out of a computer lab, what would my classroom be like? (I will share it later :) )
They were not dissappointed, and I was instantly back in my groove.
Today's Lesson: Pentominoes into Probability!!
I started with something familiar to most - a domino.
I asked what shape they saw. The answer I sometimes get and am always concerned about is "It's a square." No matter what, I have to have the square to rectangle to rhombus conversation. I feel that it's too important to leave out.
I tried several strategies. My first was to draw some rectangles and squares on my document camera. I asked what they saw and how they knew. Still, no connection to side lengths and angle measures. I pulled out the Anglegs. I asked what they thought the different colors indicated of each stick. They were quick to know same length for same colors. Then, I asked them to make a square. I watched as most chose four of the same color and a few chose pairs of different lengths that were close to congruent. I chose some for presentation and finally, the students came to understand the similarities and differences between, rectangles, squares, and rhombi.
Then, we went back to the domino. How many squares make up a domino? So, if 'do' could stand for two, then what could pentomino represent? Students were unsure until I boxed 'Pent' asked them what they knew. Instantly, a student would blurt out 'pentagon'. Ok, what is a pentagon? They knew it to be a shape with five sides. So, what would be a Pentomino? Loved this response: "An omino with five squares!"
From a box of tiles, I showed how two squares make a domino and then made a pentomino. We went over the facts of a pentomino - tiles connected with full sides, no overlapping, etc. I provided each table with a box of tiles and challenged them to find as many different pentominoes as they could.
Then came another great conversation. Is it the same pentomino if the pattern is flipped, turned, etc. So, I used myself as an example. Who am I? Ok, when I turn around, who am I? Am I still the same person? Ok, is it still the same pentomino? Yes!
After I had passed out the materials to make our summer school mini interactive books, I asked the students how many pentominoes they had and provided them with the goal of twelve pentominoes. Once, each table had discovered twelve, I asked each group to rotate to the next table and check the previous group's work.
After checking and modifying, they pushed the pentominoes aside. It was time to create our mini interactive books.
How to make a 'No Staples, Glue, or Tape Needed' Booklet:
For simple usage and to start with, you only need two pieces of paper and a pair of scissors.
1. Fold both sheets in half.
2. Taking one folded sheet of paper, measure appoximately an inch along the folded side from both edges and mark it. Do the same to the other folded sheet of paper.
3. Taking the scissors and one of the folded sheets and trim just a hair off the folded edge rom the edge to the marked spot on both sides. We will call this sheet A.
4. Taking the scissors and the other folded sheet, trim a hair off the folded side between the two markings. We will call this sheet B. The trimmings should be opposites - one inside and the other outside the markings.
5. Unfold sheet A. Bend and hold sheet A long ways and so that it resembles the shape of a taco.
6. Take sheet A and slip it through sheet B's slit, unfolding sheet A so that it locks into place.
7. Fold it over to finish making a booklet.
Kids are amazed! So simple, yet tricky. I also use this to represent intersecting planes in geometry!
After creating our mini books, we turned to page one and wrote the title - "Pentominoes"
On a gridded marker board displayed by the document camera, I had each student come up and draw one pentomino until we had all twelve displayed. After we finished copying down all pentominoes, I asked them a question.
How many total squares/tiles are on the table?
Most students thought about it and concluded to sixty.
I asked them how they figured it out. Some students showed that they counted each tile and others explained that twelve groups of five tiles yields a total of 60 tiles. Now, that was my wording. It took some guided questioning to get that final statement written with labels. They were proud of it!!
We turned to page two and labeled it probability.
While the students were creating pentominoes last Thursday, I observed that each tile was drawn at random and thought-"sweet, a transision into probability".
I asked students if they selected each color specifically for each pentomino. They said no, it was at random. I answered with "and that is probability. Now, lets take a look at our results."
I asked students what would be the best thing to do with the tiles to analyze our results. They instantly knew to group them by color.
We then defined probability using a simple shutter fold to identify the numerator and denominator. I have found that defining probability in the form of a fraction helps students connect the definition to a representation. After this we calculated the empirical probability of each color from our experiment. We finished by comparing the final empirical percentages to the theoretical probabilities and talked about which group was closest.
Yesterday, a student requested a review of subtracting fractions. This was tremendous action for this young lady, because on day one she would say one single word. I told her I would come up with something. Now, I couldn't come up with an interesting foldable or lesson and decided to consult the middle school math teacher. She gave me an excellent strategy that the students relate well to. It involves brownies. I was sold, who doesn't love brownies!? When subtracting fractions and you need to borrow from the whole number, she tells the students that it is like borrowing a pan of brownies from your neighbor. You borrow one whole pan and cut it up into the parts you need. Now you have so many brownies and can take away from it. It comes across much better when teaching kids. It worked great!!
Today, we began our exploration of prime an composite. We started by making two quarter page door flap foldables and titled the flaps prime and composite. I list several numbers for each side and asked the students for the math fact pairs. I asked them why I considered 2, 11, and 3 prime and 45 and 16 composite. We finally got down to the conclusion that prime has one math fact pair and composite has more than one math fact pair.
Venn Diagram: I then proposed the idea of categorizing the numbers from zero to fifty based on the knowledge we discussed. I held up two foam circles in the traditional overlapping Venn diagram and asked them what they saw. Most recognized the Venn diagram. I asked them if the circles always have to overlap. They were unsure. I gave each table a selection of their math fact pairs for the numbers zero through fifty. I laid the two circles on the floor and asked them what they think each circle should represent. They wrote title for the circles as prime and composite. After they categorized the numbers as prime or composite, we completed a true false flash card activity to check their selections. We found some mistakes and justified what it should be. They concluded that there was no need for the circles to overlap.
We finished the day with a ball bounce check. On the pink ball, I listed the numbers from zero to fifty and each student identified the number as prime or composite and why. What I found exhilarating was when the students started justifying without being prompted and prompted other to justify if they didn't.
We came back today with a ball bounce review. We then went back to our prime and composite foldable to write a formal definition. We googled several sites and I encourage the students to write one in their own words for the class.
I wanted to provide the students another way to look at prime and composite numbers in terms of multiples. We started with a hundreds chart that we cut down to one to fifty. We started with the first prime that they knew and looked at the multiples. They were quick to pick up that the first number is prime and the multiples are composite.
Last day! And some were actually sad a little bit.
I started by asking them to turn to the last page in our book. I told them that this was the most important page in our book. It is a way for the student to share their reflections over summer school.
Question: What did you enjoy most about summer school? Why?
I told them to keep this question in mind throughout our time today. We have a few more activities to do today. They could answer it along the way or make notes of activities they enjoyed and why.
I was surprised to see students actively writing during down time between activities. It was powerful to see students engaged in a strategy that most generally find dreadful.
This is my favorite response. This response comes from a student who is very low, shy, and quietly spoken. I quickly recognized this on day two and spent days breaking through her tough shell. I knew she was scared to be thought of as stupid. I never once let her feel stupid or like a failure. No matter what her answer was, I would take it and explain what her answer meant and ask another guiding question. No matter how long it took, we always got to the right answer.
Reflection from summer school: During summer school, I noticed that the writing with the lower level students was a much slower progression than the writing of the jr high and high school. I couldn't figure it out. I thought that it was just the grade level. The lower the grade, the slower they write and respond. I took my concern and uncertainty to the elementary couselor. She pulled out their data and as I observed the data I realized that this specific grade had a significant low writing score. Come to find out, they are not writing in the classroom. As a result, their scores are low and my approach to teach was a whole new experience! I went to the principle and shared my findings. I offered the advice and idea that journaling should be used in several subjects for these kids. I warned her that it would be tough and very slow, but the success will be tremendous in the end. I came close to begging to change grade levels just to see how much we could do for the kids. I love the challenge and more than anything, I want to see the kids become successful and empowered!!