2. The students will find the perimeter of the pre-cut construction paper. When cutting the construction paper, it may look like a square, but it should be a different width and length (e.g., 6.8 cm by 7.0 cm). Remind the students to line up the decimals as they record the measurements. If the students haven’t learned how to read a centimeter ruler or how to add decimals, it will be a great introduction before learning from the textbook. It is "applied math", and the students generally catch on quickly, especially as they work in pairs.

Materials
Each group of four students will receive:

Time Frame
Allow 15 minutes for the preliminary (introductory) measurement activity. (This may be done the day before.) The initial inquiry-based math ice investigation will take approximately one hour. The optional extended inquiry-based "Further Investigations" may take one to two weeks.

Engagement and Exploration (Student Inquiry Activity)
Introduce or Review: Circumference and Diameter:

A. Each pair of students will receive an empty Dixie cup, a black marker, a couple of kite strings (~ 21 cm long) or laminated paper strips, and a centimeter ruler. The students will learn how to find the circumference and the diameter of the empty Dixie cup before working with the ice. Since ice melts quickly, it is important for students to know this skill before working with ice.

1. Measure and record the height of the empty Dixie cup with a centimeter ruler. Record in centimeters (universal measurement).

2. Measure and record the "distance around the middle" of the cup (Use string or paper strip, then correlate marks to a centimeter ruler).

3. Introduce the term, "Circumference"- It measures the "distance around" a circle. (The students will learn that circumferenceis a special term used for measuring the perimeter of a circle, since circles do not have sides).

4. The students will also take the circumference for the top and the bottom of the cup.

5. Introduce the term, "Diameter"- It measures the "distance across" the middle of the circle. Have students measure across the top of the cup. Point out the "top" and "bottom" of the cup. The students will often confuse this when they get the ice.
(Remind students that circumference and diameter are two terms used for circles because circles do not have any sides.)

B. Pass out worksheets, "A" and "B", to each group of four students. One pair will record answers for "A", while the other pair records answers on worksheet "B".

1. Each group will set up for the ice activity by placing an absorbent pad or towel on the table. The plastic shoebox-size container will go on top of the towel along with a two-liter bottle of water. The students will pour the water into the container when instructed. It is helpful to have a roll of paper towels for each group. Each pair of students will have a centimeter ruler, a black Sharpie, and 2 strings or laminated paper strips (~21 cm long). Since ice melts quickly, it is helpful to keep the cups of ice in the cooler or freezer until students are ready to work with the ice.

2. The students will use a centimeter ruler to measure and record the perimeter of the box drawn at the top of worksheet. Measure each side, then add the decimals. Stress accurate measurements by having the student’s record measurements in decimal form (e.g., 6.5 instead of 6 and a half).

3. Give each pair of students frozen ice: one pair will get "A" (half-filled), and the other pair will get a "B" (full) 3-oz Dixie cup with frozen ice. The students will remove the paper cup after warming in their hands for five seconds. The students will record the measurements of their ice in the top section of the worksheet. The students will measure and record the height. Review and demonstrate if needed. Then have the students measure the diameter by measuring the flat surface across the top. The students will use a strip of paper or piece of string (mark with a Sharpie or pencil) to note the measurements of their ice, then correlate measurements to the centimeter ruler. The students will need to record each measurement on their worksheet before moving to the next step. The students will measure the circumference around the bottom, middle, and top. (Demonstrate and remind students to record the measurements each time.)

C. Procedure for the bottom of the first worksheet: The students will be involved with an inquiry-based "hands-on" ice investigation: "Will the dimensions of the ice (size or amount) make a difference in how ice will float in the water?"

1. Each group of four students will have a two-liter bottle filled with water and a plastic shoebox-size container. The students will fill each container two-thirds with water. 2. The students will make a prediction of how the ice will float when gently placed in water. 3. The students will draw a water line in the perimeter box at the top of the page before drawing their prediction of how they think the ice will float in the water. 4. Each pair of students will place their ice in the container of water and record their observations on their worksheets. Encourage the students to "talk" about what they are seeing. 5. The students will draw a water line and their "results" in the box at the bottom of their worksheet. 6. The students will write down their conclusion (Why do the students believe it happened? What caused it to happen?).

D. On the third worksheet, the "Ice Activity Summary Page", the students will work in groups of four. Each group will have a recorder to write down notes for this page. A representative from each group will disclose the results of their investigations.

The students will write a final conclusion based upon the comparison of the two investigations. Did the ice in the cup filled "halfway"(A) and the cup filled "to the top" (B) do the same thing when placed in the water? In this case, the one variable changed for the investigation was the amount of water (size of ice) in each cup.

Explanation (Discussing)
The observations and conclusions should be different for the ice on worksheet "A" and "B". The students should discover that the "half-filled" ice would float upright, while the ice from the "filled" cup will in most cases float on its side. The way it floats depends on the height and width of the cup. (My students later tried ice experiments with various water levels and containers to see if their predictions held true. For this comparison study, the students used milk cartons. They discovered that if the ice dimensions were almost the same, then the cube-shaped ice floated on any side, while ice from a "filled" quart-sized milk carton floated sideways.) NOTE: The biggest surface area of the ice will float on the top if the ice dimensions are significantly different. QUESTION: If one variable were changed, would the results be different? The students will answer this question in their own inquiry-based, hands-on investigations.

Elaboration (Polar Applications)
Future Experiments: This section on worksheet 3 is optional. These inquiry-based investigations may be extended class investigations, or homework. It is worth the time and effort and proved to be the best part of the activity.

1. The students will collaborate together in groups of four to think of other ice investigations. The goal is to encourage students to ask investigative-type questions. (Questions that may be answered through a hands-on investigation). On a separate sheet of paper, the students will begin by listing several "I wonder if…"questions. On day 2, they will choose one of their questions to "investigate". The investigation will lead to discovery if the students only change one variable at a time.

2. The teacher facilitates and guides the students to stay focused on their investigation. It is important not to tell ideas to the students because the "inquiry" part of the lesson is based on what they come up with on their own. Let the students discover through their own investigations. Failures or mistakes may lead to new discoveries or new questions. Learning is like a gift….Don’t open the package for the student. The joy of learning is in the discovery, and the process of learning is based on this discovery.

3. The teacher will approve and monitor the students’ investigations. The teacher’s main goal is to guide the students in keeping focused on their investigation. The students must be reminded several times that only one variable can be changed. Writing down a list of supplies and verbally discussing their plans with the teacher help keep the students focused. Remind students that they must consult teacher before making changes or adding to the approved investigation. Otherwise they tend to add extra variables or bring in other supplies, which changes their project, or keeps it from being a valid investigation.

4. The students will gather supplies, and pre-test (at home) whatever is needed. One group decided to pre-test certain liquids to see which would freeze before making a final decision on the liquids for their investigation (syrup-no, oil-yes). The students were encouraged to use different means of measurement, such as scales, thermometers, pH strips, and tape measures.

5. Presentation: For my fourth grade students, each group took a class period to share their investigations. They chose to involve their classmates in each investigation as they made their presentations by having students help record data, observe results, and discuss observations.

The "I wonder if…" investigations from my fourth grade students:

Exchange (Students Draw Conclusions)
Two summaries from my fourth grade students’ investigations.

Note of explanation for first student summary: The teacher typed the summary as the students sat at a table next to the computer with their charts and notes. One student led the discussion as the others helped to summarize the notes from various lists and charts of all observations, as well as any measurements taken during this investigation. Prior to this discussion, the students had already made charts to compare and contrast the different ice balloons. During the summarization process, the teacher was involved in the students’ learning by keeping the students focused on their task. The teacher also emphasized the need to clearly present their thoughts in complete sentences. In doing so, the students were able to summarize their investigation in their own words.

1. Student Summary of Ice Investigation Question: Would freezing different liquids make a difference in the density and the hardness of the ice? Developed by Students: Logan, Bradley, Cody, Trea, and Haley

Before we started this ice investigation, we went home and tested some of the liquids ahead of time. We wanted to see which liquids would freeze. For our investigation, we only wanted to work with liquids that would freeze. Trea checked vinegar and Ginger Ale, while Cody froze soapy water and hot water. Logan checked out shaken water (shaken for five minutes before freezing) and water with baking soda. Bradley tried to freeze vegetable oil and cold water. When he froze the oil, it was a kind of a soft freeze. It froze solid when we put the oil in the cafeteria’s freezer. Haley tried to freeze syrup, but it wouldn’t freeze.

We chose 11 liquids: water with baking soda, soapy water, vinegar, shaken water, Coke, regular water, salty water, Diet Coke, vegetable oil, hot water, and cold water . When we put the liquids into the balloons, we found that we had to use a plastic 2-Liter bottle to get enough pressure to put the liquid into the balloon. We found that we couldn’t pour the vinegar from a glass bottle into a balloon. Since it was hard to get the liquids into the balloons, we did not use the same amount of liquid for each balloon. We tried to make all the balloons look about the same size. The only balloon that looked different was the Diet Coke. We only had enough money to buy one can of Diet Coke. The balloon with the regular Coke had two cans. (We forgot to bring the Ginger Ale, so we bought Cokes from the school’s Coke machine to try instead.) We put each balloon in a small container with its name, so that we wouldn’t forget which balloon was which. Some of the ice looked alike, so it helped to keep the name with the ice balloon.

When we did the investigation, we found that we should have changed the order by putting the liquids into three separate groups. We made a chart for these liquids. One group would be "Only Water": regular water, shaken water (5 minutes), hot water, and cold water. We decided another group would be "Something Added": baking soda, soapy water, and salty water. The last group would be "No Water Added": vinegar, Coke, Diet Coke, and vegetable oil.

When we began our investigation, we checked the circumference of each balloon before we weighed them in grams. We found that even though the balloons were all different weights, they all floated about the same in the water. We were surprised that the oil, which weighed the most (1064 grams), floated the same as the others (about 90% of each ice balloon was under the water). We think the oil was the heaviest because it had fat in it. We will write the physicist at the Exploratorium to check on this. We were also surprised that the oil froze. The oil balloon started to spin in the water about two minutes after we put it in the water.

(This discussion led the students to write an email to Paul Doherty, the physicist at the Exploratorium. He told the students to go back and check their measurements again because the animal fat would not be the cause for the oil balloon to be heavier. The students went back to check their measurements, and indeed, they found that the circumference of the oil balloon was bigger than the other balloons. Without realizing, the students had put more oil in that balloon than in the balloons that contained other liquids).

If we did this again, we would have put the balloons in a different order, so we could compare the things that were alike. We noticed that the Coke and Diet Coke started eroding away quickly. It started to melt right away, and it changed the color of the water. If we did this again, we would put the ice in the order that melted the quickest. This was a problem when we tested the ice for hardness. We should have tested the Coke, Diet Coke, vinegar, and salty water ice balloons first. They were the softest and easiest to break. They were also slushy when we used a hammer and nail to test them. The soapy water, cold water, and shaken water were the hardest. We used flashlights to see inside the ice balloons. This helped us see spikes, bubbles, plates, crystals, a prism, and a few cracks.

2. Student Summary of Ice Investigation Question: Would placing an ice cube in different liquids make a difference in the density or melting rate of each ice cube? Developed by Students: Brandy, Lauren, Tucker, Paige, & Keshona:

March 8, 2002

We wanted to investigate to see how an ice cube would float in different liquids. We also wanted to see which ice cube would melt the fastest. We decided to use the same liquids as the ice balloon group, except we added corn syrup. We filled 10-oz. clear plastic cups with 12 different liquids. (We put two-thirds cup of liquid in each plastic cup.) Then we weighed each liquid in grams. We took the pH to see if each liquid was the same.

We put blue food coloring in the water and froze it in ice cube trays. We put them in the freezer in the cafeteria for three days. We thought the blue water would help us see the ice better, so that we could check the melting rate. We put the ice cubes in each liquid at 8:36 a.m. We first observed the way the ice cube floated in each liquid. We found out that they all floated about the same, except for oil. The ice cube in the oil immediately sank to the bottom. The ice in the corn syrup looked like it was stuck at the top.

The ice melted at different times. The hot water melted the fastest (two minutes). It looked darker than the cold and shaken water. The ice in vinegar disappeared slowly and didn’t let the blue liquid move when it had melted to the bottom. The ice in soapy water melted slowly and turned a little green. We could see layers or bands in the water. The ice in the salt water left a blue band at the top and salt at the bottom. The ice in the oil took one hour and seven minutes to melt. The ice in the corn syrup took almost one hour to melt.

We had to put the cup of cold water in the refrigerator to keep it cold. This ice was the last to melt. It took about a day to melt, but it started to freeze and spread out like the sea ice. We took the temperature before we stuck it in the refrigerator. The water was 44 degrees. We checked the ice again at 10:05 a.m. and noticed that the ice cube was floating at the top. We also noticed that ice crystals were beginning to form. It still looked the same at 12:20 p.m., but ice crystals were still growing. Ice was all around the edges, except for one space.

When we came back after the weekend, we checked it at 7:30 a.m., and we saw a lot of spikes and little bubbles. The bubbles were being pushed up. We measured how far the bubbles and spikes went up the cup. (It was 4 cm from the bottom). When we gently squeezed the cup, we could see tiny bubbles floating to the top. We knew that the ice wasn’t completely frozen because we could see water moving in the cup. We knew that the ice cube had melted, but we could still see blue at the bottom. So the ice cube must have melted and gone to the bottom. We kept it in the refrigerator, so that we could check it daily to see if the spikes and bubbles were changing.

Today is March 21st, and we noticed that bubbles were all over the cup. Most of the bubbles were at the bottom. There were many spikes at the bottom. The spikes were going up with small bubbles at the end of them. There were bigger bubbles in the middle, and smaller bubbles on top. When we took the ice out of the cup, we noticed that the blue had come out of the ice. It was in liquid form at the bottom. The ice is completely clear. It is a little opaque at the bottom, but the rest is transparent.

Question to ask the Exploratorium: Why did the ice cube in oil sink quickly to the bottom? (* Paul Doherty, from the Exploratorium, responded that the ice was more dense than the oil.)

Evaluation (Assessing Student Performance)
In today’s classrooms, limited time and the emphasis on testing keep many teachers and students from doing in-depth investigations. While I listed "further investigations" as optional, I saw the greatest learning take place when my fourth grade students investigated their own questions and presented it to the class. The students’ investigations far surpassed the initial inquiry-based lesson. I saw learning evolve as students went from the beginning phase as they wrote the "I wonder if….?" questions, to collecting, discussing, measuring, observing, and recording their data. They gained tremendously as they presented their investigations to the class. The questions and observations improved during each presentation. They began to learn terminology, such as transparent, translucent, opaque, spicules, grams, circumference, density, and pH as part of their vocabulary rather than vocabulary needed to pass a test. The students grew the most when they presented the investigations to the class, then reflected on the outcome of their investigation in a group conference with the teacher. As the students summarized their investigation (while the teacher typed their responses), the fourth grade students discussed what did and didn't work, and whether they would have made any changes in their project. Taking time to discuss the project proved to be beneficial in the evaluation of their project for both the students and the teacher. The students gained a greater understanding of the scientific process by reflecting and talking over the results of their investigation. When the students gathered their notes and verbalized their investigation, they began to analyze and discuss what they had learned and what they would have done differently. As a teacher, it made me realize how critical the summarization process is to move students to a higher level of learning.

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