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You are reading an older version of this FlexBook® textbook: CK-12 Modeling and Simulation for High School Teachers: Principles, Problems, and Lesson Plans Go to the latest version.

The lessons described below were developed with the POGIL method of guided inquiry instruction in mind. In each case, a model or simulation provides the mechanism for inquiry while the careful choice of questions helps the learners to construct their own understanding of what they observe. By using these lessons, the concepts and mathematical relationships found in the study of phase change and kinetic molecular theory can all be uncovered by the students in a student-centered, constructivist approach, allowing for a much richer learning experience for the students. (Note: although these lessons use a POGIL approach they have not been officially endorsed by the POGIL Project).

States of Matter with PhET

In this activity, as well as the following activity, the learners will access a simulation found on the PhET website, http://phet.colorado.edu. Through the completion of this activity, they will develop an understanding of both the physical and behavioral differences between solids, liquids, and gases. They will also uncover the various temperatures at which substances are frozen, melted, or vaporized. The numerical values are not the main focus, however. Instead, students are expected to achieve conceptual understanding of these processes: for example, that a substance could be "cold" and remain in a gaseous state or "hot" and remain in a solid state, and that a substance can change phase without changing the temperature. By having students work together and discuss their observations with their partners, the instructor can listen for the evidence that they really get the "big picture" concepts behind the nature of solids, liquids, and gases.

Information and Directions for Students

Go to the following web site and run the States of Matter simulation: http://phet.colorado.edu/en/simulation/states-of-matter

Instructions: Please read the directions that are provided with each model very carefully. This simulation will provide you with many different ways to investigate the properties of solids, liquids, and gases. You will be asked to select or move specific objects. This image will help you to know the different objects: lid, pump, heater/cooler, substance selector, and state selector. There are also controls on the panel to the right. You will be directed to select or move specific objects in order to observe certain behaviors, so follow the instructions carefully.

MODEL 1

MODEL 1: Temperature is in Kelvin (K).

  1. Select "neon." Change the states and record the characteristics observed and approximate Kelvin temperature for each state of matter for neon. Repeat this for argon, oxygen, and water, and fill in the Table below.
    Substance Solid characteristics Temp Liquid characteristics Temp Gas characteristics Temp
    Neon
    Argon
    Oxygen
    Water
  2. What temperature was the neon when you selected the solid state?
  3. What do all of the solid states have in common?
  4. What temperature was the water when you selected liquid?
  5. What do all of the liquid states have in common?
  6. What temperature was the oxygen when you selected gas?
  7. What do all of the gas states have in common?
  8. Look up the melting and boiling point for each of the substances, convert to Kelvin, and record in Table below.
    Substance Melting point Boiling point State at room temperature
    Neon
    Argon
    Oxygen
    Water
  9. Would these substances be solids, liquids, or gases at room temperature? (Record your answers in the chart above.)

MODEL 2

MODEL 2: Select the Phase Changes tab. Push on the lid to decrease the volume of the chamber to approximately half (see image below). Create a data table to record the data. Be sure to include all variables in your table. Sketch the phase diagram as it changes.

  1. Select "neon" and record the initial conditions of the chamber.
  2. Heat the neon sample until the temperature reaches about 30 K. Record the new data and observations.
  3. Use the pump handle to add about 10 pumps of gas. Record the new data and observations.
  4. Continue heating to 40 K. Record the new data and observations.
  5. What state does it appear to be at this temperature?
  6. Continue heating to 50 K. Record the new data and observations.
  7. Pull the lid back up to the top. Record the new data. Explain what you are observing.

MODEL 3

MODEL 3: Select the Phase Changes tab. Click RESET. Select "water." Create a data table to record the data. Be sure to include all variables in your table. Sketch the phase diagram as it changes.

  1. Record the initial conditions of the chamber in the data table.
  2. Cool the water sample until the temperature reaches about 30 K. Record the data.
  3. Push the lid down until the pressure starts to jump to about 15-20 atm. Record the data.
  4. Heat to 500K. Record the data.
  5. Pull the lid back up. Record the data.

Teacher Key

MODEL 1:

Substance Solid Temp Liquid Temp Gas Temp

Neon

Single atoms

Vibrating, close together 9 K Looser movement, still at bottom 26 K Move freely, all over container 55 K

Argon

Single atoms

Vibrating, close together 31 K Looser movement, still at bottom 87 K Move freely, all over container 189 K

Oxygen

diatomic

Vibrating, close together 20 K Looser movement, still at bottom 59 K Move freely, all over container 194 K

Water

Molecule with three atoms

Vibrating, not as close together as the others 102 K Looser movement, still at bottom 292 K Move freely, all over container 809 K
  1. 9 K
  2. Vibrating/tightly packed. Water is not as tightly packed.
  3. 292 K
  4. Moving around, but still on the bottom of the container.
  5. 194 K
  6. All move freely throughout the container.
  7. See Table below.
    Substance Melting point Boiling point State at room temperature
    Neon 24.5 K 27.0 K gas
    Argon 83.9 K 87.2 K gas
    Oxygen 54.8 K 90.2 K gas
    Water 273 K 373 K liquid

MODEL 2:

  1. Solid, at 9 K and 0 atm. The red dot on phase diagram is between solid and gas.
  2. For both liquid and gas, pressure increases to 2–3 atm, and temperature is 30 K. The red dot on the phase diagram is on the line between liquid and gas.
  3. Pressure increases, temperature is still 30 K, and there are still both liquid and gas.
  4. The increase in temp leads to an increase in pressure. The pressure forces the molecules into liquid state. The pressure jumped. The red dot on the phase diagram is in the middle of the liquid section.
  5. Liquid.
  6. Pressure climbs and the liquid becomes almost critical.
  7. Pressure drops rapidly and the liquid becomes a gas. Red dot is close to critical range or may end up in the critical range. This means that the liquid and the gas molecules are indistinguishable from one another.

MODEL 3: Select the Phase Changes tab. Click RESET. Select "water." Be sure to watch the phase diagram on the right side of the screen as you make changes.

  1. 102 K, 0 atm, solid.
  2. Vibrations slow down and molecules seem to lock together. The red dot is on the line between solid and gas.
  3. It stays solid, but pressure jumps around. The red dot climbs up on the phase diagram image.
  4. Changes to a liquid. The red dot moves to the right into the liquid range.
  5. The red dot drops down and the liquid changes to a gas.

Extension: Explain clearly how pressure and temperature affect the state of matter.

Solids can be melted to a liquid by increasing the temperature. Liquids can be vaporized to a gas by increasing the temperature.

Solids can also be melted by changing the pressure. In the case of water, increasing the pressure liquefies the solid ice. Liquids can be vaporized by lowering the pressure. Decreasing the pressure allows the liquid to evaporate more readily.

Gas Properties with PhET

Information and Directions for Students

The kinetic molecular theory of gases describes a gas as sample of small particles that are in constant, random motion. These particles constantly collide with each other and the walls of their container. The physical properties of gases, such as velocity, temperature, pressure, and volume can be explained by considering the motion of the molecules. These properties are interrelated and impact one another in a very predictable manner.

Go to the following web site and run the States of Matter simulation: http://phet.colorado.edu/en/simulation/gas-properties

Instructions: Please read the directions that are provided with each model very carefully. This simulation will provide you with many different ways to investigate the properties of gases. You will be asked to select or move specific objects. This image will help you to know the different objects: lid, pump, wall, heater/cooler, gas selector.

There are also controls on the panel to the right. You will be directed to select or move specific objects in order to observe certain gas behaviors, so follow the instructions carefully.

MODEL 1

For this exercise, use the data table below to help organize your observations. Each number in the table corresponds with one of the steps in the modeling activity. Record constant, increased, or decreased. Use arrows up and down for increase and decrease if you like.

Question Molecular mass (heavy or light) Pressure Temperature Volume Amount Speed
1.
2.
3.
10.
16.
26.
27.
29.
30.
34.
  1. Use the toolbar on the right to keep the temperature constant by clicking "Temperature." Next, click the tab. Select the "Stopwatch." Grab the pump handle and pump it once. Click "Start" on the stopwatch. Wait about 30 seconds and record your observations.
  2. Open the lid at the top by sliding it to the left all the way. Leave it open for about a minute. (You can reset your stopwatch here.) Record your observations. Close the lid before continuing.
  3. Grab the handle on the wall and push the wall to the right so that it aligns under the lid handle. Wait about 30 seconds and record your observations.
  4. Use the pump to inject more gas into the chamber. Raise and lower the pump SIX times. Record your observations.
  5. What does the pump control in the experiment?
  6. What does the handle on the wall control?
  7. What does the lid do?
  8. Explain what you observed in step four. Why did this happen?

MODEL 2

  1. Click to return to the original settings. On the parameter list, select "Pressure" to keep it constant.
  2. Pump in some gas. Record your observations.
  3. Why did this happen?
  4. Can you use the handle on the wall to reset the volume to the original size?
  5. What changed in this investigation?
  6. Why did it change?

MODEL 3

  1. Click to return to the original settings. On the parameter list, select "Volume" to keep it constant.
  2. Pump in some gas. Wait about 30 seconds, then record your observations.
  3. Pump in more gas and record your observations.
  4. Why did this happen?
  5. How can you change the pressure?
  6. Is there a knob for the pressure like there is for volume and temperature?
  7. Figure out a way to get the pressure to rise to 2.5 atm. What did you do?
  8. What could you do to get the pressure to return to below 1.00 atm?
  9. Can you get it to exactly 1.00 atm?
  10. Click to return to the original settings again.
  11. Pump in some gas (around 200 atoms).
  12. Add heat using the heater at the bottom of the chamber and record your observations.
  13. Remove heat using the cooler at the bottom of the chamber and record your observations.
  14. Check the "Energy histogram" under "Tools" and "Options."
  15. Add heat again using the heater at the bottom of the chamber and record your observations. Click and start over from number 25 if the top blows off.
  16. Now remove the heat using the cooler at the bottom of the chamber and record your observations.
  17. Click to return to the original settings again.
  18. Add one pump of the heavy gas, then switch to light gas using the gas selector and add another pump.
  19. Click the "More details" button on the "Energy histogram" (particle statistics) and record your observations.
  20. Which gas has a higher average speed?

Analysis Questions

  1. What are the variables or characteristics that were investigated in this activity?
  2. What happens to the pressure of a sample if more gas in injected? (Assume the volume is constant.)
  3. What happens to the volume if the pressure is held constant as the gas is injected?
  4. Is it possible to increase the pressure without adding more gas? How can this be done?
  5. What happens to the pressure if the temperature is increased? (Assume the volume is constant.)
  6. What affects the speed of the molecules?

Teacher Answers

MODEL 1:

  1. Pressure climbs to around .045 atm.
  2. Gas leaves and pressure drops to around 0.17. The amount it drops depends on how long they leave it open, etc.
  3. Pressure increases to around 0.45 atm.
  4. Top should blow off. As gas escapes, the pressure should drop back down.
  5. The amount of gas.
  6. The volume of the chamber.
  7. Prevents the gas from leaving.
  8. Because the pressure was too high.

MODEL 2:

  1. The wall moves to the left. The chamber gets bigger.
  2. Since pressure needs to stay constant, gas pushed the wall out.
  3. No, it keeps pushing back out.
  4. The amount of gas increased the volume of the container.
  5. More gas would require more space or the pressure would increase. The wall moves out to relieve the pressure.

MODEL 3:

  1. Pressure climbs up to around 0.45 atm again.
  2. Pressure builds to about 0.80 atm.
  3. More gas molecules colliding with the container wall.
  4. Continue to change the amount of gas.
  5. No.
  6. Added more gas.
  7. Open the lid to let some of the gas out.
  8. Yes, or at least really close.
  1. Increased temperature causes pressure to increase.
  2. Decreased temperature causes the pressure to decrease.
  1. The average speed increases. More molecules are traveling faster.
  2. The average speed decreases. More molecules are traveling slower.
  1. The blue graph (heavy gas) is not as spread out. The red graph (lighter gas) is broader.
  2. The red gas is faster.

Analysis Questions:

  1. Pressure, volume, temperature, amount, speed, molecular mass.
  2. It will increase.
  3. It will increase.
  4. Yes, decrease the volume or increase the temperature.
  5. The pressure will increase.
  6. The temperature and the molecular mass.

Use this data table to help organize your observations. Record constant, increased, or decreased. Use arrows up and down for increase and decrease if you would like.

Question Molecular mass (only analyzed for 34) Pressure Temperature Volume Amount Speed (only analyzed for 29-30)
1. Increase Constant Constant Increase
2. Decrease Constant Constant Decrease
3. Increase Constant Decrease Constant
10. Constant Constant Increased Increased
16. Increased Constant Constant Increased
26. Increased Increased Constant Constant
27. Decreased Decreased Constant Constant
29. Increased Increased Constant Constant Increased
30. Decreased Decreased Constant Constant Decreased
34. Light Increased

Gas Behavior

The models used in this activity are an Excel file courtesy of Scott Sinex at Prince George Community College.

Prerequisite: gas variables including P, V, n, and T

Information and Directions for Students

Ideal gases behave according to the postulates of kinetic molecular theory, which state that gas particles are very small. In fact, they are so small that the total volume of the individual gas particles is negligible compared to the volume of the container they occupy. So a container of gas is mostly empty space. These particles are in constant, random motion and collide with each other and the container walls. These collisions are perfectly elastic, meaning that the kinetic energy of the particle is conserved throughout the collision. The interactions or intermolecular forces between the particles are so small that they are considered negligible as well. The kinetic energy of the particles depends only on the temperature of the system. As ideal gases follow these rules, they are very predictable when manipulating their pressure, temperature, container volume, amount, or molecular mass.

Ideal gases are different from real gases. Ideal gases are assumed to have no intermolecular forces between the molecules, and their motion is dictated by the elastic collisions with the walls of their container.

Go to the following web site: http://academic.pgcc.edu/~ssinex/excelets/. Scroll down to the first table and click on the link to "ideal gas law." It is located in the middle column, fourth cell from the bottom.

Once opened, several tabs are available at the bottom, as shown in this image:

MODEL 1: Boyle’s Law

Select the P-V tab. Make sure that the graph has the temperature at 298 K and the amount of gas (n) at 0.00041 moles prior to beginning.

Using the graph and not manipulating, answer the following questions:

  1. What two gas variables are plotted on the graph?
  2. What is the volume at 3.0 atm?
  3. What is the volume at 2.0 atm?
  4. What is the volume at 1.0 atm?
  5. What appears to happen to the volume as pressure is decreased?
  6. What is the pressure at 20 mL?
  7. What is the pressure at 10 mL?
  8. What is the pressure at 5 mL?
  9. What appears to happen to the pressure as the volume is decreased?
  10. What type of relationship is this illustrating between pressure and volume, direct or inverse?
  11. If pressure was dropped to zero, what would the volume be?

Manipulating the temperature:

  1. Slide the bar on the temperature cell to the right. What is happening to the pressure?
  2. Slide the bar on the temperature cell to the left. What is happening to the pressure?
  3. Does anything happen to the volume?
  4. What type of relationship is this illustrating between temperature and pressure?

Reset the temperature to 298 K and now try manipulating the amount of gas (n):

  1. Type in the yellow box various values for n. Try doubling it, tripling it, halving it, etc.
  2. What change occurs when the amount of gas is doubled?
  3. What change occurs when the amount of gas is halved?
  4. Does anything happen to the volume?
  5. What type of relationship is this illustrating between amount of gas and pressure?
  6. What would happen if you removed all of the gas?
  7. What can you say about the container used in this model? (Hint: see questions 14 and 21.)

MODEL 2: Charles’ Law

Select the V-T \ (K) tab. Make sure that the graph has a pressure of 1.0 atm and an amount of gas (n) at 0.00041 moles prior to beginning.

Using the graph and not manipulating, answer the following questions:

  1. What two gas variables are plotted on the graph?
  2. What is the volume at 350 K?
  3. What is the volume at 250 K?
  4. What is the volume at 150 K?
  5. What appears to happen to the volume as temperature is decreased?
  6. What is the temperature in Kelvin at 12 mL?
  7. What is the temperature in Kelvin at 6 mL?
  8. What is the temperature in Kelvin at 3 mL?
  9. What appears to happen to the temperature as the volume is decreased?
  10. What type of relationship is this illustrating between temperature and volume, direct or inverse?
  11. If the temperature was dropped to zero, what would the volume be?
  12. Do you think that you can actually change the temperature by fluctuating the container size?
  13. What is different about the container that must have been used in this model as compared to model 1?
  14. Explain what happened in terms of the pressure in this container as you manipulated in numbers 23-30.

Manipulating the pressure:

  1. Change the pressure to 2.0 atm. What is happening to the volume? What is happening to the temperature?
  2. Change the pressure to 0.5 atm. What is happening to the volume? What is happening to the temperature?
  3. What type of relationship is this illustrating between volume and pressure?
  4. Does this correspond to what you found in Model 1?

Reset the pressure to 1.0 atm and now try manipulating the amount of gas (n):

  1. Type in the yellow box various values for n. Try doubling it, tripling it, halving it, etc.
  2. What change occurs when the amount of gas is doubled?
  3. What change occurs when the amount of gas is halved?
  4. Does anything happen to the temperature?
  5. What type of relationship is this illustrating between the amount of gas and volume?
  6. What would happen if you removed all of the gas?

MODEL 3: Avogadro’s Law

Select the V-n tab. Make sure that the graph has a pressure of 1.0 atm and the temperature is at 298 K prior to beginning.

Using the graph and not manipulating, answer the following questions:

  1. What two gas variables are plotted on the graph?
  2. What appears to happen to the volume as the amount of gas is decreased?
  3. What appears to happen to the volume as the amount of gas is increased?
  4. What type of relationship is this illustrating between amount of gas and volume, direct or inverse?
  5. Is it possible to create more gas by changing the volume?
  6. Which variable is the independent variable?
  7. Which is the dependent variable?
  8. What is different in this model as compared to Model 1 and Model 2?

Manipulating the pressure:

  1. Change the pressure to 2.0 atm. What is happening to the volume? What is happening to the amount of gas?
  2. Change the pressure to 0.5 atm. What is happening to the volume? What is happening to the amount of gas?
  3. What type of relationship is this illustrating between volume and pressure?
  4. Does this correspond to what you found in Model 1?

Reset the pressure to 1.0 atm and now try manipulating the temperature.

  1. Change the temperature using the slider. Try increasing it, decreasing it.
  2. What change occurs when the temperature is increased?
  3. What change occurs when the temperature is decreased?
  4. Does anything happen to the amount of gas?
  5. What type of relationship is this illustrating between temperature and volume?
  6. Does this correspond to what you found in Model 2?

Extension:

Sketch the graphs that represent Boyle’s Law, Charles’ Law, and Avogadro’s Law.

  1. What happens to the pressure as the volume is increased, assuming the amount of gas and temperature are constant?
  2. What happens to the volume as the temperature is increased, assuming the pressure and the amount of gas are constant?
  3. What happens to the volume if the amount of gas is decreased, assuming the pressure and the temperature are constant?

Teacher Answers

Prerequisite: gas variables including P, V, n and T

MODEL 1:

  1. Pressure and volume.
  2. About 3.5 mL.
  3. About 5 mL.
  4. About 10 mL.
  5. It increases.
  6. About 0.50 atm.
  7. About 1.00 atm.
  8. About 2.01 atm.
  9. It increases.
  10. Inverse.
  11. The volume should be infinite; it would continue to increase and form an asymptote.

Manipulating the temperature:

  1. It increases.
  2. It decreases.
  3. No, it does not change.
  4. Direct.

Reset the temperature to 298 K and now try manipulating the amount of gas (n):

  1. Type in the yellow box various values for n. Try doubling it, tripling it, halving it, etc.
  2. The pressure increases to double.
  3. The pressure decreases to half.
  4. No, the volume does not change.
  5. Direct.
  6. The pressure should drop to zero. No gas, no pressure.
  7. The fact that the volume does not change implies that the container is rigid.

MODEL 2:

  1. Volume and temperature.
  2. 11.78 mL.
  3. 8.42 mL.
  4. About 5 mL.
  5. It decreased.
  6. About 360 K.
  7. About 175 K.
  8. About 80 K.
  9. It decreased.
  10. Direct.
  11. The volume should be zero since at zero Kelvin, molecular movement should stop, and the molecules will no longer move around to fill their container.
  12. No. Temperature is the independent variable.
  13. This container must be flexible according to the data.
  14. Pressure was constant since the volume was able to change in response to the temperature changes.

Manipulating the pressure:

  1. Volume decreased and temperature stayed constant.
  2. Volume should increase and temperature should remain constant.
  3. Inverse.
  4. Yes.

Reset the pressure to 1.0 atm and now try manipulating the amount of gas (n):

  1. Type in the yellow box various values for n. Try doubling it, tripling it, halving it, etc.
  2. As the amount of gas is doubled, the volume doubles.
  3. As the amount of gas is halved, the volume is halved.
  4. No. Again, the temperature is the independent variable.
  5. Direct.
  6. The volume would drop to zero.

MODEL 3:

  1. Volume and amount of gas.
  2. It decreases.
  3. It increases.
  4. Direct.
  5. No.
  6. The amount of gas, like temperature, is the independent variable.
  7. The volume.
  8. The container is flexible like that in model 2, but the amount of gas is changing, so it must not be a closed container. Gas can be pumped in or removed.

Manipulating the pressure:

  1. The volume is decreasing and the amount of gas is constant.
  2. The volume would increase and the amount of gas would remain constant.
  3. Inverse.
  4. Yes.

Reset the pressure to 1.0 atm and now try manipulating the temperature.

  1. Change the temperature using the slider. Try increasing it, decreasing it.
  2. The volume increases.
  3. The volume decreases.
  4. No, the amount of gas is the independent variable.
  5. Direct.
  6. Yes.

Extension

Sketch the graphs that represent Boyle’s Law, Charles’ Law, and Avogadro’s Law.

  1. It decreases.
  2. It increases.
  3. It decreases.

Investigating the Height of a Stack of Rice Cakes

In this activity, students explore data collection and variations in data. Each group will need a bag containing nine mini rice cakes, a ruler, and a tenth mini rice cake that is supplied by the teacher in step three of the directions. Mini rice cakes were chosen for this activity specifically because of the variation from cake to cake. For maximum impact, stop and have the class compare results and answers after questions 4, 13, and 16.

Information and Directions for Students

  1. You have been given nine cakes. Create a stack of rice cakes using as few or as many as you wish. You will be using the measurements that you collect to predict the height of a stack of 10 cakes, so choose wisely. Use a metric ruler to measure (to the nearest 0.1 cm) the height of the cake stack and record your data in a table like the one below. Repeat this process for a total of five different measurements. Place rice cakes on a paper towel, not directly on your desk.
    Number of Rice Cakes Height of Stack (cm)
  2. Use the data collected to hypothesize what the height of 10 rice cakes would be. Briefly explain how you arrived at this number.
  3. Once you have written your hypothesis, ask your teacher for another rice cake. Measure the height of a stack of 10 cakes and record the measurement. ______ cm
  4. Was your hypothesized height for 10 cakes accepted or rejected? Explain.
  5. We will now enter the collected data into a mathematical model using Excel. Go to the following website: http://academic.pgcc.edu/~ssinex/excelets/. Scroll down to the third, mustard-colored data table and click on "Investigating the Height of a Stack of Cookies (‘Just add data’ Excel)." Open the spreadsheet.
  6. Enter your data in the table on the tab labeled "Cookie Stack Height." Leave the boxes for the data points you did not collect empty.
  7. Using the data you have supplied, the spreadsheet is able to develop a line of best fit by performing a linear regression. The blue line is the line of best fit that shows the trend of the experimental data. Describe what the graph looks like.
  8. Using this model, we are able to make predictions. Find the predicted height of a stack of 15 cakes by entering 15 in the yellow box labeled "enter number of cookies" and then press the Enter key. Repeat this process for 150 cakes. Record the predictions based on your model. Number of Cakes = 15, Predicted Height = _______________ Number of Cakes= 150, Predicted Height = _______________
  9. The computer has generated a model based on your experimental data. Which predicted height do you think is the most accurate on your model, 15 cakes or 150 cakes? Explain.
  10. Get with another group and combine your cakes to carefully form a stack of 15 cakes. Measure the height of the cakes and record the value here.
  11. Compare the predicted cake height to the actual cake height. How accurate is the model?
  12. Do you think your predictions would be more accurate with more experimental data? Explain.
  13. List at least 2 possible sources of error that might account for any differences between the predicted and experimental values.
  14. If you were given enough supplies, how could you verify the calculated prediction for the height of 150 cakes? How practical would it be to do this?
  15. How would the graph change if regular sized cakes were used instead of mini rice cakes?
  16. Scientists often use mathematical models of data in their research. What are some of the benefits and drawbacks of modeling data?

Teacher Answers

There is not one correct answer to these questions. The answers will depend on the data that the individual student collects and the class discussion when results are compared. Most students will get an average thickness of 1.2-1.4 cm for a rice cake.

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Aug 06, 2012

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Aug 21, 2014
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