Download (CO)2MP 6-8 Summer Day 7 Letter Vessels – Part 1 A series of

(CO)2MP 6-8 Summer Day 7
Letter Vessels – Part 1
A series of vessels look like letters when viewed face-on. They are 1 cm thick, and the corners of the
vessels have either whole or half cm values for their coordinates.
Water is poured slowly into each vessel through one of the holes at the top at a rate of 1 cubic
centimeter per second.
1. Seven of these vessels take the same time to fill up. Which are they?
2. Which one takes the longest to fill?
3. Which vessel would fill up fastest?
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(CO)2MP 6-8 Summer Day 7
Letter Vessels – Part 2
The height of the pool of water in one of the vessels is measured and a graph of the height as a
function of time is plotted (shown below).
4. Which vessel was used to make this graph? Explain your reasoning with specific details from the
graph and the vessel.
5. A very observant student notices that the given graph is actually slightly inaccurate: at certain
points it is an over-measurement and at certain points it is an under-measurement. Where is this
happening on the graph?
Adapted from Math Fillers, NRICH enriching mathematics. University of Cambridge.
http://nrich.maths.org/6424
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(CO)2MP 6-8 Summer Day 7
Bottle Filling – Part 1
Your group has two bottles that each hold about 240 ml. As each bottle is filled with water, what is the
relationship between the volume of the water in the bottle and the height of the water in the bottle?
Before filling the bottles with water, prepare two grids (one for each bottle) with x = the volume of the
water in the bottle (in ml) and y = the height of the water in the bottle (in cm). Then analyze each bottle
and sketch your prediction of what the graph will look like. What strategies did you use to make your
prediction?
Now, using 10 ml increments, fill each bottle with water. After each 10 ml portion is added, measure the
height of the water in each bottle to the nearest 0.1 cm. Be care to measure as accurately as possible.
Record your data in the table below. Then plot your data on the grid with your predicted graph for each
bottle.
x (ml)
y (cm)
x (ml)
y (cm)
x (ml)
y (cm)
x (ml)
10
130
10
130
20
140
20
140
30
150
30
150
40
160
40
160
50
170
50
170
60
180
60
180
70
190
70
190
80
200
80
200
90
210
90
210
100
220
100
220
110
230
110
230
120
240
120
240
y (cm)
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(CO)2MP 6-8 Summer Day 7
Bottle Filling – Part 2
Compare the graphs of the data to your predicted graphs. Which features are the same? Which features
are different?
Analyze the two bottles and their corresponding graphs.

How are the graphs for the two bottles similar and how are they different?

What do the graphs tell you about the shapes of the bottles?

Could you predict the shape of an unknown bottle based on its graph of height of water in the
bottle vs. volume of water in the bottle? If so, what strategies would you use?
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(CO)2MP 6-8 Summer Day 7
NAME:
Take a few moments to reflect on SMP’s connected to the content tasks we did today:
Name of the task and related SMP’s
Evidence for the chosen SMP’s
Jot down how you contributed to our shared community of professionals and what mathematical
and/or pedagogical knowledge you are taking away from today:
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(CO)2MP 6-8 Summer Day 7
MP Qualities
SMP #1
1. Students try to figure out for themselves the meaning of a problem.
2. Students look for entry points to a problems solution.
3. Students analyze givens, constraints, relationships, and goals.
4. Students make conjectures about the form and meaning of the solution.
5. Students plan a solution pathway.
6. Students consider analogous problems.
7. Students try special cases.
8. Students try simpler forms of the original problem.
9. Students monitor and evaluate their progress and change course if necessary.
10. Students draw diagrams or graphs of important features and relationships.
11. Students search for regularity or trends.
12. Students rely on using concrete objects or pictures to help conceptualize and solve a problem.
13. Students check their answers to problems using a different method
14. Students continually ask themselves, "Does this make sense?"
15. Students understand the approaches of others to solving complex problems and identify
correspondences between different approaches.
16. Students do not give up after the first attempt.
SMP #2
1. Students make sense of quantities in problem situations.
2. Students make sense of quantitative relationships in problem situations.
3. Students are able to decontextualize—to abstract a given situation and represent it symbolically
and manipulate the representing symbols as if they have a life of their own, without necessarily
attending to their referents.
4. Students are able to contextualize, to pause as needed during the manipulation process in order to
probe into the referents for the symbols involved.
5. Students create a coherent representation of the problem at hand.
6. Students consider the units involved in the problem.
7. Students attend to the meaning of quantities, not just how to compute them.
8. Students know and flexibly use different properties of operations and objects.
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(CO)2MP 6-8 Summer Day 7
SMP #3
1. Students understand and use stated assumptions.
2. Students understand and use definitions.
3. Students understand and use previously established results in constructing arguments.
4. Students make conjectures.
5. Students build a logical progression of statements to explore the truth of their conjectures.
6. Students are able to analyze situations by breaking them into cases.
7. Students can recognize and use counterexamples.
8. Students justify their conclusions and communicate them to others.
9. Students respond intelligently and coherently to the arguments of others.
10. Students reason inductively about data, making plausible arguments that take into account the
context from which the data arose.
11. Students are able to compare the effectiveness of two plausible arguments.
12. Students are able to distinguish correct logic or reasoning from that which is flawed, and—if there
is a flaw in an argument—explain what it is.
13. Students can construct arguments using concrete referents such as objects, drawings, diagrams,
and actions.
14. Students are able to determine domains to which an argument applies.
15. Students at all grades can listen or read the arguments of others, decide whether they make sense,
and ask useful questions to clarify or improve the arguments.
SMP #4
1. Students can apply the mathematics they know to solve problems arising in everyday life, society,
and the workplace.
2. Students can write an addition equation to describe a situation.
3. Students who can apply what they know are comfortable making assumptions and
approximations to simplify a complicated situation, realizing that these may need revision later.
4. Students are able to identify important quantities in a practical situation.
5. Students are able to map the relationships among quantities found in a problem using such tools
as diagrams, two-way tables, graphs, flowcharts and formulas.
6. Student can analyze quantitative relationships mathematically to draw conclusions.
7. Students routinely interpret their mathematical results in the context of the situation and reflect
on whether the results make sense, possibly improving the model if it has not served its purpose.
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(CO)2MP 6-8 Summer Day 7
SMP #5
1. Students consider the available tools when solving a mathematical problem.
2. Students use things like pencil and paper, concrete models, a ruler, a protractor, a calculator, a
spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software to
help them solve a problem.
3. Students are sufficiently familiar with tools appropriate for their grade or course.
4. Students are able to make sound decisions about when each of these tools might be helpful,
recognizing both the insight to be gained and their limitations.
5. Students are able to detect possible errors by strategically using estimation and other
mathematical knowledge.
6. When making mathematical models, students know that technology can enable them to visualize
the results of varying assumptions.
7. Students can utilize technology to explore consequences.
8. Students can utilize technology to compare predictions with data.
9. Students at various grade levels are able to identify relevant external mathematical resources,
such as digital content located on a website, and use them to pose or solve problems.
10. Students are able to use technological tools to explore and deepen their understanding of
concepts.
SMP #6
1. Students communicate mathematical ideas precisely to others.
2. Students use clear definitions in discussion with others and in their own reasoning.
3. Students state the meaning of the symbols they choose, including using the equal sign consistently
and appropriately.
4. Students are careful about specifying units of measure, and labeling axes to clarify the
correspondence with quantities in a problem.
5. Students calculate accurately and efficiently, express numerical answers with a degree of
precision appropriate for the problem context.
6. Students give carefully formulated explanations to each other.
7. Students are able to examine claims and make explicit use of definitions.
SMP #7
1. Students look closely to discern a quantitative pattern.
2. Students look closely to discern a structure.
3. Students notice properties arise from operations (e.g. commutativity) or that shapes have
qualities by which they can be organized.
4. Students can step back for an overview and shift perspective.
5. Students can see complicated things, such as some algebraic expressions, as single objects or as
being composed of several objects.
SMP #8
1. Students notice if calculations are repeated.
2. Students look both for general methods and for shortcuts.
3. Students maintain oversight of the process when working on a problem.
4. Students attend to the details while working on a problem.
5. Students continually evaluate the reasonableness of their intermediate results.
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(CO)2MP 6-8 Summer Day 7
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