1
Algebra I: Strand 2. Linear Functions; Topic 11. Building with Blocks; Task 2.11.1
TASK 2.11.1: BUILDING WITH BLOCKS — CHILD’S PLAY
Left Side
Right Side
Front
Part 1
1. Randy and Richie built the solid figure above. Using blocks all the same
color, build this figure.
2. Draw the top and back views.
Back view
Top view
3. Find the surface area of the figure.
The surface area is 71 square units if students don’t count the bottom; it is 84 square
units including the bottom.
4. Randy told Richie that they could add more blocks to their figure and not
change the top and back views. Do you think Randy was right? Justify your
answer.
Yes. Students can justify this by demonstrating adding one block and explaining how the
top and back views have not changed.
November 22, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The
University of Texas at Austin for the Texas Higher Education Coordinating Board.
2
Algebra I: Strand 2. Linear Functions; Topic 11. Building with Blocks; Task 2.11.1
5. Using a different color block, add one block at a time. Make sure that the top
and back view of the original figure does not change. Describe what happens
to the surface area when:
Answers may vary depending on which blocks are added and in which order. Five blocks
can be added without changing the surface area. The surface area is changed only when
the view of either the right side, left side, or front view changes.
a. 1 block is added
b. 2 blocks are added
c. 3 blocks are added
Continue adding blocks keeping the back and top views constant. Record the
surface area each time a new block is added.
6. Create a table and a scatter plot using the number of blocks added vs. the
surface area of the figure. Note: let the words “number added” be represent
with “+”. Label the axes.
Answers may vary depending on where the blocks are placed.
7. Draw the views of the right side, left side, top, and back of the figure
including all the added blocks.
Right side and left
side the same
Front will now
match the back
Top did not change
8. What is the domain and range of this data?
All domains will be 0 to 11; the range may vary depending on if they included the “floor”
of the figure. The final surface area is 90 (including the bottom); however some students
may be able to create a surface area of 94 square units before adding the 11th block.
9.
Find a section of the graph where the surface area remains constant. Write
the equation of the line passing through these points.
y = c, answer can vary depending on which blocks are added and the order
a. What would the slope of the line mean for this situation?
b. What units would represent the slope of the line?
c. What would the y-intercept mean?
For that specific domain, the surface area remains the same and a linear rule y = c
would fit this part of the data. The slope is zero and the units are square units per block
added. This means that 0 square units are added for each block added. The y-intercept
(0, c) means that if no blocks are added the surface area is c square units.
November 22, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The
University of Texas at Austin for the Texas Higher Education Coordinating Board.
3
Algebra I: Strand 2. Linear Functions; Topic 11. Building with Blocks; Task 2.11.1
Part 2
Remove the added blocks and return to the original figure.
These are sample answers, student answers may vary according to the way they remove
the blocks.
Jasmine and Yolanda strategically removed blocks from the original figure to try to
change the surface area the least and still keep the top and back views constant.
10. Record what happens to the surface area when:
a. one block is removed
b. two blocks are removed
c. three blocks are removed
Continue removing blocks keeping the back and top views constant. Record
the surface area each time a new block is removed.
There are five blocks that can be removed with no change in the surface area. Six more
blocks can be removed and there will be a loss of 2 square units each time one of these is
removed.
11. What is the greatest number of blocks that can be removed without changing
the top and back views?
11 blocks can be removed
12. Create a scatter plot that includes the data for both adding and removing
blocks. Use surface area vs. the number of blocks added (+) or removed (−).
Here are two possible graphs.
13. What is the domain and range of this data?
The domain is -11 to 11, the range will vary; however the minimum will be 72, the
maximum may be as high as 94 if you include the bottom of the figure in the surface area.
14. Compare your graph to other graphs in the class. Are all the graphs the same?
Explain the differences in the graphs
15. What point(s) must all possible graphs of this data have in common? Explain
what each point represents.
Each graph will contain (0, 84 or 71) as the surface area of the original figure, the point
(11, 90) representing the most blocks that can be added and the corresponding surface
area, and (−11, 72) the most blocks that can be removed and the least surface area
November 22, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The
University of Texas at Austin for the Texas Higher Education Coordinating Board.
4
Algebra I: Strand 2. Linear Functions; Topic 11. Building with Blocks; Task 2.11.1
16. Adding or removing certain blocks did not change the surface area of the
figure. Explain how this is possible.
If all views of the figure remain unchanged, the surface area will usually remain
unchanged. Changing blocks not on the “shell” of the figure will not affect the surface
area.
Math notes
If blocks are added so that no view changes, then there is no surface area change.
However, if added blocks change any view, then the surface area does increase.
Teaching notes
Some students may not count the bottom of the figure as part of the surface area.
They should be able to justify their reasoning; their domain and range will reflect
this.
November 22, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The
University of Texas at Austin for the Texas Higher Education Coordinating Board.
5
Algebra I: Strand 2. Linear Functions; Topic 11. Building with Blocks; Task 2.11.1
TASK 2.11.1:BUILDING WITH BLOCKS — CHILD’S PLAY
Left Side
Right Side
Part 1
Front
1. Randy and Richie built the solid figure above. Using blocks all the same color,
build the figure.
2. Draw the top and back views.
3. Find the surface area of the figure.
4. Randy told Richie that they could add more blocks to their figure and not change
the top and back views. Do you think Randy was right? Justify your answer.
November 22, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The
University of Texas at Austin for the Texas Higher Education Coordinating Board.
6
Algebra I: Strand 2. Linear Functions; Topic 11. Building with Blocks; Task 2.11.1
5. Using a different color block, add one block at a time. Make sure that the top
and back view of the original figure does not change. Describe what happens to
the surface area when:
a. 1 block is added
b. 2 blocks are added
c. 3 blocks are added
Continue adding blocks keeping the back and top views constant. Record the
surface area each time a new block is added.
6. Create a table and a scatter plot using the number of blocks added vs. the surface
area of the figure. Note: let the words “number added” be represent with “+”.
Label the axes.
7. Draw the views of the right side, left side, top, and back of the figure including
all the added blocks.
8. What is the domain and range of this data?
9.
Find a section of the graph where the surface area remains constant. Write the
equation of the line passing through these points.
a. What is the slope of this section?
b. What units would represent the slope of the line?
c. What is the y-intercept and what does it mean?
November 22, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The
University of Texas at Austin for the Texas Higher Education Coordinating Board.
7
Algebra I: Strand 2. Linear Functions; Topic 11. Building with Blocks; Task 2.11.1
Part 2
Remove the added blocks and return to the original figure.
Jasmine and Yolanda strategically remove blocks from the original figure to try to
change the surface area the least and still keep the top and back views constant.
10. Record what happens to the surface area when:
a. one block is removed
b. two blocks are removed
c. three blocks are removed
Continue removing blocks keeping the back and top views constant.
Record the surface area each time a new block is removed.
11. What is the greatest number of blocks that can be removed without changing the
top and back view?
12. Create a scatter plot that includes the data for both adding and removing blocks.
Use surface area vs. the number of blocks added (+) or removed (−).
13. What is the domain and range of this data?
14. Compare your graph to other graphs in the class. Are all the graphs the same?
Explain the differences in the graphs.
15. What point(s) must all possible graphs of this data have in common? Explain
what each point represents.
16. Adding or removing certain blocks did not change the surface area of the figure.
Explain how this is possible.
November 22, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The
University of Texas at Austin for the Texas Higher Education Coordinating Board.