Algebra I: Strand 2. Linear Functions; Topic 11. Building with Blocks; Topic Notes
1
STRAND 2: LINEAR FUNCTIONS
TOPIC 2.11: BUILDING WITH BLOCKS — CHILD’S PLAY
Topic Notes
Mathematical focus
Students will investigate surface area and graphs using changes in threedimensional figures.
Topic overview
Students will be presented with a three dimensional solid figure made of cubes.
Students will be asked to investigate changing the tower while maintaining the top
and back views. Students will graph and analyze the changes in surface area.
This topic contains one task:
Task 2.11.1: Building with Blocks – Child’s Play
TExES standards focus
TExES Standard II.006 Patterns and algebra. The teacher understands linear and
quadratic functions, analyzes their algebraic and graphical properties, and uses them to
model and solve problems. The beginning teacher:
(A) Understands the concept of slope as a rate of change and interprets the
meaning of slope and intercept in a variety of situations.
TExES Standard III.011 Geometry and Measurement. The teacher understands
measurement as a process. The beginning teacher:
(B) Applies formulas for perimeter, area, surface area, and volume of geometric
figures and shapes (e.g., polygons, pyramids, prisms, cylinders, cones,
spheres) to solve problems.
TEKS/TAKS focus
TEKS A.1 Foundations for functions. The student understands that a function
represents a dependence of one quantity on another and can be described in a
variety of ways. The student it expected to:
(A) describe independent and dependent quantities in functional
relationships;
(B) gather and record data and use data sets to determine functional
relationships between quantities;
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.
Algebra I: Strand 2. Linear Functions; Topic 11. Building with Blocks; Topic Notes
2
(C) describe functional relationships for given problem situations and
write equations or inequalities to answer questions arising from the
situations;
(D) represent relationships among quantities using concrete models,
tables, graphs, diagrams, verbal descriptions, equations, and inequalities;
and
(E) interpret and make decisions, predictions, and critical judgments from
functional relationships.
High School TAKS Objective 1: The student will describe functional
relationships in a variety of ways.
TEKS A.5 Linear functions. The student understands that linear functions can
be represented in different ways and translates among their various
representations. The student is expected to:
(A) determine whether or not given situations can be represented by linear
functions;
(B) determine the domain and range for linear functions in given
situations; and
(C) use, translate, and make connections among algebraic, tabular,
graphical, or verbal descriptions of linear functions.
TEKS A.6 Linear functions. The student understands the meaning of the slope
and intercepts of the graphs of linear functions and zeros of linear functions and
interprets and describes the effects of changes in parameters of linear functions in
real-world and mathematical situations. The student is expected to:
(B) interpret the meaning of slope and intercepts in situations using data,
symbolic representations, or graphs.
High School TAKS Objective 3: The student will demonstrate an understanding of linear
functions.
(8.7) Geometry and spatial reasoning. The student uses geometry to model and
describe the physical world. The student is expected to:
(A) Draw solids form different perspectives
(B) Use geometric concepts and properties to solve problems in fields
such as art and architecture
High School TAKS Objective 7: The student will demonstrate an understanding of twoand three-dimensional representations of geometric relationships and shapes.
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; Topic Notes
Materials
Task
2.11.1
35 -2cm cubes of one color, one set for each group
x
12 – 2cm cubes of a different color, one set for each
group
x
Procedure
Allow participants to work with partners and provide them with copies of
“Building with Blocks – Child’s Play”. Each pair should use chart paper and
record their scatter plots and drawings representing both final figures containing
the least number of blocks and the most number of blocks. Those drawings
should show all views of the figure. This activity does not specify whether or not
to include the bottom surface area. Participants should justify their choice for
including or not including this area.
When finished, work should be displayed for a summary discussion. Questions
for the summary:
• Explain each section of the graph.
• How would the graph be changed if the base area had (had not) been
included?
• Look at another group’s chart, use their graph and explain how they
removed and added blocks.
• Look at the graphs and decide which group kept the surface area
unchanged the longest. How do you know?
• Which graph shows the surface increasing the fastest?
Summary
Using perspective drawings and concrete objects, participants will develop a
deeper understanding of surface area.
Assessment/Transition to the classroom
Participants should complete the Teacher’s Journal, recording their responses and
making any modifications necessary to make the tasks completely classroom
ready.
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.
Algebra I: Strand 2. Linear Functions; Topic 11. Building with Blocks; Topic Notes
4
Math notes
Some textbooks use bold lines to represent changes in depth. The TAKS test did
not use these lines; therefore, for this task, changing depth of a figure will not
represent changes in views.
On the first part of this task, no restrictions were put on adding blocks except to
maintain the original top and back views. The location blocks are added can
cause differences in graphs and data. If a block is added and all views (right, left,
front, top, back) of the figure remain unchanged, then the surface area is usually
left unchanged. However, it is possible to change the surface area without
changing any of the views if you “skip” a space when placing a block. If any of
the views change, the surface area will change. Changing the “shell” of the figure
will change the surface area, but changing the “interior” of the figure usually
causes no change in the surface area.
In the second part of the task, the restriction of changing the surface area the least
was added. Graphs for this data should all match. The surface area should
remain the same for the first 6 blocks removed and for every block removed after
the surface area will decrease by 2 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.