few_shots: Example:
Problem/activity:
Narrative
The purpose of this activity is for students to create composite shapes and compose new shapes from composite shapes. For example, students may find that a rectangular prism can be made from four triangular prisms or from eight smaller cubes. Students then discover that they can combine two of the composed rectangular prisms to make a new rectangular prism.
Required Materials
Materials to Gather
Geoblocks
Launch
Groups of 4
Give each group geoblocks.
Display a large cube.
“This cube can be built from other shapes. See how many ways you can build a cube using other shapes.”
3 minutes: partner work time
Share responses.
Activity
“Choose a different shape. Try to find smaller shapes you can use to build that shape.”
5 minutes: partner work time
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Activity Synthesis
Display a cube and a composite cube—for example, a cube composed of four small cubes and one rectangular prism.
Demonstrate putting the shapes together.
“What new shape did I make with the cube and the cube I made?”
“Put your shapes together to make a rectangular prism.”
Display a rectangular prism and two triangular prisms.
“Find these shapes. Put them together. What new shape can you make?” (a cube, a rectangular block)


Mathematical concept/skill options: 
A. Reason with shapes and their attributes.
B. Understand similarity in terms of similarity transformations.
C. Use coordinates to prove simple geometric theorems algebraically.
D. Translate between the geometric description and the equation for a conic section.
E. Understand congruence and similarity using physical models, transparencies, or geometry software.
F. Analyze, compare, create, and compose shapes.
G. Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres).
H. Understand congruence in terms of rigid motions.
I. Classify two-dimensional figures into categories based on their properties.
J. Draw, construct, and describe geometrical figures and describe the relationships between them.
K. Explain volume formulas and use them to solve problems.
L. Prove theorems involving similarity.
M. Prove geometric theorems.
N. Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.
O. Define trigonometric ratios and solve problems involving right triangles.
P. Apply trigonometry to general triangles.
Q. Visualize relationships between two-dimensional and three-dimensional objects.
R. Draw and identify lines and angles, and classify shapes by properties of their lines and angles.
S. Make geometric constructions.
T. Apply geometric concepts in modeling situations.
U. Solve real-world and mathematical problems involving area, surface area, and volume.
V. Graph points on the coordinate plane to solve real-world and mathematical problems.
W. Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.
X. Understand and apply theorems about circles.
Y. Understand and apply the Pythagorean Theorem.
Z. Experiment with transformations in the plane.
AA. Find arc lengths and areas of sectors of circles.


Your response:
A

Now, assign the following problem or activity to one or more relevant mathematical concepts/skills it teaches.
shot_info: im_lesson_009431

1
options: Perform operations on matrices and use matrices in applications.

Perform operations on vectors.

Use properties of rational and irrational numbers.

Represent complex numbers and their operations on the complex plane.

Compute fluently with multi-digit numbers and find common factors and multiples.

Extend the properties of exponents to rational exponents.

Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.

Represent and model with vector quantities.

Perform arithmetic operations with complex numbers.

Apply and extend previous understandings of numbers to the system of rational numbers.

Use complex numbers in polynomial identities and equations.

Apply and extend previous understandings of multiplication and division to divide fractions by fractions.

Know that there are numbers that are not rational, and approximate them by rational numbers.

Reason quantitatively and use units to solve problems.
problem_activity: Activity
This activity is the third of three activities in which students investigate the value of
\(\sqrt{2}\)
. In the previous activity, students saw that
\(\frac75\)
is a pretty good approximation for
\(\sqrt2\)
.
\(\sqrt2\)
is a number when multiplied by itself equals 2, and
\(\frac75\)
multiplied by itself is pretty close to 2, it’s
\(\frac{49}{25}\)
. In the discussion of the previous activity, students also learned (or were reminded of) the definition of rational number: a fraction or its opposite.
In this activity, students have a chance to look for some more rational numbers that are close to
\(\sqrt2\)
. The idea is that they will look, for a while, for a fraction that can be multiplied by itself where the product is exactly 2. Of course, they won’t find one, because there is no such number. In the whole-class discussion that follows,
irrational number
is defined.
Launch
For this activity, it is best if students do not have access to a calculator with a square root button. A calculator that shows 9 decimal places will tell you that
\(\sqrt2=1.414213562\)
(which is not true) and that
\(1.414213562^2=2\)
(which is also not true). (To convince students that the calculator is lying to them, you have to make them multiply 1.414213562 by itself by hand, so better to sidestep the issue for now.) It
would
be handy for students to refer to a list of perfect squares while working on this activity, so consider posting such a list and drawing students’ attention to it.
Before they get started, remind students that a rational number is a fraction or its opposite, for example,
\(\frac98\)
. Let them know that terminating decimals are also rational, for example,
\(0.7 = \frac{7}{10}\)
.
As students work, it is possible they will focus on numbers in decimal form when searching for a rational number close to
\(\sqrt2\)
. If you see this, encourage students to remember their work in the previous activity and the example of
\(\frac75\)
, whose square,
\(\frac{49}{25}\)
, was very close to 2. Are there any other values like
\(\frac75\)
that might be even closer to
\(\sqrt2\)
?
Since students could search indefinitely for a solution to the last problem with no success, ask students to stop their work in order to leave 3–4 minutes for a whole-class discussion.
Representation: Develop Language and Symbols.
Display or provide charts with symbols and meanings. For example, display the term rational number with its definition and examples from the previous activity.
Supports accessibility for: Conceptual processing; Memory
Conversing, Reading: MLR2 Collect and Display.
As students work in pairs on the task, circulate and listen to as they discuss strategies for finding rational numbers that are close to
\(\sqrt{2}\)
. Record the words, phrases, and quantities students use on a visual display. Invite students to review the display, and ask questions to clarify the meaning of a word or phrase. For example, a phrase such as: “There is no number that is equal to
\(\sqrt{2}\)
” can be restated as “We could not find a rational number that is equal to
\(\sqrt{2}\)
." Encourage students to refer back to the visual display during whole-class discussions throughout the lesson and unit. This routine will provide feedback to students in a way that supports sense-making while simultaneously increasing meta-awareness of language.
Design Principle(s): Support sense-making; Maximize meta-awareness
Student Facing
A
rational number
is a fraction or its opposite (or any number equivalent to a fraction or its opposite).
Find some more rational numbers that are close to
\(\sqrt2\)
.
Can you find a rational number that is exactly
\(\sqrt2\)
?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Student Facing
Are you ready for more?
If you have an older calculator evaluate the expression
\(\displaystyle \left(\frac{577}{408}\right)^2\)
and it will tell you that the answer is 2, which might lead you to think that
\(\sqrt{2}=\frac{577}{408}\)
.
Explain why you might be suspicious of the calculator’s result.
Find an explanation for why
\(408^2\boldcdot 2\)
could not possibly equal
\(577^2\)
. How does this show that
\(\left(\frac{577}{408}\right)^2\)
could not equal 2?
Repeat these questions for
\(\displaystyle \left(\frac{1414213562375}{10000000000000}\right)^2\neq 2,\)
an equation that even many modern calculators and computers will get wrong.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Activity Synthesis
Ideally, students were given time to look for a rational number that is a solution to
\(x^2=2\)
. Select students with particularly close values to share their number with the class and what strategy they used to find it. Applaud students for their perseverance, but confess that no such number exists because it is an
irrational number
.
Display a number line for all to see such as the one shown here. Tell students that an
irrational number
is a number that is not rational. That is, it is a number that is not a fraction or its opposite.
\(\sqrt2\)
is one example of an irrational number. It has a location on the number line, and its location can be narrowed down (it’s a tiny bit to the right of
\(\frac{7}{5}\)
), but
\(\sqrt2\)
cannot be found on a number line by subdividing the unit interval into
\(b\)
parts and taking
\(a\)
of them. We have to define
\(\sqrt{2}\)
in a different way, such as the side length of a square with area 2 square units.
[IMG]
Just looking for rational solutions to
\(x^2=2\)
and not finding any is not a proof that
\(\sqrt2\)
is irrational. But for the purposes of this course, students are asked to take it as a fact that
\(\sqrt2\)
is irrational. In students’ future studies, they may have opportunities to understand or write a proof that
\(\sqrt2\)
is irrational.
While the activities focus specifically on
\(\sqrt{2}\)
, there is nothing particularly special about this example. The square root of a whole number is either a whole number or irrational, so
\(\sqrt{10}\)
,
\(\sqrt{67}\)
, etc., are all irrational. Beyond grade level but worth having on hand for the sake of discussion is that rational multiples of irrational numbers are also irrational, so numbers like
\(5\sqrt{7}\)
,
\(\text -\sqrt{45}\)
, and
\(\frac{\sqrt{5}}{3}\)
are also irrational.


correct_option_index: 12