few_shots: 
Example 1:
Concept/skill: 
Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

Problem/activity:
Find the general solution for:
$${\mathrm{cos}\left({\pi\over2}+x\right)=\mathrm{tan}{\pi\over4}}$$


Answer: 
Yes. An initial step when solving for x in this problem requires the use of the addition formula for cosine on the left side of the equation.

Example 2:
Concept/skill: 
Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

Problem/activity:
Task
Solve the quadratic equation
$$ x^2 = (2x - 9)^2 $$
using as many different methods as possible.


Answer: 
No. This task does not involve an equation in two variables. There is only one variable, x.

Example 3:
Concept/skill: 
Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.

Problem/activity:
Task
Suppose $A$ is an angle with vertex $P$, as pictured below:
[IMG]
Draw a circle with center $P$ and with  radius $r \gt 0$.  Explain why the circle
meets each ray of angle $A$ in a single point.  Label these points $Q$ and $R$ respectively.
Draw circles with centers $Q$ and $R$ respectively and radius $r$.  These circles meet at $P$ and a second point to be labelled $B$.
Show that ray $\overrightarrow{PB}$  bisects angle $P$.


Answer: 
Yes. This task asks students to draw a ray that bisects an angle. The act of drawing circles and rays may involve compass and straightedge or dynamic geometric software.

Now, determine whether the following problem or activity can enable students to understand the concepts or skills in the provided description.

shot_info: fl_problem_001894_pos_0

HS_0

im_task_000870_neg_0

HS_1

im_task_001160_pos_0

HS_2
standard_description: Use square root and cube root symbols to represent solutions to equations of the form x^(2) = p and x^(3) = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that Ö2 is irrational.
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.


aligns: yes