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: 
Solve quadratic equations in one variable.

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


Answer: 
Yes. This task shows one quadratic equation with one variable, x.

Example 3:
Concept/skill: 
Construct a tangent line from a point outside a given circle to the circle.

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: 
No. This task does not involve drawing lines that are tangent to circles. The only line that may be drawn is the ray between points P and B.

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_pos_0

HS_1

im_task_001160_neg_0

HS_2
standard_description: Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 5^(1/3) to be the cube root of 5 because we want (5^(1/3))^(3) = 5(^(1/3))^(3) to hold, so (5^(1/3))^(3) must equal 5.
problem_activity: Activity
The purpose of this task is for students to rewrite rational numbers with terminating decimal expansions in fraction form and fractions with terminating decimal expansions as decimals. This activity is the first of a series of three in which students rewrite numbers in different ways, supporting their understanding of what rational and irrational numbers are and how they can be represented.
Monitor for students who write
\(\frac15\)
and
\(\frac{2}{10}\)
for 0.2.
Launch
Arrange students in groups of 2. Do not provide access to calculators.
Remind students that all rational numbers are fractions and their opposites, and they can all be written in the form
\(\frac{a}{b}\)
where
\(a\)
and
\(b\)
are integers (with
\(b\neq 0\)
). In fact, there are many equivalent fractions that represent a single rational number. For example, 5 is equivalent to
\(\frac{10}{2}\)
and
\(\frac{15}{3}\)
.
Students should complete the problems individually, then compare with their partners and come to resolutions over any differences. Repeat this process for the second problem.
Action and Expression: Internalize Executive Functions.
To support development of organizational skills, check in with students within the first 2-3 minutes of work time. Look for students who find equivalent ways to represent fractions and decimals. Check for precision in calculations and encourage partners to seek clarification in their discussions.
Supports accessibility for: Memory; Organization
Writing: MLR3 Clarify, Critique, Correct.
Present an incorrect statement that reflects a possible misunderstanding from the class. For example, finding the decimal representations of rational numbers, an incorrect statement is: “For finding the decimal representation of
\(\frac38\)
, divide 8 by 3 using long division ”. Prompt students to identify the error, and then write a correct version. This helps students evaluate, and improve on, the written mathematical arguments of others.
Design Principle(s): Maximize meta-awareness
Student Facing
Rational numbers are fractions and their opposites. All of these numbers are rational numbers. Show that they are rational by writing them in the form
\(\frac{a}{b}\)
or
\(\text -\frac{a}{b}\)
.
0.2
\(\text -\sqrt{4}\)
0.333
\(\sqrt[3]{1000}\)
-1.000001
\(\sqrt{\frac19}\)
All rational numbers have decimal representations, too. Find the decimal representation of each of these rational numbers.
\(\frac38\)
\(\frac75\)
\(\frac{999}{1000}\)
\(\frac{111}{2}\)
\(\sqrt[3]{\frac18}\)
Student Response
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Anticipated Misconceptions
Some students may write that
\(0.333=\frac13\)
. If so, ask them to check their work by calculating the decimal representation of
\(\frac13\)
.
Activity Synthesis
The purpose of this discussion is to highlight different strategies for rewriting rational numbers in different forms. Select previously identified students to share their solutions, including one student who wrote
\(\frac15\)
for 0.2 and another who wrote
\(\frac{2}{10}\)
. For the problems with roots, the values were purposefully chosen to emphasize to students that just because a number is written with a square or cube root does not mean it is not rational.
For 0.2, draw a number line with the numbers 0, 1, and 2 with plenty of space between the integers. Subdivide the segment from 0 to 1 into 5 equal pieces, and then ask where to plot
\(\frac15\)
. Then label
\(\frac15\)
and ask how we can see that this is
\(\frac{2}{10}\)
. (Subdivide each fifth into two equal pieces—now each one is
\(\frac{1}{10}\)
.) Then label the point
\(\frac{2}{10}\)
and 0.2.
[IMG]
Remind students that the point corresponds to a rational number, and we have a lot of different ways we can represent that number.


aligns: no