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: 
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_pos_0

HS_1

im_task_001160_pos_0

HS_2
standard_description: Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.
problem_activity: Activity
This is the final task in a series that leads students toward a general procedure for dividing fractions. Students verify previous observations about the steps for dividing non-unit fractions (namely, multiplying by the denominator and dividing by the numerator) and contrast the results with those found using diagrams. They then generalize these steps as an algorithm and apply it to answer other division questions.
As students discuss in their groups, listen to their observations and explanations. Select students with clear explanations to share later.
Digital
Launch
Arrange students in groups of 2. Give students 5–7 minutes of quiet think time and 2–3 minutes to share their response with their partner. Provide access to colored pencils. Some students may find it helpful to identify whole groups and partial groups on a tape diagram by coloring.
Students using the digital materials can use an applet to investigate division of fractions.
Representation: Internalize Comprehension.
Provide additional examples and counterexamples to illustrate concepts. Students may benefit from additional examples of division problems that they can “test” to reinforce understanding of their conclusion about how to divide a number by any fraction.
Supports accessibility for: Visual-spatial processing; Organization
Conversing, Representing: MLR8 Discussion Supports.
Provide the following sentence frames for students to use when they share their response with their partner: “I know there are ___
\(\frac18\)
s in
\(\frac34\)
because….”, “I drew the diagram like this ___ because.…”, and “First, I ___ because….”. This will help students produce statements that describe how to divide a number by any fraction.
Design Principle(s): Cultivate conversation; Support sense-making
Student Facing
Work with a partner. One person works on the questions labeled “Partner A” and the other person works on those labeled “Partner B.”
Partner A:
Find the value of each expression by completing the diagram.
\(\frac 34 \div \frac 18\)
How many
\(\frac 18\)
s in
\(\frac 34\)
?
[IMG]
\(\frac {9}{10} \div \frac 35\)
How many
\(\frac 35\)
s in
\(\frac{9}{10}\)
?
[IMG]
Use the applet to confirm your answers and explore your own examples.
Partner B:
Elena said: “If I want to divide 4 by
\(\frac 25\)
, I can multiply 4 by 5 and then divide it by 2 or multiply it by
\(\frac 12\)
.”
Find the value of each expression using the strategy Elena described.
\(\frac 34 \div \frac 18\)
\(\frac{9}{10} \div \frac35\)
What do you notice about the diagrams and expressions? Discuss with your partner.
Complete this sentence based on what you noticed:
To divide a number
\(n\)
by a fraction
\(\frac {a}{b}\)
, we can multiply
\(n\)
by ________ and then divide the product by ________.
Select
all
equations that represent the statement you completed.
\(n \div \frac {a}{b} = n \boldcdot b \div a\)
\(n \div \frac {a}{b}= n \boldcdot a \div b\)
\(n \div \frac {a}{b} = n \boldcdot \frac {a}{b}\)
\(n \div \frac {a}{b} = n \boldcdot \frac {b}{a}\)
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Print
Launch
Arrange students in groups of 2. Give students 5–7 minutes of quiet think time and 2–3 minutes to share their response with their partner. Provide access to colored pencils. Some students may find it helpful to identify whole groups and partial groups on a tape diagram by coloring.
Students using the digital materials can use an applet to investigate division of fractions.
Representation: Internalize Comprehension.
Provide additional examples and counterexamples to illustrate concepts. Students may benefit from additional examples of division problems that they can “test” to reinforce understanding of their conclusion about how to divide a number by any fraction.
Supports accessibility for: Visual-spatial processing; Organization
Conversing, Representing: MLR8 Discussion Supports.
Provide the following sentence frames for students to use when they share their response with their partner: “I know there are ___
\(\frac18\)
s in
\(\frac34\)
because….”, “I drew the diagram like this ___ because.…”, and “First, I ___ because….”. This will help students produce statements that describe how to divide a number by any fraction.
Design Principle(s): Cultivate conversation; Support sense-making
Student Facing
Work with a partner. One person works on the questions labeled “Partner A” and the other person works on those labeled “Partner B.”
Partner A: Find the value of each expression by completing the diagram.
\(\frac 34 \div \frac 18\)
How many
\(\frac 18\)
s in
\(\frac 34\)
?
[IMG]
\(\frac {9}{10} \div \frac 35\)
How many
\(\frac 35\)
s in
\(\frac{9}{10}\)
?
[IMG]
Partner B:
Elena said, “If I want to divide 4 by
\(\frac 25\)
, I can multiply 4 by 5 and then divide it by 2 or multiply it by
\(\frac 12\)
.”
Find the value of each expression using the strategy Elena described.
\(\frac 34 \div \frac 18\)
\(\frac{9}{10} \div \frac35\)
What do you notice about the diagrams and expressions? Discuss with your partner.
Complete this sentence based on what you noticed:
To divide a number
\(n\)
by a fraction
\(\frac {a}{b}\)
, we can multiply
\(n\)
by ________ and then divide the product by ________.
Select
all
the equations that represent the sentence you completed.
\(n \div \frac {a}{b} = n \boldcdot b \div a\)
\(n \div \frac {a}{b}= n \boldcdot a \div b\)
\(n \div \frac {a}{b} = n \boldcdot \frac {a}{b}\)
\(n \div \frac {a}{b} = n \boldcdot \frac {b}{a}\)
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
Invite a couple of students to share their conclusion about how to divide a number by any fraction. Then, review the sequence of reasoning that led us to this conclusion using both numerical examples and algebraic statements throughout. Remind students that in the past few activities, we learned that:
Dividing by a whole number
\(n\)
is the same as multiplying by a unit fraction
\(\frac{1}{n}\)
(e.g., dividing by 5 is the same as multiplying by
\(\frac15\)
).
Dividing by a unit fraction
\(\frac{1}{n}\)
is the same as multiplying by a whole number
\(n\)
(e.g., dividing by
\(\frac 17\)
is the same as multiplying by 7).
Dividing by a fraction
\(\frac{a}{b}\)
is the same as multiplying by a unit fraction
\(\frac{1}{a}\)
and multiplying by a whole number
\(b\)
, which is the same as multiplying by
\(\frac ba\)
(e.g., dividing by
\(\frac57\)
is the same as multiplying by
\(\frac 15\)
, and then by 7. Performing these two steps gives the same result as multiplying by
\(\frac75\)
).
Finish the discussion by trying out the generalized method with other fractions such a
\(18\div\frac97\)
, or
\(\frac{15}{14}\div\frac{5}{2}\)
. Explain that although we now have a reliable and efficient method to divide any number by any fraction, sometimes it is still easier and more natural to think of the quotient in terms of a multiplication problem with a missing factor and to use diagrams to find the missing factor.


aligns: no