few_shots: 
Example 1:
Problem/activity:
Find the general solution for:
$${\mathrm{cos}\left({\pi\over2}+x\right)=\mathrm{tan}{\pi\over4}}$$


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

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


Concept/skill: 
Solve quadratic equations in one variable.

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

Example 3:
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 $P$.


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.

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, does the problem or activity shown below enable students to learn the full intent of the following concept or skill?

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: Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.
problem_activity: Activity
Students have learned several effective methods to divide a whole number by a whole number, including cases when there is a remainder. The goal of this task is to introduce a method for dividing a decimal number by a whole number. Students notice that the steps in the division process are the same as when dividing a whole number by a whole number, whether the division is done with base-ten diagrams, as in this task, or using partial quotients or the division algorithm as in future tasks. Here, students need to think even more carefully about place value and where the decimal point goes in the quotient.
Throughout this activity, students rely on their understanding of equivalent expressions to interpret the unbundling in Elena’s process. For example, to unbundle a one into ten tenths means going between the expressions 1 and
\(0.1+0.1+0.1+0.1+0.1+0.1+0.1+0.1+0.1+0.1\)
.
Launch
Give students 1–2 minutes of quiet think time for students to analyze Elena’s work. Pause and discuss with the whole class. Select a couple of students to share their analyses of what Elena had done to divide a decimal by a whole number. Then, give students 8–9 minutes to complete the questions and follow with a whole-class discussion.
Student Facing
To find
\(53.8 \div 4\)
using diagrams, Elena began by representing 53.8.
[IMG]
She placed 1 ten into each group, unbundled the remaining 1 ten into 10 ones, and went on distributing the units.
This diagram shows Elena’s initial placement of the units and the unbundling of 1 ten.
[IMG]
Complete the diagram by continuing the division process. How would you use the available units to make 4 equal groups?
As the units get placed into groups, show them accordingly and cross out those pieces from the bottom. If you unbundle a unit, draw the resulting pieces.
What value did you find for
\(53.8 \div 4\)
? Be prepared to explain your reasoning.
Use long division to find
\(53.8 \div 4\)
. Check your answer by multiplying it by the divisor 4.
Use long division to find
\(77.4 \div 5\)
. If you get stuck, you can draw diagrams or use another method.
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?
A distant, magical land uses jewels for their bartering system. The jewels are valued and ranked in order of their rarity. Each jewel is worth 3 times the jewel immediately below it in the ranking. The ranking is red, orange, yellow, green, blue, indigo, and violet. So a red jewel is worth 3 orange jewels, a green jewel is worth 3 blue jewels, and so on.
A group of 4 craftsmen are paid 1 of each jewel. If they split the jewels evenly amongst themselves, which jewels does each craftsman get?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Anticipated Misconceptions
Some students may stop dividing when they reach a remainder rather than unbundling the remainder into smaller units. Remind them that they can continue to divide the remainder by unbundling and to refer to Elena’s worked-out example or those from earlier lessons, if needed.
Activity Synthesis
Ask a student or two to display and explain their work. Ask if others performed the division the same way and if there are disagreements.
Then, focus the discussion on the connections between a division problem with a whole-number dividend (such as
\(62 \div 5\)
) and that with a decimal dividend (such as
\(53.8 \div 4\)
). Discuss:
How is the division problem
\(53.8 \div 4\)
similar to
\(62 \div 5\)
from a previous lesson?
In both problems, when we get to the final place value (tenths for
\(53.8 \div 4\)
and one for
\(62 \div 5\)
), there is still a remainder.
In both problems, to complete the division and find the quotient we need to introduce a new place value (hundredths for
\(53.8 \div 4\)
, and tenths for
\(62 \div 5\)
).
We have to unbundle at every step in both division problems.
How is the division problem
\(53.8 \div 4\)
different to
\(62 \div 5\)
from a previous lesson?
There is already a decimal in
\(53.8\)
: we had to write the decimal point for
\(62 \div 5\)
.
The quotient
\(53.8 \div 4\)
goes to the hundredths place, so there is an extra step and an additional place value.
If we were to rewrite
\(62 \div 5\)
as
\(62.0 \div 5\)
(which is what is needed in order to complete the division), then the two division problems look similar. The biggest difference between
\(53.8 \div 4\)
and
\(62 \div 5\)
is that the former problem has an answer in the hundredths while the answer to the latter only has tenths.
Representing, Speaking, Listening: MLR2 Collect and Display.
As students discuss how the division problem
\(53.8÷4\)
is related to
\(62÷5\)
from a previous lesson, create a two-column display with “similarities” and “differences” for the headers. Circulate through the groups and record student language in the appropriate column. Look for phrases such as “whole-number dividend,” “decimal dividend,” and “place value.” This will help students compare and contrast different types of division problems by recognizing the structure of the numbers.
Design Principle(s): Support sense-making; Maximize meta-awareness


aligns: no