options: Modeling: students learn to choose and use appropriate mathematics and statistics to analyze real-world empirical situations, improve decisions, and report on their conclusions and the reasoning behind them.

Functions: students learn to define, use, and evaluate functions to model relationships between quantities.

Number & Operations - Fractions: students learn to understand fractions as numbers and may work with them using addition, subtraction, multiplication, and division.

Number & Operations in Base Ten: students learn to work with the base-ten system and build place value understanding.

Number Systems and Quantity: students learn to understand the complex number system and reason quantitatively.

Ratios & Proportional Relationships: students learn to recognize, describe, represent, and reason with ratios, rates, proportional relationships, and percent.

Statistics & Probability: students learn to analyze and produce data distributions, and build understanding of univariate and bivariate data. They learn to interpret data, make statistical inferences, justify conclusions, understand rules of probability, and use probability to make decisions.

Geometry: students learn to classify geometric figures by their properties, and understand and compare the relationships between them. Students develop and use formulas to compute lengths, areas, and volumes, and they use transformations to generate new shapes and compare existing ones. Geometry can be studied with and without coordinates.

Operations & Algebra: students learn to solve problems using algebraic thinking and operations such as addition, subtraction, multiplication, and division. They may learn to identify and explain arithmetic patterns, evaluate or manipulate numerical expressions, and reason with equations and inequalities.

Measurement & Data: students learn to work with data and measure attributes such as time, money, length, area, and volume. They may learn to compare measurements to operations and convert between different units of measure.

Counting & Cardinality: students learn to know number names and the count sequence, count to tell the number of objects, and compare numbers.
problem_activity: Activity
The purpose of this task is for students to use the coordinates of a unit circle to determine information about coordinates on circles of different sizes. Returning to the context of clock hands, this activity focuses on the location of the end of a minute hand relative to the center of the clock.
While there are 3 different clock sizes in this activity, students taking advantage of the similarity between the circles (MP7) can use their solutions for the first clock to identify the solutions for the second clock that is 3 times the size of the first.
Monitor for students who express their answers using:
approximate values based on the unit circle display
exact values using
\(\cos(\theta)\)
and
\(\sin(\theta)\)
Launch
If needed, provide students with copies of separate clock faces.
Student Facing
Here is a clock face.
[IMG]
The length of the minute hand on a clock is 5 inches and the center of the clock is at
\((0,0)\)
on a coordinate plane. Determine the coordinates of the end of the minute hand at the following times. Explain or show your reasoning.
45 minutes after the hour
10 minutes after the hour
40 minutes after the hour
The minute hand on another clock, also centered at
\((0,0)\)
, has a length of 15 inches. Determine the coordinates of the end of the minute hand at the following times. Explain or show your reasoning.
45 minutes after the hour
10 minutes after the hour
40 minutes after the hour
At a certain time, the end of the minute hand of a third clock centered at
\((0,0)\)
has coordinates approximately
\((7.5,7.5)\)
. How long is the minute hand of the clock if each grid square is one inch by one inch? Explain or show your reasoning.
Student Response
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Student Facing
Are you ready for more?
The center of a clock is
\((0,0)\)
on a coordinate grid. Its hour hand is half the length of its minute hand. The coordinates of the end of the hour hand are about
\((3,0.5)\)
. What are the approximate coordinates of the end of the minute hand? Explain how you know.
Student Response
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click here to register or sign in
for free access to Extension Student Response.
Activity Synthesis
Select students to share their strategies for identifying the coordinates on the clocks with a minute hand of length 5 inches and 15 inches in the order listed in the Activity Narrative. If no students wrote their answers using exact values, such as
\(\left(5\cos\left(\frac \pi 6\right), 5\sin\left(\frac \pi 6\right)\right)\)
for the 5 inch minute hand 10 minutes after the hour, ask students to consider how they could write the exact coordinates. After a brief quiet think time, invite students to share their thinking, highlighting any callbacks to their previous work identifying exact values by using cosine and sine without approximating. If needed, here are some questions for discussion:
“What angle does the minute hand make, with the positive
\(x\)
-axis, at 10 minutes after the hour?” (
\(\frac \pi 6\)
radians.)
“If the minute hand was 1 foot long, could you use sine and cosine to express the coordinates of its tip?” (The coordinates, in feet, are
\(\left(\cos\left(\frac \pi 6\right), \sin\left(\frac \pi 6\right)\right)\)
.)
“What about when the minute hand is 5 inches long? 15 inches long?” (The coordinates, in inches, are
\(\left(5\cos\left(\frac \pi 6\right), 5\sin\left(\frac \pi 6\right)\right)\)
and
\(\left(15\cos\left(\frac \pi 6\right),1 5\sin\left(\frac \pi 6\right)\right)\)
.)
Conclude the discussion by selecting students to share how they calculated the length of the minute hand when the coordinates of the end of the minute hand are known. Make sure to highlight these strategies by inviting students to consider each and share a reason why they prefer one over the other:
use the Pythagorean Theorem
recognize the angle as
\(\frac{\pi}{4}\)
radians due to the
\(x\)
- and
\(y\)
-coordinates matching
Representation: Internalize Comprehension.
Use color and annotations to illustrate student thinking. As students share their strategies for identifying the coordinates on the clocks, scribe their thinking on a visual display. Highlight the exact coordinates (expressions using cosine and sine) and the approximate coordinates in the same color. Use annotation to draw attention to the scale factors of 5 and 15 in the expressions.
Supports accessibility for: Visual-spatial processing; Conceptual processing


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