standard_description: Construct a tangent line from a point outside a given circle to the circle.
problem_activity: Narrative
The purpose of this activity is for students to observe and use the structure of diagrams to find areas of shaded regions with non-unit fraction side lengths. Students build on what they learned in the previous activity, solidifying their understanding of why the numerator of a product of two fractions is the product of the numerators and the denominator of a product of fractions is the product of the denominators.
Action and Expression: Internalize Executive Functions.
Invite students to verbalize their strategy for writing multiplication expressions to represent the area of the shaded rectangle in each figure before they begin. Students can speak quietly to themselves, or share with a partner.
Supports accessibility for: Organization, Conceptual Processing, Language
Launch
Groups of 2
Activity
“Start working on completing the table independently. After a couple minutes, you'll work with your partner to complete the table and answer the rest of the questions.”
1–2 minutes: independent work time
5-8 minutes: partner work time
Monitor for students who:
notice the area of the shaded regions is always twentieths
write the expression
\(\frac{6}{5}\times\frac{4}{5}\)
to represent the shaded region of the last diagram in the table
explain that the expression
\(\frac{6\times4}{5\times4}\)
represents the shaded part of the last diagram in the table because
\(6\times4\)
represents the number of pieces that are shaded and
\(4 \times 5\)
represents the number of those pieces in the unit square
Student Facing
Complete the table.
|                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                diagram                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                | multiplication                       expression | shaded area                       (square units) |
|-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|-------------------------------------------------|--------------------------------------------------|
| A                |https://cms-im.s3.amazonaws.com/ui5g959sm5gv0dm8y5muaeuevuxb?response-content-disposition=attachment%3B%20filename%3D%22tikz-file-147492.svg%22%3B%20filename%2A%3DUTF-8%27%27tikz-file-147492.svg&response-content-type=image%2Fsvg%2Bxml&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240208%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240208T165553Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=da65b42438b700b09e62286557fa329bb057a4e3c43b71d5a338b3b21f3a4bad|  |https://cms-im.s3.amazonaws.com/ui5g959sm5gv0dm8y5muaeuevuxb?response-content-disposition=attachment%3B%20filename%3D%22tikz-file-147492.svg%22%3B%20filename%2A%3DUTF-8%27%27tikz-file-147492.svg&response-content-type=image%2Fsvg%2Bxml&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240208%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240208T165553Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=da65b42438b700b09e62286557fa329bb057a4e3c43b71d5a338b3b21f3a4bad| image:: https://cms-im.s3.amazonaws.com/ui5g959sm5gv0dm8y5muaeuevuxb?response-content-disposition=attachment%3B%20filename%3D%22tikz-file-147492.svg%22%3B%20filename%2A%3DUTF-8%27%27tikz-file-147492.svg&response-content-type=image%2Fsvg%2Bxml&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240208%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240208T165553Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=da65b42438b700b09e62286557fa329bb057a4e3c43b71d5a338b3b21f3a4bad |                                                 |                                                  |
| B                |https://cms-im.s3.amazonaws.com/dx240g4b4kkym1gkgs7khx0yw7ri?response-content-disposition=attachment%3B%20filename%3D%22tikz-file-149071.svg%22%3B%20filename%2A%3DUTF-8%27%27tikz-file-149071.svg&response-content-type=image%2Fsvg%2Bxml&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240208%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240208T165553Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=216cb8ca62660faf0bab04b1fb04fd987604857004a334e4036d514267493e92|  |https://cms-im.s3.amazonaws.com/dx240g4b4kkym1gkgs7khx0yw7ri?response-content-disposition=attachment%3B%20filename%3D%22tikz-file-149071.svg%22%3B%20filename%2A%3DUTF-8%27%27tikz-file-149071.svg&response-content-type=image%2Fsvg%2Bxml&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240208%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240208T165553Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=216cb8ca62660faf0bab04b1fb04fd987604857004a334e4036d514267493e92| image:: https://cms-im.s3.amazonaws.com/dx240g4b4kkym1gkgs7khx0yw7ri?response-content-disposition=attachment%3B%20filename%3D%22tikz-file-149071.svg%22%3B%20filename%2A%3DUTF-8%27%27tikz-file-149071.svg&response-content-type=image%2Fsvg%2Bxml&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240208%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240208T165553Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=216cb8ca62660faf0bab04b1fb04fd987604857004a334e4036d514267493e92 |                                                 |                                                  |
|                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                diagram                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                | multiplication                       expression | shaded area                       (square units) |
|-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|-------------------------------------------------|--------------------------------------------------|
| C                |https://cms-im.s3.amazonaws.com/9mw9803l8i0vjwwj588r8xkqowrt?response-content-disposition=attachment%3B%20filename%3D%22tikz-file-149072.svg%22%3B%20filename%2A%3DUTF-8%27%27tikz-file-149072.svg&response-content-type=image%2Fsvg%2Bxml&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240208%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240208T165553Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=80f7eec3ac67d9991858cc362c8526df4c2ac740536b8a7383a5ba2034426a7e|  |https://cms-im.s3.amazonaws.com/9mw9803l8i0vjwwj588r8xkqowrt?response-content-disposition=attachment%3B%20filename%3D%22tikz-file-149072.svg%22%3B%20filename%2A%3DUTF-8%27%27tikz-file-149072.svg&response-content-type=image%2Fsvg%2Bxml&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240208%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240208T165553Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=80f7eec3ac67d9991858cc362c8526df4c2ac740536b8a7383a5ba2034426a7e| image:: https://cms-im.s3.amazonaws.com/9mw9803l8i0vjwwj588r8xkqowrt?response-content-disposition=attachment%3B%20filename%3D%22tikz-file-149072.svg%22%3B%20filename%2A%3DUTF-8%27%27tikz-file-149072.svg&response-content-type=image%2Fsvg%2Bxml&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240208%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240208T165553Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=80f7eec3ac67d9991858cc362c8526df4c2ac740536b8a7383a5ba2034426a7e |                                                 |                                                  |
| D                |https://cms-im.s3.amazonaws.com/wy0o9qune26s9u7htzi5ilsfmts4?response-content-disposition=attachment%3B%20filename%3D%22tikz-file-149073.svg%22%3B%20filename%2A%3DUTF-8%27%27tikz-file-149073.svg&response-content-type=image%2Fsvg%2Bxml&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240208%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240208T165553Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=57e693708d062f1a9de3113218a2c369f9dbe92ca8b9a751cd2729b2d6cbb65d|  |https://cms-im.s3.amazonaws.com/wy0o9qune26s9u7htzi5ilsfmts4?response-content-disposition=attachment%3B%20filename%3D%22tikz-file-149073.svg%22%3B%20filename%2A%3DUTF-8%27%27tikz-file-149073.svg&response-content-type=image%2Fsvg%2Bxml&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240208%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240208T165553Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=57e693708d062f1a9de3113218a2c369f9dbe92ca8b9a751cd2729b2d6cbb65d| image:: https://cms-im.s3.amazonaws.com/wy0o9qune26s9u7htzi5ilsfmts4?response-content-disposition=attachment%3B%20filename%3D%22tikz-file-149073.svg%22%3B%20filename%2A%3DUTF-8%27%27tikz-file-149073.svg&response-content-type=image%2Fsvg%2Bxml&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240208%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240208T165553Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=57e693708d062f1a9de3113218a2c369f9dbe92ca8b9a751cd2729b2d6cbb65d |                                                 |                                                  |
What patterns do you notice in the table?
Explain or show how the expression
\(\frac{6\times4}{5\times4}\)
represents the last diagram in the table.
Student Response
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Advancing Student Thinking
If students do not explain that the product of the numerators represents the number of pieces in the shaded region and the product of the denominators represents the number of pieces in the whole, consider asking: “What is the same and what is different about diagrams A and B?”
Activity Synthesis
Ask previously selected students to share their reasoning.
“How do the expressions in the table represent the number of pieces shaded in and the size of the pieces shaded in?” (If we multiply the numerators, we get the number of pieces that are shaded in. If we multiply the denominators, we get the size of the pieces.)
Refer to diagrams and draw on each diagram to show how the multiplication of the numerators and denominators represents the number of shaded pieces and the size of the shaded pieces (that is, the number of those pieces in the whole).


aligns: no