few_shots: 
Example 1:
Problem/activity:
You’re mixing ingredients for cookies. The recipe says to combine 6 tablespoons, or ¾ stick, of butter with 1 cup of sugar. You accidentally mix in a whole stick of butter (8 tablespoons) with the cup of sugar. How can you fix this?


Concept/skill: 
Recognize and represent proportional relationships between quantities.

Answer: 
No. Usually problems that teach recognizing and representing proportional relationships between quantities involve deciding whether two quantities are in a proportional relationship or identifying the constant of proportionality, and then representing proportional relationships by equations or explaining proportional relationship using coordinate graphs.

Example 2:
Problem/activity:
For each pair of numbers, determine if the values are equivalent. Explain your reasoning for each one.
a.
$$-0.25$$
and
$$\frac{1}{-4}$$
b.
$$-\left ( \frac{1}{4} \right )$$
and
$$\frac{-1}{-4}$$
c.
$$\frac{-1}{4}$$
and
$$-\frac{1}{4}$$


Concept/skill: 
Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.

Answer: 
Yes. This problem involves understanding fractions and division to divide rational numbers, such as 1, -4, 4, and -1. This problem also involves an understanding of multiplication: if p and q are integers, then -(p/q) = (-p)/q = p/(-q).

Example 3:
Problem/activity:
Problem 1
The table below contains single equations in a single variable, single equations in two variables, and pairs of simultaneous equations in two variables. Check the appropriate box to show whether there is no solution, exactly 1 solution, or if there is an infinite number of solutions.
{"table": [{"_attributes": {"align": "center", "class": ["table", "table-bordered"], "style": "table-layout:fixed; width:70%"}, "tbody": [{"tr": [{"td": [{"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}, "strong": [{"_value": "No solution"}]}, {"_attributes": {"style": "text-align:center"}, "strong": [{"_value": "Exactly 1 solution"}]}, {"_attributes": {"style": "text-align:center"}, "strong": [{"_value": "Infinite number of solutions"}]}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${9=4x+7}$$"}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${y=3y+5}$$"}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${y={x\\over2}}$$"}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"p": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${3x+6y-9=0}$$"}]}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"p": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${x=y}$$"}]}, {"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${1.25x=1.25y}$$"}]}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"p": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${9.9=6x+8y}$$"}]}, {"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${9x=2.5y-8.8}$$"}]}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"p": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${11x-2y=1.5}$$"}]}, {"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${11x-2y=2.5}$$"}]}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {}]}]}]}]}
Problem 2
Kim has a small container and a large container as shown.
[IMG]
It takes 16 of the small containers to fill the large container. Three small containers leave 1.95 gallons of space in the big container.
What is the size of each of the two containers?


Concept/skill: 
Analyze and solve pairs of simultaneous linear equations.

Answer: 
Yes. The last three rows of the table in the first problem involves solving pairs of simultaneous linear equations. The second word problem can also be represented as two simultaneous linear equations.

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_000972_neg_0

6

fl_problem_001097_pos_0

7

fl_problem_001392_pos_0

8
standard_description: Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.
problem_activity: Activity
The purpose of this activity is to reinforce that some conditions define a unique polygon while others do not. Students build polygons given only a description of their side lengths. They articulate that this is not enough information to guarantee that a pair of quadrilaterals are identical copies. On the other hand, triangles have a special property that three specific side lengths result in a unique triangle. Students should notice that their recreation of Jada’s triangle is rigid; the side lengths and angles are all fixed.
Monitor for students who try putting the side lengths together in different orders to build different polygons and invite them to share during the whole-class discussion.
Digital
Launch
Arrange students in groups of 2. Remind students that a quadrilateral is a four-sided polygon. If necessary, demonstrate how to use the fasteners to connect the strips.
Distribute one set of strips and fasteners to each group. Provide access to geometry toolkit, including rulers and protractors. Encourage students to think about whether there are different shapes that would fulfill the given conditions. Give students 5–6 minutes of quiet work time followed by a whole-class discussion.
For classes using the digital materials, there is an applet for students to use to build polygons with the given side lengths. If necessary, demonstrate how to create a vertex by overlapping the endpoints of two segments. It may work best for positioning each segment to put the green endpoint in place first and then adjust the yellow endpoint as desired.
Representation: Internalize Comprehension.
Begin the activity with concrete or familiar contexts. Remind students that a polygon is a closed figure with straight sides. Demonstrate how to use fasteners to connect the slips.
Supports accessibility for: Conceptual processing; Memory
Student Facing
Diego built a quadrilateral using side lengths of 4 in, 5 in, 6 in, and 9 in.
Build such a shape.
Is your shape an identical copy of Diego’s shape? Explain your reasoning.
Jada built a triangle using side lengths of 4 in, 5 in, and 8 in.
Build such a shape.
Is your shape an identical copy of Jada’s shape? Explain your reasoning.
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. Remind students that a quadrilateral is a four-sided polygon. If necessary, demonstrate how to use the fasteners to connect the strips.
Distribute one set of strips and fasteners to each group. Provide access to geometry toolkit, including rulers and protractors. Encourage students to think about whether there are different shapes that would fulfill the given conditions. Give students 5–6 minutes of quiet work time followed by a whole-class discussion.
For classes using the digital materials, there is an applet for students to use to build polygons with the given side lengths. If necessary, demonstrate how to create a vertex by overlapping the endpoints of two segments. It may work best for positioning each segment to put the green endpoint in place first and then adjust the yellow endpoint as desired.
Representation: Internalize Comprehension.
Begin the activity with concrete or familiar contexts. Remind students that a polygon is a closed figure with straight sides. Demonstrate how to use fasteners to connect the slips.
Supports accessibility for: Conceptual processing; Memory
Student Facing
Diego built a quadrilateral using side lengths of 4 in, 5 in, 6 in, and 9 in.
Build such a shape.
Is your shape an identical copy of Diego’s shape? Explain your reasoning.
Jada built a triangle using side lengths of 4 in, 5 in, and 8 in.
Build such a shape.
Is your shape an identical copy of Jada’s shape? Explain your reasoning.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Anticipated Misconceptions
Students may think that their triangle is different from Jada’s because hers is “upside down.” Ask the student to turn their triangle around and ask them if it is now a different triangle. While there is a good debate to be had if they continue to insist they are different, let the students know that, for this unit, we will consider shapes that have been turned or flipped or moved as identical copies and thus “not different.”
Activity Synthesis
Select previously identified students to share their constructions and explanations. Display each student’s example for all to see.
If desired, reveal Diego and Jada’s shapes and display for all to see along side students’ work.
[IMG]
Ask students:
“Is this what you thought Jada and Diego’s shapes looked like?”
“Which shape did you make an identical copy of?” (Jada’s triangle.)
“Why did you not make an identical copy of Diego’s shape?” (Because you can make quadrilaterals with the same side lengths but different angle measure.)
Consider explaining to students how this finding is applied in construction projects. For stability, the internal structures of many buildings (and bridges) will include triangles. Rectangles or other polygons with more than three sides often include triangular supports on the inside, to make the construction more rigid and less floppy.
Speaking: MLR7 Compare and Connect.
Use this routine to support whole-class discussion. Ask students to consider what is the same and what is different about their quadrilateral as compared to Diego’s. Draw students' attention to the association between the order of side lengths in each quadrilateral and the angle measures in each shape. In this discussion, demonstrate the language used to make sense of the conditions required to make identical figures. Talking about and comparing the quadrilaterals will help draw students' attention to the conditions that define a unique polygon.
Design Principle(s): Maximize meta-awareness


aligns: yes