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: Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.
problem_activity: Narrative
The purpose of this activity is for students to solve problems about dividing a whole number by a unit fraction in a way that makes sense to them. The context of quilt making is used so students can visualize a strip of paper that is a whole number length being cut into fractional sized pieces. As students describe how the problems are similar and different, listen for the authentic language they use to describe division. The paper strip, or tape, is a helpful diagram to use when dividing a whole number by a unit fraction because students recognize important relationships between the divisor, dividend, and quotient (MP7). For example, if the length of the strip stays the same, but the size of the piece gets smaller, then the number of pieces will get bigger.
This activity uses
MLR2 Collect and Display.
Advances: Conversing, Reading, Writing.
Launch
Groups of 2
Refer to the picture from the warm up.
“If the blue strip of fabric under the woman’s chin is 1 meter long, about how long is the short gray strip next to it?” (
\(\frac{1}{6}\)
meter)
Activity
1–2 minutes: independent think time
8–10 minutes: partner work time
MLR2 Collect and Display
Circulate, listen for, and collect the language students use to describe what was the same and different about the strategies they used to determine the number of pieces of paper for each color. Listen for:
The size of the piece changed.
The pieces were shorter.
There were more pieces.
I made more cuts.
Record students’ words and phrases on a visual display and update it throughout the lesson.
Student Facing
Below are diagrams that show strips of different colored paper. Each strip is 2 feet long. The paper strips will be cut into different sized pieces.
The red strip will be cut into pieces that are
\(\frac{1}{2}\)
foot long. How many pieces will there be?
[IMG]
The green strip will be cut into pieces that are
\(\frac{1}{3}\)
foot long. How many pieces will there be?
[IMG]
The yellow strip will be cut into pieces that are
\(\frac{1}{4}\)
foot long. How many pieces will there be?
[IMG]
Describe what was the same about the problems you solved. Describe what was different.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Advancing Student Thinking
Students may not immediately make a connection with the situation and division, since the number of pieces that results (the quotient) is greater than the number of feet being divided (the dividend). Consider asking, “How does this situation represent division?” Allow students to recognize that the cuts are similar to partitions in a diagram, or sharing. They may also recognize that the quotient represents the number of groups, while the fraction being divided is the size of each group. Have them write a division equation for each situation, if it helps. They will write division equations in the next activity.
Activity Synthesis
“Are there any other words or phrases that are important to include on our display?”
As students share responses, update the display by adding (or replacing) language, diagrams, or annotations.
Remind students to borrow language from the display as needed.
Display:
\(2 \div \frac {1}{2} = 4\)
\(2 \div \frac {1}{3} = 6\)
\(2 \div \frac {1}{4} = 8\)
“How do these equations represent the problems about the paper strips?” (The 2 is for 2 feet of paper, and the fractions show the size of the pieces that the paper is being cut into. The 4, 6, and 8 are the number of pieces of each color of paper.)


aligns: no