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: 
Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.

Answer: 
Yes. This problem solves a real-world problem involving mixing ingredients for cookies. A student doing this problem would need to reason about equivalent ratios of butter amounts to sugar amounts.

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 to multiply a fraction or whole number by a fraction.

Answer: 
No. The expressions shown in this problem do not involve multiplying fractions with other fractions, nor whole numbers with fractions.

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: 
Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.

Answer: 
No. Completing neither of these problems requires replacing one equation by the sum of that equation and a multiple of the other equation.

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_pos_0

6

fl_problem_001097_neg_0

7

fl_problem_001392_neg_0

8
standard_description: Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding. (This standard is limited to problems posed with whole numbers and having whole number answers; students should know how to perform operations in the conventional order when there are no parentheses to specify a particular order (Order of Operations).)
problem_activity: Problem 1
How many?
[IMG]
Problem 2
Determine the value of the unknown in each equation below.
a.
$$a = 7 \times 8$$
b.
$$42 \div b =7$$
c.
$$49 = 7 \times c$$
d.
$$d\div 7 = 9$$
Problem 3
Here is the partially filled-in multiplication table from Lessons 1 and 6.
{"table": [{"_attributes": {"class": ["table", "table-bordered"], "style": "table-layout:fixed; width:100%"}, "tbody": [{"tr": [{"td": [{"_attributes": {"style": "background-color: rgb(51, 153, 255); text-align: center;"}, "b": [{"_value": "x"}]}, {"_attributes": {"style": "background-color: rgb(153, 204, 255); text-align: center;"}, "strong": [{"_value": "1"}]}, {"_attributes": {"style": "background-color: rgb(153, 204, 255); text-align: center;"}, "strong": [{"_value": "2"}]}, {"_attributes": {"style": "background-color: rgb(153, 204, 255); text-align: center;"}, "strong": [{"_value": "3"}]}, {"_attributes": {"style": "background-color: rgb(153, 204, 255); text-align: center;"}, "strong": [{"_value": "4"}]}, {"_attributes": {"style": "background-color: rgb(153, 204, 255); text-align: center;"}, "strong": [{"_value": "5"}]}, {"_attributes": {"style": "background-color: rgb(153, 204, 255); text-align: center;"}, "strong": [{"_value": "6"}]}, {"_attributes": {"style": "background-color: rgb(153, 204, 255); text-align: center;"}, "strong": [{"_value": "7"}]}, {"_attributes": {"style": "background-color: rgb(153, 204, 255); text-align: center;"}, "strong": [{"_value": "8"}]}, {"_attributes": {"style": "background-color: rgb(153, 204, 255); text-align: center;"}, "strong": [{"_value": "9"}]}, {"_attributes": {"style": "background-color: rgb(153, 204, 255); text-align: center;"}, "strong": [{"_value": "10"}]}]}, {"td": [{"_attributes": {"style": "background-color: rgb(153, 204, 255); text-align: center;"}, "strong": [{"_value": "1"}]}, {"_attributes": {"style": "background-color:#ffffff; text-align:center"}, "_value": "1"}, {"_attributes": {"style": "background-color:#ffffff; text-align:center"}, "_value": "2"}, {"_attributes": {"style": "background-color:#ffffff; text-align:center"}, "_value": "3"}, {"_attributes": {"style": "background-color:#ffffff; text-align:center"}, "_value": "4"}, {"_attributes": {"style": "background-color:#ffffff; text-align:center"}, "_value": "5"}, {"_attributes": {"style": "background-color:#ffffff; text-align:center"}, "_value": "6"}, {"_attributes": {"style": "background-color:#ffffff; text-align:center"}, "_value": "7"}, {"_attributes": {"style": "background-color:#ffffff; text-align:center"}, "_value": "8"}, {"_attributes": {"style": "background-color:#ffffff; text-align:center"}, "_value": "9"}, {"_attributes": {"style": "background-color:#ffffff; text-align:center"}, "_value": "10"}]}, {"td": [{"_attributes": {"style": "background-color: rgb(153, 204, 255); text-align: center;"}, "strong": [{"_value": "2"}]}, {"_attributes": {"style": "text-align:center"}, "_value": "2"}, {"_attributes": {"style": "text-align:center"}, "_value": "4"}, {"_attributes": {"style": "text-align:center"}, "_value": "6"}, {"_attributes": {"style": "text-align:center"}, "_value": "8"}, {"_attributes": {"style": "text-align:center"}, "_value": "10"}, {"_attributes": {"style": "text-align:center"}, "_value": "12"}, {"_attributes": {"style": "text-align:center"}, "_value": "14"}, {"_attributes": {"style": "text-align:center"}, "_value": "16"}, {"_attributes": {"style": "text-align:center"}, "_value": "18"}, {"_attributes": {"style": "text-align:center"}, "_value": "20"}]}, {"td": [{"_attributes": {"style": "background-color: rgb(153, 204, 255); text-align: center;"}, "strong": [{"_value": "3"}]}, {"_attributes": {"style": "background-color:#ffffff; text-align:center"}, "_value": "3"}, {"_attributes": {"style": "background-color:#ffffff; text-align:center"}, "_value": "6"}, {"_attributes": {"style": "background-color:#ffffff; text-align:center"}, "_value": "9"}, {"_attributes": {"style": "background-color:#ffffff; text-align:center"}, "_value": "12"}, {"_attributes": {"style": "background-color:#ffffff; text-align:center"}, "_value": "15"}, {"_attributes": {"style": "background-color:#ffffff; text-align:center"}, "_value": "18"}, {"_attributes": {"style": "background-color:#ffffff; text-align:center"}, "_value": "21"}, {"_attributes": {"style": "background-color:#ffffff; text-align:center"}, "_value": "24"}, {"_attributes": {"style": "background-color:#ffffff; text-align:center"}, "_value": "27"}, {"_attributes": {"style": "background-color:#ffffff; text-align:center"}, "_value": "30"}]}, {"td": [{"_attributes": {"style": "background-color: rgb(153, 204, 255); text-align: center;"}, "strong": [{"_value": "4"}]}, {"_attributes": {"style": "text-align:center"}, "_value": "4"}, {"_attributes": {"style": "text-align:center"}, "_value": "8"}, {"_attributes": {"style": "text-align:center"}, "_value": "12"}, {"_attributes": {"style": "text-align:center"}, "_value": "16"}, {"_attributes": {"style": "text-align:center"}, "_value": "20"}, {"_attributes": {"style": "text-align:center"}, "_value": "24"}, {"_attributes": {"style": "text-align:center"}, "_value": "28"}, {"_attributes": {"style": "text-align:center"}, "_value": "32"}, {"_attributes": {"style": "text-align:center"}, "_value": "36"}, {"_attributes": {"style": "text-align:center"}, "_value": "40"}]}, {"td": [{"_attributes": {"style": "background-color: rgb(153, 204, 255); text-align: center;"}, "strong": [{"_value": "5"}]}, {"_attributes": {"style": "background-color:#ffffff; text-align:center"}, "_value": "5"}, {"_attributes": {"style": "background-color:#ffffff; text-align:center"}, "_value": "10"}, {"_attributes": {"style": "background-color:#ffffff; text-align:center"}, "_value": "15"}, {"_attributes": {"style": "background-color:#ffffff; text-align:center"}, "_value": "20"}, {"_attributes": {"style": "background-color:#ffffff; text-align:center"}, "_value": "25"}, {"_attributes": {"style": "background-color:#ffffff; text-align:center"}, "_value": "30"}, {"_attributes": {"style": "background-color:#ffffff; text-align:center"}, "_value": "35"}, {"_attributes": {"style": "background-color:#ffffff; text-align:center"}, "_value": "40"}, {"_attributes": {"style": "background-color:#ffffff; text-align:center"}, "_value": "45"}, {"_attributes": {"style": "background-color:#ffffff; text-align:center"}, "_value": "50"}]}, {"td": [{"_attributes": {"style": "background-color: rgb(153, 204, 255); text-align: center;"}, "strong": [{"_value": "6"}]}, {"_attributes": {"style": "background-color:#ffffff; text-align:center"}, "_value": "6"}, {"_attributes": {"style": "background-color:#ffffff; text-align:center"}, "_value": "12"}, {"_attributes": {"style": "background-color:#ffffff; text-align:center"}, "_value": "18"}, {"_attributes": {"style": "background-color:#ffffff; text-align:center"}, "_value": "24"}, {"_attributes": {"style": "background-color:#ffffff; text-align:center"}, "_value": "30"}, {"_attributes": {"style": "background-color:#ffffff; text-align:center"}, "_value": "36"}, {"_attributes": {"style": "background-color:#ffffff; text-align:center"}, "_value": "42"}, {"_attributes": {"style": "background-color:#ffffff; text-align:center"}, "_value": "48"}, {"_attributes": {"style": "background-color:#ffffff; text-align:center"}, "_value": "54"}, {"_attributes": {"style": "background-color:#ffffff; text-align:center"}, "_value": "60"}]}, {"td": [{"_attributes": {"style": "background-color: rgb(153, 204, 255); text-align: center;"}, "strong": [{"_value": "7"}]}, {"_attributes": {"style": "text-align:center"}, "_value": "7"}, {"_attributes": {"style": "text-align:center"}, "_value": "14"}, {"_attributes": {"style": "text-align:center"}, "_value": "21"}, {"_attributes": {"style": "text-align:center"}, "_value": "28"}, {"_attributes": {"style": "text-align:center"}, "_value": "35"}, {"_attributes": {"style": "text-align:center"}, "_value": "42"}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}, "_value": "70"}]}, {"td": [{"_attributes": {"style": "background-color: rgb(153, 204, 255); text-align: center;"}, "strong": [{"_value": "8"}]}, {"_attributes": {"style": "text-align:center"}, "_value": "8"}, {"_attributes": {"style": "text-align:center"}, "_value": "16"}, {"_attributes": {"style": "text-align:center"}, "_value": "24"}, {"_attributes": {"style": "text-align:center"}, "_value": "32"}, {"_attributes": {"style": "text-align:center"}, "_value": "40"}, {"_attributes": {"style": "text-align:center"}, "_value": "48"}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}, "_value": "80"}]}, {"td": [{"_attributes": {"style": "background-color: rgb(153, 204, 255); text-align: center;"}, "strong": [{"_value": "9"}]}, {"_attributes": {"style": "text-align:center"}, "_value": "9"}, {"_attributes": {"style": "text-align:center"}, "_value": "18"}, {"_attributes": {"style": "text-align:center"}, "_value": "27"}, {"_attributes": {"style": "text-align:center"}, "_value": "36"}, {"_attributes": {"style": "text-align:center"}, "_value": "45"}, {"_attributes": {"style": "text-align:center"}, "_value": "54"}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}, "_value": "90"}]}, {"td": [{"_attributes": {"style": "background-color: rgb(153, 204, 255); text-align: center;"}, "strong": [{"_value": "10"}]}, {"_attributes": {"style": "text-align:center"}, "_value": "10"}, {"_attributes": {"style": "text-align:center"}, "_value": "20"}, {"_attributes": {"style": "text-align:center"}, "_value": "30"}, {"_attributes": {"style": "text-align:center"}, "_value": "40"}, {"_attributes": {"style": "text-align:center"}, "_value": "50"}, {"_attributes": {"style": "text-align:center"}, "_value": "60"}, {"_attributes": {"style": "text-align:center"}, "_value": "70"}, {"_attributes": {"style": "text-align:center"}, "_value": "80"}, {"_attributes": {"style": "text-align:center"}, "_value": "90"}, {"_attributes": {"style": "text-align:center"}, "_value": "100"}]}]}]}]}
Add the new facts you encountered in this lesson to your multiplication table.


aligns: no