few_shots: Example:
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
$$-\frac{1}{4}$$


Mathematical concept/skill options: 
A. Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
B. Reason quantitatively and use units to solve problems.
C. Apply and extend previous understandings of numbers to the system of rational numbers.
D. Represent complex numbers and their operations on the complex plane.
E. Perform arithmetic operations with complex numbers.
F. Compute fluently with multi-digit numbers and find common factors and multiples.
G. Represent and model with vector quantities.
H. Perform operations on vectors.
I. Use properties of rational and irrational numbers.
J. Use complex numbers in polynomial identities and equations.
K. Extend the properties of exponents to rational exponents.
L. Know that there are numbers that are not rational, and approximate them by rational numbers.
M. Perform operations on matrices and use matrices in applications.
N. Apply and extend previous understandings of multiplication and division to divide fractions by fractions.


Your response:
Thought: This problem involves understanding fractions, multiplication, and division.
Answer: A

Now, assign the following problem or activity to one or more relevant mathematical concepts/skills it teaches.
shot_info: fl_problem_001097

7
options: Analyze proportional relationships and use them to solve real-world and mathematical problems.

Understand ratio concepts and use ratio reasoning to solve problems.
problem_activity: Activity
Students use a realistic food recipe to find equivalent ratios that represent different numbers of batches. Students use the original recipe to form ratios of ingredients that represent double, half, five times, and one-fifth of the recipe. Then they examine given ratios of ingredients and determine how many batches they represent.
Launch
Ask students if they have ever cooked something by following a recipe. If so, ask them what they made and what some of the ingredients were.
Ask: “How might we use ratios to describe the ingredients in your recipe?” (The ratios could associate the quantities of each ingredient being used.)
Explain to students that in this task, they will think about the ratios of ingredients for a tuna casserole and how to adjust them for making different numbers of batches.
Action and Expression: Internalize Executive Functions.
To support development of organizational skills in problem solving, chunk this task into more manageable parts. For example, provide students with a task checklist which makes all the required components of the activity explicit.
Supports accessibility for: Memory; Organization
Listening: MLR8 Discussion Supports
. When asking the question, “How might we use ratios to describe the ingredients in your recipe?”, act out or use images that demonstrate the meaning of the terms "ratio", "recipe", and "ingredients" in the context of cooking. Demonstrate combining specific ingredients in their stated ratios. This will help students connect the language found in a recipe with the ratio reasoning needed for different batches of that recipe.
Design Principle(s): Support sense-making
Student Facing
Here is a recipe for tuna casserole.
Ingredients
3 cups cooked elbow-shaped pasta
6 ounce can tuna, drained
10 ounce can cream of chicken soup
1 cup shredded cheddar cheese
\(1 \frac12\)
cups French fried onions
[IMG]
Instructions
Combine the pasta, tuna, soup, and half of the cheese. Transfer into a 9 inch by 18 inch baking dish. Put the remaining cheese on top. Bake 30 minutes at 350 degrees. During the last 5 minutes, add the French fried onions. Let sit for 10 minutes before serving.
What is the ratio of the ounces of soup to the cups of shredded cheese to the cups of pasta in one batch of casserole?
How much of each of these 3 ingredients would be needed to make:
twice the amount of casserole?
half the amount of casserole?
five times the amount of casserole?
one-fifth the amount of casserole?
What is the ratio of cups of pasta to ounces of tuna in one batch of casserole?
How many batches of casserole would you make if you used the following amounts of ingredients?
9 cups of pasta and 18 ounces of tuna?
36 ounces of tuna and 18 cups of pasta?
1 cup of pasta and 2 ounces of tuna?
Student Response
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Student Facing
Are you ready for more?
The recipe says to use a 9 inch by 18 inch baking dish. Determine the length and width of a baking dish with the same height that could hold:
Twice the amount of casserole
Half the amount of casserole
Five times the amount of casserole
One-fifth the amount of casserole
Student Response
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for free access to Extension Student Response.
Anticipated Misconceptions
Students who are not yet fluent in fraction multiplication from grade 5 may have difficulty understanding how to find half or one-fifth of the recipe ingredient amounts. Likewise, they may have difficulty identifying one-third of a batch. Suggest that they draw a picture of
\(\frac12\)
of 10, remind them that finding
\(\frac12\)
of a number is the same as dividing it by 2, or remind them that
\(\frac12\)
of a number means
\(\frac12\)
times that number.
Activity Synthesis
Display the recipe for all to see. Ask students to share and explain their responses. List their responses—along with the specified number of batches—for all to see. Ask students to analyze the list and describe how the ratio of quantities relate to the number of batches in each case. Draw out the idea that each quantity within the recipe was
multiplied
by a number to obtain each batch size, and that each ingredient amount is multiplied by the
same
value.
In finding one-half and one-fifth of a batch, students may speak in terms of dividing by 2 and dividing by 5. Point out that “dividing by 2” has the same outcome as “multiplying by one-half,” and “dividing by 5” has the same outcome as “multiplying by one-fifth.” (Students multiplied a whole number by a fraction in grade 5.) Later, we will want to state our general definition of equivalent ratios as simply as possible: as
multiplying
both
\(a\)
and
\(b\)
in the ratio
\(a:b\)
by the same number (not “multiplying or dividing”).


correct_option_index: 1