Question

1. a $\frac{1}{2}$ gallon jug of milk can fill 8 cups, while 32 fluid ounces of milk can fill 4 cups. what is the relationship between number of gallons and ounces? if you get stuck, try creating a table. activity synthesis collect a copy of the 3.3 reference sheet, which includes possible equations from this activity. use it to check your own equations for problems 1 - 3, then answer the questions below. a. how could a table be helpful for reasoning about the relationships in problems 2 and 3? b. for problems 1 - 3, which quantities vary and which remain constant? how could you describe the relationships? c. why might it be helpful to represent the three situations with an equation?

Explanation:

Step1: Find cups per gallon

Since $\frac{1}{2}$ gallon fills 8 cups, 1 gallon fills $8\div\frac{1}{2}=16$ cups.

Step2: Find cups per ounce

Since 32 fluid - ounces fill 4 cups, 1 fluid - ounce fills $\frac{4}{32}=\frac{1}{8}$ cups.

Step3: Relate gallons and ounces

If 1 gallon is 16 cups and 1 cup is 8 ounces, then 1 gallon is $16\times8 = 128$ ounces. So the relationship is $y = 128x$, where $y$ is the number of ounces and $x$ is the number of gallons.

A.

A table can be helpful for reasoning about the relationships in Problems 2 and 3 because it organizes data in an orderly way. It allows for easy comparison of values, identification of patterns (such as a constant ratio between gallons and ounces), and helps in visualizing how one quantity changes with respect to another.

B.

In Problems 1 - 3, the number of gallons and the number of ounces are the varying quantities. The conversion factors (cups per gallon and cups per ounce) remain constant. The relationships can be described as proportional relationships. For example, the number of ounces is directly proportional to the number of gallons with a constant of proportionality of 128 (since 1 gallon = 128 ounces).

C.

It is helpful to represent the three situations with an equation because an equation provides a concise and general rule for the relationship. It can be used to find unknown values quickly. For example, if you know the number of gallons, you can easily find the number of ounces using the equation $y = 128x$. It also allows for algebraic manipulation and can be used to graph the relationship, which gives a visual understanding of how the quantities are related.

Answer:

The relationship between the number of gallons $x$ and ounces $y$ is $y = 128x$.
A. A table organizes data for pattern - finding and comparison.
B. Varying: number of gallons, number of ounces; Constant: conversion factors; Relationship: proportional.
C. An equation gives a concise rule for finding unknowns and allows for algebraic manipulation and graphing.