Question

lesson 3 | session 2
name:
practice understanding
proportional relationships
▶ study how the example shows a proportional relationship.
then solve problems 1 - 5.
example
the double number line shows the pounds of clay jamal uses to make
different numbers of identical bowls. is there a proportional relationship
between the pounds of clay and the number of bowls? if so, what is the
constant of proportionality for pounds of clay per bowl?
all of the ratios for clay : bowls are equivalent. so, there is a proportional relationship.
all of the ratios have the same rate, \\( \frac { 2 } { 3 } \\) pound of clay per bowl. the constant of
proportionality is \\( \frac { 2 } { 3 } \\).
1 a. look at the example. write an equation that shows the relationship
between the pounds of clay jamal uses, c, and the number of bowls he
makes, b.
b. explain how to find the pounds of clay jamal needs to make 8 bowls.
2 zara has paper rectangles of different sizes. every rectangle is 5 cm longer
than it is wide. is there a proportional relationship between the lengths
and widths of these rectangles? explain.
vocabulary
constant of
proportionality
the unit rate in a
proportional
relationship.
proportional
relationship
the relationship
between two
quantities where one
quantity is a constant
multiple of the other
quantity.
unit rate
the numerical part
a rate.

Explanation:

Problem 1a

Step1: Recall proportional relationship formula

In a proportional relationship, the formula is \( c = k \times b \), where \( k \) is the constant of proportionality. From the example, \( k=\frac{2}{3} \).

Step2: Substitute \( k \) into the formula

Substitute \( k = \frac{2}{3} \) into \( c = k \times b \), we get \( c=\frac{2}{3}b \).

Step1: Use the equation from 1a

We have the equation \( c=\frac{2}{3}b \) where \( c \) is clay in pounds and \( b \) is number of bowls.

Step2: Substitute \( b = 8 \)

Substitute \( b = 8 \) into \( c=\frac{2}{3}b \), so \( c=\frac{2}{3}\times8 \).

Step3: Calculate the value

\( \frac{2}{3}\times8=\frac{16}{3}=5\frac{1}{3} \). So we use the proportional relationship equation, substitute the number of bowls (8) into it to find the clay needed.

Let the width be \( w \) and length be \( l \). The relationship is \( l=w + 5 \). A proportional relationship is of the form \( y=kx \) (multiplicative), but here it's \( l=w + 5 \) (additive). For a proportional relationship, the ratio \( \frac{l}{w} \) should be constant. Let's test with two values: if \( w = 1 \), \( l=6 \), ratio \( \frac{6}{1}=6 \); if \( w = 2 \), \( l = 7 \), ratio \( \frac{7}{2}=3.5 \), which is not constant. So it's not a proportional relationship because the relationship is additive (\( l=w + 5 \)) not multiplicative (proportional relationships are multiplicative, \( y = kx \)).

Answer:

\( c = \frac{2}{3}b \)

Problem 1b