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 constar
multiple of the othe
quantity.
unit rate
the numerical pa
a rate.

Explanation:

1a.

Step1: Recall proportional relationship formula

A proportional relationship is of the form \( c = k \cdot b \), where \( k \) is the constant of proportionality. From the example, \( k=\frac{2}{3} \).

Step2: Write the equation

Substitute \( k = \frac{2}{3} \) into the formula. So the equation is \( c=\frac{2}{3}b \).

Step1: Use the equation from part (a)

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

A proportional relationship between two quantities \( l \) (length) and \( w \) (width) should be of the form \( l=k\cdot w \) (or \( w = k\cdot l \)) where \( k \) is a constant. Here, length \( l=w + 5 \). For a proportional relationship, the ratio \( \frac{l}{w} \) should be constant. Let's test with two different widths. Let \( w_1 = 1 \), then \( l_1=1 + 5=6 \), \( \frac{l_1}{w_1}=6 \). Let \( w_2 = 2 \), then \( l_2=2 + 5 = 7 \), \( \frac{l_2}{w_2}=\frac{7}{2}=3.5 \). Since \( 6
eq3.5 \), the ratio is not constant. Also, the relationship is \( l=w + 5 \), which is a linear relationship with a constant addition, not a constant multiple. So there is no proportional relationship.

Answer:

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

1b.