Question

lesson 3 session 3
4 lucja makes braided key chains out of cotton cord. for every 8 feet of she has, she can make 3 braided key chains.
parta use a model to represent the relationship between the number of feet of cotton cord and the number of key chains.
part b use the model to explain how to find the number of feet of cotton cord lucja needs to make 5 key chains.
5 math journal in a baking recipe, there is a proportional relationship between the number of cups of flour and the number of cups of sugar. what does that mean? tell at least two things you know are true about this relationship.
dear family,
this week your student is learning abo graphs and equations that can repre proportional relationships
one way to represent a proportio relationship is with a graph. the be a straight line that goes thr origin, or the point (0, 0).
another way is with an equ you how many y you have the equation for the pro relationship at the right
your student will solv
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Explanation:

Part A: Model the Relationship

Step 1: Identify the Proportional Relationship

We know that for every 8 feet of cotton cord, Lucia makes 3 key chains. This is a proportional relationship, so the ratio of cotton cord (let’s denote as \( x \)) to key chains (let’s denote as \( y \)) is constant. The unit rate (cotton cord per key chain) is \( \frac{8}{3} \) feet per key chain, or the rate of key chains per cotton cord is \( \frac{3}{8} \) key chains per foot.

Step 2: Represent with a Table

A table can model this relationship. Let’s list values:

Cotton Cord (ft) (\( x \))	Key Chains (\( y \))
\( \frac{8}{3} \approx 2.67 \)	1
8	3
\( \frac{16}{3} \approx 5.33 \)	2
16	6

Part B: Find Cotton Cord for 5 Key Chains

Step 1: Use the Proportionality Formula

For a proportional relationship, \( y = kx \), where \( k \) is the constant of proportionality. From the given data, when \( x = 8 \), \( y = 3 \), so \( k = \frac{y}{x} = \frac{3}{8} \) (key chains per foot) or \( k = \frac{x}{y} = \frac{8}{3} \) (feet per key chain).

To find \( x \) when \( y = 5 \), use \( x = \frac{8}{3} \times y \) (since \( x = \frac{8}{3}y \) from \( \frac{x}{y} = \frac{8}{3} \)).

Step 2: Calculate

Substitute \( y = 5 \):
\( x = \frac{8}{3} \times 5 = \frac{40}{3} \approx 13.33 \) feet.

Math Journal Question: Proportional Relationship in Baking

A proportional relationship between cups of flour (\( F \)) and cups of sugar (\( S \)) means:

1. Constant Ratio: The ratio \( \frac{F}{S} \) (or \( \frac{S}{F} \)) is constant. For example, if 2 cups of flour need 1 cup of sugar, the ratio \( \frac{F}{S} = 2 \) (flour per sugar) or \( \frac{S}{F} = \frac{1}{2} \) (sugar per flour) always holds.
2. Graph Through Origin: If we plot \( F \) on the x - axis and \( S \) on the y - axis, the graph is a straight line passing through (0, 0). Doubling the flour doubles the sugar, tripling the flour triples the sugar, etc.

Answer:

s

• Part A: The relationship is proportional, modeled by the table (or equation \( y=\frac{3}{8}x \) or \( x = \frac{8}{3}y \)).
• Part B: Lucia needs \( \boldsymbol{\frac{40}{3}} \) (or \( \approx 13.33 \)) feet of cotton cord for 5 key chains.
• Math Journal: A proportional relationship means (1) the ratio of flour to sugar (or vice versa) is constant, and (2) the graph of their relationship is a straight line through the origin.