Proportionality>Representing Problems with Ratios and Rates Using Scale Factors, Tables, Graphs, and Proportions(TEKS.Math.6.5.A)
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Texas 6th Grade Math › Proportionality>Representing Problems with Ratios and Rates Using Scale Factors, Tables, Graphs, and Proportions(TEKS.Math.6.5.A)
On a map, 1 inch represents 25 miles. The distance between two cities on the map is 3.5 inches. Let $m$ be the actual distance in miles.
Which proportion represents this situation?
$\frac{1}{25} = \frac{3.5}{m}$
$\frac{1}{25} = \frac{m}{3.5}$
$\frac{25}{1} = \frac{3.5}{m}$
$\frac{m}{25} = \frac{1}{3.5}$
Explanation
The ratio inches/miles must stay consistent: $\frac{1}{25} = \frac{3.5}{m}$. Cross-multiplying gives $m = 3.5 \times 25 = 87.5$. The same relationship can be shown with a scale factor (multiply inches by 25 to get miles), a table (Inches: 1, 2, 3.5; Miles: 25, 50, 87.5), or a graph of a line through the origin with slope 25 miles per inch.
A recipe uses 4 cups of flour to make 10 cookies.
Which table correctly shows this proportional relationship for different amounts?
Cups: 2, 4, 6, 8; Cookies: 4, 8, 12, 16
Cups: 2, 4, 6, 8; Cookies: 6, 12, 18, 24
Cups: 2, 4, 6, 8; Cookies: 5, 10, 15, 20
Cups: 5, 10, 15, 20; Cookies: 2, 4, 6, 8
Explanation
The unit rate is 10 cookies per 4 cups = 2.5 cookies per cup. The correct table keeps the ratio constant: multiply cups by 2.5 to get cookies. This can also be shown with a proportion $\frac{\text{cookies}}{\text{cups}} = \frac{10}{4} = \frac{5}{2}$, a scale factor (times 2.5), or a graph of a line through the origin with slope 2.5.
A shop rents bikes for 7 dollars per hour.
Which graph description best represents the relationship between hours ($x$) and cost ($y$)?
A line through the origin passing through points (1,5), (2,10), (3,15)
A horizontal line at $y=7$
A line that starts at (0,7) and passes through (1,14)
A line through the origin passing through points (1,7), (2,14), (3,21)
Explanation
The cost is proportional to time with rate 7 dollars per hour, so $y=7x$. A correct graph is a line through the origin with slope 7, hitting (1,7), (2,14), (3,21). A matching table would list those pairs, and a proportion would be $\frac{y}{x}=7$.
A blueprint uses a scale of 1 cm for every 2.5 m in real life. A wall is 7.5 m long. Let $b$ be the length on the blueprint in centimeters.
Which proportion correctly represents this situation to find $b$?
$\frac{1}{2.5} = \frac{7.5}{b}$
$\frac{1}{2.5} = \frac{b}{7.5}$
$\frac{2.5}{1} = \frac{b}{7.5}$
$\frac{b}{1} = \frac{7.5}{2.5}$
Explanation
Keep drawing/real consistent: $\frac{1,\text{cm}}{2.5,\text{m}} = \frac{b}{7.5}$. So $\frac{1}{2.5}=\frac{b}{7.5}$ and $b=7.5\times\frac{1}{2.5}=3$ cm. This is the same as using a scale factor (multiply meters by $\frac{1}{2.5}$ to get cm), making a table (Real m: 2.5, 5, 7.5; Drawing cm: 1, 2, 3), or graphing a line through the origin with slope 0.4 cm per meter.
A faucet fills 3 liters of water in 4 minutes. Let $x$ be minutes and let $y$ be liters.
Which equation in the form $y=kx$ models this proportional relationship?
$y = 0.75x$
$y = \frac{4}{3}x$
$y = 3x + 4$
$x = 0.75y$
Explanation
The unit rate is $\frac{3}{4}$ liter per minute, so $k=\frac{3}{4}=0.75$ and $y=0.75x$. A table would include (4,3), (8,6), (12,9). A proportion is $\frac{y}{x}=\frac{3}{4}$. The graph is a line through the origin with slope 0.75.