A focused Grade 6 learning session for Graphing Ratios using visual models, worked examples, strategy guidance, and a short mastery check.
Student goal: I can solve problems involving graphing ratios.
Graphing Ratios means reading, creating, and explaining displays so data can answer real questions.
Question: The coordinate plane shows ordered pairs representing the ratio of hours worked to dollars earned. Based on the graph, how much is earned per hour?
Why it works: Look at the y-axis label: each tick on the dollars axis stands for $10. The point \((1,1)\) means \(1\) hour and \(1\times10=$10\), so the rate is \($10\) per hour.
Answer: \($10\) per hour
Question: A store sells oranges at a constant rate. The graph below shows the relationship between the number of oranges and the total cost in dollars. What is the unit rate (cost per orange)?
Answer: \($0.50\) per orange
Question: This table shows the relationship between gallons of water used and plants watered at a greenhouse. If this relationship is graphed with gallons on the x-axis and plants on the y-axis, what does the point \((1, 4)\) represent?
| Gallons of Water | Plants Watered |
|---|---|
| 3 | 12 |
| 5 | 20 |
| 7 | 28 |
Answer: 1 gallon waters 4 plants
Question: Which scenario best matches the graph showing hours on the x-axis and total distance in kilometers on the y-axis?
Why it works: Using two points on the line, e.g., \((1,2)\) and \((2,4)\): slope \(=\frac{4-2}{2-1}=2\) km/hr. Verify with \((4,8)\): \(8\div 4=2\) km/hr. The constant rate is \(2\) km per hour.
Answer: Traveling at 2 km per hour
Graphing Ratios becomes easier when students connect the question to a model, use clear steps, and explain why the answer fits.
Read the scale before reading the answer.
Use scratch paper. The explanation matters more than simply seeing whether the choice is correct.
The coordinate plane shows ordered pairs representing the ratio of hours worked to dollars earned. Based on the graph, how much is earned per hour?
Answer C: \($10\) per hour
Look at the y-axis label: each tick on the dollars axis stands for $10. The point \((1,1)\) means \(1\) hour and \(1\times10=$10\), so the rate is \($10\) per hour.
A ratio table shows the relationship between hours and miles traveled at a constant speed. Which graph correctly represents the data from this table? Looking at the coordinate plane below, which set of points matches this ratio?
Answer A: \((1,1), (2,2), (3,3)\)
The vertical axis is scaled in groups of \(50\) miles. So \(50\) miles is shown as \(1\), \(100\) miles as \(2\), and \(150\) miles as \(3\), giving the points \((1,1)\), \((2,2)\), and \((3,3)\).
A store sells oranges at a constant rate. The graph below shows the relationship between the number of oranges and the total cost in dollars. What is the unit rate (cost per orange)?
Answer D: \($0.50\) per orange
At the point where oranges \(= 2\), cost \(= $1\), giving a ratio of \($1 \div 2 = $0.50\) per orange. Or use any point: at \((8, 4)\), the rate is \($4 \div 8 = $0.50\) per orange.
This table shows the relationship between gallons of water used and plants watered at a greenhouse. If this relationship is graphed with gallons on the x-axis and plants on the y-axis, what does the point \((1, 4)\) represent?
Answer A: 1 gallon waters 4 plants
On the graph with gallons on the x-axis and plants on the y-axis, the point \((1, 4)\) means 1 gallon (x-value) corresponds to 4 plants watered (y-value). This shows the unit rate: 4 plants per gallon.
A student created this ratio table about babysitting rates: If the student plots \((0, 0)\), \((2, 24)\), \((4, 48)\), and \((6, 72)\) on a coordinate plane, which statement about the points is true?
Answer B: The points form a diagonal line through the origin
All these points lie on a straight line passing through \((0, 0)\) because the ratio is constant (\(12\) dollars per hour). Proportional relationships always graph as straight lines through the origin.
Which scenario best matches the graph showing hours on the x-axis and total distance in kilometers on the y-axis?
Answer A: Traveling at 2 km per hour
Using two points on the line, e.g., \((1,2)\) and \((2,4)\): slope \(=\frac{4-2}{2-1}=2\) km/hr. Verify with \((4,8)\): \(8\div 4=2\) km/hr. The constant rate is \(2\) km per hour.
A recipe uses a 3:2 ratio of flour to sugar. The graph below shows ordered pairs of this ratio. Which point should also be plotted to continue this ratio pattern?
Answer B: \((9, 6)\)
The ratio is 3:2 (flour:sugar). Points on this line satisfy \(\text{sugar} = \frac{2}{3} \times \text{flour}\). Check: \((3, 2)\) gives \(2 = \frac{2}{3}(3)\) ✓; \((6, 4)\) gives \(4 = \frac{2}{3}(6)\) ✓; \((9, 6)\) gives \(6 = \frac{2}{3}(9)\) ✓. The other options do not maintain this ratio.
A ratio table shows the number of apples and oranges in a fruit basket. If this ratio is graphed with apples on the x-axis, what is the y-coordinate when the x-coordinate is 8?
Answer D: 20
The ratio is apples to oranges at 2:5. For every 2 apples, there are 5 oranges. When apples = 8, the multiplier is \(8 \div 2 = 4\). So oranges = \(5 \times 4 = 20\).
The graph shows the relationship between minutes studied and test score points earned. What is the unit rate (points per minute)?
Answer B: \(0.5\) points per minute
Using the point \((2, 1)\): unit rate \(= 1 \div 2 = 0.5\) points per minute. Or from \((10, 5)\): unit rate \(= 5 \div 10 = 0.5\) points per minute.
A cell phone plan charges a constant rate per gigabyte of data used. The graph below shows this relationship. What does the point \((1, 3)\) represent?
Answer A: 1 GB costs 3 dollars
On a graph with gigabytes on the x-axis and cost on the y-axis, the point \((1, 3)\) means 1 gigabyte (x-value) costs $3 dollars (y-value).
A fruit vendor sells bananas at a constant price. This table shows the relationship: Which equation represents the relationship between pounds (\(p\)) and cost (\(c\))?
Answer B: \(c = 0.75p\)
For 2 pounds, cost is \($1.50\), so the unit rate is \(1.50 \div 2 = 0.75\) per pound. Check: \(c = 0.75p\) gives \(c = 0.75(2) = 1.50\) ✓ and \(c = 0.75(4) = 3.00\) ✓.
