Mathematics
Grade 4
15 min
Write equations of hyperbolas in standard form using properties
Write equations of hyperbolas in standard form using properties
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1
Introduction & Learning Objectives
Learning Objectives
Identify the 'center time' (h) and 'center place' (k) from a story problem.
Find the 'morning steps' (a) and 'afternoon steps' (b) in a time story.
Determine if a time story is 'Sun-First' (horizontal) or 'Moon-First' (vertical).
Choose the correct standard form equation for a 'Sun-First' or 'Moon-First' story.
Correctly write a full time hyperbola equation by placing h, k, a, and b into the standard form.
Explain that the equation tells a story about two parts of a day moving away from a center point in time.
Have you ever noticed how the morning and the afternoon seem like two separate adventures with lunchtime in the middle? ⏳ Let's learn how to write a special math s...
2
Key Concepts & Vocabulary
TermDefinitionExample
Time HyperbolaA special shape that looks like two curves opening away from each other. We use it to tell a math story about two parts of time, like morning and afternoon.Imagine a story about a trip to the zoo. The morning part is visiting the lions, and the afternoon part is visiting the monkeys. The hyperbola shows both parts of the trip.
Center (h, k)The middle point of our time story. 'h' is the middle time on the clock, and 'k' is the middle place or activity number.If lunchtime is at 12 o'clock (h=12) at the main picnic area (k=1), the center is (12, 1).
VerticesThe most important moments in our time story. They are the start of the morning fun and the start of the afternoon fun.If the fun starts 2 hours before lunch and 2 hours after l...
3
Core Formulas
Sun-First Story Equation (Horizontal)
\frac{(Time - h)^2}{a^2} - \frac{(Activity - k)^2}{b^2} = 1
Use this rule when the story is mainly about time moving forward, like the sun moving across the sky. The 'Time' part comes first. 'h' is the center time, 'a' is the morning steps, 'k' is the center activity, and 'b' is the afternoon steps.
Moon-First Story Equation (Vertical)
\frac{(Activity - k)^2}{b^2} - \frac{(Time - h)^2}{a^2} = 1
Use this rule when the story is mainly about changing activities, like climbing up and down a big play structure. The 'Activity' part comes first. 'k' is the center activity, 'b' is the afternoon steps, 'h' is the center time, and 'a' is the morning...
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Challenging
Story: 'On Tuesday, a 6-hour science fair project was centered on 3 o'clock at Table #4. The morning research took 2 hours. In the afternoon, there were 5 judges to impress. The project won 1st prize.' This is a Sun-First story. Find the correct equation.
A.\frac{(Time - 3)^2}{2^2} - \frac{(Activity - 4)^2}{5^2} = 1
B.\frac{(Time - 3)^2}{6^2} - \frac{(Activity - 4)^2}{5^2} = 1
C.\frac{(Time - 6)^2}{2^2} - \frac{(Activity - 1)^2}{5^2} = 1
D.\frac{(Time - 3)^2}{2^2} - \frac{(Activity - 4)^2}{1^2} = 1
Challenging
Which story below best matches the equation \frac{(Activity - 5)^2}{2^2} - \frac{(Time - 12)^2}{4^2} = 1 ?
A.day centered at 12 o'clock and Activity #5. The morning was 2 hours long and there were 4 afternoon choices. It was a Sun-First day.
B.day centered at 5 o'clock and Activity #12. The morning was 4 hours long and there were 2 afternoon choices. It was a Moon-First day.
C.day centered at 12 o'clock and Activity #5. The morning was 4 hours long and there were 2 afternoon choices. It was a Moon-First day.
D.day centered at 12 o'clock and Activity #5. The morning was 4 hours long and there were 2 afternoon choices. It was a Sun-First day.
Challenging
A story about a garden is centered at 10 o'clock (h) and Row #8 (k). The morning part of the day was 3 hours long (a). The equation for the day is \frac{(Time - 10)^2}{3^2} - \frac{(Activity - 8)^2}{6^2} = 1. What must the value of 'b' (afternoon steps) be?
A.3
B.6
C.8
D.10
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