Occasion for an Equation

An Occasion for An Equation

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Occasion for an Equation Header

Grade 6-8 | 10 (45 min) Classes

In this unit, students will use algebraic equations and expressions to make sense of what effect different masses have on the velocity and acceleration of different model rockets. They will answer the question, “How are algebraic equations used to calculate model rocket acceleration?” and will learn how to calculate altitude, velocity, and acceleration using basic algebraic equations of motion.

 

For their final assessment, students will create a bar graph of the average acceleration for each of the three rockets (Alpha, Generic E2X, and Viking) and reflect on the effect of the mass of the rocket on acceleration.

Standards

Targeted Performance Expectation(s):

Common Core Standards - Math

GRADE 6

Write and evaluate numerical expressions involving whole-number exponents.

Write, read, and evaluate expressions in which letters stand for numbers.

Write expressions that record operations with numbers and with letters standing for numbers.

Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity.

Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations).

Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.

Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.

Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.

Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables and relate these to the equation.

GRADE 7

Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.

Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies.

Use variables to represent quantities in a real-world or mathematical problem and construct simple equations and inequalities to solve problems by reasoning about the quantities.

Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach.

GRADE 8

Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways.

Solve linear equations in one variable.

Common Core Standards - ELA

Follow precisely a multistep procedure when carrying out experiments, taking measurements, or performing technical tasks.

Vocabulary

ACCELERATION

How much an object’s velocity changes over a certain time, or the rate at which an object increases its speed

ALGEBRAIC EXPRESSION

A mathematical phrase that contains ordinary numbers, variables (like x or y) and mathematical operations

ALGEBRAIC EQUATION

A mathematical sentence with an equal sign where the value of the variable cannot be changed

APOGEE

The peak altitude or highest point of a rocket’s flight

CONSTANT

A value or number that does not change

DISTANCE

How far something has gone

DISPLACEMENT

How far something has gone from where it started (includes direction)

LIKE TERMS

When terms have the same algebraic factors and are raised to the same power

MASS

The amount of matter in an object

SPEED

The rate of change of position or how far something went in a certain amount of time

TERM

A number, variable, or quotient of numbers and variables in an algebraic expression or equation

THRUST

The propulsive force that moves something forward

UNLIKE TERMS

When terms have different algebraic factors or are raised to different powers

VARIABLE

A placeholder for a quantity that may change in an algebraic expression or equation

VELOCITY

The speed and direction of an object

Teacher Background

Algebraic Expressions and Equations

Algebraic expressions are created when addition, subtraction, multiplication, or division are operated upon by a variable. Variables are any symbol that does not have a fixed value, meaning the number inputted for the variable can change, whereas constants are symbols that have a fixed value, meaning the number cannot change. An algebraic expression combines variables and constants using the operational symbols (+,-,x, ÷) but does not include an equal sign.

 

Algebraic expression: 2x + 5

 

Algebraic equations, on the other hand, include an equal sign and are composed of algebraic expressions. An easy way to remember this is that algebraic EQUAtions have EQUAL signs. For example,

 

Algebraic equation: 2x + 5 = 11

 

Simplifying Algebraic Expressions

To simplify an algebraic expression, combine like terms (ones that have the same variables and are raised to the same power). For example, 3×2 and 4×2 are like terms because they have the same variable (x) and are raised to the same power (2), but x and x2 are not like terms even though they have the same variable, because they are not raised to the same power.

 

Types of Algebraic Expressions

There are many types of algebraic expressions and the key to understanding them is to understand the building block. A monomial is an expression that has a single term, with variables and coefficients. An example of a monomial expression would be 3xy because it is a single term, with 2 variables (x,y) and a coefficient (3). An expression with two terms is called a binomial (4x – 2y2) and one with three terms is a trinomial (4x-3y+z). Binomials, trinomials, and expressions with more than three terms can all be classified as polynomials.

 

Solving Algebraic Equations

To solve an equation, place the variables on one side of the equal sign and the constants on the other side. Make sure the variable stands alone such as x = __ or a = ___ and not 2x = ___ or 5y = _____.

Speed, Velocity, and Acceleration

Speed is the distance traveled divided by the amount of time it took to move. Distance is how far the object traveled. Velocity is similar but is a vector quantity (has magnitude and direction), so it uses displacement which is the distance the object has traveled from where it started. Velocity is displacement divided by time. This is an important distinction to make for students.  
Speed = distance/time
S = d/t
Velocity = displacement/time
V = d/t

Acceleration is also a vector as its formula is change in velocity divided by change in time. This is written as “delta v over delta t” or final velocity minus initial velocity divided by the change in time.

How does a Rocket Fly?

Students should be familiar with how a model rocket launches and all safety procedures that should be followed. The safety requirements can be found in the Model Rocket Safety Code of the National Association of Rocketry (NAR).

A Typical Model Rocket Flight

Thrust is the upward force that makes a rocket move off the launch pad. This is a demonstration of Newton’s Third Law of Motion: “For every action there is an equal and opposite reaction.” The action of the gas escaping through the engine nozzle leads to the reaction of the rocket moving in the opposite direction. The casing of a model rocket engine contains the propellant. At the base of the engine is the nozzle which is made of a heat-resistant, rigid material. The igniter in the rocket engine nozzle is heated by an electric current supplied by a battery-powered launch controller. 

The hot igniter ignites the solid rocket propellant inside the engine which produces gas while it is being consumed. This gas causes pressure inside the rocket engine, which must escape through the nozzle. The gas escapes at a high speed and produces thrust. Located above the propellant is the smoke-tracking and delay element. Once the propellant is used up, the engine’s time delay is activated.

 

The engine’s time delay produces a visible smoke trail used in tracking, but no thrust. The fast-moving rocket now begins to decelerate (slow down) as it coasts upward toward peak altitude (apogee). The rocket slows down due to the pull of gravity and the friction created as it moves through the atmosphere. The effect of this atmospheric friction is called drag. When the rocket has slowed enough, it will stop going up and begin to arc over and head downward. This high point or peak altitude is the apogee. At this point the engine’s time delay is used up and the ejection charge is activated. The ejection charge is above the delay element. It produces hot gases that expand and blow away the cap at the top of the engine. The ejection charge generates a large volume of gas that expands forward and pushes the recovery system (parachute, streamer, helicopter blades) out of the top of the rocket. The recovery system is activated and provides a slow, gentle and soft landing. The rocket can now be prepared for another launch.

 

To summarize, the steps of the Flight Sequence of a Model Rocket are:

  1. Electrically ignited model rocket engines provide rocket liftoff.
  2. Model rocket accelerates and gains altitude.
  3. Engine burns out and the rocket continues to climb during the coast phase.
  4. Engine generates tracking smoke during the delay/coast phase.
  5. Rocket reaches peak altitude (apogee). Model rocket ejection charge activates the recovery system.
  6. Recovery systems are deployed. Parachutes and streamers are the most popular recovery systems used.
  7. Rocket returns to Earth.
  8. Rocket touchdown! Replace the engine, igniter, igniter plug and recovery wadding. Rocket is ready to launch again!

Materials

Each Student Needs:

Student Design Portfolio

Safety Goggles

Scissors

Calculator

The Class Needs:

Unit Plan

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Student Portfolio

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Slippery Slope

Slippery Slope

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Unit Plan
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Slippery Slope Unit Plan

Grade 6-8 | 10 (45 min) Classes

In this unit students will answer the question: How is slope represented in a model rocket launch?

 

Students will learn about the different types of slope and practice calculation methods to find slope from a graph. Students will then analyze launch data to identify and calculate where a rocket has a positive, negative and/or neutral slope on its flight path.

 

For their final assessment students will be creating a visual of the four types of slope that shows how slope is seen on a model rocket’s flight path.

Standards

Targeted Performance Expectation(s):

Common Core Standards - Math

GRADE 7

Recognize and represent proportional relationships between quantities.

Decide whether two quantities are in a proportional relationship.

Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. 

GRADE 8

Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear.

Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.

High School

For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship

Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.

Vocabulary

APOGEE

The peak altitude or highest point of a rocket’s flight

HORIZONTAL

Goes left to right, parallel to the horizon

VERTICAL

Goes up and down, perpendicular to the horizon

PAYLOAD

The object being carried by an aircraft or launch vehicle

RATIO

A comparison of two or more numbers or measurements

SLOPE

Shows the rate of change, steepness, and direction of a line

X-AXIS

Horizontal axis, on a graph it shows the dependent variable

Y-AXIS

Vertical axis, on a graph it shows the independent variable

Teacher Background

Slope

Slope measures the steepness of a line. A line that is steep will have a bigger slope than a line that is flatter. Slope is calculated as the ratio of the amount of vertical change to horizontal change. Basically, it shows a rate of change. This is often referred to as “rise over run.” The rise is the y-axis and the run is the x-axis.

There are four types of slope.

1. Positive

  • The line goes up from left to right.
  • The y-axis values are increasing as the x-axis values are increasing.
  • Example: The funnier the cat videos are that I post on Facebook, the number of likes goes up.

2. Negative
  • The line goes down from left to right.
  • The y-axis values are decreasing as the x-axis values are increasing.
  • Example: If I increase my time watching Netflix, the time I have left to study will decrease
3. Zero
  • The line is horizontal.
  • The y-axis values do not change as the x-axis values are increasing.
  • Example: If I stand in one place and don’t move, time will continue to increase but my distance travelled will not.
4. Undefined
  • The line is vertical.
  • The x-axis values do change as the y-axis values are increasing.
  • Example: An elevator can go up and down, but does not move horizontally.

How does a Rocket Fly?

Students should be familiar with how a model rocket launches and all safety procedures that should be followed. The safety requirements can be found in the Model Rocket Safety Code of the National Association of Rocketry (NAR).

A Typical Model Rocket Flight

Thrust is the upward force that makes a rocket move off the launch pad. This is a demonstration of Newton’s Third Law of Motion: “For every action there is an equal and opposite reaction.” The action of the gas escaping through the engine nozzle leads to the reaction of the rocket moving in the opposite direction. The casing of a model rocket engine contains the propellant. At the base of the engine is the nozzle which is made of a heat-resistant, rigid material. The igniter in the rocket engine nozzle is heated by an electric current supplied by a battery-powered launch controller. 

The hot igniter ignites the solid rocket propellant inside the engine which produces gas while it is being consumed. This gas causes pressure inside the rocket engine, which must escape through the nozzle. The gas escapes at a high speed and produces thrust. Located above the propellant is the smoke-tracking and delay element. Once the propellant is used up, the engine’s time delay is activated.

 

The engine’s time delay produces a visible smoke trail used in tracking, but no thrust. The fast-moving rocket now begins to decelerate (slow down) as it coasts upward toward peak altitude (apogee). The rocket slows down due to the pull of gravity and the friction created as it moves through the atmosphere. The effect of this atmospheric friction is called drag. When the rocket has slowed enough, it will stop going up and begin to arc over and head downward. This high point or peak altitude is the apogee. At this point the engine’s time delay is used up and the ejection charge is activated. The ejection charge is above the delay element. It produces hot gases that expand and blow away the cap at the top of the engine. The ejection charge generates a large volume of gas that expands forward and pushes the recovery system (parachute, streamer, helicopter blades) out of the top of the rocket. The recovery system is activated and provides a slow, gentle and soft landing. The rocket can now be prepared for another launch.

 

To summarize, the steps of the Flight Sequence of a Model Rocket are:

  1. Electrically ignited model rocket engines provide rocket liftoff.
  2. Model rocket accelerates and gains altitude.
  3. Engine burns out and the rocket continues to climb during the coast phase.
  4. Engine generates tracking smoke during the delay/coast phase.
  5. Rocket reaches peak altitude (apogee). Model rocket ejection charge activates the recovery system.
  6. Recovery systems are deployed. Parachutes and streamers are the most popular recovery systems used.
  7. Rocket returns to Earth.
  8. Rocket touchdown! Replace the engine, igniter, igniter plug and recovery wadding. Rocket is ready to launch again!

Engine Types

Rocket engines are governed by the National Association of Rocketry (NAR) and follow the same alphanumeric coding:

B6-4 Engine
  • Letters: Each successive letter has TWICE as much power as the previous letter (a B engine would have twice as much impulse power as an A engine).
  • First number: Average Thrust: This is a measure of how slowly or quickly the engine delivers its total energy. Engines with low numbers will have a weak thrust for a long duration. Engines with high numbers will have a high thrust over a short duration (high numbers are good for heavy rockets).
  • Second number: This number tells you the delay (in seconds). In other words, the time between the end of thrust and the ejection charge. The ejection charge is activated after the delay. It produces a quick puff of gas that pressurizes the inside of the body tube to push out the recovery system.
  • Color
    • Green: Single stage engineEngines in Hand
    • Purple: High performance single stage/final stage multistage engine
    • Red: Booster-stage engine
    • Blue: Special use engine

Materials

Each Student Needs:

Student Design Portfolio

Safety Goggles

Tape

Scissors

The Class Needs:

Unit Plan

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Student Portfolio

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Grade 6-8 | 10 (45 min) Classes

In this lesson, students will answer the question, how does gravitational potential energy transform to kinetic energy in a model rocket flight? The students will read a story about humans living on Mars in the year 2343 who wish to launch rockets to the moons of Mars. They will anticipate how the gravitational force on Mars will affect their exploration.

 

Students will hypothesize how the gravitational force on Mars will affect the gravitational potential energy of a model rocket. After a review of potential and kinetic energy, students will practice what they learned and build and test a rubber band rocket. They will then review the data and apply their learnings to build a model rocket in groups to analyze its transformation of energy.

 

After the flight, the students will combine their data into a class data chart. They will compute how their rockets would be affected if they were being launched from Mars.
The student’s final product will be to complete a Claims- Evidence- Reasoning writing piece supporting their final conclusions. A traditional multiple-choice quiz is included for use if desired.

 

In 1949, a clandestine group of government scientists met at a secret airbase in Nevada to form Project Star Hopper. The goal was to produce a fast and maneuverable piloted vehicle to compete with the unidentified objects commonly referred to as “flying saucers.” With the nation’s best engineers on the task, plans were drawn up for a sleek and functional atomic-powered vessel that could be launched quickly to intercept the aggressors. The result was the Star Hopper – the world’s first interplanetary spacecraft. A small fleet was constructed and tested, and by 1955, they were ready to protect the skies from alien invaders. Or so we were told…

 

It was absolutely crucial that the Star Hoppers were able to land accurately on the stars if they were to be successful in identifying the aliens. The engineers at Project Star Hopper need your help to determine how to make the rocket land accurately for the pilots navigating space. The focus of your research will be on the recovery system and adjusting the length of the streamer

Standards

Targeted Performance Expectation(s):

Next Generation Science Standards (NGSS)

MS-PS3-2

Develop a model to describe that when the arrangement of objects interacting at a distance changes, different amounts of potential energy are stored in the system.

MS-PS3-5

Construct, use, and present arguments to support the claim that when the kinetic energy of an object changes, energy is transferred to or from the object.

MS-ETS1-2

Evaluate competing design solutions based on jointly developed and agreed-upon design criteria using a systematic process to determine how well they meet the criteria and constraints of the problem.

MS-ETS1-3

Analyze data from tests to determine similarities and differences among several design solutions to identify the best characteristics of each that can be combined into a new solution to better meet the criteria for success.

Common Core Standards - English

CC SS.ELA-LITERACY.RST.6-8.3

Follow precisely a multistep procedure when carrying out experiments, taking measurements, or performing technical tasks.

Common Core Standards - Math

CCSS.MATH.CONTENT.6.EE.B.6

Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.

CCSS.MATH.CONTENT.6.EE.B.8

Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.

CCSS.MATH.CONTENT.6.RP.A.1

Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.

CCSS.MATH.CONTENT.6.RP.A.3

Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.

CCSS.MATH.CONTENT.7.RP.A.2

Recognize and represent proportional relationships between quantities.

Vocabulary

APOGEE

The peak altitude or highest point of a rocket’s flight.

GRAVITY POTENTIAL ENERGY

Energy that is stored due to the gravitational force of the Earth, dependent on the object’s mass and height, and measured in Joules.

GRAVITY

Force that pulls everything down toward the center of the Earth.

STARTER

Device used to ignite a rocket engine

JOULES

Unit of work or energy, abbreviated as J

KINETIC ENERGY

Energy of motion that is dependent on mass and velocity, measured in Joules

DIRECT RELATIONSHIP

Relationship between two variables such that when one variable changes, the second variable changes in the same manner.

INVERSE RELATIONSHIP

Relationship between two variables such that when one variable changes, the second variable changes in the opposite manner.

Teacher Background

Energy Transformation

Potential energy is energy that is stored in an object and is dependent on its position. While there are several different types of potential energy, Gravitational Potential Energy is the focus in this lesson. Gravitational Potential Energy (GPE) is the energy that is stored because of the object’s height. It is a result of the gravitational force of the Earth. GPE is calculated by multiplying the mass of the object by the gravitational force (on Earth, this is 9.8 m/s2) by the height (or distance that the object can fall). It is written:

 

 

Gravitational Potential Energy = m x g x h
m = mass (kg); h = distance the object can fall (m); g = acceleration due to gravity (9.8 m/s2) 

 

Thus, a heavy object will have a greater GPE than a lighter object. The higher the object is (in other words, the farther away the object is from the center of Earth), the greater the GPE. The unit of measurement for GPE is the Joule, abbreviated J.


Since GPE depends on gravitational force, an object on a planet other than Earth will have a different GPE. As an example, the gravitational force on Mars is 3.7 m/s2. An object on Mars would have less GPE compared to its GPE on Earth, assuming the same mass and distance from the planets.


In a model rocket, the transformation of energy is related to the momentum of the rocket. The Law of Conservation of Energy states that energy is neither created nor destroyed, it is transformed. In a model rocket, the GPE is transformed into kinetic energy. Kinetic Energy (KE) is the energy of motion. KE is calculated by multiplying two variables – mass and velocity. The equation for KE is as follows:


Kinetic Energy = 1/2 x m x v2
m = mass (kg); v = velocity (m/s)

KE is a scalar quantity. Since it does not have direction, KE is described in magnitude. The unit of measurement for KE, like the unit of measurement for PE, is the Joule, abbreviated J.

 

Energy in Rocketry

Review the steps of the rocket flight sequence, alongside the energy conversion:

Step Flight Sequence Energy Conversion
1 Electrically ignited model rocket engines provide rocket liftoff. Since there is nothing moving, the rocket’s KE = 0 and the GPE =0 since its height is 0
2 Model rocket accelerates and gains altitude.
3 Engine burns out and the rocket continues to climb during the coast phase. The rocket is gaining both speed and height, so GPE and KE are both increasing. Right before it coasts, the KE is the highest
4 Rocket reaches peak altitude (apogee). Model rocket ejection charge activates the recovery system. The rocket has the greatest height therefore the greatest GPE and there is no KE.
5 Recovery system is deployed. Parachutes and streamers are the most popular recovery systems used. Rocket returns to Earth. As the rocket falls, GPE is converted to KE.
6 Rocket touchdown! Right before landing the KE is greater than the GPE

Using the Estes Altimeter

Altimeter Functions: The altimeter will record the highest point that the rocket reaches. This is called apogee.

 

  1. To turn on and use the altimeter,
    1. Install the battery.
    2. Using a pen or screwdriver, slide the switch to ON.
    3. The Altimeter will display 0 feet or meters. (When not in use, always turn it off.)
    4. With the altimeter on, press and hold the button until the required function is displayed, then release it to provide access to that function.
  2. To change units from ft to m, press and hold button until UNIT is displayed, and then release the button. Press button until 0000 is displayed and altimeter is ready for flight.
  3. To clear the altimeter from the previous launch, press and hold the button until 0000 is displayed, then release. This will clear the display, but the altitude will still be stored in memory.  Data for up to 10 flights will be saved.
  4. To view recorded flights, press and hold the button until REC0 is displayed then released. Press and release the button quickly while in the REC0 mode to view each of the 1-10 recorded flights in order of “last flight first”.
  5. To exit REC0 mode, press and hold the button until the REC0 display starts to flash and release the button. Press the button until 0000 is displayed.
  6. To clear flight data memory, press and hold the button until the CLER mode is displayed and then released. Press and release once more to clear the memory of all the launch data. (Be sure you have written it down first!)

Installing the Altimeter: Attach the altimeter to the base of the nose cone with the clip. If launching the altimeter inside a rocket body tube, pack recovery wading and the parachute with sufficient room for the altimeter to fit easily.

Materials

Each Student Needs:

Student Design Portfolio

Safety Goggles

Clipboard

Calculator

Scissors

Ruler

Meter Stick

Rubber Band

Bamboo Skewer

Straw

Masking Tape

Pennies

Permanent Marker

Stopwatch (1 per group)

The Class Needs:

Potential and Kinetic Energy Slide Presentation (Available as part of the Unit Plan Download)

Green Eggs Rocket Kit

C11-3 Engines

Lifetime Launch System

Estes Altimeter

Glue

Exacto Knife

Camera (optional)

Unit Plan

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