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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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