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High School: Algebra

Creating Equations HSA-CED.A.4

4. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm's Law V = IR to highlight resistance R.

Students can rearrange formulas until their pencils whittle down to toothpicks, but it won't help them one bit if the formulas they use won't enable them actually solve the problem. That means students should be able to match commonly encountered formulas to context in word problems as well as rearranging them to solve for whatever value they want.

Matching Formulas to Create Equations

In addition to being able to translate word problems into equations, students also need to be able to identify when a common formula is needed for the given context. These are most commonly geometric formulas (like perimeter, area, or volume of various shapes) or physical formulas (such as F = ma, p = mv, V = IR, v = dt, KE = ½mv2, or GPE = mgh).

It is important to note that this assumes that students are already familiar with the relevant formulas from previous learning. Students who are not already familiar with the formulas need to be supported in understanding them before they will be able to match them to contexts. Matching formulas may be treated as a variation from the creating equations process. When attempting to write down the equality or inequality, students will notice that there isn't enough information. A general relationship might be implied, but no specific equality or inequality is described. What ever will they do?

Well, students need to identify the formula that describes that relationship. They can look for clues such as appropriate units (volume, for example, is always in cubic units). Once the correct formula is matched to the contextual relationship, the students can continue solving the problem as usual.

Rearranging Formulas

Once a formula is written algebraically, students can manipulate it however they want (within mathematical reason). The process for rearranging it is identical to the process of rearranging any equation or system of equations. It involves simplifying expressions and solving equations, which students should already know how to do.

The best way to rearrange questions when looking for a particular value is to isolate that value. Students should rearrange the equations so that the desired value is on one side of the equal sign, and there's a whole big mess of stuff on the other. That way, they'll be able to plug in the values they know and end up with whatever they want equals some number.

Drills

1. The Addams family builds a rectangular deck outside their creepy mansion that is 12 feet long and 8 feet wide. Which of these formulas could be utilized to calculate the area of the deck?

A = lw

The Addams family's deck is a rectangle with the specifications 12 feet long and 8 feet wide. Area A may be calculated as lw, where l = length and w = width.

2. The Addams family builds a rectangular deck outside their creepy mansion that is 12 feet long and 8 feet wide. What is the area of the deck?

96 ft2

The equation A = lw may be used to find the area of the Addams family's deck. Substitute 12 ft for length l and 8 ft for width w to get A = 12 ft × 8 ft. This gives A = 96 ft2.

3. The Addams family also wants to build a hot tub with a radius of 3 ft. The depth is 4 ft in the middle. All around the edge, a step extends 1 ft into the tub at a depth of 2 ft. Which equation could be used to calculate the volume of water that the hot tub will contain?

V = πr2h

Since the Addams family's hot tub is in the shape of a cylinder, we need to calculate the volume of a cylinder, essentially. The formula for the volume of a cylinder requires two pieces of information: r, the radius of the cylinder's base and h, the height of the cylinder itself. The volume for a cylinder is expressed by V = πr2h.

4. The Addams family also wants to build a hot tub with a radius of 3 ft. The depth is 4 ft in the middle. All around the edge, a step extends 1 ft into the tub at a depth of 2 ft. How can the formula for the volume be manipulated to find the volume of this hot tub specifically?

V = πr12h1 + πr22h1

The hot tub can be thought of as two joined cylinders: the first is the top portion from the surface of the water to the top of the step and the second is from the step to the bottom of the pool. If we add these two volumes together, can find the volume of the hot tub. That means the addition of two cylinders, each with its own height and radius. The formula that properly expresses this is V = πr12h1r22h1.

5. The Addams family also wants to build a hot tub with a radius of 3 ft. The depth is 4 ft in the middle. All around the edge, a step extends 1 ft into the tub at a depth of 2 ft. What is the volume of the hot tub in ft3?

81.7 ft3

Since the Addams family's hot tub can be thought of as two cylinders, we can substitute the values for each one's radius and height. The first cylinder, from the surface of the hot tub down to the first step, has a radius of 3 ft and a height of 2ft. That gives V1 = π × (3 ft)2 × 2 ft for the first cylinder. The second cylinder has a radius of 3 ft – 1 ft = 2 ft and a height of 4 ft – 2 ft = 2 ft. If we substitute these values, we bet V2 = π × (2 ft)2 × 2 ft. Adding these two volumes gives V = 26π ≈ 81.7 ft3.

6. Lurch and the Thing are playing a game of baseball together. The Thing throws the baseball at a speed of 50 meters per second. If the kinetic energy of the baseball is 175 Joules, which equation correctly expresses the value of the baseball's mass?

Now that we know which equation to use, we need to rearrange it to solve for the value we want. We have  and we want to find the value of m. To do that, we can start by multiplying both sides of the equation by 2. This give 2KE = mv2. Then we can divide the equation by v2, which will give us the final equation of .

7. Lurch and the Thing are playing a game of baseball together. The Thing throws the baseball at a speed of 50 meters per second. If the kinetic energy of the baseball is 175 Joules, what is the mass of the baseball in kilograms?

0.14 kg

Equipped with the equation , we can now substitute in the values we know and solve for m. Since KE is 175 Joules and the speed v equals 50 meters per second, we will have . This is the same as . Plugging this into a calculator, we end up with m = 0.14 kilograms.

8. If the baseball that the Thing throws travels at 50 m/s and reaches Lurch after 2.3 seconds, which formula can be used to find the distance the baseball traveled?

We're looking for an equation that relates speed (or velocity), time, and distance. The problem is that all the answer choices relate time t, distance d, and speed v to each other in different ways. We need to find the correct relationship. Using our knowledge that speed is the amount of distance covered in a given length of time (usually in units of miles per hour, meters per second, and so on), we can deduce that the speed equals distance over time. That leads us to our answer of .

9. If the baseball that the Thing throws travels at 50 m/s and reaches Lurch after 2.3 seconds, which equation correctly expresses the value of the distance the baseball traveled?

d = vt

Now that we know which equation to use, we need to rearrange it to solve for the value we want. We want the distance d, so we can rearrange the equation  by multiplying both sides by t. If we do that, we'll end up with the equation tv = d, or d = tv.

10. If the baseball that the Thing throws travels at 50 m/s and reaches Lurch after 2.3 seconds, what distance does the baseball travel in meters?