# High School: Algebra

### Creating Equations HSA-CED.A.2

2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

This standard has two significant components. The first is translating word problems into equations with two or more variables. The more the merrier. Well, maybe not in this case. Translating word problems to create simple equations with two or more variables is not that different conceptually from creating equations with one variable.

The main difference is that more complicated mathematical relationships such as systems of equations, functions, and proportions may develop (along with nausea, headaches, and spontaneous yodeling). In any case, this aspect of this standard should be taught with the previous one.

The second component is creating graphs of equations on coordinate axes, which incorporates multiple skills such as visual perception, interpreting data, and synthesizing information. Such graphs relate to equations with multiple equations by relating one variable to another.

Take lines, for example. In the form y = mx + b, we can look at either x or y and any defined value for x will give us a defined value for y, and vice versa. Graphs can help visualize these relationships between variables and facilitate the connection of equations to the graphs that represent them. Yearnin' for more graphin'? Don't worry. There'll be more down the line.

#### Drills

1. The ratio of nut chocolates to cordials in an assortment box is 2:1. Which equality describes the contents of an assortment box?

Ratio of nut chocolates to cordials = 2:1

The relationship 2:1 is already expressed as a ratio; so, this is just a simple equality of ratio of nut chocolates to cordials = 2:1.

2. The ratio of nut chocolates to cordials in an assortment box is 2:1. Which relation describes the contents of an assortment box?

The key word "ratio" indicates division. The two ratios involved are nut chocolates to cordials and 2:1. Dividing in each case gives .

3. The ratio of nut chocolates to cordials in an assortment box is 2:1. Which relation describes the contents of an assortment box? (Let n = number of nut chocolates, and c = number of cordials.)

Substituting in variables for the nut chocolates (n) and the cordials (c) give . The  simplifies to 2. This gives the equation , where n = number of nut chocolates, and c = number of cordials.

4. The ratio of nut chocolates to cordials in an assortment box is 2:1. If there are 5 cordials in an assortment box, how many nut chocolates are there?

10

The equation  describes the ratio of nuts to cordials in an assortment box. The assortment box has 5 cordials so, c = 5. Substituting in, this gives . Multiplying both sides by 5 gives n = 2 × 5 = 10. There are 10 nut chocolates in the box.

5. The Fuzzlegump School for Gifted and Not-So-Gifted Children requires that three chaperones go on any field trip. More chaperones are required if there are more than 30 students on the trip. For every 12 additional students, another chaperone is required. Which of these inequalities describes the number of chaperones required?

Chaperones required ≥ three chaperones for 30 students and another chaperone for every 12 additional students

There are two different requirements for chaperones. Three are required for any field trip with up to 30 students. Then, for every 12 students over the 30-student mark, an additional chaperone is required. These two requirements need to be combined to calculate the minimum number of chaperones for a trip. This gives the inequality chaperones requiredthree chaperones for 30 students and another chaperone for every 12 additional students.

6. The Fuzzlegump School for Gifted and Not-So-Gifted Children requires that three chaperones go on any field trip. More chaperones are required if there are more than 30 students on the trip. For every 12 additional students, another chaperone is required. Which relation describes the minimum number of chaperones required?

Chaperones required ≥ three chaperones +

The inequality is chaperones required ≥ three chaperones for 30 students and another chaperone for every 12 additional students. The "and" indicates addition to link three chaperones with the additional chaperone for every 12 students over the initial 30. The key is then recognizing that the, "every 12 additional students," holds only for more than 30 students. The key phrase, "more than" indicates subtraction to get how many more than 30 students there are. And the key words, "for every" indicates division. These give the relation chaperones requiredthree chaperones.

7. The Fuzzlegump School for Gifted and Not-So-Gifted Children requires that three chaperones go on any field trip. More chaperones are required if there are more than 30 students on the trip. For every 12 additional students, another chaperone is required. Which equation describes the minimum number of chaperones required? (Let c = chaperones and s = students.)

The relation forming the basis for the equation is chaperones requiredthree chaperones. Translating in the variables c = chaperones and s = students gives the equation .

8. The Fuzzlegump School for Gifted and Not-So-Gifted Children requires that three chaperones go on any field trip. More chaperones are required if there are more than 30 students on the trip. For every 12 additional students, another chaperone is required. If 60 students attend the field trip, how many chaperones are required?

6

The equation  describes the relationship between number of chaperones (c) and number of students (s). Substituting in 60 students with s = 60 gives . Simplifying the calculation gives  and then c ≥ 3 + 2.5 so c ≥ 5.5. Since there can't be half a chaperone (unless you count robots), they need 6 chaperones on the trip.

9. Which graph represents x = 1?

We need our x value to be 1 regardless of the y value. For any and every y value we choose, we should have x = 1. No matter where we travel along the y-axis, we'll always end up moving one unit to the right of it and drawing a point there. This creates a perfectly vertical line that intersects with the x-axis at 1.

10. Which graph represents y = 2x + 3?