Lesson 7.1.4
HW: 7-45 to 7-50
Learning Target: Scholars will learn multiple ways to solve equations with fractional coefficients and decimal coefficients.
In this lesson, you will develop strategies to solve a new type of equation: equations with coefficients that are fractions. Recall that coefficients are the numerical part of a term, such as the 5 in 5xand the in . You will choose between different strategies based on how the problem is represented, what diagram you draw, or whether you represent the situation with an equation. As you explore solving strategies today, keep these questions in mind:
· How can I represent it?
· How are the representations related?
· What is the best approach for this equation?
7-39. ARE WE THERE YET?
The Sutton family took a trip to see the mountains in Rocky Mountain National Park. Linda and her brother, Lee, kept asking, “Are we there yet?” At one point, their mother answered, “No, but what I can tell you is that we have driven 100 miles and we are aboutof the way there.”
Linda turned to Lee and asked, “How long is this trip, anyway?” They each started thinking about whether they could determine the length of the trip from the information they were given.
1. Explain how both Linda’s work and Lee’s work illustrate the situation described by their mother. What does x represent in each diagram?
2. What equation could represent this situation? Use the scale factor (multiplier) to represent this situation in an equation. Let x represent the total distance inmiles.
3. Is the answer going to be more or less than 100 miles? Explain your thinking.
7-40.Linda remarked to Lee, “When I look at my diagram, I see that the total distance is two-and-a-half times the distance we’ve driven.”
4. Do you agree? How can you add labels to Linda’s diagram to show 2times the distance?
5. How long is the trip? Show how you know.
7-41. Lee thinks what he knows about similar triangles can help.
6. Review problem 7-22, which involved similar triangles. How are the scale factors related?
7. How can the relationship of the scale factors between similar shapes help Lee? Find the missing scale factor in his diagram at right. That is, what could he multiply 100 by to solve for x?
8. Use the new scale factor to find x. Does your answer agree with the one you found in part(b) of problem 7-40?
7-42.As Linda and Lee were talking about the problem (from problem 7-39), their mother overheard and offered them another strategy: “Here is how I would start solving the problem.” She showed them the work at right.
·
1. “I see a Giant One!”exclaimed Lee. Where is the Giant One? Help rewrite the left side of the equation.
2. One way of making sense of is as 100 ÷ = ?.
Thiscould be read, “How many twofifths are in one hundred wholes?” With your team, find one way to explain how you could figure out how many two-fifths are in100.
7-43.Lee began to wonder how his diagram could help him solve his mother’s equation (from problem 7-42). He showed his work below.
3. How does Lee’s work relate to the similar-triangles diagram?What is equal to?
4. Finish Lee’s work to solve for x. Then check your solution.
7-44. Linda and Lee wondered how these new equation-solving strategies would work with different equations. They made up more equations to try to solve. Copy the equations below on your paper and solve each equation using one of the strategies from this lesson. How did you decide which strategy touse?
5.
6.
7.
7-45. Mr. Anderson’s doctor has advised him to go on a diet. He must reduce his caloric intake by 15%. He currently eats 2800 calories per day. Calculate his new daily caloric intake rate in two different ways, using two different multipliers.
7-46. Last month, a dwarf lemon tree grew half as much as a semi-dwarf lemon tree. A full-size lemon tree grew three times as much as the semi-dwarf lemon. Together, the three trees grew 27 inches. Write and solve an equation to determine how much each tree grew. Make sure you define your variable.
7-47. Enrique is saving money to buy a graphing calculator. So far he has saved $30. His math teacher told him he has saved 40% of what he will need. How much does the calculator
cost? 7-48.A principal made the histogram at right to analyze how many years teachers had been teaching at her school.
1. How many teachers work at herschool?
2. If the principal randomly chose one teacher to represent the school at a conference, what is the probability that the teacher would have been teaching at the school for more than 10 years? Write the probability in two different ways.
3. What is the probability that a teacher on the staff has been there for fewer than 5years?
7-49.Lue is rolling a random number cube. The cube has six sides, and each one is labeled with a different number 1 through 6.
4. What is the probability that he will roll a 5 or a 3 on one roll?
5. What is the probability that he will roll a 5 and then a 3 in two rolls?
6. What is the probability that he will roll a sum of 12 in two rolls?
7-50.This problemis a checkpoint for simplifying expressions. It will be referred to as Checkpoint 7A.
7. 4x2 + 3x– 7 + (–2x2)– 2x + (–3)
8. –3x2– 2x + 5 + 4x2– 7x + 6
Lesson 7.1.4
· 7-39.See below:
1. x represents the total distance; the 100 miles represents the distance they have traveled so far, and the represents the portion of the trip they have traveled so far.
2. x = 100; See the “Suggested Lesson Activity” notes for other possible answers.
3. The distance is going to be more than 100 miles, as they have already driven 100 miles and they are only part of the way there.
· 7-40.See below:
1. See diagram below.
2. 2.5(100) = 250 miles
· 7-41.See below:
1. The scale factors,andare reciprocals.
2.
3. 100·= 250 miles. Yes.
· 7-42.See below:
1. The fractionis 1, so the left side becomes 1x or x.
2. x = 250; see the “Suggested Lesson Activity” for ways to reason about this.
· 7-43.See below:
1. The is the multiplier to go from x to 100, and is the multiplier to scale 100 to x; therefore, the two multipliers are inverses of each other and their product is 1, leaving only one x on the left side.
2. Possible student work might be: x = (100)== 250 miles
· 7-44.See below:
1. x = 6
2. x = 42
3. x =−28
· 7-45.Method1: (0.15)(2800) = 420, 2800− 420 = 2380calories per day, Method2: 100%− 15% = 85%, (2800)(0.85) - 2380calories per day
· 7-46. x =growth of the semi-dwarf tree; x + 0.5x + 3x = 27; the dwarf grew 3 inches, the semi-dwarf grew 6inches, and the full-size tree grew 18 inches.
· 7-47.$75
· 7-48.See below:
1. 24
2. or≈ 29%
3.
· 7-49. See below:
1. P(5 or 3) ==
2. P(5 and 3) =·=
3. P(12) =
· 7-50.See below:
1. 2x2 + x− 10
2. x2− 9x + 11