lesson 5 skills practice problem solving strategies answer key

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• Lesson 5: Problem-Solving Strategies

Hi Everyone!

On this page you will find some material about Lesson 5. Read through the material below, watch the videos, and follow up with your instructor if you have questions.

Table of Contents

In this section you will find some important information about the specific resources related to this lesson:

• the learning outcomes,
• the section in the textbook,
• the homework,
• supporting video.

Learning Outcomes. (extracted from the textbook)

• State the four steps in the basic problem-solving procedure.
• Divide radical expressions.
• Solve problems using a diagram.
• Solve problems using trial and error.
• Solve problems involving money.
• Solve problems using calculation.

Topic . This lesson covers

Section 1.3: Problem-Solving Strategies

pages 29-35, ex. 1-6.

Practice Homework:

page 36: 19, 21, 25, 35, 38, 41, 43

ALEKS Assignment

Warmup Questions

These are questions on fundamental concepts that you need to know before you can embark on this lesson. Don’t skip them! Take your time to do them, and check your answer by clicking on the “Show Answer” tab.

Warmup Question 1

$$(2\sqrt{5a})(-4\sqrt{5a}).$$

Show Answer 1

$$(2\sqrt{5a})(-4\sqrt{5a})= -8\sqrt{(5a)^2}=-8(5a)=-40a$$

Warmup Question 2

$$(\sqrt 3-6)(\sqrt 3+6).$$

Show Answer 2

$$(\sqrt 3-6)(\sqrt 3+6)= (\sqrt 3)^2-6^2 = 3 – 36 = -33$$

If you are not comfortable with the Warmup Questions, don’t give up! Click on the indicated lesson for a quick catchup. A brief review will help you boost your confidence to start the new lesson, and that’s perfectly fine.

Need a review? Check Lesson 6 .

Quick Intro

This is like a mini-lesson with an overview of the main objects of study. It will often contain a list of key words, definitions and properties – all that is new in this lesson. We will use this opportunity to make connections with other concepts. It can be also used as a review of the lesson.

A Quick Intro to Division of Radicals and Rationalization

Key Words. Radicals, division of radicals, simplified form, rationalization, conjugate.

When dividing radical terms, the following property can be very helpful.

Division Property

$$\dfrac{\sqrt[n]a}{\sqrt[n]b}=\sqrt[n]{\dfrac{a}{b}}$$

If not, it may be necessary to rationalize the denominator. On Lesson 5 we listed three conditions for a radical expression to be in simplified form. The third one is:

There should be no radicals in the denominator of a fraction.

$\bigstar$ We gave $\dfrac{1}{\sqrt 2}$ as an example that fails this condition. To simplify it, we multiply both the numerator and the denominator by $\sqrt 2$.

$$\underbrace{\dfrac{1}{\sqrt 2}}_{\text{radical in the denominator}}=\dfrac{1}{\sqrt 2}\cdot\dfrac{\sqrt 2}{\sqrt 2}=\dfrac{1\cdot\sqrt 2}{\sqrt 2\cdot\sqrt 2}=\underbrace{\dfrac{\sqrt 2}{2}}_{\text{no radical in the denominator}}$$

$\bigstar$ The process of removing a radical from the denominator is called rationalization .

$\bigstar$ The key idea was to multiply the original denominator by another copy of it, since squaring eliminates the radical.

$$(\sqrt a)(\sqrt a) = (\sqrt a)^2=a.$$

$\bigstar$ But what if we have $\dfrac{1}{\sqrt 3 -6}$? Squaring $\sqrt{3}-6$ will not help (try it!). In the Warmup Question #2 we saw that

$$(\sqrt 3 -6)(\sqrt 3 +6)=-33$$

results in a number free of radical. So

$$\underbrace{\dfrac{1}{\sqrt 3 -6}}_{\text{radical in the denominator}}=\dfrac{1}{\sqrt 3 -6}\cdot\dfrac{\sqrt 3 +6}{\sqrt 3 +6}$$

$$=\dfrac{1\cdot(\sqrt 3 +6)}{(\sqrt 3 -6)\cdot(\sqrt 3 +6)}=\underbrace{-\dfrac{\sqrt 3 +6}{33}}_{\text{no radical in the denominator}}$$

$\bigstar$ This happens because the above product is a difference of squares

$$(a-b)(a+b)=a^2-b^2,$$

and squaring a single radical eliminates the radical.

$\bigstar$ We say that $a-b$ and $a+b$ are conjugates . So if the denominator is $\sqrt 3 -6$, we rationalize it by multiplying the numerator and the denominator by its conjugate $\sqrt 3+6$.

Video Lesson

Many times the mini-lesson will not be enough for you to start working on the problems. You need to see someone explaining the material to you. In the video you will find a variety of examples, solved step-by-step – starting from a simple one to a more complex one. Feel free to play them as many times as you need. Pause, rewind, replay, stop… follow your pace!

A description of the video

In the video you will see the following radical expressions.

• $\dfrac{3\sqrt x+1}{2\sqrt x}$
• $\dfrac{3\sqrt 5 + 1}{2\sqrt 7 -3}$

Try Questions

Now that you have read the material and watched the video, it is your turn to put in practice what you have learned. We encourage you to try the Try Questions on your own. When you are done, click on the “Show answer” tab to see if you got the correct answer.

Try Question 1

Simplify $$\dfrac{\sqrt{15}}{5\sqrt{20}}.$$

$$\dfrac{\sqrt{15}}{5\sqrt{20}} = \dfrac{\sqrt{15}}{5\sqrt{20}}\cdot \dfrac{\sqrt{20}}{\sqrt{20}}$$

$$= \dfrac{\sqrt{15}\sqrt{20}}{5\cdot 20} = \dfrac{\sqrt{15} \cdot 2\sqrt{5}}{5\cdot 20}$$

$$= \dfrac{\sqrt {5\cdot 15}}{5\cdot 10} = \dfrac{5\sqrt {3}}{50} = \dfrac{\sqrt 3}{10}$$

Try Question 2

Simplify $$\dfrac{3}{4+2\sqrt 5}.$$

$$\dfrac{3}{4+2\sqrt 5}=\dfrac{3}{4+2\sqrt 5}\cdot \dfrac{4-2\sqrt 5}{4-2\sqrt 5}$$

$$= \dfrac{3(4-2\sqrt 5)}{(4+2\sqrt 5)(4-2\sqrt 5)}= \dfrac{12-6\sqrt 5}{16-20}$$

$$=\dfrac{12-6\sqrt 5}{-4} = \dfrac{-6+3\sqrt 5}{2} $$

Try Question 3

Rationalize and simplify

$$\dfrac{1+3\sqrt 2}{1-\sqrt 2}+\sqrt 2.$$

Show Answer 3

$$\dfrac{1+3\sqrt 2}{1-\sqrt 2}+\sqrt 2= \dfrac{1+3\sqrt 2}{1-\sqrt 2}\cdot\dfrac{1+\sqrt 2}{1+\sqrt 2}+\sqrt 2$$

$$=\dfrac{(1+3\sqrt 2)(1+\sqrt 2)}{(1-\sqrt 2)(1+\sqrt 2)} +\sqrt 2 = \dfrac{1+\sqrt 2+3\sqrt 2 +3\cdot 2}{1-2}+\sqrt 2 $$

$$=\dfrac{7+4\sqrt 2}{-1}+\sqrt 2 = -7-4\sqrt 2+\sqrt 2 = -7-3\sqrt 2$$

You should now be ready to start working on the homework problems. Doing the homework is an essential part of learning. It will help you practice the lesson and reinforce your knowledge.

It is time to do the homework on WeBWork:

RationalizeDenominators

When you are done, come back to this page for the Exit Questions.

Exit Questions

After doing the WeBWorK problems, come back to this page. The Exit Questions include vocabulary checking and conceptual questions. Knowing the vocabulary accurately is important for us to communicate. You will also find one last problem. All these questions will give you an idea as to whether or not you have mastered the material. Remember: the “Show Answer” tab is there for you to check your work!

• What is the goal in rationalizing the denominator? 
• Why does the ‘conjugate’ play a role in accomplishing this?

$\bigstar$ Simplify

(a) $\dfrac{3-3\sqrt{3a}}{4\sqrt{8a}}$

(b) $\dfrac{\sqrt{5}+3}{4-\sqrt 5}$

Show Answer

(a) $$\dfrac{3-3\sqrt{3a}}{4\sqrt{8a}}=\dfrac{3-3\sqrt{3a}}{4\cdot 2 \sqrt{2a}} $$

$$= \dfrac{3-3\sqrt{3a}}{8\sqrt{2a}}\cdot\dfrac{\sqrt{2a}}{\sqrt{2a}} =\dfrac{(3-3\sqrt{3a}) \sqrt{2a}}{8\cdot 2a} $$

$$= \dfrac{3\sqrt{2a}-3\sqrt{6a^2}}{16a} =\dfrac{3\sqrt{2a}-3a\sqrt 6}{16a}$$

(b) $$\dfrac{\sqrt{5}+3}{4-\sqrt 5}= \dfrac{\sqrt{5}+3}{4-\sqrt 5}\cdot \dfrac{4+\sqrt 5}{4+\sqrt 5} $$

$$= \dfrac{(\sqrt{5}+3)(4+\sqrt 5)}{(4-\sqrt 5)(4-\sqrt 5)}=\dfrac{4\sqrt{5}+5+12+3\sqrt 5}{16-5}$$

$$ =\dfrac{7\sqrt 5+17}{11}$$

Need more help?

Don’t wait too long to do the following.

• Watch the additional video resources.
• Talk to your instructor.
• Form a study group.
• Visit a tutor. For more information, check the tutoring page .

Lessons Menu

• Lesson 1: Applications of Linear Equations
• Lesson 2: Ratio, Proportion, and Variation
• Lesson 3: The Nature of Mathematical Reasoning
• Lesson 4: Estimation and Interpreting Graphs
• Lesson 6: Statement and Quantifiers
• Lesson 7: Measures of Length: Converting Units and the Metric System
• Lesson 8: Measures of Area, Volume, and Capacity
• Lesson 9: Measures of Weight & Temperature
• Lesson 10: Percents
• Lesson 11: Simple Interest
• Lesson 12: Compound Interest
• Lesson 13: Basic Concepts of Probability
• Lesson 14: Tree Diagrams, Tables and Sample Spaces
• Lesson 15: Gathering and Organizing Data/Picturing Data
• Lesson 16: Measures of Average
• Lesson 17: Measures of Variation
• Lesson 18: Measures of Position
• Lesson 19: The Normal Distribution/Applications of the Normal Distribution
• Lesson 20: Correlation and Regression Analysis
• Lesson 21: Points, Lines, Planes & Angles
• Lesson 22: Triangles
• Lesson 23: Polygons and Perimeter/Areas of Polygons and Circles

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McGraw Hill My Math Grade 5 Chapter 2 Lesson 5 Answer Key Problem-Solving Investigation: Make a Table

All the solutions provided in McGraw Hill Math Grade 5 Answer Key PDF Chapter 2 Lesson 5 Problem-Solving Investigation: Make a Table  will give you a clear idea of the concepts.

McGraw-Hill My Math Grade 5 Answer Key Chapter 2 Lesson 5 Problem-Solving Investigation: Make a Table

Answer: Sixty-three out of every 100 households owned at least one pet. We need to find households owned at least one pet. households out of ten thousand owned at least one pet.

2. Plan I will make a ___ to solve the problem. Answer: I will make a table to solve the problem.

4. Check Is my answer reasonable ? Explain. Use mental math to multiply. 63 × 100 = _____ Answer: The answer is 6300.

Explanation: Given multiplicand is 63 and the multiplier is 100 By multiplying both multiplicand 63 and multiplier 100, we get the value as The multiplication of 63 × 100 is 6300.

Practice the Strategy Nestor is saving money to buy a new camping tent. Each week he doubles the amount he saved the previous week If he saves $1 the first week, how much money will Nestor save in 7 weeks?

1. Understand

What facts do you know? ________________ Answer: Nestor is saving money to buy a new camping tent. He saves $1 the first week.

What do you need to find? ________________ Answer: We need to find out how much money will Nestor save in 7 weeks.

2. Plan ________________ Answer: By adding money every week.

3. Solve ________________ Answer: $127

Explanation: 1 + 2 + 4 + 8 + 16 + 32 + 64 = $127

Is my answer reasonable? Explain. ________________ Answer: Yes, the answer is reasonable.

Explanation: He saves $1 the first week Each week he doubles the amount he saved the previous week. $1 + $2 + $4 + $8 + $16 + $32 + $64 = $127

Apply the Strategy

Solve each problem by making a table.

Question 1. Betsy is saving to buy a bird cage. She saves $1 the first week, $3 the second week, $9 the third week, and so on. How much money will she save in 5 weeks? Answer: 81 dollars

Explanation: Betsy saves $1 for the first week $3 for the second week $9 for the third week $27 for the fourth week $81 for the fifth week

Question 2. Kendall is planning to buy a laptop for $1,200. Each month she doubles the amount she saved the previous month. If she saves $20 the first mönth, in how many months will Kendall have enough money to buy the laptop? Answer: 6 months

Explanation: Kendall is planning to buy a laptop for $1,200 Each month she doubles the amount she saved the previous month If she saves $20 the first month. In 1st month he saved $20 2nd month he saved $20 + $40 = $60 3rd month he saved $80 + $60 = $140 4th month he saved $160 + $140 = $300 5th month he saved $300 + $320 = $620 6th month he saved $620 + $640 = $1280 Hence after 6 months she can save enough to buy the laptop.

Question 3. Mrs. Piant’s yearly salary is $42,000 and increases $2,000 per year. Mr. Piant’s yearly salary is $37,000 and increases $3,000 per year. In how many years will Mr. and Mrs. Piant make the same salary? Answer: 5 years

Explanation: Let us assume the x years will Mr. and Mrs. Piant make the same salary. 42000 + 2000x = 37000 + 3000x 3000x – 2000x = 42000 – 37000 1000x = 5000 x = 5 Hence in 5 years will Mr. and Mrs. Piant make the same salary.

Review the Strategies

A card shop recorded how many packs of trading cards it sold each day. Use the table to solve Exercises 5—7.

Use any strategy to solve each problem.

• Make a table.
• Use the four-step plan.

Question 5. In which week did they sell the most packs of cards? Answer: In week 2 they sold the most packs of cards.

Explanation: The week 2 trading cards sold are 48 + 43 + 45 + 41 + 39 = 216 178 216 180

Question 6. In which week did they sell the least amount? Answer: In week 1 they sold the least amount.

Explanation: The week 1 trading card sold are 28 + 32 + 38 + 44 + 36 = 178

Question 7. How many more packs did they sell in Week 2 than in Week 3? Answer: There are 36 more packs they sold in week 2 than in week 3.

Explanation: In week 2 trading cards sold are 216 In week 3 trading cards sold are 180 The difference between week 2 in week 3 is 216 – 180 = 36

Question 8. A putt-putt course offers a deal that when you purchase 10 rounds of golf, you get 1 round for free. If you played a total of 35 rounds, how many rounds did you purchase? Answer: 32 rounds.

Explanation: A putt-putt course offers a deal that when you purchase 10 rounds of golf, you get 1 round for free. 10 + 1 =11 rounds you played a total of 35 rounds. This means you played 35 ÷ 11 3.1 = 3 times of 11 rounds + 2 rounds in 3 times of 11 rounds we purchased 30 rounds and got 3 free rounds plus 2 more rounds. Total round 35 rounds =  3(10 rounds) + 3 free rounds + 2 more rounds You purchased 30 +  2 = 32 rounds and got 3 free rounds.

Explanation: Given, The amount she saved in the first week = $24 Each week she saves $6 Let us assume she saved x weeks after the first week = 24 + 6x if x = 5 Her total saving in 6 weeks is 24 + 6 (5) 24 + 30 = $54

McGraw Hill My Math Grade 5 Chapter 2 Lesson 5 My Homework Answer Key

Problem Solving

Question 1. Mathematical PRACTICE 8 Look for a Pattern In a recent year, one United States dollar was equal to about 82 Japanese yen. How many Japanese yen are equal to $100? $1,000? $10,000? Answer: 82 yens, 8200 yens, 82000 yens, 820000 yens.

Explanation: 1 dollar = 82 yens $100 = 100 × 82 = 8200 yens $1,000 = 1000 × 82 = 82000 yens $10, 000 = 10,000 × 82 = 820000 yens.

Question 2. A local restaurant offers a deal if you purchase 3 medium pizzas, you get 2 side dishes for free. If you get a total of 8 side dishes, how many pizzas did you buy? Answer: 12 medium pizzas.

Explanation: He purchased 3 medium pizzas, he got 2 side dishes for free. for buying 6 medium pizzas we will get 4 side dishes free for buying 9 medium pizzas we will get 6 side dishes free for buying 12 medium pizzas we will get 8 side dishes free. Hence to get a total of 8 side dishes, we would buy 12 medium pizzas.

Question 3. A recipe for potato salad calls for one teaspoon of vinegar for every 2 teaspoons of mayonnaise. How many teaspoons of vinegar are needed for 16 teaspoons of mayonnaise? Answer: 8 spoons of vinegar.

Explanation: 1 teaspoon of vinegar is equal to 2 teaspoons of mayonnaise.

Question 4. A package of 4 mechanical pencils comes with 2 free erasers. If you get a total of 1 2 free erasers, how many packages of pencils did you buy? Answer: 6 packages.

Explanation: There are 6 packages of pencils.

Explanation: Veronica is saving money to buy a saddle for her horse that costs $175. The amount increased every month by $5. Veronica saved in the first month = $10 $15 and $20 175 = n ÷ 2 (2 × 10 + (n-1)5) 350 = 5n 2 + 15n 5n 2 + 15n – 350 = 0 n 2 + 3n – 70 = 0 n = -10, 7 Hence it will take 7 months for veronica to save $175 for the saddle.

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