lesson 8 problem solving use logical reasoning

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McGraw Hill My Math Grade 4 Chapter 8 Lesson 8 Answer Key Problem-Solving Investigation: Use Logical Reasoning

All the solutions provided in  McGraw Hill My Math Grade 4 Answer Key PDF Chapter 8 Lesson 8 Problem-Solving Investigation: Use Logical Reasoning will give you a clear idea of the concepts.

McGraw-Hill My Math Grade 4 Answer Key Chapter 8 Lesson 8 Problem-Solving Investigation: Use Logical Reasoning

Learn the Strategy

Maria used flour, sugar, and brown sugar to make a treat. Use the clues below to find the amount of each ingredient she used.

• The amounts were \(\frac{3}{4}\) cup, \(\frac{1}{4}\) cup, and \(\frac{2}{3}\) cup.
• She used more flour than sugar.

1. Understand What facts do you know? Maria used more _____________ than sugar and more sugar than _______________. What do you need to find? the amount of each ______________ she used

2. Plan I can use logical reasoning to solve the problem.

3. Solve The order of the ingredients from greatest to least is _____________ , ______________ and ______________. The greatest fraction is \(\frac{3}{4}\). So, Maria used \(\frac{3}{4}\) cup of flour. \(\frac{2}{3}\) > \(\frac{1}{4}\) So, she used \(\frac{2}{3}\) cup of sugar and \(\frac{1}{4}\) cup of brown sugar.

4. Check Does your answer make sense? Explain. Answer: Number of cups of flour she used = \(\frac{3}{4}\) Number of cups of sugar she used = \(\frac{2}{3}\) Number of cups of brown sugar she used = \(\frac{1}{4}\) Yes, my answer is correct because it fulfills all the rules given in the question.

• The amounts are: \(\frac{1}{2}\) cup, \(\frac{3}{4}\) cup, and \(\frac{1}{3}\) cup.
• He is using more tomatoes than black beans and more black beans than onions.

1. Understand What facts do you know? What do you need to find?

4. Check Does your answer make sense? Explain. Answer: Number of cups of tomatoes he used = \(\frac{3}{4}\) Number of cups of black beans he used = \(\frac{1}{2}\) Number of cups of onions he used = \(\frac{1}{3}\) Yes, my answer makes sense because it fulfills all the rules given in the question.

Apply the Strategy Solve each problem by using logical reasoning. Question 1. Sophia is making a salad with tomatoes, cucumbers, and mozzarella cheese. Use the clues to find the amount of each ingredient.

• The amounts are \(\frac{3}{6}\) cup, \(\frac{2}{5}\) cup, and \(\frac{3}{4}\) cup.
• There is a less amount of tomatoes than cucumbers.
• There is a less amount of cheese than tomatoes.

Answer: Number of cups of tomatoes she used = \(\frac{3}{4}\) Number of cups of cheese she used = \(\frac{3}{6}\) Number of cups of cucumbers she used = \(\frac{2}{5}\)

Question 2. Mason walked on Monday, Wednesday, and Friday. Use the clues to find how far he walked each day.

• The distances were \(\frac{6}{8}\) mile, \(\frac{1}{4}\) mile, and \(\frac{1}{6}\) mile.
• He did not walk the farthest on Monday.
• He walked less on Friday than Monday.

Answer: Number of miles he ran on Wednesday = \(\frac{6}{8}\) Number of miles he ran on Monday = \(\frac{6}{8}\) Number of miles he ran on Friday = \(\frac{6}{8}\)

Question 3. Mathematical PRACTICE Plan Your Solution Emma saw birds, fish, and bears at the zoo. One-third of the animals did not have feathers or fur. One-half of the animals have four legs. There were one-sixth of the animals left. What fraction of the animals were birds? Answer: Fraction of number of birds = One-sixth.

Explanation: Emma saw birds, fish, and bears at the zoo. Fraction of the animals did not have feathers or fur = One-third. => Fraction of number of Fishes = One-third. Fraction of the animals have four legs = One-half. => Fraction of number of bears = One-half. Fraction of the animals left = One-sixth . => Fraction of number of birds = One-sixth.

Review the Strategies Use any strategy to solve each problem.

• Work backward.
• Make a table.
• Make a model.
• Look for a pattern.

Explanation: Total number of flowers Emil bought his mom = 12. Number of flowers are yellow = 4. Number of flowers are red = 6. Number of flowers are white = Total number of flowers Emil bought his mom – (Number of flowers are yellow + Number of flowers are red) = 12 – (6 + 4) = 12 – 10 = 2. Simplest form: Fraction of flowers are white = Number of flowers are white ÷ Total number of flowers Emil bought his mom = 2 ÷ 12 = 1 ÷ 6 or \(\frac{1}{6}\) Equivalent fraction of \(\frac{1}{6}\): \(\frac{1}{6}\) = \(\frac{1}{6}\) × \(\frac{3}{3}\)  = \(\frac{3}{18}\)

Question 5. Dirk ran eight-tenths of a mile at track practice. Leslie ran \(\frac{80}{100}\) of a mile. Who ran farther? Explain. Answer: Dirk and Leslie ran same distance.

Explanation: Number of miles Dirk ran at track practice = eight-tenths or \(\frac{8}{10}\) Number of miles Leslie ran at track practice = \(\frac{80}{100}\) Simplest form: \(\frac{8}{10}\) = \(\frac{8}{10}\) ÷ \(\frac{2}{2}\) = \(\frac{4}{5}\) \(\frac{80}{100}\) = \(\frac{80}{100}\) ÷ \(\frac{10}{10}\) = \(\frac{8}{10}\) ÷ \(\frac{2}{2}\) = \(\frac{4}{5}\) => Both are same with distance they ran.

Question 6. Mrs. Keys has 36 pens. Yesterday she had half that amount plus 2. How many pens did she have? Answer: Total number of pens Mrs. Keys has = 56.

Explanation: Number of pens Mrs. Keys has = 36. Yesterday she had half that amount plus 2. => Number of pens she had added yesterday = (36 half)+ 2 = (36 ÷ 2) + 2 = 18 + 2 = 20. => Total number of pens Mrs. Keys has = Number of pens Mrs. Keys has + Number of pens she had added yesterday = 36 + 20 = 56.

Question 7. Mathematical PRACTICE Draw a Conclusion There are bananas, pears, and peaches in a bag. One-eighth of the fruits are pears. One half are bananas. Are there more pears or bananas? Answer: => Number of the fruits are bananas are more than number of the fruits are pears.

McGraw Hill My Math Grade 4 Chapter 8 Lesson 8 My Homework Answer Key

Problem Solving Solve each problem by using logical reasoning. Question 1. Mathematical PRACTICE Stop and Reflect Ryan has his artwork displayed at the library, the mall, and the bank. Use the clues to find the fraction of his art that is displayed at each place.

• \(\frac{1}{4}\) of the art is at one location, \(\frac{1}{8}\) of the art is at the second location, and \(\frac{5}{8}\) of the art is at the third.
• There is more of Ryan’s art at the library than the mall.

Question 2. Benjamin made a fruit salad with strawberries, blueberries, and kiwi. Use the clues to find the amounts of each ingredient.

• The amounts were \(\frac{3}{4}\) cup, \(\frac{2}{8}\) cup, and \(\frac{1}{2}\) cup.
• Benjamin used more blueberries than strawberries.
• Benjamin used more strawberries than kiwi.

Answer: Number of cups of blueberries Benjamin used = \(\frac{3}{4}\) Number of cups of strawberries Benjamin used = \(\frac{1}{2}\) Number of cups of kiwi Benjamin used = \(\frac{2}{8}\)

Question 3. Layla wrote a report about insects. She listed the lengths of tiger beetles, carpenter ants, and aphids. The lengths were \(\frac{1}{2}\) inch, \(\frac{5}{8}\) inch, and \(\frac{1}{8}\) inch. A tiger beetle is bigger than a carpenter ant. A carpenter ant is bigger than an aphid. List the sizes of each insect. Answer: Length of tiger beetles = \(\frac{5}{8}\) inch. Length of carpenter ants = \(\frac{1}{2}\) inch. Length of an aphids = \(\frac{1}{8}\) inch.

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My Math 4 Volume 2 Common Core, Grade: 4 Publisher: McGraw-Hill

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Course: LSAT   >   Unit 1

Getting started with logical reasoning.

• Introduction to arguments
• Catalog of question types
• Types of conclusions
• Types of evidence
• Types of flaws
• Identify the conclusion | Quick guide
• Identify the conclusion | Learn more
• Identify the conclusion | Examples
• Identify an entailment | Quick guide
• Identify an entailment | Learn more
• Strongly supported inferences | Quick guide
• Strongly supported inferences | Learn more
• Disputes | Quick guide
• Disputes | Learn more
• Identify the technique | Quick guide
• Identify the technique | Learn more
• Identify the role | Quick guide
• Identify the role | learn more
• Identify the principle | Quick guide
• Identify the principle | Learn more
• Match structure | Quick guide
• Match structure | Learn more
• Match principles | Quick guide
• Match principles | Learn more
• Identify a flaw | Quick guide
• Identify a flaw | Learn more
• Match a flaw | Quick guide
• Match a flaw | Learn more
• Necessary assumptions | Quick guide
• Necessary assumptions | Learn more
• Sufficient assumptions | Quick guide
• Sufficient assumptions | Learn more
• Strengthen and weaken | Quick guide
• Strengthen and weaken | Learn more
• Helpful to know | Quick guide
• Helpful to know | learn more
• Explain or resolve | Quick guide
• Explain or resolve | Learn more

Logical Reasoning overview

• Two scored sections with 24-26 questions each
• Logical Reasoning makes up roughly half of your total points .

Anatomy of a Logical Reasoning question

• Passage/stimulus: This text is where we’ll find the argument or the information that forms the basis for answering the question. Sometimes there will be two arguments, if two people are presented as speakers.
• Question/task: This text, found beneath the stimulus, poses a question. For example, it may ask what assumption is necessary to the argument, or what must be true based on the statements above.
• Choices: You’ll be presented with five choices, of which you may select only one. You’ll see us refer to the correct choice as the “answer” throughout Khan Academy’s LSAT practice.

What can I do to tackle the Logical Reasoning section most effectively?

Dos and don’ts.

• Don’t panic: You’re not obligated to do the questions in any order, or even to do a given question at all. Many students find success maximizing their score by skipping a select handful of questions entirely, either because they know a question will take too long to solve, or because they just don’t know how to solve it.
• Don’t be influenced by your own views, knowledge, or experience about an issue or topic: The LSAT doesn’t require any outside expertise. All of the information that you need will be presented in the passage. When you add your own unwarranted assumptions, you’re moving away from the precision of the test’s language and toward more errors. This is one of the most common mistakes that students make on the LSAT!
• Don’t time yourself too early on: When learning a new skill, it’s good policy to avoid introducing time considerations until you’re ready. If you were learning piano, you wouldn’t play a piece at full-speed before you’d practiced the passages very slowly, and then less slowly, and then less slowly still. Give yourself time and room to build your skill and confidence. Only when you’re feeling good about the mechanics of your approach should you introduce a stopwatch.
• Do read with your pencil: Active reading strategies can help you better understand logical reasoning arguments and prevent you from “zoning out” while you read. Active readers like to underline or bracket an argument’s conclusion when they find it. They also like to circle keywords, such as “however”, “therefore”, “likely”, “all”, and many others that you’ll learn throughout your studies with us. If you’re reading with your pencil, you’re much less likely to wonder what you just read in the last minute.
• Do learn all of the question types: An effective approach to a necessary assumption question is very different from an effective approach to an explain question, even though the passage will look very similar in both. In fact, the same argument passage could theoretically be used to ask you a question about the conclusion, its assumptions or vulnerabilities to criticism, its technique, the role of one of its statements, a principle it displays, or what new info might strengthen or weaken it!
• Do spend time on the fundamentals: The temptation to churn through a high volume of questions can be strong, but strong LSAT-takers carefully and patiently learn the basics. For example, you’ll need to be able to identify a conclusion quickly and accurately before you’ll be able to progress with assumptions or flaws (identifying gaps in arguments). Similarly, a firm understanding of basic conditional reasoning will be invaluable as you approach many challenging questions. Be patient with yourself!

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Problem Solving Using Logic Reasoning and Bar Modeling – March Math Calendar 2016 Problem

We received some comments on one of the questions in the March Math Calendar . We thought we might share some strategies that we use in this blog post.

The question is for March-31st 2016, and is as follows:

This is a very “versatile” question. The answer is specially designed to be a simple round number so that the question can be attempted by students in three different stages.

The first group of students whom we usually give this problem to are the 4th graders. Without specific instructions on how to find the answers, the students usually rely on logic reasoning and Guess-and-Check to find out what B is. Students derive great satisfaction in knowing they are able to solve a problem that looks formidable at first. Here is a video clip of how one of our students solved the question using logic reasoning, with Guess-and-Check.

For more advanced students, we introduce an intermediate method using bar-models. This provides a link between the Guess-and-Check method used by Pre-Algebra students and the formal introduction of solving the System of Linear Equations in middle schools.

Below is another video on how to solve the same problem using logic reasoning and bar models.

It should be noted there are many approaches for this method. For example, we could have equivalently used the pair for variables A and B to solve for B , starting with the last two equations. The key is the use of bar models to visualize the relationships between two variables and deduce the value of B from the given clues.

The last group of students are 8th graders, who are starting to work on Systems of Linear Equations. They would apply the standard method of solving the set of equations and simultaneously derive the values of A, B and C.

What do you think of the use of bar-models as an intermediate step between the Guess-and-Check stage and formal introduction of Systems of Linear Equations? Let us know in the comments below!

Related Resources

For more related resources, please refer to our Bar Models page.

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Using Folktales to Teach Logical Reasoning

Having young students write a pourquoi—an origin folktale—is a great way to cultivate problem-solving and divergent thinking.

Logical reasoning is more than interpreting patterns, number sequences, and the relationships between shapes in math—it can also be successfully integrated into language arts. The underlying skill of divergent thinking can be developed in young students through the process of writing a creative pourquoi—a folktale that explains how or why something came to be.

There’s a strong need for educators to take a multidisciplinary approach to teaching logical reasoning skills to students starting at a young age. American students fall significantly behind their peers in other developed nations when it comes to their ability to problem-solve and use divergent thinking skills: On the creative problem-solving portion of the Programme for International Student Assessment in 2012, nearly 10 percent of students in Singapore and Korea performed well enough to be “highly skilled problem solvers,” while only 2.7 percent of American students earned that ranking.

Using Pourquoi Tales to Foster Logical Reasoning

A pourquoi (French for “why”) is a terrific way to cultivate problem-solving and divergent thinking—core skills of logical reasoning—in the student learning framework. Logical reasoning is associated with the capacity to generate alternative ideas, and pourquois involve creating alternative ideas to explain a natural phenomenon in the world (e.g., why stars shine at night or how zebras got stripes).

Pourquois are further a fun and playful way for students to apply inductive and deductive reasoning skills—they can identify conditional reasoning premises like “if P, then Q” in these tales (e.g., if the lion fires an arrow at the raincloud, rain will fall down).

Students can also examine how a single cause leads to multiple far-reaching effects (e.g., banging a drum causes the earth to quake, which in turn causes animals to scamper).

To introduce a pourquoi unit, create a “Wonderings” anchor chart upon which students can brainstorm natural phenomena they wonder about, activating their prior knowledge. This empowers students by embracing their curiosity, and it also enables them to make powerful real-world connections.

Next, explain that pourquois fall into the folktale genre and have distinguishing elements, including nameless characters, a subtle moral or lesson, use of personification, and more.

There are outstanding multicultural pourquoi tales you can read aloud to students, as well as incorporate into guided reading groups, literacy centers, or written assignments. Such tales can foster open-mindedness and inclusiveness toward other cultures, helping to shape students who are world citizens and globally minded.

Read-aloud pourquoi tales:

• Why The Sky Is Far Away by Mary-Joan Gerson and Carla Golembe
• Why Mosquitos Buzz in People’s Ears by Verna Aardema and Leo and Diane Dillon
• How Snake Got His Hiss by Marguerite Davol and Mercedes McDonald
• How the Elephant Got Its Trunk by Jean Richards and Norman Gorbaty
• How Tiger Got His Stripes by Rob Cleveland and Baird Hoffmire

Formative and Summative Assessment

There are a variety of formative assessments targeting higher-order thinking skills that you can use in a pourquoi unit. Have students map out the “if P, then Q” premises in a pourquoi onto a cause and effect graphic organizer. This teaches students how to make connections between conditional syllogisms, which is paramount to logical reasoning.

You can also have students compare and contrast elements in two or more pourquois. A third assessment is to have students make text-to-self connections using a pourquoi (e.g., “I am similar to the lion because my anger caused me to act rashly”).

As a summative assessment, students can plan, draft, and publish original pourquois. First, students should identify what natural phenomenon in the world they wish to explain. Next, they should identify their characters, setting, a problem, and a solution. Their tales should contain a beginning, middle, and ending. For enrichment or extension opportunities, have students research and select a culture upon which to base their tale.

Their writing will be enhanced if you routinely share student samples that demonstrate rich writing skills, including use of similes, metaphors, dialogue, and descriptive language.

As a best practice, incorporate one-on-one conferencing and peer editing into the writing process. End the unit with a publishing party to celebrate the authors and their process of arriving at an alternative explanation through use of logical reasoning skills.

Through designing and implementing a comprehensive, structured pourquoi unit, teachers can effectively integrate logical reasoning into the language arts discipline.

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How to Solve Math Problems Using Logical Reasoning

How to Do a Simple Calorimeter Experiment

Logical reasoning is a useful tool in many areas, including solving math problems. Logical reasoning is the process of using rational, systemic steps, based on mathematical procedure, to arrive at a conclusion about a problem. You can draw conclusions based on given facts and mathematical principles. Once you master the skill in solving math problems, you can use logical reasoning in a wide array of real-world situations.

Read and understand the problem. For example, let’s say Bob grilled hot dogs to sell at the concession stand during the hockey game. By the end of the first period, Bob had sold one-third of the hot dogs. During the second period, Bob sold 10 more hot dogs and continued selling hot dogs through the third period. When the game ended, Bob sold half the remaining grilled hot dogs. If given the information that 10 grilled hot dogs didn’t sell, how many hot dogs did Bob grill before the game started?

Make a plan to solve the problem backward using critical thinking and logic. In the concession stand example, you know that Bob had 10 unsold grilled hot dogs when the game ended.

Working backward, start with the known quantity of 10 unsold, grilled hot dogs. You were also told that Bob sold half the remaining hot dogs when the game ended. Therefore, the second half of unsold hot dogs totals 10. Multiply 10 by 2 = 20 hot dogs. Earlier, Bob had sold an additional 10 hot dogs to equal a running total of 30 hot dogs. Continuing to work backward, you recall that Bob sold one-third of his hot dogs in the first period, meaning that two-thirds remained, which equals 30. Now that you’ve determined that two-thirds equal 30 hot dogs, you can surmise that one-third equals 15. Add 15 + 30 = 45. Your final calculation reveals that Bob grilled 45 hot dogs before the game started.

To check the accuracy of your work, do the problem in reverse using logical reasoning. Start with your final answer – 45 hot dogs grilled before the game started. This time, however, work forward. Bob sold one-third of his hot dogs during the first period of the hockey game. Make the calculation. Divide 45 by three, which equals 15. When you subtract 15 from 45, the answer is 30. Because Bob sold 10 more hot dogs during the second period, subtract 10 from 30, which is 20. Half of 20 is 10, which is the number of hot dogs remaining. Arriving at this solution confirms your logical reasoning abilities.

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• Education.com: The Strategies of Working Backward and Using Logical Reasoning in Word Problems: Study Guide

About the Author

Nicholas Smith has written political articles for SmithonPolitics.com, "The Daily Californian" and other publications since 2004. He is a former commissioner with the city of Berkeley, Calif. He holds a Bachelor of Arts in political science from the University of California-Berkeley and a Juris Doctor from St. John's University School of Law.

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Chapter 11, Lesson 4: Problem-Solving Investigation: Use Logical Reasoning

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Problem Solving Strategy: Logical Reasoning

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