Cracked CLock

Starter:

Problem Solving

ACT it OUT

Act it out / Make a model

What Is It?

Two adults and two children have been stranded on an island in a river.They must cross the wide river to safety, but they only have one canoe.The canoe can either take one adult or two children at a time. How can they safely reach the other side?

Understanding the problem

WHAT DO WE KNOW?

There is one canoe. It can hold one adult or two children at a time. There are two adults and two children.

WHAT DO WE NEED TO FIND OUT?

Questioning: How can they safely leave the island? How many trips will it take?

Planning and communicating a solution

WHAT WE DID

Students should label a block for each member of the family – A for each adult and C for each child. Demonstrate for the students how to write down each step, as below,so that steps are not duplicated. Students should create three areas. One for the island, the second for crossing the river and the third for the safe river bank.

Reflecting and generalising

Using blocks to represent the movements of the family members made it easier to work through the steps of the problem and arrive at an accurate answer.

Extension

What if there were five people on the island instead of four (three children and two adults)? What if an adult and a child could fit in the canoe at the same time? How would this affect the speed of moving people off the island?

Independent Task:

WALT - Solve a problem by acting it out or using materials.

GOT it Resource - Problem Sheet 7

Logical Reasoning

Logical Reasoning

WHAT IS IT?

Logical reasoning is universal and it is used just about every time a person makes a logical decision. It comes up in school, at work, and even in normal life. Once you master the art of solving logical reasoning questions, then you can apply logical reasoning to your arguments and sway people to agree with you.

Logical reasoning is the process of using a rational series of steps based on mathematical procedures, and given statements to arrive at a conclusion. It is generally a crucial section in aptitude and IQ tests. Having better developed logical reasoning skills make you able to understand, analyze, and question arguments based on statements or questions. These skills can be used to make an argument stronger or weaker.

Logical reasoning questions come in various formats including number series, verbal reasoning, analogies, and logical problems. I will offer a general approach to each problem type, which might be a step by step analysis, or a drawing. Some techniques may be more helpful to some than others. Test different approaches to see what works best for you.

Basic logical reasoning questions involving number series:

General Approach: First look at the numbers, do they increase or decrease? Then, if they do change, then by how much? For example, if the question asks what the next number in the series is and begins “2, 4, 6, 8, 10…,” then I would see that the numbers are increasing by 2 each time. So, I would add 2 to the last number and the correct answer would be 12.

1. 4, 2, 1, ½, ¼…what should be the next number in the series?

• ⅓
• ⅛
• 1/14
• 1/12

The pattern is being cut in half, so the next number in the series after ¼ should be ⅛.

2. 7, 10, 8, 11, 9, 12…what should be the next number in the series?

• 6
• 8
• 10
• 13

The pattern is increasing by 3, then decreasing by 2, so the next number in the series after 12 should be 10.

3. 8, 16, 12, 24, 20, 40, 36… what should be the next number in the series?

• 8
• 54
• 60
• 72

The pattern is doubling and then decreasing by 4, so the next number in the series after 36 should be 72.

Basic logical reasoning questions involving verbal reasoning:

General Approach: You must assume each statement to be true and reason through the problem, step by step. Do not infer anything that is not supported factually. Sometimes the answer may be obvious because it is logic that you naturally go through.

4. Evan is 11 years old. For 2 years, he has been asking his parents for a puppy. His parents have told him that the believe a dog would not be happy in an apartment, but they have given him permission to have a bird. Evan has not yet decided what kind of bird he would like to have.

• Evan’s parents like birds better than they like dogs
• Evan does not like birds
• Evan and his parents live in an apartment
• Evan and his parents would like to move

Since Evan’s parents believe that a dog would not be happy in an apartment, we can reasonably conclude that the family lives in an apartment. We do not know how Evan’s parents feel about birds or how Evan feels about birds.

5. Jennifer is older than her cousin Maria. Maria’s brother Ben is older than Jennifer. When Maria and Ben are visiting with Jennifer, all three like to play Monopoly. Maria wins more often than Jennifer does.

• When Bart plays Monopoly with Maria and Jennifer, he often loses.
• Of the three, Jennifer is the oldest.
• Jennifer hates to lose at Monopoly.
• Of the three, Maria is the youngest.

Since we know that Ben is older than Jennifer and Jennifer is older than Maria, Maria must be the youngest of the three. We know nothing about how often Ben wins or loses and we do not know whether Jennifer hates losing or not.

Analogies:

General Approach: If you can create a statement that explains the relationship between the first two objects, you can apply the same general idea to the third object and the answer. For example, “if it says apple is to fruit as broccoli is to,” then your statement might be that an apple is a type of fruit and broccoli is a type of vegetable. The correct answer would be vegetable.

6. Odometer is to mileage as compass is to

• direction
• speed
• needle
• sailing

The correct answer is direction. Odometer is an instrument that measures mileage. A compass is an instrument that determines direction.

7. Marathon is to race as hibernation is to

• nap
• winter
• sleep
• bear

The correct answer is sleep. A marathon is a long type of race. Hibernation is a long period of sleep.

8. Window is to pane as book is to

• glass
• cover
• novel
• page

The correct answer is page. A window is made up of one or more panes. A book is made up of one or more pages.

9. Optimist is to cheerful as pessimist is to

• gloomy
• petty
• mean
• helpful

The correct answer is gloomy. Optimistic people tend to be cheerful and see the positive side to things, whereas pessimistic people tend to be gloomy and see the negative side. While pessimists might be mean, that word does not have the same relation to pessimists as cheerful does to optimists.

10. Embarrassed is to humiliated as frightened is to

• courageous
• terrified
• agitated
• angry

The correct answer is terrified. A person is humiliated when he or she has been extremely embarrassed. A person is terrified when he or she has been extremely frightened.

Logical problems:

General Approach: Each problem consists of 3 statements. Based on the first two statements, you must determine if the third statement is true, false, or uncertain. These problems can generally be drawn out in one way or another to show how the items in each statement are interrelated. For example, if it says that “1. Abe is faster than Bill. 2.Cate is faster than Abe. 3. Cate is faster than Bill.” The correct answer would be that #3 is true. If you use X > Y to represent who is faster, then A > B and C > A, so C > A > B.

11. Tim is older than Elena.

Curtis is older than Tim.

Elena is older than Curtis.

If the first two statements are true, the third statement is

• True
• False
• Uncertain

The correct answer is that the third statement is false. If T > E and C > T, then C > T > E. Elena is the youngest of the three, so the statement is false.

12. Apartments in the Runnyvale Manor cost less than apartments in The Grapevine Commons.

Apartments in the Livingwell Gate cost more than apartments in the The Grapevine Commons.

Of the three apartment buildings, the Livingwell Gate costs the most.

If the first two statements are true, the third statement is

• True
• False
• Uncertain

The correct answer is that the third statement is true. If GC > RM and LG > GC, then           LG > GC > RM. Livingwell Gate is the most expensive apartment complex.

13. During the past year, Josh saw more movies than Stephen.

Stephen saw fewer movies than Darren.

Darren saw more movies than Josh.

If the first two statements are true, the third statement is

• True
• False
• Uncertain

The correct answer is that the third statement is uncertain. If J > S and D > S, then both J and D saw more movies than S and there are no facts to say who saw the most.

Independent Task:

WALT - Solve a problem by using logical reasoning.

GOT it Resource - Problem Sheet 8

Make an Organised List

Make an organised list

What Is It?

Make a List is a problem-solving strategy that you can use to solve mathematical word problems by writing the information in a more organised format. Here is an example of a problem that can be solved by making a list:

Doug has 2 pairs of pants: a black pair and a green pair. He has 4 shirts: a white shirt, a red shirt, a grey shirt, and a striped shirt. How many different outfits can he put together?

UNDERSTANDING THE PROBLEM

• How many pairs of pants does Doug have? (2)
• How many shirts does Doug have? (4)

PLANNING A SOLUTION

• Suppose Doug wears his black pants. What colour shirt can he wear? (white, red, grey, or striped)
• If Doug wears his striped shirt, how many different outfits can he wear? (2: striped—black and striped—green)
• If Doug wears the green pants, can he wear all 4 shirts? (yes).

Finding the Answer

Make an Organised List

Black—White

Black—Red

Black—Grey

Black—Striped

Green—White

Green—Red

Green—Grey

Green—Striped

Answer: Doug can make 8 different outfits.

Draw a Picture

Draw a picture strategy

The draw a picture strategy is a problem-solving technique in which you make a visual representation of the problem. For example, the following problem could be solved by drawing a picture:

A frog is at the bottom of a 10-meter well. Each day he climbs up 3 meters. Each night he slides down 1 meter. On what day will he reach the top of the well and escape?

Problem 1:

Marah is putting up a tent for a family reunion. The tent is 16 feet by 5 feet. Each 4-foot section of tent needs a post except the sides that are 5 feet. How many posts will she need?

Problem Solving Make an Organised List

Make an organised list

What Is It?

Make a List is a problem-solving strategy that you can use to solve mathematical word problems by writing the information in a more organised format. Here is an example of a problem that can be solved by making a list:

Doug has 2 pairs of pants: a black pair and a green pair. He has 4 shirts: a white shirt, a red shirt, a grey shirt, and a striped shirt. How many different outfits can he put together?

UNDERSTANDING THE PROBLEM

• How many pairs of pants does Doug have? (2)
• How many shirts does Doug have? (4)

PLANNING A SOLUTION

• Suppose Doug wears his black pants. What colour shirt can he wear? (white, red, grey, or striped)
• If Doug wears his striped shirt, how many different outfits can he wear? (2: striped—black and striped—green)
• If Doug wears the green pants, can he wear all 4 shirts? (yes).

Finding the Answer

Make an Organised List

Black—White

Black—Red

Black—Grey

Black—Striped

Green—White

Green—Red

Green—Grey

Green—Striped

Answer: Doug can make 8 different outfits.

Independent Task:

WALT - Make an organised list to solve a problem.

GOT it Resource - Problem Sheet 4 and 5

Make an Organised list

Numeracy - Problem Solving - Thursday 2nd February 2017

Guess and Check Strategy

"Guess and Check" is a problem-solving strategy that you can use to solve mathematical problems by guessing the answer and then checking that the guess fits the conditions of the problem. For example, the following problem would be best solved using guess and check:

Of 25 rounds at the regional spelling contest, the Mighty Brains tied 3 rounds and won 2 more than they lost. How many rounds did the Mighty Brains win?

Problem 1:

Ben knows 100 baseball players by name. Ten are Red Sox. The rest are Blue Jays and Diamondbacks. He knows the names of twice as many Diamondbacks as Blue Jays. How many Blue Jays does he know by name?

When you use the strategy of guess and check, you should keep a record of what you have done. It might be helpful to use a chart or table.

Steps to take to solve this problem:

1. Understand the problem

Find key pieces of information needed to solve the problem - this could mean that you need to read the original problem several times or put it into your own words.

For example, "I know there are twice as many Diamondbacks as Blue Jays. There are 10 Red Sox. The number of Blue Jays and Diamondbacks should equal 90."

2. Choose a strategy

3. Solve the problem

Now, solve the problem. You may want to set up a table to record the guesses.

Guess Number	Blue Jays	Diamondbacks	Red Sox	Total
First Guess	10	20	10	40

Guess a greater number of Blue Jays.

Guess Number	Blue Jays	Diamondbacks	Red Sox	Total
First Guess	10	20	10	40
Second Guess	20	40	10	70

Now guess a greater number of Blue Jays.

Guess Number	Blue Jays	Diamondbacks	Red Sox	Total
First Guess	10	20	10	40
Second Guess	20	40	10	70
Third Guess	40	80	10	130

Now guess a number lesser than 40 and greater than 20.

Guess Number	Blue Jays	Diamondbacks	Red Sox	Total
First Guess	10	20	10	40
Second Guess	20	40	10	70
Third Guess	40	80	10	130
Fourth Guess	30	60	10	100

That is the answer.

4. Check

Read the problem again to be sure the question was answered.

Yes, I found the number of Blue Jays.

Check the math to be sure it is correct.

30 doubled is 60. 30 + 60 + 10 = 100

Determine if the best strategy was chosen for this problem, or if there was another way to solve the problem.

Guess and check was a good way to solve this problem.

5. Explain

The last step is explaining how you found the answer. Write a paragraph describing the steps and how decisions were made throughout the process. Justify your answers.

YOUR TURN:

Try to solve this problem using the strategy of Guess and Check:

Of 25 rounds at the regional spelling contest, the Mighty Brains tied 3 rounds and won 2 more than they lost. How many rounds did the Mighty Brains win?

Independent Task:

WALT - Guess and check

GOT it Resource - Problem Sheets 1