Making a Drawing(Visual representation)

Making a Drawing (Visual representation)

• Use of drawing to solve when a visual representation is not the usual approach

• Facilitating some problem situations

Making a Drawing(Visual representation)

• Everyday – life examples

– Maps– Manuals– Newspaper– Strategy in sports

Problem 1

• The train trip between Boston and New York takes exactly 5 hours. A train leaves Boston for New York every hour on the hour, while a train leaves New York for Boston every hour on the half-hour. A student at Harvard takes the New York-bound train from Boston at 10:00 a.m. How many Boston-bound trains will she pass before she arrives in New York?

Problem 1

Step 1: Understanding the problem• Given

– Time of the trip is 5 hours– From Boston to New York is every hour on the hour– From New York to Boston is every hour on the half-

hour– She departs from Boston at 10:00 a.m.

• Asked– How many Boston-bound trains did she pass?

Problem 1

Step 2: Making a plan

Let’s draw a timeline

Problem 1

Step 3: Carring out the plan

New York

Time left New York

9:30 8:30 7:30 6:30 5:30

Boston

Time left New York

2:30 1:30 12:30 11:30 10:30

She will pass 10 trains

Problem 1

Step 4: Checking the solution / looking backBy making a list

Trains passed

Arrives Boston Leaves New York

1 10:30 a.m. 5:30 a.m.

2 11:30 a.m. 6:30 a.m.

3 12:30 p.m. 7:30 a.m.

4 1:30 p.m. 8:30 a.m.

5 2:30 p.m. 9:30 a.m.

6 3:30 p.m. 10:30 a.m.

7 4:30 p.m. 11:30 a.m.

8 5:30 p.m. 12:30 p.m.

9 6:30 p.m. 1:30 p.m.

10 7:30 p.m. 02:30 p.m.

Problem 1Step 5: Extending the problemThe train trip between Boston and New York takes

exactly 5 hours. A train leaves Boston for New York every hour on the hour, while a train leaves New York for Boston every hour on the half-hour. A student at Harvard takes the New York-bound train from Boston at 10:00 a.m. How many Boston-bound trains will she pass before she arrives in New York?

What if we want to know how many trains are on the tracks between New York and Boston during the 5-hour trip?

Problem 1

Step 5: Extending the problem (carry out plan)

• Boston – New York bound (let’s begin with 9:00 a.m.)

• All these trains are on tracks between 9:00 a.m. – 2:00 p.m (6 trains)

Time left Boston

9:00 10:00 11:00 12:00 1:00 2:00

Problem 1

Step 5: Extending the problem (carry out plan)

• New York – Boston bound

• All these trains are on tracks between 9:00 a.m. – 2:00 p.m (6 trains)

• 6 + 6 = 12 trains

Arrives Boston

9:30 10:30 11:30 12:30 1:30 2:30

Problem 2

• A pastry cook makes 10 cookies from a ball of dough by using cookie cutter. After the cookies are cut by cookie cutter the remaining dough is kneaded to form another ball of dough. The remaining dough of 20 cookies is kneaded to form a ball of dough. Then how many cookies can be made from 4 balls of dough, if the cook does not use pieces smaller than a ball of dough?

Problem 2

Step 1: Understanding the problem• Given

– A pastry cook makes 10 cookies from a ball of dough– the remaining dough is kneaded to form another ball of

dough– remaining dough of 20 cookies is kneaded to form a ball of

dough– cook does not use pieces smaller than a ball of dough?

• Asked– how many cookies can be made from 4 balls of dough?

Problem 2

Step 2: Making a plan

• Let’s use visual representation to see the work of pastry cook

Problem 2

Step 3: Carrying out the plan

40

20

10

40 +20 +10 = 70 cookies

Problem 2

Step 4: Checking the solution / looking back

• 10 cookies from each ball of dough4 x 10 = 40 cookies

• remaining dough of 20 cookies is kneaded to form a ball of dough40 : 20 = 2 balls of dough2 x 10 = 20 cookies20 : 20 = 1 ball of dough1 x 10 = 10 cookies

40 + 20 + 10 = 70 cookies

Problem 2

Step 4: Checking the solution / looking back (another way)

• 1 ball 10 cookies

• 2 balls 30 cookies

• 3 balls 50 cookies

• Then the formula is (2n-1).10

• If there are 4 balls – (2n - 1).10 = (2x4 - 1).10 = 70 cookies

Problem 2

Step 5: Extending the problem• A pastry cook makes 10 cookies from a ball of

dough by using cookie cutter. After the cookies are cut by cookie cutter the remaining dough is kneaded to form another ball of dough. The remaining dough of 20 cookies is kneaded to form a ball of dough. Then how many cookies can be made from 4 balls of dough, if the cook uses pieces smaller than a ball of dough but does not use smaller pieces than a cookie?

Problem 2

Step 5: Extending the problem (carry out plan)

• A half ball comes from 10 cookies• This half ball forms 5 cookies• A one-quarter ball comes from 5 cookies• This one-quarter ball forms 2.5 cookies• A 1/8 ball comes from 2.5 cookies• This 1/8 ball forms 1.25 cookies• If the cook does not use smaller pieces;

– So 70 + 2 + 1 = 78 cookies

Problem 3

• A 1000-liter pool has a small leak and along with evaporation loses 50 liters of water each day. Every three days it is added 100 liter of water to the pool. After 21 days the pool is filled. How much water is used to fill the pool on the 21 st day?

Problem 3

Step 1: Understanding the problem• Given

– A 1000-liter pool– loses 50 liters of water each day– Every 3 days it is added 100 liters of water– After 21 days the pool is filled

• Asked– How much water is used to fill the pool on the

21 st day?

Problem 3

Step 2: Making a plan

Let’s draw a graph of the situation.

Problem 3

Step 3: Carrying out the plan

Graph

• 1000 – 550 = 450 liters

Problem 3

Step 4: Checking the solution / looking back• 50 x 3 = 150 liters

150 – 100 = 50 liters net lose every 3 days

21 : 3 = 7

50 x 7 = 350 liters

350 + 100 = 450 liters

Problem 3

Step 5: Extending the problem

• There is a frog at the bottom of a well that is 100 ft deep. The frog climbs upward 5 ft during the daytime. However, at night, he falls asleep and slips back 4 ft. At this rate, how many days will it take the frog to get out of the well?

Problem 3

Step 5: Extending the problem (carry out plan)

• A graph may be drawn

• Another method is– A net gain of 1 ft per day

– The frog has reached a height of 95 ft at the end of day 95 (5 ft from the top of the well)

– During the 96 th day it climbs 5 ft more and out of the well

Problem 4

• There are 6 control points on a linear racetrack. The first point is at the beginning and the sixth one is at the end of it. It takes the first racer 20 sec to go from the first control point to the third one whereas it takes the second racer 22 sec. What is the time difference between the first and the second racer to complete the whole racetrack?

Problem 5

• Selen and Burak work as student assistants in the library. Selen works two consecutive days and then two days off before she works again. Burak works one day and one day off before he works again. The library is open seven days a week. If Burak begins working on Friday September 1 st and Selen begins working on Saturday September 2 nd, on which days in September do they work together?