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9.8: Problem-Solving Strategy: Look for a Pattern; Use a Venn Diagram

Created by: CK-12

Introduction

Skate park Construction

Isabelle, Isaac and Marc are very excited that their plans were accepted! The skate park builders are about to begin construction. The local lumber store has agreed to donate all of the wood that the team needs. To figure out what to ask for, the trio has collected a list of materials needed for the half pipe and the quarter pipe. Here is the list.

Half Pipe

42 8’ 2 \times 6’s

5 8’ 2 \times 4’s

4 8’ 2 \times 8’s

12 8’ 1 \times 6’s

4 8’ 4 \times 4’s

2 \frac{3}{4}” sheets of plywood

12 \frac{3}{8}” sheets of plywood

Quarter Pipe

13 8’ 2 \times 4’s

4 8’ 2 \times 6’s

1 8’ 4 \times 4’s

2 \frac{3}{4}” sheets of plywood

4 \frac{3}{8}” sheets of plywood

The team needs to organize the materials to figure out what materials both ramps have in common, and then which materials are unique to each ramp. By doing this, they can provide the lumber company with a list of materials needed for both ramps.

Isabelle thinks that it would be a good idea to design a Venn diagram, but Marc and Isaac aren’t sure how to do it. Isabelle thinks that she knows, but she isn’t sure.

In this lesson, you will learn how to draw a Venn diagram to organize these materials. Pay close attention and you will have a chance to complete this diagram at the end of the lesson.

What You Will Learn

In this lesson you will learn to problem solve with the following skills:

  • Read and understand given problem situations.
  • Develop and use the strategy: Look for a Pattern.
  • Develop and use the strategy: Use a Venn Diagram.
  • Plan and Compare alternative approaches to solving problems.
  • Solve real-world problems using selected strategies as a part of a plan.

Teaching Time

I. Read and Understand Given Problem Situations

When faced with a word problem, the first thing that you need to do is to read the problem. Reading the problem will help you to identify any given information as well as the information needed to solve the problem.

This lesson is about using patterns and organizing information to solve a problem. To do this, you can choose between two different methods, one is to find a pattern and one is to draw a Venn Diagram.

Both strategies have benefits to them. When you read a problem that has common elements such as common numbers, you can use patterns and Venn Diagrams with these problems. Because both strategies rely on common things, you must have something in common in the problem to work with.

Let’s look at solving problems with Greatest Common Factor using patterns and Venn Diagrams.

II. Develop and Use the Strategy: Look for a Pattern

When solving problems that involve greatest common factors, we can use patterns to help us. The strategy “look for a pattern” is just that. What pattern can be seen in the numbers that we are working with? How does the pattern appear?

Let’s look at an example.

Example

There are 280 girls and 260 boys playing on soccer teams this fall. If each team has the same number of girls and the same number of boys, what is the greatest number of teams that can be formed?

To solve this, we need to find the prime factors of 280 and 260. Then, we can figure out the greatest common factor which is the largest number that divides into both groups. Once we have this factor, we will know the number of teams. The greatest common factor is also the number of teams that can be formed.

We start by factoring 280.

Next, we factor 260

If we look at what is common here, we can see that 5 and two 2's are common.

5 \times 2 \times 2 = 20

There are 20 possible teams.

By looking for patterns, we could use 10 as a factor. Right in the beginning, we have 10 as one of the factors, then we just had to find any other factors. This gave us our answer.

III. Develop and Use the Strategy: Use a Venn Diagram

We just used patterns to solve a word problem where common elements are featured. What about Venn Diagrams? What is a Venn Diagram?

A Venn Diagram is shows the common numbers in two sets of objects or numbers by using overlapping circles.

Let’s look at the example again.

Example

There are 280 girls and 260 boys playing on soccer teams this fall. If each team has the same number of girls and the same number of boys, what is the greatest number of teams that can be formed?

Now a Venn Diagram is used to show things that are common and things that are different. For this example, we can write the prime factors of 280 in one circle, the prime factors of 260 in the other circle and the common factors in the part of the circle that overlaps.

By looking at this diagram, you can see that the common factors between 280 and 260 are 5, 2, and 2. If we multiply these together, we will have the total number of groups possible.

5 \times 2 \times 2 = 20

There are 20 possible groups.

A Venn Diagram helps you to organize data in a visual way to notice patterns and solve for the answer.

IV. Plan and Compare Alternative Approaches to Solving Problems

We can work on solving problems in many different ways. In the last two sections, we discussed solving a problem using patterns, prime factorization and Venn diagrams. Let’s look at another example and think about ways that we could solve the problem.

Example

1, 1, 2, 3, 5, 8, 13, 21, _____

What is the next number in this sequence?

If we were going to solve this problem, we would need to look for a pattern in the numbers. A Venn diagram wouldn’t really help us here-we have one set of data and we aren’t comparing anything. We are looking to figure out the next number.

How can we figure this out?

We can look for different ways to get the numbers using different operations. Were these numbers multiplied?

1 \times 1 & = 1 \\1 \times 2 & = 2\\ 2 \times 3 & = 6

That’s right, it doesn’t work. You have to use a different operation.

Let’s think about addition.

Do you see a pattern?

The pattern here is to find the sum of the two previous numbers. That sum is the next number in the pattern.

Let’s see if this works.

Example

1, 1, 2, 3, 5, 8, 13, 21, _____

1 + 1 & = 2 \\1 + 2 & = 3 \\2 + 3 & = 5 \\5 + 3 & = 8 \\5 + 8 & = 13 \\8 + 13 & = 21 \\13 + 21 & = 34

Our answer is 34.

Selecting a different strategy was a key in finding the answer!!

Real Life Example Completed

Skate Park Construction

Here is the problem once again. Reread it and then draw a Venn diagram to organize the materials for the two ramps.

Isabelle, Isaac and Marc are very excited that their plans were accepted! The skate park are about to begin construction. The local lumber store has agreed to donate all of the wood that the team needs. To figure out what to ask for, the trio has collected a list of materials needed for the half pipe and the quarter pipe. Here is the list.

Half Pipe

42 8’ 2 \times 6’s

5 8’ 2 \times 4’s

4 8’ 2 \times 8’s

12 8’ 1 \times 6’s

4 8’ 4 \times 4’s

2 \frac{3}{4}” sheets of plywood

12 \frac{3}{8}” sheets of plywood

Quarter Pipe

13 8’ 2 \times 4’s

4 8’ 2 \times 6’s

1 8’ 4 \times 4’s

2 \frac{3}{4}” sheets of plywood

4 \frac{3}{8}” sheets of plywood

The team needs to organize the materials to figure out what materials both ramps have in common, and then which materials are unique to each ramp. By doing this, they can provide the lumber company with a list of materials needed for both ramps.

Isabelle thinks that it would be a good idea to design a Venn diagram, but Marc and Isaac aren’t sure how to do it. Isabelle thinks that she knows, but she isn’t sure.

If Isabelle draws a Venn diagram, what would it look like? A Venn diagram shows the common elements of two different things. In this case, the Venn diagram would have two circles. One circle would be for the half pipe and one for the quarter pipe. We can begin by listing all of the needed materials in each circle.

By organizing the date in this way, the students will be able to keep track of the lumber that has been donated. They can also be sure to request an accurate amount so that none is wasted. Using the Venn diagram has simplified the work for our skateboarding trio!!

Information in this problem is courtesy of http://www.xtremeskater.com/ and their free skateboard ramp plans!!

Technology Integration

James Sousa, Set Operations and Venn Diagrams Part 1

James Sousa, Set Operations and Venn Diagrams Part 2

James Sousa, Solving Problems with Venn Diagrams

Other Videos:

  1. http://www.linkslearning.org/Kids/1_Math/2_Illustrated_Lessons/5_Patterns/index.html – This is a GREAT video on finding number patterns and identifying pattern. It is animated.

Time to Practice

Directions: Figure out the pattern in each of the following problems. Then write in the next number in each pattern.

1. 2, 4, 6, 8, 10 ____

2. 20, 17, 14, 11, ____

3. 4, 8, 16, 32, ____

4. 200, 100, 50 ____

5. 120, 60, 30, 15, ____

6. 22, 33, 44, 55, 66, ____

7. 4, 12, 36, _____

8. 5, 10, 6, 12, 8, 15, 11, _____

9. 6, 4, 8, 5, 10, 6, 12, _____

10. 5, 11, 6, 13, 7, 15, 8, _____

Directions: Create a Venn diagram for the following data.

11. The factors of 20 and 30.

12. The factors of 45 and 55

13. The factors of 67 and 17

14. The factors of 54 and 18

15. The factors of 27 and 81  

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CK.MAT.ENG.SE.1.Math-Grade-6.9.8

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