Mathematics

Lesson 3.1

Introduction

Course Objectives

This lesson will address the following course outcomes:

· 11. Distinguish between variables and constants. Represent real-world problem situations using variables and constants. Construct equations to represent relationships between unknown quantities.

· 25. Use functional models to make predictions and solve problems.

Specific Objectives

Students will understand

· There are multiple ways to “see” and describe a pattern

Students will be able to

· Form an expression to describe a pattern

· Use that expression to evaluate and solve

In this lesson, you are going to work on seeing patterns and representing them algebraically.

To do this, we’re going to look at a progression of “steps” of pattern, and try to write down what we see, then find an expression that explains it.

https://s3.amazonaws.com/wamapdata/ufiles/2/96-3-1-ex.JPG Example:  Consider the three steps to the right.  How many blocks would be in Step 4?  Step 10?  Step n?

We’ll look at two different student’s approaches.

Student 1 notices that all three steps shown have a single dot on the far left and far right, so that’s 2 dots.  There’s a top row and bottom row of dots, each of which is increasing by 1 each time.

So in step 1, we have 2 dots + 2 rows of 1 dot each:  2 + 2·1.  In step 2, we have 2 dots + 2 rows of 2 dots each:  2 + 2·2

We jot this down, and note the pattern, which we can then extend:

Step What I See Here Number of dots
1 2 + 2 · 1 4
2 2 + 2 · 2 6
3 2 + 2 · 3 8
4 2 + 2 · 4 10
10 2 + 2 · 10 22
n 2 + 2 ·  n 2 + 2n

 

Student 2 notices that we start with 4 dots, and add 2 dots each time.  So, in Step 2, we have 4 dots + 2 more.  In step 3 we have 4 dots + 4 more, which is 2 more twice: 2·2, or 4 + 2·2

We jot this down, and note the pattern, which we can then extend:

Step What I See Here Number of dots
1 4, or 4 + 2 · 0 4
2 4 + 2 · 1 6
3 4 + 2 · 2 8
4 4 + 2 · 3 10
10 4 + 2 · 9 22
n 4 + 2 · ( n – 1) 4 + 2(n  1)

 

Is one of the students wrong?  Or are their answers the same?

We can check by simplifying Student 2’s answer:

4 + 2(n  1)                             Distributing 4 + 2n – 2                               Combining like terms, 4 – 2 = 2 2 + 2n

The answers are the same, just written differently.

#1 Points possible: 10. Total attempts: 5

Watch this video to see 4 different students’ approaches to finding a formula for a pattern.

Which of the approaches/formulas in the video are valid?

· Student 1: 3(n+1)+2n3(n+1)+2n

· Student 2: (2n+1)⋅2+(n+1)(2n+1)⋅2+(n+1)

· Student 3: 8+5(n−1)8+5(n-1)

· Student 4: 3(3+2(n−1))−n3(3+2(n-1))-n

Start with one of the valid expressions, and simplify it was much as possible.  Your answer should be an expression involving n.

 

Set 1

Now it’s your turn.

Use the pattern shown for the next set of questions.

https://s3.amazonaws.com/wamapdata/ufiles/2/96-3-1-q1.JPG

#2 Points possible: 5. Total attempts: 5

Complete the table.

Stage Number of dots
1
2
3
4
10

#3 Points possible: 5. Total attempts: 5

How many dots will there be in stage n?  Write an expression involving n.

 

#4 Points possible: 5. Total attempts: 5

How many dots will there be in stage 20?

dots

#5 Points possible: 5. Total attempts: 5

What stage will have 137 dots?

Stage

#6 Points possible: 5. Total attempts: 5

Which best describes this pattern?

· Linear

· Quadratic

· Exponential

· Other

Set 2

Use the pattern shown to answer the next set of questions:

https://s3.amazonaws.com/wamapdata/ufiles/2/96-3-1-q2.JPG

#7 Points possible: 5. Total attempts: 5

Complete the table.

Stage Number of dots
1
2
3
4
10

#8 Points possible: 5. Total attempts: 5

How many dots will there be in stage n?  Write an expression involving n.

 

#9 Points possible: 5. Total attempts: 5

How many dots will there be in stage 20?

dots

#10 Points possible: 5. Total attempts: 5

What stage will have 144 boxes?

Stage

#11 Points possible: 5. Total attempts: 5

Which best describes this pattern?

· Linear

· Quadratic

· Exponential

· Other

Set 3

One more.  Use the pattern shown to answer the next set of questions:

https://s3.amazonaws.com/wamapdata/ufiles/2/96-3-1-q3.JPG

#12 Points possible: 5. Total attempts: 5

Complete the table.

Stage Number of dots
1
2
3
4
10

#13 Points possible: 5. Total attempts: 5

How many dots will there be in stage n?  Write an expression involving n.

 

#14 Points possible: 5. Total attempts: 5

Which best describes this pattern?

· Linear

· Quadratic

· Exponential

· Other

HW 3.1

#1 Points possible: 8. Total attempts: 5

https://wamaps3.s3.amazonaws.com/qimages/sqrpattern.gif The first four stages of a pattern are shown in the diagram above. If the pattern continues,  a) How many red boxes will there be in stage 10 of the pattern?  b) Write an expression for the number of red boxes in stage nn of the pattern.

#2 Points possible: 8. Total attempts: 5

The table below shows the number of boxes in each stage of a pattern

Stage Boxes
1 4
2 7
3 10
4 13

 

a) How many boxes will there be in stage 10 of the pattern?

b) Write an expression for the number of boxes in stage nn of the pattern.

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