Math 6 - Bid Is Non Negotiable (Read first before bidding!!!)
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.
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.
#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:
#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:
#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
#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.