Patterns and Relationships in Tables
Easiest: Complete the table for the expression n+2:
Output: 3, 4, 5, 6, 7
What is the pattern rule if the number of squares in Figure 1 is one and increases by 2 for each consecutive figure?
The pattern rule is add 2 for each figure. If Figure 1 has one square, then Figure 2 has three squares, Figure 3 has five squares
Answer: The pattern rule is "Start with 1 square and add 2 for each subsequent figure."
Easiest: Which of the following is an example of the commutative property?
a) 5 + 7 and 7 + 5
b) 6 x 10 and 10 x 6
Both a) and b) are examples of the commutative property (addition and multiplication, as they show that the order of addition does not affect the sum.
Plot the points (1,2), (2,4), and (3,6) on a graph. What type of relationship do these points represent? (shape, line, etc.)
The points (1,2), (2,4), and (3,6) represent a linear relationship, as they form a straight line when plotted on a graph.
For the pattern 3, 6, 9, 12, what is the relation? Write an expression that represents this pattern.
Answer: 3s
Input: 1, 2, 3, 4
Output: 3, 6, 9, 12
3(1) = 3
3(2) = 6
If Figure 3 has eight squares and Figure 4 has twelve squares, how many squares are in Figure 5? (Provide an expression to explain your answer.)
If Figure 4 has twelve squares, then Figure 5 will have sixteen squares (12 + 4 = 16).
Answer: Figure 5 has sixteen squares.
Is the equation 3 + 8 = 10 balanced? Explain how you know.
No it is not balanced because 3 + 8 =11 and not 10
Complete the table for the expression 2n+3:
Answer: 5, 6, 9, 11, 13
2(1) + 3=5
2(2) + 3=7
etc.
If n=5, evaluate the expression 4n−3 and state the output.
Answer: The output is 17
4(5) - 3
20 - 3
= 17
Complete the following table based on the given pattern:
Describe the relation/algebraic expression
Answers:
20, 28
8n - 4
Rewrite the expression using the commutative property: 8 + 3.
Using the commutative property, 8 + 3 can be rewritten as 3 + 8.
Answer: 8 + 3 = 3 + 8.
11 = 11
Graph the relation from the completed table of "n" and 5n−4. What is the slope of the line you graphed?
Answer: The slope of the line graphed for 5n−4 is 5, as the output increases by 5 for each increase of 1 in the input.
A sequence starts at 10 and increases by 4 each time. Write a relation for this pattern. What is the 7th term?
Answer: 4t + 6 and the 7th term would be 34
Input: 1, 2, 3, 4
Output: 10, 14, 18, 22
4(1) + 6 = 10
4(2) + 6 = 14
Graph the pattern:
F1: 3
F2: 5
F3: 7
F4: 9
Explain how you can determine the number of squares in Figure 10 based on the graph.
To graph the pattern, plot the points: (1,3), (2,5), (3,7), (4,9). To determine the number of squares in F10, use the pattern rule (adding 2 each time):
Answer: F10 = 2n + 1
= 2(10) + 1
= 20 + 1
= 21
If 48 ÷ 8 = 6, what can you say about 2 × 3 in relation to this equation?
If 48 ÷ 8 = 6, then 2 × 3 also equals 6. This demonstrates that both expressions yield the same result.
Answer: Yes, because both equal 6, confirming that 48 ÷ 8 = 2 × 3.
6 = 6
If the input is 6 in the relation 3n+1, what is the output? Plot this point on a graph and describe the relationship.
If the input is 6 in the relation 3n+1:
Given the pattern represented by the relation 5n+2, complete the following table:
Answer: 7, 12, 17, 22, 27, 32, 37, 42
5(6) + 2=32
5(7) + 2=37
5(8) + 2=42
If F1 has six triangles and F2 has thirteen triangles, what is the relation/expression and how many triangles would F52 have?
7x - 1
= 7(52) - 1
= 364 - 1
= 363
Given the equation 3 + x = 7, what is the value of x, and how would you preserve the equality if you added 2 to both sides of the equation?
To solve 3 + x = 7, subtract 3 from both sides:
x = 7 - 3 → x = 4.
If you add 2 to both sides, the equation becomes 3 + x + 2 = 7 + 2 or 5 + x = 9, preserving the equality
Answer: x = 4, and the equivalent equation after adding 2 is 5 + x = 9.
The students wash cars for $3 each. Write a relation for the money collected in relation to the number of cars washed. Graph this relation and interpret the slope of the line.
The relation for the money collected based on the number of cars washed is 3c (where "c" is the number of cars).